Modern Navigation, Guidance, And Control Processing

Modem Navigation, Guidance, and Control Processing b y Ching-Fang L i n Modeling, Design, Analysis, Simulation, and ...

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Modem Navigation, Guidance, and Control Processing

b y Ching-Fang L i n

Modeling, Design, Analysis, Simulation, and Evaluation ( M D A S E ) Modern Navigation, Guidance, and Control Processing Advanced Control Systems Design Integrated, Adaptive, and Intelligent Navigation, Guidance, and Control Systems Design Digital Navigation, Guidance, and Control Systems Design

Modem Navigation, Guidance, and Control Processing

Ching-Fang Lin American Gh'C Corporation

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Prentice Hall, Englewood Cliffs, New Jersey 07632

L i b r a r y of C o n g r e s s Catalo~lng-ln-Publlcatlon D a t a

Lln. Chlng-Fang. M o d e r n n a v l g a t l o n , guldance. and c o n t r o l processing / by C h l n g -Fang Lln. p. CI. ( S e r l e s In a d v a n c e d n a v l g a t l o n , guldance. a n d c o n t ~ o l . a n d t h e l r a p p l l c a t l o n s : b k . 2) I n c l u d e s b l b l l o g r a p h t c a l r e f e r e n c e s and index. I S B N 0-13-596230-7 1. F l l g h t c o n t r o l . 2. G u i d e d ntsslles--Control s y s t e m s . I. T l t l e . 11. S e r l e s . TL589.4.L55 1991 90-43009 629.1--0C20 CIP

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Editoriallproduction supervision and interior design: Brendan M . Stewart Cover design: Bruce Kenselaar Manufacturing buyers: Kelly Behr and Susan Brunke Acquisitions editors: Bernard Goodwin and Michael Hays

O 1991 by Prentice-Hall, Inc. A Simpn & Schuster Company Englewood Cliffs. New Jersey 07632

Thii book can be made available to businesses and organizations at a special discount when ordered in large quantities. For more information, please eontad: PrenticeHall, Inc., Special Sales and Markets, College Division, Englewood Cliffs, NJ 07632 All rights reserved. N o part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher Printed in the United States of America 1 0 9 8 7 6 5 4 3 2 1

ISBN 0-13-59b230-7 Prenticc-Hall International (UK) Limited, Lotidon Prentice-Hall of Australia Pty. Limited, Sydney Prentice-Hall Canada Inc., Toronto Prenticc-Hall Hispanoamericana. S.A.. Mexico Prentice-Hall of India Private Limited, ,Vru, Drlhi Prentice-Ha11 ofJapan. Inc.. Tokyo Simon & Schuster Asia Pte. Ltd.. Singapore Editora Prenticc-Hall do Brasil. Ltda.. Rio de Jatreiro

To my family for their love, understanding, and support throtrghout.

Contents Series Foreword Preface Acknowledgments 1 Introduction 1.1 Overview, 1 1.2 Outline/Scope,

8

2 Modeling-Design-Analysis-SimulationEvaluation (MDASE) of NGC

Processing 2.1 LinearlNonlinear Intercept NGC System, 13 2.1.1 LinearlNonlinear Intercept N G C Processing, 14

Contents

2.1.2 2.1.3 2.1.4

Modeling and Simulation, 18 Guidance System Classification, 21 Flight Control System (FCS) and FCS Sensing, 22 2.2 Target Signal Processing, 23 2.2.1 Targeting, 30 2.2.2 Kinematic/Relative Geometry, 34 2.2.3 Targeting Sensor Dynamics, 37 2.3 N G C System Design and Analysis, 45 2.3.1 Guidance Filtering/Processing, 45 2.3.2 N G C Stability and Pevfovmance Analysis, 52 2.4 Target Tracking State Modeling, 68 2.4.1 Target Noise and Target Maneuver Modeling, 68 2.4.2 Two-Dimensional Target Tracking State Modeling, 72 2.4.3 Three-Dimensional lntercept State Modeling, 74

3 Modem Multivariable Control Analysis Singular Value Analysis, 78 Sensitivity and Complementary Sensitivity Functions, 82 Design Requirements, 84 Structured Singular Value, 86 General Robustness Analysis, 89 Robustness of Real Perturbations, 90 3.6.1 State-Space Model for Additive Uncertainty, 91 3.6.2 Slate-Spare Sing~rlarI/allres, 92 3.6.3 Root Loctrs, 92 3.6.4 A Monte Carlo Comnputation, 94 3.6.5 A Monte Carlo Analysis.for SecondOrder Systetns, 95 3.6.6 Stability Margin Comttp~rtatiot~, 98 Monte Carlo Analysis, 100 Covariance Analysis, 102 Adjoint Method, 106 3.9.1 Adjoint P/~ilosophy,106 3.9.2 Applications, 108

Contents 3.10 Statistical Lzncarization, 11 1 and Tools for Statistical 3.10.1 l'c~clrtriq~rc~s L~trcartzation, 116 3.10.2 Statistical Litlearization with Adjoint hlrtlrod, 1 18 3.10.3 Applicatiot~s, 120 3.11 Qualitativc Comparison, 121 3.12 Other l'erformancc Analysts, 124

4 Modern Filtering and Estimation

Techniques 4.1

4.2

4.3

4

4.5

4.6

Linear Minimum Variance Estimation: Kalman Filter, 128 4.1.1 Discrete Kalman Filter, 129 4.1.2 Contirl~rortsKalmatr Filter, 129 Nonlinear Filtering, 133 4.2.1 Estetrded Kalmaw Filter (EKF), 133 4.2.2 Sratisrical Lit~earization Teclzrtiqrre, 135 4.2.3 ~tfrrltipleModel Estirnadon, 137 Prediction and Smoothing, 138 4.3.1 Prediction, 139 4.3.2 Strroothirrg, 140 Kalman Filter Design and Performance Analysis, 140 4.4.1 Kalrnar~Filter Desigtl Process, 142 4.4.2 Selectiort of Noise Itltensity Matrices, 142 4.4.3 Error Analysis, 146 Operational Considerations, 148 4.5.1 Filter Pevfarmance, 149 4.5.2 Computational Reqtrirements, 149 4.5.3 Absence o f A Priori Information, 153 4.5.4 Filter Divergence, 153 4.5.5 &lodeling Process Noise and Biases, 156 Combined Complementary/Kalman Filter Approach to Estimator Design, 158 4.6.1 First-Order Complert~entary/Kalman Filter, 158 4.6.2 Second-Order Complementary/Kalman Filter, 163 4.6.3 Third-Order Complernentary/Kalman Filter, 170

Contents

5 Inertial Navigation 5.1 Introduction, 176 5.2 Common Requirements for Inertial Navigators, 181 5.3 Navigation Computation and Error Modeling, 184 5.3.1 Coordinate Systems, 184 5.3.2 Position and Velocity Generation, 186 5.3.3 Data Processing, 188 5.3.4 IA'S Error Analysis and Modeling, 188 5.4 Gimballed Inertial Navigation System (INS), 192 5.4.1 Ginzbal Mechanism, 192 '5.4.2 Iitertial Setisors on the Stable PlatJorrn, 193 5.4.3 Plarfornt Aligrlriient Modes, 193 5.4.4 nhvigatioit Mode, 194 5.4.5 Systein Ititernal arid External Integaces, 195 5.4.6 Navigatioii Mechanizatiorz and Error Model, 195 5.5 Strapdown Inertial Navigation System (INS), 199 5.5.1 Generalized Strapdowti Mechai~izatiori, 199 r, 5.5.2 Strapdouwi Navixatiori ~ o r n ~ u t e 201 5.5.3 Strapdown Coniputer Requireineiits, 201 5.5.4 Strapdowrl Error Model and Attalysis, 204 5.5.5 Operation Flow Diagram, 208 Comparison and Analysis of Gimbal Versus 5.6 Strapdown, 209 5.6.1 Fcatitrc Coiiiparisoii, 209 5.6.2 h'ai~igatiori Errors Cori~parisoil,209 5.7 External Navigation Aids, 213 5.7.1 Aided liierti~l,Vavkatioir Meckatiizatior?, 213 5.7.2 Global Positioi~iiigSysterri ( G P S ) , 215 5.7.3 Tactical A i r hlavigatiorl ( T A C A N ) , 277 5.7.4 L o r ~ gRange Navigation ( L O R A N ) , 220

176

Contents

5.7.5

Terrain Contour Matching (TEHCO.M), 220 5.7.6 Doppler Radar, 225 5.7.7 Star Trackers, 225 5.7.8 Kaln~anFiltering, 226 5.7.9 Kaltnon Filtering Performance, 228 5.8 Integrated Inertial Navigation System (IINS). 229 5.8.1 I t ~ t e ~ ~ r a rSensittcq/Flight ed Control Reference Systrtn (ISFCRS), 231 Sensory Subsystetn 5.8.2 it~te~qrated ( I S S ) , 231 5.8.3 Itltecqrared Inertial Sensing Assembly ( I I S A ) , 232 5.8.4 Helicopter Integrated Inertial h'avigation Systern (HIINS), 241 5.8.5 integrated Missile Guidance Systerns, 245

6 Guidance Processing 6.1 Guidance Processors, 252 6.2 Guidance Mission and Performance, 268 6.2.1 Guidance Pevformance, 269 6.2.2 Pltases of Flight, 279 6.2.3 Operation, 280 6.3 Multiple Mode Guidance Modeling, 298 6.3.1 M~dcourseGuidance, 298 6.3.2 Terminal Guidance, 302 6.3.3 Error Analysis Model Development, 308 6.4 Guidance Algorithm, 310 6.4.1 Preset Guidance, 310 6.4.2 Direct Guidance Methods, 312 6.5 Guidance Law, 347 6.5.1 LOS An3le Guidance, 348 6.5.2 LOS Rate Guidance, 348 6.5.3 Sensitivity and Cornparison of G~ridarrceLaws, 373 6.6 Single-Mode, Dual-Mode, and Multimode Guidance, 379 6.6.1 Single-Mode Guidance, 379 6.6.2 Dual-Mode Guidance, 380 6.6.3 Multitnode Guidance Applications, 384

Contents 6.7

Defense and Offense Systems, 392 6.7.1 Performance Paratneters, 392 6.7.2 Low-Altitude Air Defense Systems, 397 6.8 Future Guidance Processing, 404 6.8.1 Areas for Teclztiological Advartces in Signal Processitig, 405 6.8.2 Future Signal Processing for Missile Guidatlce, 407

7 Navigation and Guidance Filtering

Design 7.1 Target 7.1.1 7.1.2 7.1.3 7.1.4 7.1.5 7.1.6

State Estimation, 414 Target Tracking Filter Suiizmary, 416 The Wiener Filter, 420 Kalmatl Filter, 420 The Sitiiplified Kalnlari Filter, 420 Alpha-Beta-Canima Filter, 421 Modified Maxitn~rm-Likelihood Filter, 424 7.1.7 Two-Point Extrapolator, 425 7.1.8 Coriiparisori of Targct Trackirrg Filters, 425 7.2 Practical Navigation and Guidance Filtcr Design, 427 7.2.1 Guidance Tracking Filtcr, 427 7.2.2 Navigation and Guidance Filteringfor Position Estiriiatc, 430 7.2.3 Navigatiorr arid Cuidatice Filteritlgfor Position and Velocity Estitilate, 431 7.2.4 Navigation atrd Cuidarrcc Filterit~gfor Position, I/clociry, and Accrleratiot~ Estiriiatc, 434 7.2.5 Advatrccd Guidance Filter, 440 7.3 Radar Tracking, 442 7.4 Spacccraft Attitudc Estimation, 444 7.5 Advanccd Navigation System Design, 445 7.5.1 Global Positioning Systetir (GPS) Acc~rracyItnpro~~etttcrit, 445 7.q.2 Integrated CPSIIR'S, 457

Contents

8 Advanced Guidance System Design 8.1

8.2

8.3 8.4

8.5

8.6

Advanced Guidancc Laws, 468 8.1.1 O~itirttnlCrridatrce Law Survey, 469 8.1.2 Atrnlyticnl Solrrtiotr of Optirnal Filters and Optitnal G~ridatrceLaw, 474 Complenicntary/Kalman Filtered Proportional Navigation: Biased PNG and Complcmcntary I'NG, 481 8.2.1 Corttliitted Seeker-Czridatrce Filtering it1 a Cottrplettret~taryFilter, 483 8.2.2 Biased PAYC ( B P N G ) Algorithm, 486 8.2.3 Cort~plettret~tary PNG (CPNG) Alyoritl~trr, 492 8.2.4 Terrtr inal G~ridatlceSystem Analysis, 501 Other Terminal Guidance Laws, 510 Radome Error Calibration and Compensation, 517 8.4.1 Radorne Error Cotnpensntion, 51 7 8.4.2 Gtridowce Perforrnat~ceAnalysis with In-Fl!qht Radotne Error Calibratiotr, 520 8.4.3 Des[gtt Equations for P N C with Parasitic Feedback, 528 Command Versus Semiactive Homing Guidance System Design and Analysis, 544 8.5.1 Modeling, 544 8.5.2 Miss-Distance Analysis for Command Guidance, 550 8.5.3 Analysis of Optimal Command Ccridance Versus Optimal Semiactive Homirlg C~ridance,557 Analytical S o l ~ ~ t i oonf Optimal Trajectory Shaping for Combined Midcourse and Terminal Guidance, 562 8.6.1 Optitnal Trajectory Sllaping Guidance, 562 8.6.2 Problem Formulation, 563 8.6.3 Attalytic Optimal Gtridance Law, 566 8.6.4 Real-Time lmplemetztation and Pevformance, 578 8.6.5 Discussiotls, 583

Series Foreword The role played by modern navigation, guidance, and control (NGC) in the development and advancement of such areas as commercial and military aviation, to name only two, has continually expanded since its earliest inception in the 1950's. As this field began to grow and take on added importance, many books were written dealing mainly with the theoretical aspects of NGC, but most of these were confined to the earlier years of N G C development. Currently, although N G C system applications continue to take on an ever-increasing importance, the availability of reference books, especially textbooks suitable for graduate level and advanced undergraduate students as well as those who practice in the field, has not kept pace. This series emphasizing NGC systems and their applications is long overdue; in fact, it has been 30 years since such a series dealing with N G C systems has been written and available to the academic and ~rofessionalcommunities. Moreover. it is the first ever such series to thoroughly discuss the advanced control system design (modern multivariable control analysis; robust control; estimation; adaptive control; nonlinear control; intelligent control; etc.). It comes at a time when concern over issues such as the status of education and the decreasing number of trained, qualified professionals in this country is a t an all-time high. It is against such a background that the present series was conceived to assess state-of-the-art systems and control theories, and engineering applications of advanced N G C systems. Another purpose of the series is to develop future research agenda and at the same time encourage A

xuiii

Series Foreword

discussions in those areas that do not always find the systems and control community in complete agreement. The series provides a comprehensive coverage of the latest N G C technology as follows. The first book begins by introducing the various applications of N G C systems, after which it provides a thorough, fundamental treatment of what is considered the five most important stages in N G C system development: modeling, design, analysis, simulation, and evaluation (MDASE). The second book in the series takes up the subject of advanced estimation and guidance systems design, as well as N G C processing. The third book is concerned with the subject of advanced control system design, with particular emphasis placed on the topic of flight control system (FCS) design. The topics that constitute the fourth book include integrated, adaptive, and intelligent N G C systems design, while the fifth book is devoted completely to digital N G C systems design. Although most of the material in these five books is self-contained, there is a natural progression in the series as a whole toward more advanced topics. For example, much of the material in the second book actually serves as a prelude to the third, fourth, and fifth books. These books are the result of several years of experience gained on the part of the authorleditor both as a professor at the university level and as a practitioner in the field. It is believed that this series will provide invaluable insight and instruction to students, mainly at the graduate level but also to advanced urldcrgraduate students, as well as to those engineers who work directly or indirectly in the field of N G C system design and applications. In addition, this series is intended to provide both engineers and managers with the advanced N G C knowledge and concepts necessary to make correct decisions concerning the best N G C system design in a particular situation.

Preface It is very likely that few people who labor in any scientific discipline are unaware of the contributions of advanced navigation, guidance, and control (NGC) theory to aerospace-related programs. It is, however, equally unlikely that many are aware of the dramatic impact of this field on such diverse areas as medicine, industrial manufacturing, energy management, and chemical engineering. While its broad range of applications would at first appear to indicate that advanced N G C theory is enjoying an immense popularity in scientific and academic settings in general, this unfortunately turns out not to be the case. It is felt by many experts that many of those in the aerospace field in particular either are content to rest on the laurels surrounding the success of NGC theory developed in the 1950's and 1960's or have become so conservative in their design philosophies as to be unduly apprehensive about using advanced N G C theory. The latter appears to be especially true in the area of aviation. The author is quick to point out, however, that the NGC field itself is somewhat responsible for many of the misperceptions on the part of those w h o are not convinced of the usefulness of advanced N G C theory. More than a mere shadow of doubt has been cast on the usefulness of this theory as a result of its having taken

xx

Preface

a much too mathematically-oriented turn almost immediately after the theory was first applied in the solution of practical problems. It is the author's opinion that, while N G C theorv is built around a rather beautiful framework of mathematics. its primary emphasis'must nonetheless always bk placed on solving engineering problems of great practical importance. N G C technology has always been the focal point of aerospace engineering and automation research and development. A combination of theoretical concepts, the rapid evolution of computer and microelectronics technology, and the continued refinement of sensor and actuator technology has contributed to its advances. This book examines the role of modern N G C processing in the design of advanced N G C systems. This volume places major emphasis on the practical applications of advanced NGC systems, treating the subject more from an engineering than a mathematical perspective. Nevertheless, theoretical and mathematical concepts are introduced and adequately developed to make the book a self-sufficient source of instruction for readers. The intent of this book is to enable readers to achieve a level of competence that will permit their participation in the practical applications of modeling, design, analysis, simulation, and evaluation (MDASE) to advanced N G C systems. The book presents basic as well as advanced algorithms. A wide range of examples culled from various applications are provided to meet the needs of the different levels and types of readers, extending from issues requiring only a rudimentary knowledge to those involving avant-garde research. Morever, problems in the text span from those that concern only N G C to those that are interdisciplinary, and ultimately to those that encompass the entire systems and control field. An outline of the topics presented in this book is given in Fig. 1. Following the Introduction (Chap. I ) , the text is organized according to five principal topics: MDASE of N G C Processing (Chap. 2), Modern Multivariablc Control Analysis (Chap. 3), Design Algorithms for Advanced NGC Systems Design (Chap. 4). Fundamentals of N G C Processing (Chaps. 5 and 6), and Advanced N G C Systems Design (Chaps. 7 and 8). Each of these is in turn divided into a number of subtopics that discuss the relevant theories, algorithms, and computing tools related to their applications in advanced NGC systems. Referring to the outline organizing the five principal topics covered in this book, only those subtopics that are connected by solid lines are treated in this book. Those subtopics that are connected by broken lines are treated specifically in the books referenced under them. The numerous examples that are included in this book are supplemented by a liberal use of rcferenccs, thus making it easier for the reader to get access to a tremendous body of literature it] this field. As noted by one rcvicwer, there are n o comparable books currently on the market that prcscnt in a usable format such a complete collection of practical tools that are applicable to real world problems. Moreover, the organization of thc material coupled with comprchensive examples make this book well suited to self-teaching, bridging the gap betwccn thc theoretical and the practical.

Preface

Introduction

Modeling-Design-Analysis-Simulation-Evaluation (MDASE) of NGC Processing

-----

I

Hodern Multivariable Control Analysis

1

Performance

-----

rn Advanced

Modern Filtering and Estimation Techniques (Chap. 4)

Design for Advanced NGC Systems Design

Adaptive NGC I Systems: Detection I and Identification (Book 4)

Multivariable Control Design Techniques

1

Knowledge-Based and Neural-Network Approaches to Systems Control (Book 4)

- - I- - - - 1 --Fundamentals of NGC Processing

Guidance Processing

1

Flight Control Processing (Bogk 3)

--7

Navigation and Guidance Filtering Design (Chap. 7)

+

Advanced Guidance Systems Design

Design

+

Advanced Flight Control System Design (Book 3)

I

r

7

- - -C - - - 1

Integrated, Adaptive, and Intelligent NGC Systems Design (Book 4)

Digital NGC Systems Design (Book 5)

Figure 1. Modern Navigation, Guidance, and Control (NGC) Processing

Acknowledgments The number of people who have lent assistance in one way or another to this book is too great to allow me to thank each one individually. 1 am indebted to each person who has contributed to making this project a success, but I feel that I must acknowledge a few individuals by name for their support and effort during the countless hours spent on this project. My first acknowledgement goes out to my staff at American GNC Corporation (AGNC) who have assisted me as follows: Chun Yang and Jerry Juang have provided important technical input, while Jim Wright has spent many hours editing the manuscript. Special thanks are due to Kylie Hsu and Janet Young for their involvement in designing, planning, coordinating, and preparing the entire manuscript, including word processing and numerous original pieces of computer and manual artwork. I am also grateful to William R. Yueh for his technical advice. Several of my colleagues and friends have provided invaluable technical assistance. I specifically want to acknowledge William H. Gilbert of Martin Marietta Electronic Systems who, through his ardent support and encouragement of this project, made this impossible task appear less so at times. I am deeply touched by his unselfish giving of his time in the overall guidance and input to this project. I am

xxiv

Acknowledgments

am also indebted to J. Stanley Ausman of Litton Guidance and Control Systems Division who has greatly improved the quality of Chapter 5. Dr. Ausman is an authority on the subject of this chapter and has co-authored a book on Inertial Guidance (Wiley, 1962). I would also like to thank him for granting me permission to adopt his lecture notes in Tactical Aircraft Weapon Delivery Systems. I also want to express my gratitude to Gary Hewer of Naval Weapons Center, Hsi-Han Yeh of Wright-Patterson Flight Dynamics Lab., and Keqin Gu of Southern Illinois University for their painstaking effort in reviewing Chapter 3. T. Sen Lee of MIT Lincoln Lab., is also appreciated for his effort in reviewing Chapters 4 and 5. A very special note of thanks goes out to Bernard Goodwin, Vice-President, Professional and Technical Reference Publishing of Prentice Hall, who has taken on the task of publishing this book and has continued to support enthusiastically throughout this project. I would also like to extend my appreciation to Michael Hays, Executive Editor and Assistant Vice-President, Professional and Technical Reference Publishing of Prentice Hall, for his constructive advice and his special interest in this project. Also deserving special mention is Brendan Stewart, Production Editor of Prentice Hall, for his excellent job throughout the production process. I am equally obliged to other Prentice Hall personnel who have participated behind the scenes in the different stages of this book project. Finally, I would like to acknowledge those individuals and organizations that have permitted me to reprintladapt portions of their outstanding work in order to make this book a well-rounded source of information. They include (in alphabetical order): AACC, AGARD, AIAA, Arthur Gelb of TASC, Frederick W. Hardy of Hughes Missile Systems Group, Robert J. Heaston of GACIAC, Kaz Hiroshige formerly of General Dynamics Convair Division, IEEE, Johns Hopkins APL Tcchnical Digest, Robert J. Kelly of Allied-Signal Aerospace Co., Litton. James A. McLean of the U.S. Army Missile Command, Pergamon I'rcss, Rockwell International Collins Avionics Division, and SCS International.

Modern Navigation, Guidance, and Control Processing

Introduction This chapter provides an overview of advanced navigation, guidance, and control (NGC) design. The text is oriented to the applied rather than the theoretical aspects of the subject matter. Although advanced techniques are discussed, the contents are presented in such a manner as to also provide a simple and interesting picture of the central issues underlying both classical and advanced control theory and the practice of the NGC modeling, design, analysis, simulation, and evaluation (MDASE) process.

1.1 OVERVIEW The objective of this book is to provide both engineers and managers with the advanced NGC knowledge and concepts necessary to make sensible decisions concerning the best NGC system design in a particular situation. It will hopefully serve as a useful source of information for NGC systems designers by providing them with ideas for the solutions of current problems and future designs, as well as information about the problems encountered with microprocessor-based systems, microelectronics, aerodynamics, structures, propulsion, sensing, actuation, target acquisition, and weapon systems. In an attempt to achieve such an objective, this book demonstrates practical tools that are realistically applicable in the work area.

2

Introduction

Chap. 1

The development and applications of present-day systems and control theory were spurred on by the challenge of unsolved aerospace problems, especially by the series of events that has occurred since the late 1950s. Therefore, it is beneficial to review the development of systems and control theory since that time. The jolting success of the Soviet Union's satellite technology during that time inspired the United States to excel in aerospace technology, thus giving birth to an entire new generation of support for the field of systems and control. This in turn led to success with regard to solving urgent aerospace problems. The emergence of the Apollo program in the 1960s restored confidence in systems and control research and provided opportunities for conceptualized systems and control theories to be transformed into actual practical NGC system designs. Among the more well-established concepts that found great applicability to solving real engineering/control problems in the 1960s are the recursive minimum variance estimator, often referred to as the Kalman-Bucy filter, the LQG technique, the time domain concepts related to the fields of linear algebra and probability, and the Kelley-Bryson variational optimization procedure. Research into this last area proved particularly fruitful, as it resulted in the successful design of optimal trajectories for several space missions. In particular, the Apollo program and space shuttle flights that followed benefitted greatly from the application of optimal control theory. The contributions of Kalnlan filtering to the Apollo program represent yet another nlilestone for the emerging control theory of the late 1950s and early 1960s. The Kalman filter was implemented using a square-root algorithm in the measurement of stars taken by the astronauts with the aid of a sextant. Both optimal control theory and Kalman filtering provide tangible examples of the cruc~alrole played by control science in important programs such as the Apollo project. As a result of its early success in solving mainly aerospace-related problems, NGC theory soon found application in such diverse areas as mcdicine, industrial manufacturing, and energy management. For example, two French researchers werc able to formulate the problem of treating certain cerebral edemas or malignant brain tumors by the simultaneous administration of vasopressin and cortisone into a nonlinear rnultivariable control problem. The treatment resulting from their work is currently used in the neurosurgery clinic of the Hbpital de la Pitie in Paris [Fleming, 19881. Although examples of this type demonstrate the profound impact of control theory in general, and are not altogether rare, it remains true that the majority of progress in N G C research and development continues to find its greatest application in aerospace. The preceding historical account is provided to illustrate the emergence of advanced NGC systems and their applications. The emergence of the various N G C methodologies contributed significantly to progress achieved in the development of state-of-the-art systems and control theories in the 1960s. Unfortunately, it was not long before the development of the NGC system became too mathematically oriented in spite of the pressing need to solve many remaining practical problems. It would appear reasonable that the obligation of the NGC engineer should be first to commit to solving practical NGC problems [Ho, 19871. In the course of carrying

Sec. 1.1

Overview

3

out such an obligation, thc cngincer will oftcn discover relevant theorics that surface spontaneously. While theoretical discovcries of this type often allow thc engineer to conduct furthcr systcm research, it remains impcrativc for professional engineers to commit themselves to solving what are considered to bc rcal-world problems. Very oftcn, practicing engineers encounter difficulty in just understanding the theorics presented at conferences and in journals, not to mention applying them. What is at issue here is not a debate of the merits of theory versus those of applied technology. Rathcr, thc issue is onc of determining an effcctivc way to build a practical N G C systcm; understanding an csotcric N G C theory is of sccondary importance. While there is currently an abundance of N G C problems waiting to be solved, along with a whole assortment of tools capable of attacking these problems, diminished funding in this area hinders further development of N G C technology. In order to recapture the strong financial support for NGC research of the 1960s, the N G C engineering community must first be seriously committed to attacking those immediate ~racticalNGC ~ r o b l e m sthat remain outstanding. " In the absence of such a first step, any practical system and control theory is not likely to be developed, as evidenced by the slow progress of N G C technology in aerospace. T o highlight this fact, N G C tcchnologv used in the Apollo project was later directly applicd to the space shuttle with hardly any new development. As a result of the diminishing research effort in the NGC discipline since the Apollo era, scarcely any systems or control theories have proven valuable enough for practical use in N G C systems. The extent to which system and control theories have facilitated the clarification of various N G C problems and issues cannot be overemphasized. However, the N G C engineering field still faces several practical problems, one of which involves the question of identifying the basic reason for the gap between theory and practice and recommending a means to close this gap for the sake of advancement in the N G C field. In light of this, the impetus behind this book is an attempt to bridge the gap between theoretical arguments and the practical needs of the N G C community by providing detailed discussions and practical examples of MDASE in this area. A more detailed treatment of the MDASE cycle is given in Book 1 of the series. Because experimental and theoretical aspects of NGC system research both represent integral parts of systems and control science, it is essential to consider N G C theories and physical systems together. If an N G C system of a flight vehicle or robotic system is carefully modeled, it usually serves as a strong foundation for the design process which, in turn, leads to analysis and simulation for validation. The final step of the procedure involves evaluation. Thus, the MDASE cycle is an important process in yielding practical results. Through demonstrations of the various techniques and examples in this book, the author hopes to shed light on and unravel complicated systems and control theories, and to translate them into practice. Before they can be brought into the operational stage, advanced N G C algorithms must pass through a developmental stage involving several years of studies and experiments. Through organizing sessions, panel discussions [Lin, 1983-1985; Lin et al., 1986; Lin and Speyer, 19851, and workshops [Lin, 1987; Lin and Franklin,

4

Introduction

Chap. 1

19901 on aerospace vehicle N G C systems for the Amcrican Control Conference and the IEEE Conference on Decision and Control over a period of several years, the author has been able to keep abreast of the current trends in N G C technology and to understand its practical needs in different situations. Thus, the author is able to provide a diverse range of advanced NGC techniques in the book which the reader can then apply to various areas of NGC with competence, developing the best intuitive design. Since all flight vehicles and robotic systems share the same N G C techniques, the techniques presented in the book are applicable to these complex dynamic systems. Applications presented throughout the book and techniques for these applications are summarized in Book 1 of the series. In addition, appropriate applications are embedded throughout the text to enhance the reader's understanding of the techniques presented.

Importance of advanced NGC concepts and their impact.

Advanced N G C systems, although designed so that machines might interact effectively with the elements of nature, rely on human elements for their continued progress and success. The beginning of this chapter highlights the role played by control theory in the success of the Apollo project. The successful development of modern fly-by-wire aircraft such as the F-16 fighter jet can also be credited largely to control theory. And while modern manufacturing is becoming increasingly dependcnt on highly accurate process and machine control, it is an unfortunate circumstance that the continuous contributions that are made in N G C technology and the resulting achievements go largely unappreciated by the broader scientific and engil~ccring communities. The way it1 which N G C technology is isolated from the othcr technological disciplines in governmental, industrial, and academic settings is vcry apparent [Speyer, 19871. Consequently, the acrospacc industry has not fully rcalizcd the potctltial of N G C tech11010gy it1 acrospace systems, and progrcss in this area especially continues to be delayed. This situation will continue unabated so long as those involved fail to grasp both the importance of making progress in the area of advanced N G C technology and the relation of this tcchnology to vital functions in certain acrospace projects. For example, in certain commercial aircraft projects, advocates of new methodologies for enhancing safety and ride quality performances are for the most part ignored. While this can be partially attributed to thosc who prefer conventional dcsign approaches such as the root locus or othcr classical techniques, some of the blame must be laid on inadequately trained pcrsonncl, many of whom lack a complcte understanding of thc basic principles of modcrn and advanced control tcchniqucs. At~othcrreason for this is that thc designers of these commercial transports dcpcnd entirely on the aerodynamic dcsign to improve vehicle stability and efficiency, and totally ignorc the merits of advanccd NGC systems design. Those who are usually quick to criticize techt~ologicaladvance have always been skeptical about the complex and unreliabic nature of new technology, doubtirlg the wisdom of investing so much trust in the capabilities of ncw tcchnology. However, competition and survival dcpend on technological advatlcc. It xvould bc a grave

Sec. 1.1

Overview

5

error to halt technological advance just becausc some inconvcnicnces arise at the beginning of each new era. Despite the fact that any criticisms can generally be found to be true in the short term, the long-term positive contributions that advanced technology has to offer should be emphasized. After a new era of technology has set in, new systems perform more complex tasks and become less expensive than their archaic counterparts. The reliability factor, that is, the mean time between failures, also improves several times over that of older technology of similar complexity. The cost does not increase as much as the cr~ticsargue, as evidenced in the recent shift between the technologies of the 1970s and 1980s. Spccifically, in the weapon systems field, the gains have measured up to the cost. Therefore, critics can discard the notion that new technological concepts are not important [Deitchman, 19871. In the realm of aerospace, unlike other technologies such as aerodynamics, structures, and propulsion, advanced NGC technology has yet to distinguish itself as an essential discipline. There are two main factors that account for this predicament. First, N G C technology in aircraft development has been traditionally accorded a secondary role. This is because many of those who direct large aerospace programs are unacquainted with the important role of advanced NGC technology and its impact on aerospace. Second, no hardware or other tangible entities are produced by N G C technology; whereas, shapes are formed by aerodynamics, delivered by structures, and driven by propulsion [Speyer, 19871. Therefore, NGC technology has barely been utilized in any of the aerospace vehicle design processes with the exception of the Apollo project. According to the author's knowledge, only a few newly developed European commercial airplanes employ advanced control laws. Faulty instrumentation and inadequate computational capabilities reinforced the negative view of NGC technology in the past, but even with the rapid improvements existing in the present this view still persists. T o reap the benefits of advanced technology, timely investments in the pursuit thereof are crucial. Most people cite only the recent progress in military technology. However, new technology must be applied to the commercial arena as well if economic dominance is to be possible. For example, airlines tend to purchase ultramodern airplanes from companies employing more advanced technical sophistication. From this viewpoint, Airbus, the European company, is a commercial success story. It is actively involved in investigating N G C issues and incorporating new N G C technology into its products as well as its human resources. Thus, research and development (R&D) of advanced N G C concepts cannot be stressed enough. It is important that advanced NGC design concepts enter early in the design phase. The earlier engineering errors are detected, the less severe future costs and schedule delays will be. An example of this type of situation is brought into focus in the now famous article entitled, "Probing Boeing's Crossed Connections" by Karen Fitzgerald in the May 1989 issue of IEEE Spectrum, in which the author writes: "By the end of Uanuary], the [FAA] had received 12 more reports of crossed wires, not only on 757s, but on 737s, and not only on cargo fire-extinguishers, but on engine fire extinguishers, and one case of crossed wires from engine temperature sensors. . . . As of March 17, the FAA had received 66 more reports of bungled ~

~

Introduction

6

Chap. 1

wiring and plumbing in fire-protection systems on all four classes U737, J747, J757, J7671 of aircraft."' If it is possible to make such errors in those systems which are by n o means as complicated as an NGC system, then errors in more complicated systems will certainly result unless advanced system concepts coupled with the proper quality control techniques are utilized. As shown in Figure 1-la, the overall cost resulting from engineering error increases dramatically as time progresses. Therefore, errors detected early can drastically reduce the overall cost. Although engineering R&D design and analysis constitutes only approximately 15 percent of the entire project cost, as high as 90 percent of the overall cost of the project may rely on R&D results, as illustrated in Figure 1-lb. Therefore, inadequacy or errors in the R&D design and analysis due to ina)

Cost

Detection of )Engineering Error 1 2 (Normalized Year) Cost Resulting From Engineering Error Increases Dramatically As Time Progresses Therefore Errors Detected Early Can Drastically Cut Down The Overall Cost

b,

Cost

OPS: Operating Cost MAN: Manpower cost R&D: Research 8: Development Cost

Life Cycle CDst = OPS t MAN t R&D Figure 1-1 I

Cost lnvolvemcnt

Karen Fitzgerald. "Probing Bocing's Crosscd Connections." IEEE Spccfrurn. May 1989. p 33.

experience or other conservative factors can often lead to a drastic increase in the overall time and cost. Quoting from the same article, the author writes, "A Boeing engineer who asked that he not be identified said a too ambitious schedule for the new 747-400 aircraft has caused wiring errors so extensive that a prototype had to be completely rewired last year, a $1 million job. . . . In a technology that can tolerate few errors, the crossed connections raise the specter of undetected errors in other parts of aircraft. Each new II&D assignment presents additional challenges which will tax the skill and ingenuity of ~ ~ C - N G design C engineer and will only be solved through hard work, experimentation, and tests. Hence, each new assignment must be analyzed on its own merits, and past practices and methods must not be allowed to stifle new ideas and concepts. Both the past and the present must never forget the common denominator for all designs, which is the human element. Unfortunately, in advanced technology, undue conservatism often hinders progress. Ironically, the objective of any technology should be to go beyond the current state o f design concepts. The NGC field, although relatively new, is currently in an excellent state. I t is responsible €or substantial contributions to engineering, science, and economics as well as the standard of living in the United States and other countries. One of the main contributions of this text is a new perspective of the N G C system whereby the NGC system's interaction with other disciplines will establish the basis for truly innovative problem formulations and methodologies. Methodologies used throughout the book are practical and have been employed in the designing a n d testing of NGC systems. In the past, classical designs were generally used for NGC of unaugmented dynamic systems. Thus, some designs covered in the book are inevitably related to the classical approach. However, modern state-of-the-art N G C system design for highly augmented dynamic systems, which is rapidly gaining popularity, plays the principal role in the design methodologies used in the book. Integration of N G C designs with various engineering automation and signal processing systems are the ultimate goals of this advanced technology. The intention, therefore, of this book and Books 3-5 of the series is to meet this challenge.

"'

Advanced NGC systems. It is apparent that the usefulness of advanced NGC systems has infiltrated the modern world. In the modern aerospace field, NGC designers must be thoroughly equipped to handle N G C systems problems and able to draw on a rather sophisticated knowledge base encompassing many diverse fields; in particular, they must draw on their in-depth knowledge of modern dynamics and MDASE techniques (see Book 1 of the series). They must also understand the finer interactions between systems and components in their nonlinear operational range. Thus, it is natural for them to appreciate the fact that an NGC system should be designed as an integrated system. Additionally, they must consider the wide range of applications of modern digital, analog, and hybrid computers. Very often,

* Ibid.. p 34-35.

8

Introduction

Chap. 1

more sophisticated analytical and computational tools are required to properly model N G C systems. The advantage of new technologies can be utilized to the fullest extent through the advances of new mathematics, analysis, and computation. For example, N G C system scientists and engineers are now more actively involved with the designing and manufacturing of microprocessor/microelectronic chips and computers in addition to interfacing with t h m d y n a m i c s , structure, propulsion, sensing, actuation, target acquisition, and weapon systems.

1.2 OUTLINWSCOPE As shown in the Preface, in Figure 1 labeled Modern Navigation, Guidance, and Control (NGC) Processing, the five primary topics that make up this book are: (1) MDASE of N G C processing (Chapter 2); (2) modern multivariable control analysis (Chapter 3); (3) design algorithms for advanced N G C system design (Chapter 4); (4) fundamentals of N G C processing (Chapters 5 and 6); and (5) advanced N G C systems design (Chapters 7 and 8). O f the subtopics that appear alongside each of these five main categories in the figure, only those that are connected by solid lines are treated in this book. Those subtopics that are connected by broken lines are treated specifically in the books referenced after them.

MDASE of NGC processing. For the purpose of continuity, the first of the five main topics, as shown in Figure 1, deals with MDASE of N G C processing in Chapter 2, highlighting some ofthe key features of Book 1. In particular, Chaprer 2 summarizes the major stages in the evolution of the design and development process. Modem multivariable control analysis. Modern multivariable control analysis, which is the second of the five primary topics as shown in Figure 1, is mainly concerned with the analyses of robustness, performance, dynamics, and stability. Chapter 3 deals specifically with robustness and performance analysis. Robustness analysis is particularly important for examining multivariable control systems design. Performance analysis, which is taken up in the later part of Chapter 3, includes the various statistical methods used in the analysis and synthesis of any modern NGC system. The subject of stability and dynamics analysis, which is threaded throughout this book, is one of the principal topics in Book 1 of the series. The performance predicted by advanced control theory is rarcly achieved when designing and developing an N G C system. A detailed analysis of N G C systems must therefore include a treatment of the actual hardware equipment, citing those characteristics that tend to limit system performance. The most difficult aspects of analyzing N G C loops (at least, in missile applications) include, but are not lirnitcd to, estimating their stability, determining their accuracy, and finding the trajcctory of the pursuer vehicle along with the normal and lateral accclcrations necessary to achieve that trajcctory for varying types of targct motion. Thc theoretical and cx-

perimental techniques for treating these problems are presented in Chapters 2 and 3 in this book, and in Book 1 of the series. The methods that together constitute the third of the five main topics listed in Figure 1, design algorithms for advanced N G C systems design, arc listed with this category in the same figure. The first of these, modern filtering and estimation techniques, is covered in Chapter 4, while advanced multivariablc control systems design techniques are the subject of Book 3. The last two topics in this category, adaptive N G C systems (detection and identification) and knowledge-based and neural network approaches to system control are given special treatmcnt in Book 4 of the series. I t is worthwhile at this point to examine the hierarchical and interactive relationships among the different design techniques mentioned previously that make up the design algorithms for advanced NGC systems. These techniques provide a methodology for selecting successful N G C systems and integrating them according to the mission requirements of a particular vehicle. The primary focus of this particular treatmcnt of design algorithms is to describe present-day components, systems, and synthesis techniques from the system-integration point of view. The goal is to demonstrate how to analyze and select an N G C system to meet a set of performance requirements when the vehicle maneuver and dynamic environment are specified. The various techniques which make up the design algorithms are arranged in hierarchical order in Figure 1-2a. From the figure, it can be seen that modern

Design algorithms for advanced NGC systems.

Modern Filering and Estimation

+

Advanced Multivariable

Control Design Techniques (Book3)

(Chap. 4)

+

Adaptive NGC System Design Twhniques

CBook 4)

v v Knowledge-Eased and Neural-Network Awroacha to Svstems ConUol

Figure 1-2 Design Algorithms for Advanced NGC Systems (a) Hierarchical Relationship (b) Advanced NGC Systems

10

Introduction

Chap. 1

filtering and estimation techniques represent the starting point in the use of design algorithms. Results from the application of these techniques are then used for both advanced multivariable control and adaptive N G C system designs; however, adaptive N G C techniques are also used to iterate or improve upon the multivariable control design. The results from the application of these last two techniques serve as input to the design of a knowledge-based and neural network intelligent system. The interactive relationships between components and techniques are depicted in Figure 1-2b. This figure illustrates how the design algorithms are made more intelligent by incorporating a knowledge-based and neural network approach to N G C system design. Ultimately, an N G C system is designed to accurately control the outputs of a system whose dynamics contain significant uncertainties. This involves the following fundamental processes and their associated algorithms: (1) modeling ofthe system based on physical laws (see Book 1 ofthe series); (2) systern identification based on experimental data (presented in Book 4 of the series); (3) signal processing of the output by filtering, prediction, state estimation, and detection (modern filtering and state estimation are presented in Chapter 4, while adaptive N G C systems detection is presented in Book 4 of the series, and digital processing is presented in Book 5 of the series); and (4) synthesizing the control input and applying it to the system (advanced control theory is presented in Book 3).

NGC processing. Referring again to Figure 1 of the Preface, the fourth of the five main subjects which comprise this book deals with N G C processing. As indicated in the figure, N G C processing includes as subtopics navigation. guidance, and flight control processing. The first of these, navigation processing, is covered in Chapter 5 . Guidance processing is treated in Chapter 6, while flight control processing is presented in Book 3 of the series. In the course of the last three decades, N G C systems have evolved from a state in which they existed only in the imagination to their current state in which they are very much a part of reality. Both military and commercial aviation, as well as other fields including automation and manufacturing, owe a good deal of their success and technical advancement to the parallel advancement in the area of N G C systems. It can readily be said that the development of N G C systems and techniques, which in their ,--infancy often led to unpredictable accuracy and reliability performance, has been most successful, especially when examined in light of the often extreme environmental conditions to which the components of these systems are typically subjected. Figure 1-3 gives a block diagram representation of all the signal processing elements that are needed to perform the functions listed in the previous paragraphs. Each block in the diagram additionally lists the corresponding chapter number where the individual elements are treated in more detail. It can be seen from the figure that N G C proccssing involves several processors, each of which must function individually as well as in unison with others. Advanced NGC systems design. This constitutes the last of the five main topics covered in this book. Looking once again at Figure 1 of the Preface, it

Feedback Sensing Information

I

I

I Tracking Sensing Information

+

+

I

Target Tracking (Chap. 7) I

Guidance Algorithms (Chap. 6)

---

Advanced Navigation Guidance System 4 System (Chap. 5) (Chap. 8)

Control Signal

Reconstruction/Estimation

State Estimate

(Book 1 )

-----Vehicle Motion Estimator (Chap. 4; Books 1, 3,4, 5) Guidance Commands

I

I Navigation Information Data

Advanced

night Control System (Books 1, 3,4, 5)

b

Steering Signal and Attitude Information

Man-in-the-Loop Commands (Controls & Displays) Figure 1-3

NGC Processing

Control Surface

Deflection Commands

12

Inhroductlon

Chap. 1

can be seen that this topic is concerned with the following subjects: navigation and guidance filter design (Chapter 7); advanced guidance systems design (Chapter 8); and advanced flight control system design (Book 3). These subjects prepare the reader for the actual design of integrated, adaptive, and intelligent N G C systems design, presented in Book 4 of the series, and of digital NGC systems design, presented in Book 5 of the series.

Modeling-Design-

Analysis-SimulationEvaluation (MDASE) of NGC Processing It is brought to the reader's attention at this point that a very broad picture of the modeling-design-analysis-simulation-evaluation (MDASE) cycle of the NGC system has already been put forth previously in Book 1 of the series. In order to allow for smooth discussions of the modern NGC processing tasks and for highlighting the way in which NGC processing gets involved in the MDASE cycle, advanced NGC concepts must first be examined. The examples chosen in the following discussions serve to introduce various parameters and functions of NGC processing with respect to particular applications.

2.1 LINEARINONLINEAR INTERCEtT NGC SYSTEM The conventional approach taken by design engineers to the avionics has been to break up the avionics into three distinct and independent systems: the navigation, guidance (midcourse andlor terminal), and flight control systems.

Navigation system. The navigation system functions to provide position, velocity, and attitude of the vehicle with respect to a reference coordinate frame. Using high-accuracy gyros and accelerometers, it is conventionally configured as an inertial system in either a gimballed or strapdown mode.

14

MDASE of NGC Processing

Chap. 2

Guidance. From the perspective of a control system, guidance is a matter of finding the appropriate compensation network to place in series with the plant in order to accomplish an intercept. In order for the pursuer to impact a maneuverable target with little miss distance, guidance uses the principles of feedback control. The purpose of the guidance system is to determine appropriate pursuer flight path dynamics such that some pursuer objective might be achieved efficiently. The guidance system decides the best trajectory (physical action) for the pursuer based on its knowledge of the pursucr's capability, target capability, and desired objectives. In many applications, the guidance system is designed so that it makes use of an inertially stabilized tracker (for example, seeker) that directly measures the angular rates between the pursuer and its target in a fixed coordinate frame. The function of the guidance computer is to mathematically integrate the separate functions of navigation and the flight control system (FCS).

FCS. The function of the FCS is to control the pursucr in pitch, yaw, and roll motion. The FCS executes the guidance commands and stabilizes the pursuer in flight. The FCS, upon receiving commands from the guidance law, then issues its own commands to the appropriate aerodynamic andlor thrust controls of the pursuer so that the guidance command can be properly executed. It is usually configured as a system equipped with low-accuracy inertial components (gyros and accelerometers). Small tactical missiles and most large transport vehicles often have open-loop control instead of the more complex FCS control. 2.1.1 LinearlNonlinear Intercept NGC Processing

Guided weapons or missilcs are normally guided from shortly after launch until target intcrccption. The NGC systcrn supplics stccring commands to aerodynamic control surfaces or corrccting clemcnts of the thrust vector subsystem to maneuver the weapon to its targct and to make it possible for the weapon to intercept moving targets. The guidance process, which is ruled by signal proccssing algorithms implcmented in the N G C system, a prcset flight program, or both, is essentially a feedback control systcrn where the pursuer-target engagcmcnt is considered part of the N G C loop. Thc N G C loop consists of the guidance systcnl together with dynamic controls. Elc~ncntsof this loop include an information subsystem, control clcmcnts, an opcrator, and pursucr dynamics. The components for the N G C systcrn arc shown in Figurc 2-1 in which the ovcrall control of thc pursucr is dividcd into two or lnorc loops. The main control loop in thc diagram is thc guidancc loop, which is thc outcr loop that controls translational dcgrccs of frccdom, whilc the inner controlIFCS loop controls pursucr attitude. ,The guidancc loop contains guidancc scnsors for sensing pursucr motion, targct motion, or rclativc motion of thc targct with rcspcct to the pursucr. This information is uscd in thc guidance computer or the corrccting nctworks, togcthcr with information conccrning thc intended flight profilc, to gcncratc guidance (latcral acceleration) commands for the FCS. Thc FCS and actuation in turn direct control surface dcflcctions to altcr thc pursucr's trajcctory. The body ratcs and accclcrations arc fed back to thc incrtial sensors to closc the FCS loop.

16

MDASE of NGC Processing

Chap. 2

The guidance and control laws used in current tactical missiles are based largely on classical control design techniques. These control laws took birth in the 1950s and have evolved into fairly standard design procedures. Proportional feedback is generally used to correct missile course in the guidance loop, which is commonly referred to as proportional navigation guidance (PNG), and is quite successful against nonmaneuvering targets. The controller for a homing missile, in general, is a closedloop system known as an autopilot, which is a minor loop inside the main guidance loop. In addition to the control surface and servon~echanism,the autopilot consists of mainly the acceleron~etersand/or (rate) gyros to provide additional feedback into the missile servos for missile motion modifications. Advanced sensors may measure other variables. N o explicit state estimators are used and the signals are filtered to reject high-frequency noise. Broadly speaking, autopilots control either the motion in the pitch and yaw planes (lateral autopilots), or the motion around the missile axis (roll autopilots). In general, the roll: pitch, and yaw channels are uncoupled and are typically controlled independently of each other. All commands are amplitude or torque constrained to ensure autopilot and n~issilcstability. Classical controllers have two major advantages, simplicity in design and simplicity in implementation, but they also have several problems. A pursuer is designed to complete the basic homing loop, which requires a sensor (a seeker in the case of a guided missile) to track the targct, a noise filter to reduce the effect of noise, a guidance law to generate thc dcsircd guidance (acccleration) commands to home on the target. an FCS to rcccibc thc dcsircd accelcration commandfrom the guidance system and to generate the required acceleration capability to insure interception of the maneuvering targct, and, finally, a good understanding of the physics of the homing engagement itsclf. Figure 7-2 depicts a miss-distance analysis model conlposed of elemenrs rcprcscnting the intercept kinematics plus clclncnts affecting the intcrccpt guidancc dynamics (that is. pursuer dynamics and the NGC system), and illustrates the intcrdcpcndcncc bct~veensystcnl elements. Inasmuch as thc total system perfornlancc is affcctcd by the individual characteristics of evcry elcmcnt, the system cnginecr is grcatly conccrned with this system interaction. The kinematic part inside the lowcr left dottcd box generates the LOS angle in terms of pursucr and target motion. Based on the LOS anglc, the N G C system generates a latcral acceleration for the pursuer, whilc trying to bring the projectcd miss to zero. In the model, a rclativc position ) I , is added to a glint noisc y,, thc result of which is multiplied by the inverse of thc range R to produce a true LOS anglc corruptcd by glint noise. u, in inertial coordinates. A mcasured LOS angle u is thcn obtaincd by adding ro (5. a radomc crror coupling u,and the following angular noisc tcrms: rangc-depcndcnt (thermal) noisc o,,,, rangc-indcpcdcnt (fading) noisc a,, and cluttcr noisc o,.It thcn scrvcs as input to thc scckcr dynamics to producc a mcasured LOS ratc 6 that is itsclf corruptcd by bias u l , and g sensitivity. The output from thc scckcr u is the input to thc guidancc filter. Many signal processing tcchniqucs arc uscd to discrin~inatcthctargct from its background, other targcts, and decoys. The guidancc law functions to gcncratc midcoursc and/or

MDASE of NGC Processing

18

Chap. 2

terminal steering commands u, based on the guidance filter output. Before being sent to the autopilot, u, is limited. In general (except for coordinated turn vehicle), the methods used to generate the guidance command signals u, for guiding a pursuer in each of two mutually perpendicular planes are identical. Consequently, the guidance algorithm needs to be determined forwnly one of the planes. In the treatment presented here, this plane will be the missile s horizontal plane. In Figure 2-2, u, is normally taken to be the lateral acceleration command. As long as saturation effects are ignored, the model shown in Figure 2-2 is a linear time-varying system driven by stochastic inputs. The inclusion of acceleration saturation effects causes the model to become nonlinear.

2.1.2 Modeling and Simulation The digital simulation technique in which the equations of motion are represented by' a set of first-order nonlinear differential equations is widely recognized. The modeling vector form is given as (see Figure 2-3): x(t) = f(x(t), u(t), t) with initial conditions x(tO) = xg

(2- 1a)

where the function f(x, u ) includes all of the model equations and NGC algorithms. Equation (2-la) corresponds to Figure 2-3, in the absence of any noise inputs. Depending on the specific application, the initial conditions are given, and the terminal time tf is to be determined. For intercept flight simulation, rf is determined by either the point of closest approach of the two vehicles or the time at which the range rate passes through zero for the first time. Numerical integration techniques are then used to integrate these equations with respect to time so that the time history of the state from an initial condition x(t0) to a terminal state x(tf) can be obtained. The output of the simulation is typically the time history of the state, from which a performance criterion can be formulated as a function of the terminal state. The flight vehicle to be designed determines the performance criteria in the simulation. In the case of a guided missile, good performance is represented by short miss distance or short range between the two vehicles at t,. The overall simulation block diagram, shown in Figure 2-4, defines all of the state equations. Referring again to

Nonlinear Function

t

-

f(x(t).uo),t) Figure 2-3

Nonlinear Dynamic Systems

MDASE of N G C Processing

20

Chap. 2

Figure 2-3, the modeling vector form, when a noise input vector is considered as shown in the figure, can be written k(t) = f(x(t), 4 t h t )

+

w(t)

(2-1 b)

where w(t) usually, although not necessarily, .denotes a white-noise input vector. A more general case is considered in Figure 2-4, in which the model includes certain error sources resulting from imperfect measurements or unmodeled dynamics. Incorporating these error sources in Equation (2-la) gives k = f(x,

U)

+ b + n ~ ( x t), + m(t)

(2- lc)

where b is the constant bias term; n l (x, t) is a state-dependent random vector; and nz(t) is a state-independent random vector. Equation (2-lc) could also apply to Figure 2-3, in which case w ( t ) is simply the sum of 6, nl(x, t), and n2(t). If M I or n2 represents a colored-noise process, a shaping-filter technique can be used to model either or both so that the new state vector can be augmented to Equation (2-lc). The resulting augmented state equation is driven by the white-noise characteristics only. A Monte Carlo technique, covariance analysis method, or an adjoint method can be used to properly evaluate the selected performance criteria. These techniques are described in Chapter 3. In closed-loop control system analysis, it must be decided what level of detail will be used to represent each element in the guidance loop. If it is desired to study high-frequency system instabilities, then one must employ a representation that is accurate at those frequencies. The elements that may be required to construct a sufficiently accurate representation in this case include modeling the dynamics of the nonrigid system, knowledge of the full nonlinear aerodynamic characteristics, and detailed modeling of the seeker track loop, guidance signal processing, autopilot, and noise sources. Conversely, a complicated model of this type would not be required in the case of a preliminary parametric study of homing trends since the main effccts of interest are found at low frequencies. Rathcr, a simplified representation of the more complex system capable of incorporating the available trim aerodynamic data and the low-frequency approximations to the different subsystcms that make up the guidance kinematic loop would be more appropriate. After being validated, this simplified representation can be used to study the relative performance of various missile configurations as well as the result of varying system paramctcrs such as time constants and limits [Reichert, 19811. In problems involving dynamic modeling and controls, a state model is usually specified in addition to thc nieasuremcnt equation, relating the state and measurement vcctors as shown in Table 2-1. Bcfore implctnenting thc preceding N G C simulation, it is important to understand the roles of thc pursucr system, the targets, and thc cr~virot~ment so that appropriate modcls in Figure 2-4 can be devclopcd for a particular pursuer-targct engagcmcnt scenario. Essential guidance and control softwarc modulcs arc as follows: intcrfacc, flight path planning, strapdown device, navigation, guidance, stabilization and control, fin actuation, target tracking, and self-test. Their relationships are dcscribcd in Figure 2-5.

LinearlNonlinear Intercept NGC System

Sec. 2.1

TABLE 2-1

-

21

STATE MODELING

D t f ~ n xg ( m l ) state vcclor. y

-

(pxl) system output vcclor. z = (1x1) measured output vcetor.

u = (mxl) mnlml vmor, I, i(qil) cxogenram inpu v m o r .?(I)w d a i n s all thc variables that can be measured and arc fed back to thc controller. Thc enuies of y(t) arc the variables lo be conmllcd. up) is h c output of ihs mnuollsr. lc(l] wnsisu of thc reference inputs, the disturbance inpuu (e.g.. process noise w(t)), and the rcnror M i s s ~(1).Note that y(t) and z(t) may have some wmponenu in common. The wnlinuous-lime nonlinear system is dcscribcd by

i t

=x

I

I t

I

Z ( I ) s h(x(1). I) t

,

X

"(I),

I

=x

Q ~ I= 0. I ~ 4q1)*TI

T

I = at)b(t-r)

T Qv(t)] = 0. Qv(l) "(1) ] = R(t) b(1-z)

The continuous-time nonlinear system with d*crete.time nonlinear mcasuremcnls is repnscnled by: i(1) = f(X(1). Nt), 11 + *I) T

(la) ., (lb)

(2=4

q k ) = h[x(tkh kl + v(k); qv(kl1 = 0. Qv(k) v(k) I = Q

Gb) w h e n the parameter k is considered to be a timc t and the measurements an sampled at discrete timc intcwak. The discrete form of the state equation (la) is as follows: k =k l1 k l Qqk)l = 0, q W k ) wT(k)l = or (2c) h applying many MDASE uchniqucs. the system quations (1)need to be linearized as:

where u, w, and v arc defined in Eqs. (1). Eq.(3) is equivalently expressed as i(t) = &(I) + Bqf) t Br i(1) and y = Cx + h ~,

The state equation in Egs. (4) is generally a nonhomogenwm differential quation with forcing term ~ .t f. (where ~ ) f(l) is a command vcctor anUor an anticipated forcing function mulling from a system nonlinearity andior an estimated dinurbance. In the output quation of Eq. (4), h consisls of the measurement nonlinearilia and uncenainties. The discrete form of the system quations (3) is: x(k) = Wk.1) x(k-1) + W(k-1) qk.1) + wk-1) qk-1)

2.1.3 Guidance System Classification

There are several solutions that can be applied to the problem of guiding a missile to an eventual near miss or collision with a target using only the observed motion of the missile-target line of sight (LOS). In one of these, the missile is constantly flying . - directly toward the current target location. This method, termed simple pursuit, suffers from the fact that the missile must generally undergo rather severe maneuvers as intercept is approached. A second solution concerns a constant-bearing trajectory wherein the missile leads the target, much like the way in which a projectile is traditionally fired at a moving object. A collision invariably occurs as long as the missile flies a course that keeps the relative missile-target velocity aligned with the LOS. As shown in Figure 2-2, the missile trajectory becomes straight line when the missile speed, target speed, and course are constant. PNG is a steering law that is designed to produce a constant-bearing course even in the case of a maneuvering target. T o accomplish this, one makes the rate of change of the missile heading directly proportional to the LOS rate. While this technique presupposes that the missile is able to respond instantaneously to changes in the LOS direction,

MDASE of NGC Processing

22

Chap. 2

-

StrapdownAFtU Guidance Navigation o initialization o Midcourse & Terminal o State Estimation o Euler o Position Transformation _* 4 + o Power On Mode o Power Off Mode o Velocity o Bias Compensation o Vertical Launch o Acceleration o Coordinate E m r s o Separation Mode o Estimate Transformation

Autopilot o Rate Stabilization o Acceleration Control o Aerodynamic Coupling Stability

*

t I

Seeker

I

t Relative

Interface o Aim Point o Initialization 0 Status

4-

Self Test o Test Sequence 4-W

-

1

Right Path Planning o Stored Mission Profile o Change Profile

Actuation o Fin Rate Limitation o Fin Angle Limitation

RF Environment & Clutter ECM: Target Kinematics

Figure 2-5

Sensors o Rate Gyro o Accelerometer

,

Dynamics o Equations of Motion o Engagement o Environment

Pursuer Target Tracking Processing Software Modules

there is in practice some delay in returning to a constant-bearing trajectory subsequent to a target maneuver as a result of finite missile responsiveness and filtering introduced to diminish the effects of noise. Figure 2-6 defines the conventional PNG configuration, in which the measured LOS rate u is filtered through a noise filter to produce a smoothed LOS rate estimate u, and u, is made proportional to u A similar idea is used in modern guidance approaches based on optimization techniques. If measurement noise is to be suppressed, then consideration must be given to a low-pass noise filter. PNG systems generally employ a first-order lowpass filter, which has a transfer function as shown in Figure 2-6. The missile guidante computer, using the preceding relationships, computes steering commands. Besides noise filters, com~ensationsare introduced to offset such factors as variations in the missile's velocity and radome effects. Also, in order to maintain aerodynamic stability along with structural integrity, steering command limits are imposed [Witte and McDonald, 19811.

2.1.4 Flight Control System (FCS) and FCS Sensing That part o f the system comprising the FCS also includes the inertial reference unit (IRU)to measure A,,,. A,,, is the acceleration of the pursuer in response to the guidance command, which is applied to the FCS and results in operation of the pursuer's control surfaces. The control surfaces can be caused by aerodynamic cross coupling, dynamics nonlinearity, and structural vibration to deflect at different orientations from the desired guidance input, thus contributing to miss distance. Au-

Target Signal Processing

Set. 2.2

vmf

Mi& velocity vector

23

--

MiSc without

, , R

Missile maneuver

~ V target ~ velocity vector

The basic proportional navlgation equation

Seeker Guidance filter

Am

a

V,,,? = AV,b

/

Autopilot

-*-

/ 1\

Gs(S)

GF(S)

GA(S)

Missile ~uidance.res~onse / \ maneuver Navigation clos/nng Line sight ratio velocity = Velocity I line of sight

,

rate

Range

g(S)

* k = Ik, Noise Ntcr bandwiith

V, Closing vebcily A Effcaive navigalioo mtw

s+t

Figure 2-6

Proportional Navigation (Courtesy o j K . Hiroshige, 1984)

topilot, actuator, structural aeroelasticity, sensors, and nonlinear airframe dynamics are part of the actual mode of an FCS. Details of the FCS are presented in Book 3 of the series. The useful models for NGC design are summarized in Figure 2-7, Table 2-2. and Table 2-3. 2.2 TARGET SIGNAL PROCESSING

A mathematical model is used to embody pursueritarget kinematics in the form of parameters of the LOS as shown in Figure 2-8, which shows what type of processing is required. A complete simulated engagement of pursuer target tracking begins with the launching of the missile, is followed by a midcourse and a terminal phase, and ends in an intercept and the scoring of missile-to-target miss distance. In active homing, after launch the missile seeker transmits radio-frequency (RF) energy to the target. The target converts this to a synthesized return and radiates it back to the seeker radar. Multiple target returns, which include different ranges, velocities, amplitudes, angular noise, and environments, can be generated simultaneously with or without ECM. The returns are tracked by the seeker. As shown in the relative geometry box in Figures 2-2 and 2-8, the homing loop is closed by calculating the missile-to-target geometry as well as using the RF array to update the relative angular

. MDASE of NGC Processing

Chap. 2

N [NORMAL FORCE1 X AXIS

(PITCH TOTAL MOMENT) REFERENCE Fx

(YAW TOTAL MOMENT1

(AXIAL FORCE) V

Z AXIS (A)

XCG XACC

-

center of gravity lacation arcelermeter location

(B)

6

-

-

c~ TOTAL CN 0 NORMAL FORCE COEFFICIENT

MAX (TRIM1

0

MOMENT COEFFICIENT

-

MACH =CONSTANT ALTITUDE = CONSTANT

k@ya MACH CONSTANT ALTITUDE =CONSTANT

.30°

OMAX ITRIM)

& = . 3 0 ° = 6MAX

6 = 00 b = -100

6 = .20°

TOTAL MOMENT COEFFICIENT CM VERSUS ANGLE OF ATTACK n (D)

Figure 2-7 (a) Horning Missile Axis System and Forces and Moments (b) Forces Due t o Angle of Attack a and Control Fin Uetlection S in a Tail-Controlled Missile (c) Normal Force Coefficient CN VS. Angle of Attack a (d) Total Moment Coefficicnr CM 1,s. Angle of Attack a (Froin [n'erlincs, 1984(a)] u,irk prnnisrio~tfi.otnAACC)

position. As shown in the endgame computation and miss-distance computation boxes, immediatcly before interception, the characteristics of the endgame including the miss distance are determined by extrapolation. Finally, the LOS rates in the pitch and yaw axes are computed and sent to the missile seeker as pitch and yawhead radar errors to fully close the guidance loop. E C M J E C C M modeling involves digital computer simulations of an operating radarlsensing system including the effects from an enen~v'sE C M . This allows the sensine " unit to be assessed under field conditions as to whether it would perform satisfactorily so as to support the tactical control system nlission. A sensing detection model is required to obtain signal-to-noise ratio data as a function of target range. The model must be able to allow these data to be obtained in both clear and noise-jamming environn~cnts. Jamming can be produced by a self-protecting or stand-offjammer operating against the radar signal in either the main lobe or the side lobe. Enemy threats may be taking placc in ascending, level, or descending flight profiles. Multiparh and terrain (terrain-masking and clutter) effects may also need to be considered.

TABLE 2-2

LINeARIZED AIRPRAME RESPONSE

Hjrmal Auxlemlion: A,

= v,?,

? = Aa + I36

"

I1ilch Anxllrr - Aoxlcrdlion: 0 = -(h-1%

6

(

6

s2

-= -

'liansfcr l'maions:

-=

G

Dl

~- I i~ -C HI^

- lib; u = 0 - y - 1*i2)

+ (A+#>&+ <: + All

-

(8)

(1)

kl

w m lag

(2)

c')

-=

-0= .

(5)

6

v

m

- I SkI(~+~lls+~12s2) (4) (6) s2 t (A+l>)s + C + N) k ( I + )&I' (W. - HC + 1%) - 3 (6) s2 + (A+l))s + (: + N) "1 (")

-

Ik,r 1;nil mnlml or c;tn:~rrlnirfr;mw with negligihlc liII from ctmlq,l s u t t ~ i.c.. ~ , 1% 0. knrdynamic lift: A,,, = Vm?

A = -7ff

- -at

-

Vm Atz

CN, aftcr boas1 (thrwl = 0 )

n = -i.,= -a, cNa C

= A Vm (0-Y);

= -Ma = -a,CM,

O =(I

+ s/A)Y

(8)

-

-

D = -Mi, = -ao (
6 = fin &fleaion or thnnt A, =achieved normal ameleralion a1 missile Iznfer o f gravity prrprndicllar lo missile body A,,," = auzleration measured al the amelemmeler 6 rate P = Vm ( H ( z - a l k, = acceleration gain umstant = k 2 = (J((!+/U)), 2 ( 8 1 ~ k 3 = L I P m = MaI(l&u)d) Ul = 2 W w d = (A+I))/(C+AD) if M a < ( ) &- h m p i n g ralio of airframe o natual frequrncy of airframe = Ma > 0 and 2Sdwd i s small d B., = 11o:" = lllC+r\o, . Ta = 4 missile l m i n g rate time mnslanl = -P/(BC-AE) = 111% lIA(if B 0 ) u b = low f r e w n c y lead

4

--

.

& .

-

-

-

rA Courtesy

of [ H i r o s h i g e , 19861 and from [Neslines, 1')84(a)l w i t h pernlissioll f r o t n AACC

-

AIRFRAME IS T H E HISULT O F TRADE-OFFS TO REDUCE 1/A &INCREASE MANEUVERABILITY WHILE MEhTlNG PHYSICALCONSTRAINTS BO kn &n!s%h W c w Wnpw-5253.

I

TABLE 2-3

HIOH ~ R E Q ~ E~ N ~~ ~1 l From - s [Neslines. 19851 with permission from AACC

a(~) Aaua10r: -4(5)

-

1

25,

s2

"A

"1

I+-st-

4 ~ s ) 1 Acaleromaer: -= 2t*m %(S) lt-st"xc

-

-

Rate Gyro: ~ G ( S )-

(1)

0 )

(3)

s2

Suuclural Filter:

1

2h

1+-s+"G

02 1t- s2

6 (s)

"sf

&= US)

1 +2 hsf + "sf

w& ~~~

(2)

s2

~

Nominal Values for a Typical Missile: UG = UW) &% LU, = 220 radkec,

= 0.65.

fA=

(4)

s2

"fF

0.65,

The mode shape cnvclopes a n shown below.

.

MODE

5 .-

2

MODE

FREOUENCY

5

GENERALtYD MASS t l b / ~/in) 0.0517

SECOND MODE

0.6. 0.4 i 0.2

4.6

0.0373

Gym Location

)X

-

LOCATION

bl Normalized Bodv Bendinn Sham

a) Fint and E m n d Mode Shapes The detailed flcxiblc body transfer Function from b

LO body

rae is

(av% &

- 24

1

( I v l s2

-

~ ~ I Y Y &

-. &B

6

-6 <-

Xffi

,1 2Ei l-+-s+l) w1' Wi

Nominal values typically used in thc above equalion are k damping of i t h v i b r a t i o n a l mode 9 t h e v i b r a t i o n a l frequency of t h e i t h .ode n o r n a l i z e d .lop a t r a t s g y r o a t a t i o n 10.00503 rad/in., -0.01986 rad/in.) % n o r u l i a e d airvane deflection 10.315 i n . / i n . , -0.1485 i n . / i n . ) normalized a i r v a n e s l o p e 1-0.02 r a d / i n . , 0.03308 r a d / i n . ) lt. e f f e c t i v e mass 40.0514 l b - s e c 2 / i n . , -0.03131 lb-sec / i n . )

Defining

%mi Klsi

XBL

I n

Xm

%

% I,

-

-*

m i s s i l e molaent of i n e r t i a 11.75 (lb-ft-sec2) l o c a t i o n of missile ffi (5.885 f t ) l o c a t i o n of missile hinge l i n e (11.3108 f t l mass of one f i n (0.0489 slupm) d i s t a n c e between ffi of on8 f i n and hinge l i n e 10.084 I t ) m e n t of i n e r t i a of m e f i n (0.00683 lb-ft-s.c2)

g a i n of s t r u c t u r a l p a t h of the I t h moda

qu*dIatlC a s r o Of the i t h .Dde a t r u c t u r s l path

the general transfer funaion from b lo body rate for each mode is given by

404%

- 24 41"

Sec. 2.2

Target Signal Processing ECM Target Dynamicl&

+ Sewor bra

HITMU

1

Relative ~ c o r n c n y Compurarion

-Target Stale

4 Sellsor Models -el Filtering &A Estimation

Esbres

t Target Starc Pmdicnon

T

rare Esnmator lnrduaoo

Target Tracking Processing

Figure 2-8

The most common measurements of target tracking systems consist of the range R, range rate R, azimuth u, or elevation u, angles (see Figure 2-9). The three coordinates to specify the position of the target are:

1. The x , y , z components of the tracking sensor-target range R 2. The azimuth u, elevation u,, and range R. 3. Direction cosines of the vector R and its length. A tracking sensor or seeker mounted on a homing missile or an aircraft is a device to detectltrack the relative position vector of a target with respect to the missile or the aircraft. The function of the tracking sensor is to receive external signals that are sent to the pursuer vehicle for the purpose of directing its course. Because of its role in providing an essential communication link to the external environment, the sensor becomes the key element of the entire vehicle system. In fact, a vehicle is given a particular identity by the choice ofsensor. The source ofenergy responsible for the signal that is picked up by the sensor may be energy reflected from the target, energy emitted from the target, or energy from the target's own environment. Assuming at this point that the sensor is capable of recognizing hot targets, or targets that emit infrared energy, then the sensor will be selected on the basis of a certain frequency or frequency band emitted by the target and will be optimized to respond to the target signature in question. Given the relation of the sensor to the successful operation of a vehicle, it becomes apparent that the aforementioned optimization to a great extent affects the design of all smart weapons. Although there are other important components besides the sensor'in the vehiclelmissile, the ability to neutralize the sensor reduces the vehiclelmissile to something less than a smart weapon [Heaston and Smoots, 19831. One of the requirements of a sensorltracker is to acquire and track all modes of target day or night. T o do this, an air-to-air tracker uses a laser-auadrant tracker and a centroid track. whereas an air-to-ground tracker uses a scene track, target track, and track adjust. A sensorltracker must accurately update target position relative to IP, OAP, and VRP,

Tracking sensor/guidance comparisons. L

MDASE of NGC PrOCesSlng

I

ELEVAT 1ON VIEW

Chap. 2

PLAN VIEW

(A)

ALTITUDE (BAR0 OR RADAR ALTIMETER) DIRECT

ANGLE RATE ( E - J TPACKER)

Figure 2-9 (a) Visual Line-of-Sight (LOS) Angle Measurements Determine T w o Coordinates of Aircraft-to-Target Position (b) Determination of Slant Range (c) External Grid Reference Targeting (Courfesy 0jJ.S.Ausman, 1986)

permitting target identification day or night. Low-altitude ranging sensors for NGC algorithms include redundant range measurement and accurate range measurement during terminal maneuver. Sensors used in the capacity of seeker, depending on their operating wavelengths, are typically characterized, depending on the sensor's operating wavelength,

SIX. 2.2

Target Signal Proceulng

29

MULTllATERATlON OR TDOA FROM 3 OR MORE GROUND-BASED, AIRBORNE, OR SATELLITE STATIONS GROUND-BASED RADAR

AIRCRAFT TO RGET

/

=

\

X

Figure 2-9

TARGET

(Continued)

as optical sensor, infrared (IR) sensor, synthetic aperture radar (SAR), millimeter

wave (MMW), laser radar, television (TV)ivideo, microwave, antiradiation homing (ARH), acoustic sensors, and multiple sensors. While . ~. of good .quality . . . . .and high resolution, an optical sensor nonetheless can suffer degraded-performance due to weather conditions or time-of-day effects. An IR sensor is designed for use at day or night, in conditions of rain and/or smoke, and is capable of hot-spot detection. I t is also well suited for use against armor. and high-value and ship targets, and has reasonably good resistance to ECM. However, this all comes at a high relative cost. Like the SAR, the MMW sensor is designed for all-weather operation with the capability of cloud penetration, and is particularly well suited to bad weather conditions. MMW sensors can be used against armor and high-value targets, but are not designed tor use against ship targets. They have a moderate resistance to ECM, and are not unduly expensive to build. The laser radar, which utilizes three-dimensional distance information and surface reflectance, is desigaedfouse.atday or night against armor and high-value targets. While not designed for use against ship targets, it has a high resistance to ECM and comes at a relatively low cost. Acoustic sensors are designed to function in underwater-erivironments. A T V sensor cannot function at night and is designed primarily tor use against armor targets. It .,

,

MDASE of NGC Processing

30

Chap. 2

is not very useful against high-value or ship --- targets. -.Although TV sensors have a low resistance to E C M , - & ~are ~ relitiVdy inexpensive. While a forward-looking IR (FLIR) imaging sensor provides a different mode of target detection and recognition, a laser provides a different mode of target range and track. A video tracker microprocessor FLlR consists of sensors and ancillary electronics as well as video processing. Microwave sensors can function in all types of weather conditions and are designed for use against high-value and ship targets, but not against armor targets. They have low resistance to ECM and are relatively expensive to build. ARH sensors are designed for all types of weather conditions and are only moderately expensive to build [Hardy, 19861. Finally, multiple sensors act to process multisensor information. The interceptor/missile uses sensor data that are processed by the seeker, which also acts to ensure that the sensor is receiving as much target information as possible. Related to this last function, the seeker actually orients the sensor so that it can survey, acquire, and then lock-on and track the target. The important components making up the seeker include [Heaston and Smoots, 19831: energy-gathering system, that is, antenna for radio frequency or lens/mirror for IR, visible, and ultraviolet spectral regions, including radome o r IRdomel~vindow stable platform and its associated control system (body-fixed seekers employ inertial sensors for an equivalent function) sensor to convert the received energy into a more usable signal for processing signal processing system used to produce a signal to point the antennalmirror at the target and/or to provide signals to ultimately control the missile's trajectory Rather than focusing too much attention on any particular tracking sensor or seeker, it is more advantageous to consider their fundamental properties. When, however, the problem is that of conducting a precise system analysis, then a specific dynamic model is assumed for a particular seeker, and this is integrated with that o f the pursuer. Three homing configurations are assumed: passive, active, and semiactive. In the passive case, only the LOS angle is available through an 1R seeker. In the active homing case, both range and closing-velocity radar information are available through measurements, but in the semiactive mode, these measurements are bistatic.

2.2.1 Targeting The presentation on targeting in this section follows an excellent lecture by [Ausman, 19861. Position sensor data must be available for targeting so as to be able to locate the pursuer with respect to the target. One or more coordinates of pursuer-to-target relative position is measured with a targeting sensor. This device may also be used

Target Signal Processing

Sec. 2.2

31

to locate the pursuer with respect to a pseudo-target o r offset aim point, the position of which with respect to the real target is known beforehand. Various types of targeting sensors with examples of each are listed in Table 2-4. A particularly common typc of targeting sensor is one that measures the azimuth angle and depression angle of the pursuer-to-target LOS. Examples of this typc of sensor system include gunsights and headup displays (HUD) in combination with an attitude reference. As shown in Figure 2-9a, these sensor systems measure just two angular position coordinatcs. A slant-range sensor such as an air-to-ground ranging radar or a laser range tinder can be used to directly measure the third coordinate of pursuer-totarget position. This is illustrated in Figure 2-9b, which also shows two additional more indirect techniques for determining the third (slant-range) coordinate. Pursuer altitude h can be measured using a barometric altimeter. Subtracting from this the target's altitude hT, assuming knowledge of the latter, results in the quantity h - h ~ A. radar altimeter, however, can measure h - hT directly in a target overfly. If it is not possible to realize a preparatory target overfly, the pursuer must accept thc difference in terrain elevation between the target and the spot below the aircraft at release or pickle as an additional measurement error. Slant range can also be obtained indirectly from LOS angle ratc measurements. One example of an angle ratc measuring device is the Norden bomb sight used in World War 11, which relied on manual tracking during a long, straight, and level bombing approach. By using EO trackers, modern angle rate sensors avoid the long, straight and level runin. The last entry in Table 2-4 corresponds to the external reference type of targeting sensor, which is illustrated in Figure 2-9c. The typical application of this sensor type is a blind bombing of known coordinates. As will be shown in Chapter 5, examples of external references that can locate the pursuer with respect to the target, assuming the same reference system is used to specify target coordinates, include ground-based radar, LORAN, TACAN, DME, and GPS.

TABLE 2-4

TARGETING SENSORS

Generic Tyve Line-of-Sight ( L a ) Angles Azimuth o and Elevation o,

Range R Altitude h LOS Rate b, bc Range Rate k = -Vc External Reference Courtesy 0fJ.S. Ausman, 1986

Exam~les unsight, T kpping ~ a d a r , ' ~ Sonar. ~. Optical, Video Tracker Radar, Laser, Sonar, Optical Instruments, FCS, Passive Ranging Processor, Onboard Active Sensor Baro-Altimeter, Radar-Altimeter EO Tracker Doppler Radar Ground Radar Director LORAN, TACAN, DME, GPS

MDASE of NGC Processing

Chap. 2

Line-of-sight (U)S)sensors.

Common to all LOS sensors is that some type of image of the target area is displayed to the weapon system operator. The operator then selects the target by positioning a marker or pair of crossed reticles over the target and depresses a target designating button. The computer responds by recording the azimuth and depression angles of the target marker (pipper). What the operator sees may be the real visual scene, or it may be an electronic reproduction using TV, IR, or radar sensor subsystems. The resolution and accuracy ofthe shorter wavelength systems (optical) are generally better than that of other systems. Accuracy may be improved by using scene magnification, which can also aid in target identification.

Range sensors. A range sensor, as shown in Table 2-4, provides t w o means of measuring slant range from the pursuer to the target. For weapon delivery, it is not necessary to have extremely accurate range measurement since the bomb miss distance for typical dive deliveries is on the order of one fourth of the range error responsible for the miss. That is, a range measurement error of 100 feet causes a miss distance of only 25 feet, typically. What may at first appear to be an inconsistency is actually a consequence of the fact that the range error causes correlated errors in ground range and in altitude that partially compensate each other. As an example, a very large slant-range measurement causes the computer to estimate the aircraft to be further from the target (short impact) but at a higher altitude (long impact). It should be noted that the range R, the quality of information concerning which would be very important if it were being directly utilized in the solution of the NGC problem, is not an easily measured quantity. For missile applications, the explicit range information may be obtained from either the launching platform or launching shiplground defense site, as when the missile and target are illuminated by the launching apparatus, or from the target sensors listed in Table 2-4. It still may happen, however, that range information is not readily available, or that the price in terms of space andlor dollars associated with the tracking and guidance hardware to acquire reliable range information is prohibitively great. Such can be the case with smaller missiles. Thus, it is advantageous to derive guidance algorithms that do not require range information. Altitude sensors.

Inasmuch as it is a component of the pursuer's flight instrument package, the barometric altimeter provides a convenient targeting sensor. However, by reason of its rather doubtful accuracy, it usually serves as a backup to be used in the event that the slant range sensor ceases to function correctly. Assuming that local barometric pressure is accounted for, the largcst altimetry errors are typically those arising from uncompensated stat~cpressure defect, which is a bias type error that is on the order of a few pcrccnt of the dynamic pressure, and to uncertainty in lapse rate, which is a scale factor typc error that is on the order of a few percent of the pursuer's altitude above mean sea level. Besides altimetry errors, any unccrtainty in the target altitude will result in an error in the computed value of the pursucr's altitude above the target. While this unccrtainty is removed

Sec. 2.2

Target Signal Processing

33

on a bombing range, it can be significant in actual operational usage. Consequently, one expects a certain amount of degradation in accuracy between bombing range results and operational results with this type of bombing system. It becomes unnecessary to know target altitude with a radar altimeter since this measures altitude above terrain directly. Given that the pursuer is not usually situated directly above the target at the instant of release, the flatness of the tcrrain between the release point and the target to a large extent determines the bombing accuracy associated with a radar altimeter. For this reason, it is advantagcous to make a dry pass over the target for a relative altitude update before proceeding to the attack. This, however, is only possible if the threat environment permits.

LOS angular rate sensors. In an angular rate targeting scheme, the relative accuracy of the slant range computation is directly proportional to the relative accuracy of the angular rate measurement. A 1 percent error in measuring angular rate therefore results in a 1 percent error in slant range. For a typical depression angular rate of 20 milliradians per sec, this accuracy translates to 0.2 milliradians per sec, a level not easily achieved with current state-of-the-art trackers. The great majority of angular rate sensing systems that have been built so far are of the T V contrast tracker type. In addition to requiring three-axis stabilization, these sensors have a response time that is scan rate limited. Owing to this last limitation, these devices can measure angular rates only so accurately. The T V contrast tracker is also characterized by the requirement of a single, high-contrast point to track.

Range rate (closing velocity). Determination of the closing velocity V , is done with the aid of a Doppler tracking device. Doppler frequencies, which are one of the principal outputs of the radar sensor, are proportional to V,. Continuouswave-illumination homing systems can exploit the Doppler frequency of the target return to generate a good estimate of V,.

External references. Each of these targeting sensors is a navigation aid, which locates the pursuer in an electronically generated grid or coordinate system. The rest of the targeting task centers around locating the target in this same electronic grid or set o f coordinates. One way in which to implement this type of targeting is to use radar ground control to direct the pursuer's flight, which requires that the pursuer be within LOS of the ground radar. The pursuer, in order to maintain this LOS, must fly at relatively high altitude, particularly when it is very far from the radar. Moreover, accuracy degrades in proportion to the distance from the radar. Other LOS positioning systems include TACAN and DME, which generally rely on at least two ground stations and use triangulation and/or multilateration to establish the pursuer's position. T o a large extent, accuracy is determined by the relative geometry between the pursuer and the several ground stations, also known as geometric dilution of position (GDOP). G D O P has a greater influence on the altitude coordinate than on the horizontal coordinates, unless the pursuer is situated directly overhead a ground station. This is not likely to occur, however, during a

MDASE of NGC Processing

34

Chap. 2

bombing run. The LORAN system is capable of obtaining coverage beyond the LOS and at low altitudes because it uses ground-wave transmission oflow-frequency radio waves. However, since it is only a two-dimensional positioning system, LORAN must rely on another method such as a baro-altimeter to obtain vertical targeting information. When GPS is operational, it will be able to position lowflying pursuers in all three dimensions since its reference stations are satellites. With the GPS system, each satellite can maintain LOS to almost half the earth. 2.2.2 Kinematic/Relative Geometry

Because only kinematic equations are considered in the kinematic method, the pursuer may be treated as a geometric point as illustrated in Figure 2-10. The corresponding guidance is therefore ideal, and the character of the pursuer trajectory can be determined approximately. Moreover, the required accelerations may be estimated. The kinematics of a pursueritarget scenario are contained in the change in time of the vector from the pursuer to the target R. In other practical applications, the kinematics are characterized by two pairs of kinematic equations, one of which determines the relative motion of the centers of mass of the pursuer and target in the horizontal plane, and the other of which determines this motion in the vertical plane. The geometry of the homing guidance in the horizontal plane is described by the following relations:

where, for convenience of discussion in this section, the subscript true in u,,, is omitted in this section from this point on. The nomenclature is displayed in Figure 2-10 and the measurable variables are 2 I12

R = (x? + y,)

,

o = tan-' y , / x ,

(2-3)

The closing velocity V , = - R of the pursuer and target is determined by Equation (2-2a), while the LOS rate u is related to the parameters of motion of the pursuer and target by Equation (2-2b). A set of analogous equations is used for the vertical plane. Differentiation of Equation (2-2b) and substitution of Equation (2-2a) yield R6

+ 2db

=

-

VT

sin(u

- yT) +

V T cos(u

- ~T)+T

(2-4)

+ i', sin(u ,, - y) - V,,, COS(U - y)+ Because kinematic Equations (2-2) and (2-4) are nonlinear differential equations, they render it a difficult process to study and analyze guidance loops. The analysis can be facilitated by linearizing the kinematic equations relative to the reference trajectory of the pursuer, using approximation methods. With initial values given by YT, and yo, theangles y~ and y are perturbed during the engagement by y, and y, where y, and y,,, are very small compared to yr,, and yo, respectively. Consequently, the instantaneous angles of the velocity vectors are y r = y ~ + , y, and

MDASE of NGC Processing

36

Chap. 2

+ yn,. A

l y simde -case, which is of course linear as well, occurs when sin a zz u , cos u = 1, s i n y,, cos y, = 1, sin y, = y,,, and cos y,, = 1. The L O S angle in Figure 2-10 can be approximated as:

y = yo

where t, = tf - t = time to go

and

t f =- flight time

For analysis purposes, assuming the intercept geometry is simplified to a head-on engagement with constant closing velocity, the range is determined from R

= Ro -

VCtZZ V,(tf - t) = I/,!,

(2- 5b)

where R, = V C t fThe relative velocity and acceleration are j/r

=

YT y,

- jrn = VT Y I cos Y T ~- V,n ~m cos =

yT-

YO

y,,, = AT" - A m y = V T j T - V,,,?

(2-SC) (2-Sd)

where

AT^

=

y ~ =. actual target lateral acceleration = V T jT cos y ~ , = ! VT jT

A,,

=

,

=

actual pursuer acceleration = V,,,j cos yo

--

V,, j

(2-5el

Equations (2-5) are linear equations representing the dynamics of the engagement. The signal that results from subtracting A,,,,, from the simulated AT, is integrated twice to generate y,. With this information, Equation (2-Sa) can then be used to obtain the true L O S angle a,,,,. Assuming VT and V,,, are constant, and hence V, is constant, differentiating Equation (2-Sa) leads to the following formula for the L O S rate u: Rewriting the equation with the help of the geometry yields yr = Vr&u - Vru =

~ J T

jm = VT y, cos y ~ "- V,,, y,,, cos yo

(2-7)

or in block diagram form shown in Figure 2-lla. Differentiating Equation (2-7) yields

y,

= V,t,U

- 21/,u = AT,. - A,., = VT jr cos y n , - V,,,j cos yo

(2-8a)

or, in a more general form, y , = RU

-

~ V , U= AT" - A","

(2-8b)

By applying the assumption made so far, Equation (2-4) can be reduced to Equation (2-8b), while Equation (2-7) can be derived from Equation (2-2). The component of the system for y , to I?, based on Equation (2-8b),can be modeled with an in-

Figure 2-11 (a) Kinematic Eqi~ations (b) Guidance Loop

tegrator contained in the positive feedback loop, as shown in Figure 2-11b. The kinematic element, however, becomes unstable on account of the presence of positive feedback. That is, the output signal u grows continuously for a constant input y,. A negative feedback (A,,,) loop is consequently added to the model in order to counter the effects of the positive feedback. The guidance law, FCS, and the pursuer dynamics and propulsion are all contained in this loop. From Equation (2-8), AT, is given by

AT, = (Rs

- 2V,)u + A ,,,, =

V<(t,?s - 2)u

+ A,,

(2-9)

2.2.3 Targeting Sensor Dynamics For the active RF seeker and the passive IR seeker, two gimbals (that is, azimuth gimbal and elevation gimbal) are required. In order to describe the generalized seeker tracking dynamics, the RF seeker with an antenna dish is used as an example. Figure 2-10 shows the essential seeker angles and geometrical quantities. As shown in Figure 2-12, due to the presence of lags in the seeker tracking loop and radome refraction effects, the seeker is not pointing directly at the actual target [Murray,* 19841. Figure 2-13a illustrates the tracking loop that describes the dish dynamics in the absence of radome effects and having the seeker bandwidth w, = 117,. In the case of small track loop time constant T , , the seeker dish rate command u; and the LOS rate are approximately equal. Care must be taken to ensure that T, is adequately small to be capable of tracking highly maneuvering targets, but not so small as to induce excessive noise transmission or a stability problem. The stabilization represents a closed-loop system encompassing the dish rate feedback loop.

MDASE of NQC Processing

-

Chap. 2

RADOh ERROR

Figure 2-12 Radome Error Distorts Boresight Error (From [Munay, 19841 with permission . t o m AACC)

Radome coupling and compensation loops. The increased miss distance associated with radar-guided air defense missiles can be blamed on radome refraction error. Miss-distance predictions for nonlinear radome error are historically less than that predicted for a linear radome error with constant slope [Yost et al., 19801. As shown in Figure 2-12, the effect of radome error, a,,is to distort boresight error measurement E. a,,which is the amount by which the direction of the apparent Total LOS Angle Noise v,

Gyro Acceleration

(B) Figure 2-13 Effect

rsuer laation

Sensitivity

Seeker Block Diagram (a) Without Kadomc Effect (b) With Radomc

Sec. 2.2

Target Signal Processing

39

target and that of the true target differ, is a nonlinear function of the gimbal angle q,. Differentiating u, with respect to u, yields the radome slope r, which is not a constant but varies inp predictably with angle-off and from dome to dome. u, itself depends on both pitch and yaw gimbal angle [Murray et al., 19761. Here, the effect of u, on miss distance can be shown quite well using only a single-plane guidance model [Murray, 19841. The radome error a, arises because RF energy passing through the radorne to the seeker antenna is refracted by the dome material. The magnitude of refraction during homing depends on many factors including radome shape, fiticness ratio, thickness, material, tcnipcrature, operating frequency, and polarization of the target 'echo signal. . Other sources of error such as standing waves inside the dome, reflection from gimbals, receiver phasing, and processing errors also contribute to the radome error. All of these factors differ with each flight and therefore cannot be specifically accounted for in the computer algorithm. As a result, the radome error magnitude can neither be precisely measured nor predicted. This is not to say that it cannot be modeled. The most serious aspect is that the magnitude of refraction error is also a function of where the target energy impinges the radome. This causes in-plane and cross-plane radomc errors to become look-angle dependent with the slopes also depending on the roll attitude of the in~erceptor.Specification of radome characteristics involves the radome slope r, which . ~. is ~shown to be-the main parameter in the homing loop. The most optimistic situation is o i i in which the designer would be able to specify the manufacturing tolerances and the limits on the allowable variations of r. As shown in Figure 2-2, if the nonlinear kinematic guidance loop is to include the radome error coupling loop, then the missile body pitch angle 0 must be generated by Equation (7) of Table 2-2. In addicion to being used to generate 0 in Figure 2-2, A,,,l is also applied to both the guidance filtering and guidance algorithms. In this study of targeting sensor dynamics, a PNG system is assumed where the missile's antenna provides a measure of the target bearing u as follows. Looking at Figure 2-10, there is an angle a d called seeker dish made by the seeker antenna's center line with the inertial reference. The geometry tracking error without radome error is ~ , , r f e ~ ~= ulrue- ud (see Figure 2-13a). The radar converts this error into a voltage, which is then filtered. This is done in order to achieve the proper driving signal that will precess the seeker antenna in the direction that acts to reduce ~,,rfe,, to zero. The seeker dish angle and the LOS angle are equal in the case the seeker tracks the target precisely. However, the actual angle tracking error E = ~ ~ ~ +f u, v, (see Figure 2-13b, v, = measurement noise) is the angle between the apparent target LOS and the seeker center line measured by the gimballed seeker dish. The seeker center line is positioned by the executive computer system and periodically updated based on the reconstructed LOS angle 6. The pitch angle 0, measured by the IRU, has added to it the actual gimbal angle u2measured by the gimballed pickoff. This provides UJ (that is, ud = 0 + a,), which is then added to E to obtain the measured LOS angle u as follows: ~~

~

+

u = e

+ ud = brmr+ u, +

= LOS angle measurement

(2-10)

i . ~

MDASE of NGC Processing

40

Chap. 2

where the apparent LOS angle u as seen by the homing eye is equal to the true LOS angle plus the aberration angle u,, as shown in Figure 2-10. In order to simplify the analysis here, it is assumed that u, is a linear function of the gimbal angle u, such that

Making a correction to include radome effects in Equation (2-lo), the tracking error measurement (see Figure 2-10) then becomes - ~ -

r , e - e - ~ g + ~ r + V , = ~ ~ - u , + ~ , + ~ ,

Equation (2-11) is substituted into Equation (2-12) to yield

The angular relationship of Equation (2-13) and the dish dynamics from Figure 213a are employed to develop a seeker block diagram with radome effects shown in Figure 2-13b. Since the radome-induced error here depends not only on the LOS angle but also on 0 as well, 0 is used in the radome compensation algorithms for stabilizing the (negative) radome error slope. T o enlarge the stability region for the positive error slope and for cross-plane coupling slopes, the radome compensation algorithm is inserted at the output rather than the input side of the guidance processing algorithms [Yueh, 1983(a); Yueh and Lin, 1984, 1985(a)]. Integrating 0 to get 0, the latter quantity is then multiplied by the radome error r and used in the radome coupling block of Figure 2-2 to compute the LOS angle measurement. Combining Equations (2-10) and (2-ll), another simplified equation that can be used to calculate the LOS angle measurement is

where a,,,,,is given by Equation (2-5a). The look angle is defined as Assuming radome aberration error u, is explicitly dependent on the look angle U L , then the time derivative of the measured LOS [Equation (2-lo)] to the first-order linearization approximation can be written as

where the radome local slope is The importance of the seeker as a missile subsystem becomes evident when one considers what it must be able to do. The seeker must be able to track a target, remain spatially stabilized or go into a search mode if target track is lost, and measure

Sec. 2.2

Target Signal Processing

41

a guidance signal. Most seekers have a stabilization loop (see Figure 2-13b) to isolate body rotations and track loop to maintain the seeker axis along the missile-target direction. These loops, in conjunction with the appropriate signal processing, permit the measurement of the guidance signal. The knowledge of the seeker dynamics together with information about the radome characteristics and noise sources would be required to develop a detailed model of this measurement process. In order for the tracking and stabilization functions to work properly, the seeker slew-rate capability for a BTT missile must be compatible with thc missile's banking motion, being ablc to deal with potctltially large look angles. It is important also to assess the impact of banking maneuvers on track loop requirements. A detailed seeker analysis must include modeling of the inertial coupling between the two seeker channels and the effects of friction. When analyzing missile banking maneuvers, one must also consider the effects of polarization. In the case of a semiactive, linearly polarized seeker with linearly polarized radiation, the seeker gain becomes zero when the polarizations are 90 deg out of phase. This situation can arise, for example, when the missile undergoes a 90-dcg bank. Also, as the relative polarization of the seeker and incoming radiation change, there can be serious problems having to do with changes in radome aberration error. That is, if the radome is designed for a particular polarization, a change in this angle may produce larger aberration errors. An active seeker may also be subject to polarization effects when used against a large target such as a ship, for example. If the target's reflectivity is a function of the signal's polarization, the aim point may be altered as the missile banks. Should the guidance system mistakenly interpret this rapid change in aim point as a spike in the LOS rate, it may cause the missile to miss the target [Riedel, 19801

Target tracker (seeker) modeling. The LOS rate u is corrupted by bias ub and gyro sensitivity as shown in Figure 2-2. It is convenient at this point to consider the tracking loop time constant T,, the stabilization loop gain K,, and the gyro sensitivity gain K?,as shown in Figure 2-13b. In general, if the tracking loop time constant T, is too large, the seeker will be unable to track highly maneuvering targets. If, on the other hand, this constant is too small, there will be problems associated with excessive noise transmission and, hence, a stability problem. T h e tracking loop time constant must therefore be chosen somewhere between these two limiting values. he stabilization loop gain K, which is the loop crossover frequency should be large (that is, perfect seeker dynamics) so as to speed up the guidance system, with a practical upper bound placed on it so as to stabilize the seeker well. Because it will have almost no influence on the seeker track loop dynamics if K, is large, the seeker dynamics can be approximated by

+

Without radome effect (see Fig. 2-13a) u'dlu = W,S/(S w,) With radome effect (see Fig. 2-13b)

u'd

= [(I

+ r)u,,,

- re] w,/(s

+ w,)

(2- 16)

that is, a,,,, is passed through a target tracker (for example, seeker), composed of a differentiator followed by a first-order lag, to produce a seeker dish rate command which is approximately the LOS rate I?.

MDASE of NGC Processing

42

Chap. 2

Tracking error guidance system seeker model. The transfer function of the seeker as it appears in Equation (2-16) is more representative of a tracking error guidance system shown in Figure 2-13b. Figure 2-14a illustrates a typical tracking error guidance seeker loop in which an error signal is produced by the angle error detector and is proportional to the misalignment of measured target angular tracking error E. This error signal'is then input to a proportional plus integral amplifier whose transfer function is A rate gyro feedback must be included in the design in order to improve the stability margin in the Nyquist sense and to help reject outside disturbances [Garnell, 19801. The output of gimbal dynamics plus pitch rate 8 is the antenna (dish) rate ed in body coordinates, and, since the feedback is defined as dish angle and not rate, the

Total LOS

Angle Noise

v-

Anele Error

Power

Amalifier

Rate Gym

1

-,---~ Sensitivity

(B) Figure 2-14 (a) Typical Tracking Error Guidance Seeker Loop (b) LOS Rate Guidance System Seeker Model

Sec. 2.2

Target Slgnal Rocesslng

43

loop is closed by the implied integration of u d . An evaluation is now made of the quantitics T I and K I K2K3 which fall under the designer's control. The generalized tracking guidancc system seeker model is illustrated in Figure 2-13b in which the tracking loop time constant T, is proportional to the torque loop gain TI of the seeker. The PNG guidance system transfer function is approximated by u

=

V [ (

+ T , s ) ( ~ + TJ)]

(3-17b)

LOS rate guidance system seeker model. An alternative to the tracking error guidance system sccker model is the LOS rate guidancc system seeker model which measures LOS rate u. The target is tracked by the seeker antenna assembly, which is stabilized with respect to missile body motion. The measured LOS rate is given by the sum of the time derivative of the angular tracking error E and the antenna rotation rate u d throughout Equation (2-10). That is, Thus, to obtain a measurement of u, the signal processor provides i ( s u,) and seeker-mounted rate gyros provide u d . The measured LOS rate u is actually very close to the geometric u,,,,. There are roll-stabilized radar seekers that generate the LOS rate u for each axis using Equation (2-183). While it can be shown that the measurement of u becomes independent of the seeker track loop dynamics under this operation, it is also true that this method may lead to noise amplification owing to the process of differentiation. Moreover, the differentiation is electronic and is performed on a signal measured in seeker coordinates. The process of generating u using Equation (2-18a) together with all seeker low-frequency signal processing must be examined in the context of being suitable when the missile and seeker cease to be roll stabilized, but can bank at a rate of a few hundred degrees per second. A LOS rate guidance system seeker model is presented in Figure 2-14b. The PNG system transfer function is approximated by

Parasitic feedbacks.

As illustrated in Figures 2-2 and 2-13b, bodyrotation rates always have the effect of corrupting LOS rate measurements. Imperfect sensor stabilization or unmatched signal processing gains are only two factors that may act to produce undesirable perturbations on the LOS rate. A parasitic loop may develop in which body-rate induced perturbations on the guidance signal cause a perturbation in the commanded and achieved accelerations and, hence, additional body rate. The result of this situation is that the system can become unstable and may suffer degraded performance [Riedel, 19801. Both the body rate and body acceleration are parasitic feedbacks, owing to the necessity of an aerodynamic missile to pitch to an angle of attack to be able to maneuver. Because of the radome refraction effects, the autopilot and seeker dynamics are coupled through the missile bodyrate feedback signal. This creates an attitude feedback loop in which the missile responds to a target LOS change by maneuvering. The resulting rotation of the

44

MDASE of NGC Processing

Chap. 2

missile body causes an apparent additional change in the LOS angle, which closes the loop. This feedback increases as the altitude increases. When the values o f the radome slope I. are large, they represent the worst effect of an undesirable feedback path through the gimbal angle as shown in Equation (2-13). Beside the radome effects, the stabilization loop is fed with a false angular rate due to missile maneuvers. The gyro acceleration sensitivity drift produces a direct unwanted feedback path with a gain K, from the achieved acceleration to the measured antenna rate in space. If the gyro mechanism is not perfectly balanced about the output axis, any linear acceleration normal to that axis to which the gyro is subjected will cause an erroneous rate signal to be generated. Such a signal will result in a tracking error that in turn commands an additional lateral acceleration to close the loop. This effect was first brought to light during a flight test in which the missile flew a helical path to an excessive miss distance. After conducting an analysis, improved mass balance procedures were put into effect. In addition, the gyro was mounted in a direction that minimized the miss-distance impact [Fossier, 19841. The preceding two parasitic feedbacks become important at high closing velocities occurring in short-range or high-altitude flights. Thus, the design of the seeker must take into account the seeker's own performance characteristics and provide sufficient stability margins to handle unwanted feedback paths. As will be shown later in Examples 3-3 and 2-4, the effect of radome error feedback on the control system is determined by the sign of the radome error. The effect of a degenerative error (positive), which is one that tends to reduce the input LOS rate, is to slow the system response and to reduce the autopilot's stability margin. A regenerative (negative) error may bring about guidance instability at very low frequency. Although both can result in increased miss distance, a regenerative error is of greater concern inasmuch as low-frequency oscillations are characterized by large displacements of thc missile. In a situation somewhat similar to what occurs with the radome error, a feedback loop is formed by in~perfectionsin the antenna stabilization system (sec Figure 2-13b). This causes a change in the antenna direction as the missile body attitude, or missile acceleration, changes. In early systems. the stabilization loop consisted of rate gyros mounted on thc back of the antenna, with their outputs electronically integrated to drive a hydraulically actuated gimbal system. Owing to the servo's finitc frcqucncy response, the antenna is unable to kccp perfectly stationary for body motions a t frcqucncies of primary concern, and the resulting motion gcncratcs a borcsight error cquivalcnt to that caused by positivc radon~eslopc. One aspcct of the solution to this problem involved improving thc hydraulic valve rcsponsc so that the head stabilization loop could be closed to a high enough frcqucncy to rcducc this effect to acceptable icvcls. Another aspect of thc solution was clcctronic in nature and centered around the closed-out frequcncy of the stabilization loop. It was realized that this frequency mattcrcd only to the extent that it affects thc gain in the stabilization loop at frequencics of interest, typically 1 to 3 Hz, and that it could bc increased by changing the elcctronic integrator to provide a - 2 slopc instcad of a - 1 slopc, keeping in mind loop stability considerations. The resemblance to radome slope was brought into grcatcr focus when it was discovcred that purposely limiting the DC gain of

Sec. 2.3

NGC System Deslgn and Analysk

45

the stabilization loop, as opposed to allowing perfect integration of the rate gyro, created the same cffect as a positive radome slope. Thus, one could compensate for the negative radome slope at the very low frequencies that produced large miss distances. A rather serious accuracy problem was alleviated by the implementation of this simple compensation scheme. Chapter 8.4.3 presents the design cquations for PNG with thc previously-mentioned parasitic feedback. Thc cffect of body bending is yet another type of parasitic feedback. In contrast to bcing a separate parasitic loop, this cffect is simply a high-frequency autopilot instability in which body bcnding is detected by the autopilot as a motion ofthe missile. In early systems, the autopilot rate gyros were mounted as closely as possible to the nonrotational point for the first bending mode. This in conjunction with the electronic filtering employed made it possible to avoid autopilot instability [Fossier, 19841. Details of this structural vibration control are presented in Book 3. 2.3 NGC SYSTEM DESIGN AND ANALYSIS The N G C analyst works with all members of the design team. He or she must be able to work from simple to very complex analyses and simulations, and to provide preliminary and firm requirements and designs.

2.3.1 Guidance FilteringlProcessing

Simplified pursuer dynamics. A functional block diagram of the PNG kinematic loop is shown in Figure 2-13a with the guidance computer defined previously in Figure 2-6. Seeker dynamics and corruption of the LOS rates due to body motion coupling through the radome are modeled in Figures 2-lja(iii) and 2-15a(iv). Guidance Jdnematic loop. Consider the block diagram corresponding to Figure 2-15b which is simplified from Figure 2-2. The target position measurement is (2- 19) "' = Yr + Yg where y , corresponds to the target maneuver, which is generally considered as a random variable, and y, is the target glint/scintillation noise (generated by the input equivalent radar noise), which is a property of target radar return signal and arises from sources that physically enter the system. Noise inputs. The two target noise inputs in Figure 2-15b are denoted by y , and U ~ (angle N noise of the tracker). These noises, which corrupt the observation, induce miss distance, and impact pursuer acceleration requirements, are assumed to be uncorrelated white-noise processes. In Figures 2-15b and 2-2, u ~ ~ ( t ) is equal to u,,,(t) + u f ( t ) + uc(t).Represented as a linear displacement at the target, y , has a PSD of and, like miss distance, is divided by rhe range to become an angular glint noise u,,i,,, with PSD

+,,

MDASE of NGC Processing

Chap. 2

a) Simplified P u r s ~ Dynamics r TNE U S Angle

i)

I I qne I I

Guidance & N D ~ Gs ('1 Total LOS Angle N o i s

Guidance Conlpuvr G , ($1 Air+c

I *kc,

Dirh

FCS & Pmpukion

I I

Ratc Command

b;l-a

I

Pursuer mnamicr

Simplified Punuer D y o a k (See F i e a)

Figure 2-15 (a) Sirnplificd I'ursuer Ilynarnics (h) Trajectory Ilynarnics Model (c) Equivalent Trajectory Dynamics Model

to be input to the scckcr. It is pointed out here that, evcn under the assumption o f stationary scintillation noisc at the target, this becomes nonstationary at the seeker by virtue of division by R ( t ) . Target scintillation is commonly assumed to be aptly represented as stationary band-limited noise, thc amplitude and bandwidth being functions of the physical dimensions and motion of thc target. Inasmuch as band-

NGC System DesIgn and Analysis

Sec. 2.3

47

limited noise and filtered white noise possess the same statistical properties, the simulation can be performed by introducing a simple lag filter with ~ C L T= l/wg as a correlation time constant between w, and y , as shown in Figure 2-2. The PSD of y , is +,\.

= 27CLTrCLT, ~ C L T= glint noise variance

(2-2Ob)

The filter was excluded in Figure 2-15b as a matter of simplicity because it falls outside thc homing loop and has no effect on the system adjoint [Peterson, 19611. The net return signal from a target with many radar reflectors can be modeled as a movement of the apparent radar target position. It is typically modeled as a Gaussian random variable only with zero mean and variance rcLT. that is,

where (r~gET)is 112 to 118 the target's dimension perpendicular to the LOS. That IS, rg;T = WJa, where a = 5 is the rule o f thumb and can vary between 2 a n d 8 according to data, and W , is the wing span of the target [Alpert, 19881. Rangeindependent noise (KIN) uf is the noise inherent in the receiver. It is independent of target return power and is caused by many sources, only some of which include signal processing effects, cross polarization of returns, quantization effects, and gimbal servo drive inaccuracies. uf is modeled as of

- iV(0, rf) with equivalent white RIN PSD +f

= 2?frf

(2-20d)

where ~f is the RIN correlation time constant. Thermal noise a,,,is the receiver noise, which is a function of the strength of the target return, and hence the signalto-noise ratio. u,,, is modeled as

- N(0, r,,)

u,,,

with equivalent white receiver noise PSD +,,, = 27,,r,,,

(2-20e)

where T,,, is the correlation time constant of u,,,and r,, is the range-dependent noise variance governed by (Semiactive)

~,,,sA

= r,,,~(RIRO)',

(Active) r , , ~ = r,,~ ( R I R o ) ~ (2-200

where Ro is referred to as the target-to-pursuer reference range. The clutter noise u, is modeled as u,

- N(0, r,) with equivalent white clutter noise PSD +, = 2~,r,

(2-20g)

where 7, is the correlation time constant of a,. Glint noise, receiver noise, and RIN are the important contributions to tracking accuracy when radar is used. Decreasing the range to go increases the glint noise but decreases the receiver noise. The variance ~ T N and the spectral input + T N for the angle noise arNare determined by the characteristics of the seeker and are of the following forms, respectively:

MDASE of NGC Processing

48

Chap. 2

The total LOS angle measurement noise is then Vm

=

UTN

+

uglinr

=

urn

+ uf

f uc

+

Ug*llittr

(2-22)

The angle noise variance and spectral input are of the following forms: v, = v,,,

+ rf + r, + rCLTIRZ

In addition, the tracker noise, or.., can be added to target position noise, y,, so that the measurement equation is modified from Equation (2-19) as: z(t) = y T

+n

=

target position measurement

where n = target position noise = o , R = y, of n are of the following forms, respectively: r,, = +tt=

YTN

R2

+ R U T N , and the variance and PSD '

+ YCLT = [rm + r, + v , ] R 2 +

~ T , v R+~ 4 s =

[@r,z

(2-24)

+ 4f

f

+c]R2

+

ICLT

(2-25)

Thus, Figure 2-15b has been modified to become equivalent to Figure 2-15c.

Nonnoise inputs. The initial condition at the start of homing (launch, transient, and so on) is the nonnoise inputs to the system. The initial seeker error and the initial turning rate of the missile are both assumed to be constant random variables. In certain instances, either of these two errors may be zero. For example, the seeker initial error input becomes zero if the seeker is lockcd on and settled out at the start of homing. Similarly, the initial turning rate input becomes zero if the missile is not maneuvering when homing begins. Other quantities assumed to be constant random variables include initial heading error and detcrn~inistictarget maneuvering inputs [Peterson, 19611. NGC system design requirements and design considerations. Consider the possibility of choosing a guidance and control law such that the system within the dotted box of Figure 2-1% looks like a guidance filter. Then, pursuer lateral position output y,,, is controlled by the guidance computer and FCS to minT y , ) T ( y T - Y , ) ] , that is, minimum miss. The guidance and control imize E [ ( ~ system must get the missile close enough to kill the target when the warhead explodes. Therefore, an appropriate measure of guidance performance is miss distance. which is defined as the minimum distance between the missile and target. Besides minimizing miss, other goals that the NGC system must attempt to accomplish include maximizing the following: the range of operating conditions, the allowable environmental disturbances, the reliability, and the effectiveness. At the same time, the N G C system must be designed so that both cost and sensitivity to target tactics are minimized. When viewed from the standpoint of meeting these objectives, the guidance and control problem becomcs a matter of designing a compensation filter that optimizes system performance with respect to accuracy, stability, speed of response, and external or environmental disturbances. A high relative stability is

Sec. 2.3

NGC System Deslgn and Analysls

49

measured in terms of good gain and phase margins. Concerning speed o f response, the system must be suitably fast so as to wipe out the nonnoise inputs with small guidance time constant T,,. As for environmental disturbances, the filter must be able to filter whatever noise inputs arc present to acceptable levels [Axelband, 19861.

Advanced multivariable control system design. T o begin with, consider Figure 2-16 which describes a standard procedure for designing a robust, integrated control law (see Books 3, 4, and 5 of the series). After a nonlincar simulation modcl is developed and the requirements for robustness and performance are determined, the results are used to conduct an open-loop analysis. As shown in Book 1 of the series, the purpose of the open-loop analysis is to determine the significant dynamic characteristics and interactions of the airframe, propulsion, and control surfaces. It consists of nonlinear and linear simulation, as well as stability, covariance, observability, controllability, and frequency response analyses. The results of these analyses are used to define the design model. Once the design model is defined, control law synthesis can be performed to develop control law algorithms. Control law synthesis consists of: (1) feedforward controller design to enhance command response; (2) feedback controller design to satisfy stability and robustness requirements; and (3) controller order reduction to minimize implementation complexity. These results are then used to conduct closed-loop analysis which, as Figure 2-16 shows, also considers directly robustness and performance requirements. The closed-loop analysis, which makes use of the control law combined with the openloop simulation model, consists of nonliner and linear simulation, as well as stability, covariance, and robustness analyses. This is done to ensure that the control law

.

P

Model helupmnent ' Wind tunnel test L CFD. CSM * Physical laws SyPtem identification Model validillim

Design Specifieatim Military speciliatiow * Flyingqualities Stability requirements 4-M i u distance 4Performance

'

I c

Open L m p Analysh Design model Model reduction * System limitation

C

,

Conlml Syslem D e i g n CACSD Advanced guidanceand control laws 'Design knowledge h w

Simulation and Performance Evaluation l i m e and frequency responses I.inearand nonlinear Simulatim

Implcmentatia, Fast prototype Envirol~mmlled

Figure 2-16 Robust Integrated Control Law Design Procedure

50

MDASE of NGC Processing

Chap. 2

design satisfies control functional requirements. If these requirements are not adequately met, it may happen that further open-loop analysis is required. However, it may be that the algorithms developed in the control law synthesis stage simply need to be refined, in which case the design model does not have to be altered. Some of the methods available for advanced multivariable control law design are shown in Book 3 of the series.

Classical versus modem guidance and control. As shown in Figs. 2-6 and 2-17a, low-pass filters attenuate the noise component of the sensor signal, given the frequency characteristics of target signal and noise. Also, as can be seen from Figure 2-17a, separation principle allows one to design the optimal estimator and the advanced guidance law independently. Optimal estimators such as Kalman filters ideally separate the target signal from the noise by using information about target (and pursuer) dynamics and noise covariances. The objective of an optimal estimator is to provide accurate estimates of all states and model parameters required for the advanced guidance law without significantly increasing the sensor requirements (and therefore cost). The advanced guidance law provides the optimal steering commands assuming a noise-free environment. Figure 2-17b is a functional comparison of how the current modern approach differs from the classical approach to missile guidance and control. Classical approaches have become firmly entrenched in guided missile designs because they worked well in the rather benign environment of past target engagements and were easily implemented with analog circuitry. Because of such precedence, missile designers have often tried to satisfy the increased performance requirements of modern missiles by increasing the complexity of associated hardware such as seekers, gyros, accelerometers, airframes, and engines. Improving performance nearly always involves a trade-off between more sophisticated hardware or more sophisticated software. While such approaches in many cases have improved performance, the increased hardware sophistication almost always results in increased costs. In fact, the resulting cost has often been so high that the systems either were never developed for operational use or were purchased in small quantities [Gonzalez, 1979(a), (b)]. In spite of many classical terminal guidance and control laws which are still being used for tactical missiles, each characterized by varying degrees of performance, complexity, and seeker-sensor requirements, the increascd accuracy requirements and morc dynamic tactics of modern warfare render contemporary guidance and control laws unsatisfactory in many applications. Whilc many modern guidance and control system des~gntechniques have yiclded better solutions than those obta~nedfrom a classical approach, these remain nevertheless overly complex and are not well undcrstood by industry as a whole. As a result, advanced guidance and control sysfetn design, as extensively discussed in this book, is an ideal approach which combines the most attractive features of both classical and modern guidance and control system design techniques. This advanced approach, coupled with the advent of new theoretical methods and low-costlhigh-speed microprocessing techniques, makes it possible to provide better effectiveness, reliability, tremendous increases in missile performance, and greater leeway in other subsystem performance for the same or less power, weight, and cost.

See. 2.3

NOC System Design and Analysls

c

LINEONIOW1 U T E

Xl'W

C

FILTERED LINEOF. LlOn M T Z

Low C A N FILTER

C

CRWAV LAW

KCELERATIW COMMAND

-

FILTERED LlNEOFalOHT

MTE

OTHER SENSORS

LINEOFJIOHT RAT€

-

/L=.

L1

XltW

nus rson~orr~~

EIIIMATU

u

EITIMATOR

bDVU(CED OUlOANCE

U X E6ETlER LERATIW COYUAND

U W

TARGET

C L W K A L LIIROhCH NOISE

OUIDANCE LAW ~mo-wAv)

SEEKER

AUTOPILOT

AIRFRAME

MISSILE KINEMATICS

TARGET KINEMATICS

ESTIMATOR

MISSILE KINEMATICS

Figure 2-17 Classical vs. Modern Guidance and Control (a) Guidance Filtering and Processing (b) Functional Comparison ( ( a ) j o r n [Conzalez, 1979(b)] withpennissionjom AGARD ( b j j o m [Conzalez, 1979(a)], CC 1979 IEEE)

52

MDASE of

NGC Processing

Chap. 2

2.3.2 NGC Stability and Performance Analysis The guidance analysis model shown in all parts of Figure 2-15 is composed of elements representing the homing kinematics plus other systems that affect the homing guidance dynamics. The root-locus method and frequency response analysis can be applied to guidance system analysis. Performance analysis techniques are used to evaluate the NGC performance. The guidance system transfer function from a,,,, to A,,, is shown in Figure 2-15a. The stability analysis pertains only to the guidance system transfer function, whereas the miss-distance analysis will apply to the entire homing loop of Figure 2-15b. The homing loop shown in Figure 2-15b can be used to obtain a clear understanding of the requirements of an analysis model, and to obtain quantitative comparisons of miss distance as a function of the significant guidance system paran~eters.Such information is then used to evaluate the effects of N G C parameters on missile performance. Miss distance is brought about by the dynamics of the guidance system transfer function, the simplest representation for which is shown to be a fifth-order binomial [Nesline and Zarchan, 1984(a)]. The desir.ed/actual effective navigation ratio A, which varies due to changes in the flight condition, is one major determinant of homing system performance. Another major determinant of homing performance is the guidance system tinte constant T,, which is the sum of system constants. The ejfertivegrridance systenl titne constant, T;, is defined as the guidance system time constant with radome effect. Examples 2-3 through 2-5 later will discuss radome error and its severity on the guidance and control responsiveness of homing missiles. As will be shown in Example 2-4, the overall system stability needs to be studied before computing the time constant of the FCS because the timc constant is the major factor that affects the miss distance. The trend of the effect of cach feedback loop on the linear, time-\.arying gain systcm is clarified by the preceding discussion using root locus. The adjoint simulation is required because all the transient behavior in the miss-distancc response cannot bc predicted by only the simple root structure. The most efficient method of parametric study can be realized through a combination of the root-locus and the adjoint simulation techniques. The former is initially used to idcntify the region of parameters for system stability, while the lattcr is used to check thc conclusions dcrived from the root-locus study, concentrating on the optimum choice of N G C gains. -

Example 2-1: Guidance kinematic loop stability analysis. A vcry good guidancc stability analysis has already bccn put forth prcviously in Book 1 of thc scrics. Thc opcn-loop phase and gain frcqucncy rcsponsc charactcristics arc gcneratcd so that the stability charactcristics of thc guidance kincrnatic loop can be examincd. Once specific values arc assigned to cach paramctcr, the contribution to this responsc by the linear clcmcnts in this loop arc easily evaluatcd. The opcn-loop transfer function in Figurc 2-l5b with the simplified pursucr dynamics defined in Figurc 2- 15a(i) is

Substituting jo for s in Equation (2-26) for sinusoidal input yields for magnitudc

Sec. 2.3

NGC System Deslgn and Analysls

53

and phase

where GA(w) is the autopilot gain and L$A(o) is the autopilot phase response. Although the linear mode of the autopilot can easily be solvcd, it is difficult to analytically obtain the contribution to this response by the nonlinear autopilot. Thus, the nonlinear autopilot phase and gain characteristics are obtained through computer simulation.

Example 2-2: Comparison of statistical digital simulation methods. This example begins with the simplified model of an intercept homing loop shown in Figures 2-15a(ii) and 2-15b. The FCS and the airframe and propulsion dynamics are approximately represented by a single tag (7, = llw,) transfer function CA(S) = A,,lu, = 1/(7,s + 1) (although a second-order representation seems more reasonable) to include their effects in the guidance gains. The guidance system trans~ 5 T) , = T, + fer function without radome effect is A,,lu = AV, s/(l + ~ ~ where 7, + T,. Included in this simplified model are: (1) effective navigation ratio A = 3; (2) closing velocity V, = 3,000 ftlsec; (3) pursuer velocity V,, = 2,300 ftlsec; (4) 7, = l / w y = 0.1 sec; (5) seeker tracking time constant 7, = l/w, = 0.05 sec; (6) noise filter bandwidth k = l / i , = 10 radlsec; and (7) flight time t f = 3 sec [Zarchan, 1979(a) & 19881. The target maneuver model is assumed to be a Poisson jinking maneuver shown later in Figure 2-21e where B = RMS target acceleration = 161 ft/sec2, and A, = the target maneuver bandwidth = 0.2 sec-'. The PSD of the white glint noise u, and the white range-independent fading noise uy are denoted = 4 (ft2/Hz) and +f = 1 x 10-16 (rad2/Hz), respectively. This example is by used in Chapter 3 to demonstrate the utility of each of the performance analysis techniques via digital simulation.

+,

Example 2-3: Radome error-induced miss-distance predictions. Following Example 2-2 and Figures 2-15a(iii) and 2-15b, included in this simplified model are: (1) effective navigation ratio A = 3.5; (2) closing velocity V, = 4,500 ftlsec; (3) pursuer velocity V, = 2,500 ft/sec; (4) T, = l / o , = 0.45 sec; (5) seeker tracking time constant T, = l/w, = 0.1 sec; (6) T, = l l k = 0 sec infinity bandwidth; (7) aerodynamic turning rate constant T , = 2 sec; (8) flight time tf = 5 sec. The model developed by Murray [I9841 and presented here begins with the fact that system disturbances that cause miss distance result also in gimbal angle motion. The effective slopes r are reduced by this motion, while the effective noise q is increased. This example, largely fdllowing the excellent article by Murray [1984], looks at a linear radome error model that spans the gap between nonlinear and linear radome error miss-distance predictions. Included in this model are an effective slope and an effective noise variance. Gimbal angle motion, which is a function of guidance system disturbances such as target maneuver and range-independent noise (RIN), determines the effective slope and effective noise. In the treatment presented here,

MDASE of NGC Processing

54

Chap. 2

analytical expressions for effective slope and effective noise are derived for a sinusoidal radome error. The guidance system transfer function with radome effect is: 7;

'2 4 rn U

= effecti4e guidance system constant

A' Vcs l + Tis ' A! =

=

=1 + ~ A V / V , ) ' ( T T , ) I I (+~ A V A = effective navigation ratio 1 + rAV,/V,

V (2-28)

+ +

where T~ = T, T, T~ Figure 2-15a(iii) illustrates how the radome slope changes the missile system dynamics. The radome slope r appears twice in A,,,/u. A' is altered in terms of r effect by a factor of 1 over the quantity 1 + rAV,/V,,. For positive radome slope, A' is reduced and the response time of the system is increased, the effect of which is to increase the miss distance due to target maneuver. T h e opposite occurs for negative radome slope, the result of which is to increase the effect of A' and thereby increase the miss distance due to RIN [Fossier and Hall, 19671. System damping is reduced in the case of a negative radome error when higher-order terms are concerned. Figure 2-18a shows a typical miss-distance sensitivity to r. The figure shows that there exists a negative threshold value of r, below which miss rapidly escalates. This value depends on the flight condition through a/-$ and on the control system design through the value of T,. Given the variation of a/-$ with altitude, the value of T, must be increased appropriately so as to maintain acceptable performance [Fossier, 19841. The radome slope r also enters in the denominator of Arn,lu, where the time constant (s coefficient) T; is modified by a factor of r T,A V,/ V,,,. The effect of a positive r is to increase 7i, thereby increasing the miss distance due to target maneuver. A negative r., if it is sufficiently large, will cause T: to go negative, resulting in an unstable guidance system. The cffect of r on 7; is of greater concern at high altitude where T , is large. Given the parameter values, the missile system in Figures 2-lja(iii) and 2-15b gocs unstable for r = -0.045. This second-order missile system remains stable for all values of positivc r, a fact whlch is not significant for the following reason. By adding to the model a typical third-order autopilot as will be given later in Examples 2-4 and 2-5, together with a first-order noise filter, r = 0.095 is required for missile system instability. However, fcr positive slope values much less than 0.095, the miss distance due to target maneuver becomes unduly high.

Miss due to radome slope. Target maneuver and RIN are thc two disturbanccs evaluated. As Figure 2-18b shows, the miss distance increases with raFigure 2-18 (a) Representative Effect of Radome Boresight Error Slope on Miss Distance (b) Miss Increases with Radonie Slope (c) Miss Is Less with l'eriodic Radoinc (d) Target Maneuver Miss Is Less with Periodic lladome (e) Gimbal Angle Motion Deterniines Effective Slope and Effective Noise (f) Target Maneuver and Noise Cause Gimbal Angle Motion (g) Effective Slopes and Effective Noise Are Derived by Mininiizing Mean Square Error ((aJ,fmitr /Fossirr, 1984/, 8 1984 AIAA (h)-(g) jam IMuwoy, 19841 urith permissiotr j o m A A C C )

.-

~

89

CONSTANT SLWE I?

8. 4.

8 2. 5 0.

.2. -4.

3

2

1

TARGET MANEUVERTIME TOGO i

3 9 TARGET MANEUVER 7. mred RANGE INDEPENDENT NOISE

4

rl

(D)

TOTAL

-----.O.W

.0.02

0

0.02

0.04

0.06

0.08

RADOME SLOPE

q = r,

SINUSOIDAL RADOME ERROR:

-

TARGET MANEUVER:

- ...-

- . .-

RAWME

0, !0.08.o.m

- 0 . ' ~ .o.b2

o

0.b2

0.04

MAXIMUM RADOME SLOPE

0.05

I 0.08

RANGE INDEPENDENT NOISE:

(C)

t

Effective Slope for Target Maneuver Gimbal Angle Due to Target Manewer

Gimbal Angle Due to

Range Independent Noise

Oggf

Effec~iveSlope for Range Independent Noise

sin (+(up

Effective Bias b

*))

d

'

MDASE of NGC Processing

56

Chap. 2

dome slope. This miss due to target maneuver increases with positive slope, whereas the miss due to RIN increases with negative slope. For this example, the initiation time of the 3-g step target maneuver is uniformly distributed with 25 samples over the 5-sec terminal flight time. The amplitude standard deviation of the RIN is 2 mrad within a bandwidth of 1.8 radfsec (7 = 0.55 sec). In Figure 2-18b, the RMS value for zero radome slope is 1.0 because all RMS miss distances in this example are normalized by the same factor.

Miss i s less with periodic radome. Actual radome is periodic with gimbal angle [Murray, 19841. As shown in Figure 2-18c, periodic radome error results in smaller miss distance than occurs with constant maximum radome slope. The figure shows that miss is significantly reduced for a sinusoidal radome error with a 30-deg period. Specifically, the miss distance due to target maneuver is less for a periodic radome error. Miss distance is plotted against target maneuver time in Figure 2-18d for three types of radome error. The first corresponds to r = 0, the second to r = - 0.045, and the third to a 30-deg period sinusoidal radome with a maximum slope of 1 0.045 1. At the initiation time of the target maneuver, the gimbal angle is such that the radome slope has the maximum negative value of -0.045. While the RMS target-induced miss is 0.7 for r = 0, it is only slightly higher with a value of 1.1 for the case of the 30-deg period sinusoidal radome with a 10.045 1 maximum slope. These ~ a l u e scan be compared with the RMS targetinduced miss for the case of constant -0.045 radorne slope, which is much higher at 3.4. Actually, this adjoint continues to increase with increasing flight time owing to the fact that the missile system is marginally stable for r = -0.043. Effective slope and effective noise. As demonstrated in Figure 2-18e, the effective slope and effective noise of a radonle error are determined by gimbal angle motion. In this figure, the effective slope r due to gimbal angle motion is less than the maximum slope. The difference between the nonlinear radome error, u,, and the effective slope approximation, r u,, gives the effective noise, q. Gimbal angle motion is brought about as a result of missile system disturbances such as target maneuver and RIN. Gimbal angle motion, particularly that component which owes itself to target maneuver, can be a substantial fraction of a radome error. This example looks at a sinusoidal radome error, target maneuver, and RIN disturbances as shown in Figure 2-18f. The sinusoidal radome is characterizcd by amplitude, period, and phase of ro, P, and @, respectively. The gimbal angle is incremented as a rcsult of target maneuver by the value u,l over the flight time. A uniform distribution with a (1/3)"2u,,/2 standard deviation is assumed. Figure 2-18g shows the two effective slopes, one for each of the two nlissile system disturbances. Minimizing the mean-square error, q, yields r~ = ur(aE)* u ~ ~ l ( u , ~ ~ l l 2r, ) ,= u,(ux) *

02

= u,Z

-

~,~f14.

r+u:I/12

- r34,

(1-29)

b =i

Ssc. 2.3

NGC System Design and Analysls

57

Evaluating Equations (2-29) for the sinusoidal radome given in Figure 2-18f results in thc following normalized effective slopcs and effective noisc variance:

-r-T ~.WAX

-0s(

- rf- -

)

(y)

[[sin

COS

~.\IAX

where r ~ f . 4= ~2 ~ r , ~ and /P

(J,ro

sin

2 ~ 4 ~ (Y) sin

P

Because it is not significant for long flight times, the bias is neglected. Equations (2-30b and 2-30c) are plotted in Figure 2-19a for zero phase a. As the target maneuver and RINs increase. the effective sloves decrease, whereas the effective noise increases. The effective slopes are much less than the maximum slope, M MAX, as the gimbal angle increment, r,~, due to target maneuver approaches and exceeds the radome error period, P. As this occurs, the effective noise approaches 0.7 of the radome error amplitudes. The greater gimbal angle oscillation, + ' I 2 , due to RIN accelerates this effect of reduced effective slopes and increased effective noise.

Linear radome miss prediction. Equation (2-31) represents the linear radome miss-distance prediction, which is the RSS of the components due to system disturbances and the component due to effective noise RangeRMS Miss

RMS Miss

Target Maneuver RMS Miss: fM(r = rT); Effective Noise RMS Miss: fN(r = 0)

RMS Miss

RIN RMS Miss: fN(r = rf)

(2-31)

MDASE of NGC Processing

Chap. 2

SlNUSOlDAL RADOME DENOTES EFFECTIVE SLOPE V A L U E GIMBAL ANGLE STANDARD D E V I A T I O N ldegl I

-

EFFECTIVE

SLOPE, rf

TRAJECTORY DYNAMIC MODEL WITH 2 mrad RANGE INDEPENDENT NOISE

TARGET MANEUVER EFFECTIVE

1---

7.5 deg .0.06 -0.04

.0.02

0

0.02

0.04

R A D O M E SLOPE

(B) 3 0 d q PERIODSINUSOIDAL R A D W E 0 --180d.p INDEPENDENT EFFECTIVE SLOPE. rf

EFFECTIVE NOISE STANDARD DEVIATION 1rnr.d)

:;;,,

.0.0(5

.

o

.

m

h

,

0

0

5

10

15

0

GIMBAL ANGLE STANDARD DEVIATION Id41

5

10

15

GIMBAL ANGLE STANDARD DEVIATION 1-1

(C)

CONSTANT SLOPE RAWME

30dW PERIODSINUSOIDAL RADOME 2 rnrad RANGE INDEPENDENT NOISE

'J

EFFECTIVE I SLOPE AND \

rmr MISS

4-

3-

(A)

SlNUSOlDAL RADDME

0

1

.o.a . 0 . b

.o.b2

o

0.b2

0.k

o.'a 0.h

MAXIMUM RADOME SLOPE

Figure 2-19 (a) Effective Slopes Decrease and Effective Noise Increases with Target Maneuver and Range Independent Noise (b) Effective Slopes are Reduced with Range Independent Noise (c) Effective Slope and Noise are Defined by Gimbal Angle Standard Deviation (d) Effective Slope and Effective Noise Predict Range Independent Noise Induced Miss (e) Target Maneuver Causes Gimbal Angle Motion (1) Effective Slope Due to Noise Is Reduced with

Sec. 2.3

NGC System Design and Analyski 3 9 TARGET MANEUVER

SIMULATION

ANALYTICAL

1

Y

%

SIMULATION

1

TARGETMANEUVERAT

5wTOW

-40 0

TIME TO GO lul

TARGET MANEUVER AT 3ucTOGO

O

1 5

TIME TOGO (ucl

(E) 30 deg PERIOD SlNUSOlDAL RADOME 3 9 TARGET MANEUVER . 2 mrad RANGE INDEPENDENT NOISE

30 deg PFRIOD SlNUSOlDAL RADOME DENOTES EFFECTIVE SLOPE VALUE GIMBAL ANGLE STANDARD DEVIATION (deg)

I

-CONSTANT 1 RADOME

SLOPE

SlNUSOlDAL RADOME

TRAJECTORY

INDEPENDENT SLOPE A N 0 NOISE RADOME I I - 0 . E -0.@4

I

I

-0.02

I

0

0.02

1

I

0.04

0.06 0

1

rf = .0.004

RAOOME SLOPE

0.02 0.04 0.06 0.08 MAXIMUM RADOME SLOPE

(H) 30 dog PERIOD SlNUSOlDAL RADOME (0.045( MAXIMUM SLOPE

4

30 deg PERIODSlNUSOlDAL RADOME

2 mrad RANGE INDEPENDENT NOISE EFFECTIVE TARGET MANEUVER SLOPE. '.IRANGE INDEPENUtN'l

EFFECTIVE NOISE STANDARD DEVIATION (mradl

3 SlNUSOlDAL RADOME MISS

0.038

0.014

6 TARGET MANEUVER (91

0

3

(G)

6

TARGET MANEUVER 191

EFFECTIVE SLOPE AN0 NOISE RADOME

0

1

2

3

4

TARGET MANEUVER ( g ) (1)

Target Maneuver and Range Independent Noise (g) Effective Slope Due to Target Maneuver Is Reduced and Effective Noise Increased with Target Maneuver and Range Independent Noise (h) Effective Slopes and Effective Noise Predict Miss (i) Effective Slopes and Effective Noise Predict Target Maneuver Miss (From [Murray, 19841 with permissionfrom AACC)

5

60

MDASE of NGC Processing

Chap. 2

Range-independent npise (RIN) miss.

The result of RIN is gimbal angle motion, and the oscillation amplitude is greater for a negative radome slope. here fore, in addition to the miss distance being greater with negative slope ( ~ i ~ i r e 2-18b), the gimbal angle standard deviation is larger as well. The result is a lower effective slope and thus a reduction in miss with a periodic radome from that with a constant slope radome as demonstrated later. Figure 2-19b shows the gimbal angle standard deviation. The figure also shows the effective radome slope versus gimbal angle standard deviation for a 1-0.045 1 maximum slope radome. These effective slopes for periods of 7.5, 30, and 120 deg are obtained from Figure 2-19a. The value of the effective slope due to RIN is given by the intersection of these two sets of curves. For @ = 180 deg, corresponding to a -0.045 maximum slope (Figure 2-18f), the effective slope, rf, is -0.028 for a 30-deg period sinusoidal radome as shown in Figure 2-19c. The resultant effective noise standard deviation is 1.0 mrad. The RIN induced miss with a sinusoidal radome is predicted by the effective slope and effective noise, as shown in Figure 2-19d. When Equation (2-31) and the effective slope and effective noise are used to compute miss, this correlates with the miss for the sinusoidal radome. The reason for which the miss computed this way agrees so much better than the unrealistically large miss with a constant slope radome is not difficult to understand. In addition to increasing RIN miss, negative radome slope also increases the gimbal angle standard deviation. The result of this is to lower the magnitude of the effective negative slope, which in turn reduces the miss. Consider a linear radome miss vrediction model for RIN-induced miss as follows: a 30deg period sinusoidal radome, a -0.045 maximum slope at 0-deg gimbal angle, and a 2-mrad RIN. Based on this model, the RMS RIN-induced miss is 1.5, while the RMS miss component due to effective noise is 0.3. The RSS for this model is therefore 1.5. This can be compared to a value of 1.2 for the RSS of the sinusoidal radome error model. In this examvle the RIN comvonent for a -0.028 effective slope is dominant. The effective noise component is relatively small.

Noise and target maneuver miss. Gimbal angle motion is induced as a result of target maneuver as shown in Figure 2-19e. Here, both simulation and analytical results are displayed for target maneuvers initiated at 3 and 5 sec to go. The gimbal angle increment, a,,, due to target maneuver can be expressed analyticall y as

An arialytical expression for a missile acceleration, A,,, due to target maneuver. AT,, for a zero time lag missile system can be found in Fossier and Hall [I9671 and Jerger [1960]. Equation (2-32) is obtained for computing 8, given A,,,, and assumirlg perfect seeker stabilization and therefore that q, = 8. This equation is not highly

Sec. 2.3

NGC System Design and Analysls

61

sensitive to radome slope, r. There is a variation of 12 deg, from 22 deg to 34 deg, in u , ~for , r varying from -0.045 to +0.045 with a value of 27 deg for r equal to 0.0. The results that follow therefore neglect r in Equation (2-32). That is, A, = A. As can be seen in Figure 2-19f, the effective slope due to RIN is dramatically reduced with target maneuver and RIN. This reduction is primarily a consequence of the 3-g target maneuver, which results in a 27-deg gimbal angle increment, u,~,. This caused the effective slope, r,, to decrease from -0.028 (no maneuver) to -0.004 for the 30-deg period sinusoidal radome. As shown in Figure 2-19g. there is also a big falloff in the effective slope due to target maneuver as target maneuver is increased. Thc 27-deg gimbal angle increment, q,,, owing to a 3-g target ma, fall off from a value of 0.038 to 0.014 for neuver causes the effective slope, r ~ . to the 30-dcg period sinusoidal radome. The effective slopes and effective noise predict the miss with a sinusoidal radome as shown in Figure 2-19h. Using Equation (2-31) together with the effective slopes and effective noise to compute miss gives values that correlate well with the miss with the sinusoidal radome. These miss distances are the RMS of four sets of 25 flights each, with the sets having the initial gimbal anglc at the radome phase angle, 0,of 6.0, 13.5, 21.0, and 28.5 deg. Thus, these miss distances are the RMS over RIN, uniformly distributed target maneuver, and gimbal angle. Consider a linear radome miss-distance prediction model as follows: a 30-deg period sinusoidal radome; a 1 0.043 / maximum slope; a 3-g target maneuver; a 2-mrad RIN; and an RMS miss of four radome phases. Using this linear radome miss-distance prediction model, the RMS miss-distance component due to target maneuver is 0.8, while that due to RIN is 0.8 and that due to effective noise is 0.8. With these values, the RSS is 1.4, which can be compared with a value of 1.5 for the sinusoidal radome. In this example, the effective noise component is equal to the target maneuver and RIN components. The effective slopes and effective noise predict the miss with increasing target maneuver as shown in Figure 2-19i.

Empirical radome slope calculation. It is not difficult to find in the literature empirical relationships for radome slope [Donatelli and Fleeman, 1982; Giragosian, 1979; Travers, 1982; Youngren, 19611. That found in Travers [I9821 is examined here as an example to illustrate certain key points. where p is the radome fineness ratio (radome length divided by radome diameter), E is the dielectric constant, f~ is the deviation from design frequency, A is the signal wavelength, K , is the performance measure, and Dd is the dish diameter. The dielectric constant is a function of both temperature and radome material. Walton [I9701 gives the dielectric constant for different radome materials at various temperatures. Generally, radome slope error decreases with increasing seeker dish diameter, with decreasing fineness ratio, and with decreasing signal wavelength. The thickness of the radome is usually constructed to be one half of the wavelength to avoid multiple reflections. The performance measure k, reflects how sophisticated the manufacturing process is, higher numbers indicating an advanced radome grind-

62

MDASE of NGC Processing

Chap. 2

ing process. While a hemisphere ( p = 0.5) possesses ideal electromagnetic properties, it is possible to achieve lower drag with shapes having a higher fineness ratio. Therefore, a compromise must be struck between these opposing factors. Radome slope is related to the total radome swing by r = + r ~ / 2which is treated as a maximum expected value for 90 percent of the slopes for a given radome [Nesline and Zarchan, 1984(b)].

Higher-order system analysis. chan [1984(b)] consider

In Figure 2-15a(iii), Nesline and Zar-

The guidance system transfer functions are

The effective guidance time constant, T;, can be approximated as the coefficient of the s term in the denominator of ~ ~ u a t i o(2-35) n anh is the same as that in Equation (2-28). Here, as in the case of the first-order system, the effective navigation ratio, A', is modified by radome slope according to Equation (2-28). It is easily seen that rand T, have an effect on the guidance time constant and effective navigation ratio the same as occurs in the first-order system of this example. It is possible to achieve larger stability regions if one can keep small any or all of the following: the navigation ratio, the ratio of the turning rate to guidance time constant, the ratio of closing to missile velocity, and the radome slope [Nesline and Zarchan, 1984(b); Peterson, 19611. Because for a given radome, increasing A acts to destabilize the system, the designer must strike a balance between having A to be sufficiently small so as to satisfy stability requirements and at the same time sufficiently large so as to ensure effective homing. For a given higher Ta, radome effects were responsible for stability problems as well as an increase in miss distance. Higher T, is found at higher altitude and lower missile velocity. While T, is usually increased at higher altitudes to improve stability, this brings about a corresponding increase in miss distance. Consequently, the designer is faced with attempting to make 7 ,as small as possible to reduce the miss distancc without sacrificing too much stability. O f course thosc components of miss distancc caused by saturation and radomc/parasitic effects will not be affected by lowering 7,. The minimum achievable guidance time constant, (~~),i,,,given T, and the expected r, can be determined on the basis of a useful stability criterion originally developed by Nesline and Zarchan [1984(b)] as - 0.79 < r h,(VJ V,,,)(Ta/rg) < 2.07. The left-hand side of the preceding inequality gives the lower limit of the allowable negative radome slope, while the right-hand side establishes the positive upper limit of radome slope. If the guidance time constant is increased, there is a slight degradation in miss accompanied by a decrease in the system scnsitivity to radome swing. If the guidance time constant is decreased below

Sec. 2.3

NGC System Design and Analysis

63

the minimum predicted by the stability analysis, it becomcs impossible to attain satisfactory performancc over the expected radomc swing range. When thc resultant performance estimatc is not compatible with the allowable warhcad size, eithcr r or T, must be reduccd. Reducing thc former may involvc blunting the nose, using compensation, or somc other techniquc, while reducing the latter can be done by increasing the lifting surface area, reducing the maximum operational altitudc, increasing missile velocity, reducing missilc weight, and so on [Ncslinc and Zarchan, 1984(b); l'etcrson. 19611.

Example 2-4: Homing missile guidance and control analysis. Trimming thc missile at high altitudc may require that the fin be deflected through large angles. If the fin angle is limitcd, the missile's acceleration capability will also be limited. Determining the fin dcflection required for deterministic inputs can be aided through the use o f zero-lag guidance system results. Neglecting guidance dynamics in Equation (4) o f Table 2-2, fin angle can be expressed according to Nesline and Zarchan [1984(a)] as Achieving in a rapid fashion the large control fin angles necessary to trim the missile at high altitudc requires that the control fin move at high rates. The result of an inability on the part of the control fin actuator to attain these rates can be instability. Combining Equation (2-35) and Equation (4) of Table 2-2, the fin rate response due to LOS rate for the fifth-order binominal guidance system becomes

As was true in the case of fin angle, zero-lag guidance system results are very useful for determining fin rate response due to deterministic inputs. When guidance system dynamics are neglected, Equation (2-37) becomes [Nesline and Zarchan, 1984(a)]

Following Examples 2-2 and 2-3, if the structural filter, actuator, accelerometer, and gyro dynamics are relatively fast, the FCS can be represented by the ideal thirdorder transfer function described by

where T, 5, and o are the FCS desired time constant, desired damping, and desired natural frequency, respectively. The values for the zeros are determined by the aerodynamic configuration (the way in which these values are determined i~ pre-

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Chap. 2

sented in Book 1 of the series). The FCS gain is represented by K.This third-order model of the autopilot dynamics is useful in generating a nominal system design. The model supposes only two homing loop disturbances with white glint noise with PSD +, in addition to a uniformly distributed sinusoidal target maneuver shown later in Figure 2-21d. These are two important factors that contribute to high miss distance in a radar-guided PNG system. An adjoint model similar to that to be presented in Chapter 3 for Example 2-2 is constructed based on the homing loop shown in Figures 2-15a(iii) and 2-ljb, which follows the adjoint rules presented in Chapter 3. From such an adjoint model, an adjoint simulation is created using representative values for error sources and system parameters, the results of which are shown in Figure 2-20. Neglecting parasitic effects, the guidance system transfer function is given by

TOTAL rmr

TARGET MANEUVER CONTRIBUTION

0

0.2

0.4

0.6

0.8

NOISE CONTRIBUTION

1.0

FLIGHT CONTROL SYSTEM T l M E CONSTANT k c )

(B)

o

i

2

6

E F F E C T I V E NAVIGATION RATIO, A (A)

NORMALIZED RADOME SLOPE-

NORMALIZED RADOME SLOPE

(D)

Figure 2-20 (a) Effective Navigation Ratio Influences System Performance (b) Flight Control System Time Constant Influences System Performance (c) Decreasing Flight Control System Time Constant Makes System Performance More Sensitive to Radome Slope (d) Decreasing Flight Control System Damping Makes System Performance More Sensitive to Radomc Slope (From /Neslines, 1984(a)] and /Nesline G Zarchan, 19821 utith permissions from AACC and ACARD)

NaC System Design and Analysls

Sec. 2.3

65

-

where A,, = KA is the desiredlactual effective navigation ratio. Since the gain K is varied around unity, the actual and the desired navigation ratios are almost identical. 140, which varies due to changes in the flight condition, is one major determinant of homing system performance. Typical performance results displayed in Figure 220a show that if A,, is too high, miss distance increases due to noise and, if it is too low, miss distance increases due to target maneuver. Therefore, A. has an optimal value based on the error sources and system parameters, and in no case should exceed certain bounds. This imposes limitations on the allowable variation in K,as a variafrom tion in gain is equivalently a variation in A,]. For example, a variation in unity along with the corresponding variation in A,, makes it impossible to attain optimal homing performance. Neglecting parasitic effects, T , ~can be approximated as the coefficient of the first-order term in the denominator of Equation (2-40a) as

Therefore, increasing the FCS time constant T increases T,,, while decreasing T de~ and Zarchan, 19821. The system performance is also substantially creases T , [Nesline affected by T.

Autopilot response characteristics necessary to achieve homing. The speed of response of an FCS can be measured to first-order accuracy by 7. The maximum system time constant permissible to achieve successful homing depends primarily on the maneuverability of the targets the missile is expected to engage and the reauired miss distance necessarv to intercept these targets. Seven-tenths of " a second would be an acceptable maximum for a moderately difficult homing engagement. Figure 2-20b shows the effect of T on miss distance [Neslines, 1984(a)]. As shown in this figure, in which it is assumed that the time constants T,, T,, and 2510 are all known, if 7 is decreased, the miss due to target maneuver will also decrease, but that due to noise will increase. Increasing 7 produces the opposite result. For the set of inputs considered it appears from Figure 2-20b that T can be made arbitrarily small to optimize the miss distance, but this cannot be done because of radome refraction effects. When the parasitic radome loop is considered, the guidance transfer function becomes A*,,

-ir

cr

(1

+ ?,$)(I + T , s ) ( ~ +

KAV,(l Ails AI2s2)s rs)(l 2[s/w + s2/w2) + (i7i\vCr/v,,,)(l

+

+

+

+

T,s)

where 7; and A' are the same as those in Equation (2-28) except that T, is given by Equation (2-40b). The effect of radome slope error on system stability is itself strongly dependent on T,. At high altitudes, where T, is large, the guidance transfer function can become unstable due to either excessively positive or negative radome slopes. Thus, small T yields a large guidance system sensitivity to radome slopes as shown in Figure 2-20c [Nesline and Zarchan, 19821. Therefore, the optimal value

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Chap. 2

of 7 is bounded from above by target maneuver (especially at low altitude) considerations, and from below by radome refraction effects (especially at high altitude). The minimum amount of damping required in order to maintain performance, given a reasonable radome. should be determined usingu a detailed complete missile model: however, a damping ratio of 0.3 is a reasonable estimate of the minimum value that a radar-guided missile could tolerate. Figure 2-20d shows the effect of reduced damping and radome slope on miss distance [Neslines, 1984(a)]. The figure also shows typical high-altitude performance for a FCS with high and low damping. As shown in this figure, the FCS damping becomes more crucial when radome effects are considered. A system with low damping is far more sensitive to radome effects than one with higher damping. This means that if the unaugmented airframe is to be used in a radar homing application, a FCS must improve the low damping.

Example 2-5: Radome error compensation. 2-4 and Figures 2-15a(iv) and 2-15b, assuming 7, and assuming

7y

-

=

Following Example Ilk = 0 sec infinity bandwidth,

effective FCS time constant

;---

27A

(2-42a)

Combining the guidance equation (2-16) and autopilot Equation (2-42a) then gives

A,, -

1

+

A V
Substituting for

6 yields the guidance system transfer function

~ where 7; and A' are the same as those in Equation (2-28) except that 7, = 2 7 + 7,. With AV,/V,,, = 6, r = 5.02, and T , = 10, A V,rT,I V,,, can be & 1.2 sec and hence the guidance system becomes unstable (that is, 7,;<0 ). The radome crror slope can be the dominant factor in the guidance response time [Hiroshige, 19861. A guidance and control system is designed to compensate for the stability problcm caused by the radome error. To alleviate the severe radome stability problem for a high-altitude engagement scenario, one of the commonly used engineering "fixes" is to introduce pitch rate compensation G,0 with gain G , feedback to either before or after the guidance filtering network as shown in Figure 2-2. It can somewhat symmetrize the positive and negative slope stability region at the cost of slowing down the autopilot response [Yueh and Lin, 1984, 1985(a)]. Adding the pitch ratc compensation and the guidance filtering (1/(1 7.5)) to the guidance cquatiorl

+

NOC System Design and Analysis

Sec. 2.3

67

(2-42c). The guidance system transfer function bccomcs

-A,,,, --

(I

urn,<. s(l

+ r)AV<s

+ T ~ ~ ) [ ~ / G , ,+( SG,AV,(I ) + Tas)lV,,,llGs(s) + r A V 4 + ?,s)/Vm

The transfcr function for thc FCS, G.d(s) can be modeled as a third-order transfcr function as described in Equation (2-3'9) or (3-42a). If thc nonlinear elcmcnts are not included in the analysis, Equation (2-43a) based on the linear FCS model is easily solved. Howevcr, if it bccomcs necessary to employ the nonlinear FCS model, it is difficult to obtain an analytical solution using the classical frequency-response approach. Assuming the input to the saturated nonlinear elements is a sinusoid, then the nonlinearity can be represented by a variable gain N obtained from the describing function of the nonlinear element to be presented in Chap. 3. The transfer function ) N(ao + als + azr2)l(bo + bls + bzs2 for the FCS would then be given by G . 4 ( ~ = + b3s3) where the parameters are specified according to a particular condition. If G,(s) = G , = constant, C s ( s ) = 1/(1 7,s) have been assumed, then from Equation (2-43a).

+

T,;

(1 '=

+ G,A V,i C',,,)(T, + T,) + 7, + (G, + r) T,A 1 + (G, + r)AV,/V,,, A' = (I

+ r)Al[l + (G, + r)AVclV,,,]

V,I V,,

(2-43b) (2-43c)

where the effective autopilot time constant 7, -- ~ T Athe , guidance filter constant T,, and the radome compensation gain G , can be selected to compensate for the stability problem caused by the radome error. In comparing Equations (2-28) and (2-43), one can see that the radome compensation G,0 has reduced A' and increased 7;, both of which tend to stabilize the system of a given radome but at the expense of decreasing the guidance response time. The root-locus technique is applicable only to a linear, time-invariant system. The root locus for a linear, time-varying system of Equation (2-43a) can be obtained with the roots at different t,. The evaluation can be performed on the transfer function given in Equation (2-43a) for T,, T,, and G,, which vary with respect to t,.

Example 2-6: Gyro sensitivity gain design. In this section, a method of approximating aeroelastic effects on the dynamic behavior of a rate gyro is developed. More detailed derivations of a flexible body transfer function from control surface deflections to body rotation rates have been presented in Book 1 of the series. A gyro sensitivity gain expression of body-mounted rate gyro coming from body acceleration due to the aeroelastic vibration is considered here. The following assumptions are made in this derivation: (1) the first mode of vibration considered here is excited by body accelerations, (2) only longitudinal and lateral vibrations are considered, and (3) superposition of the aeroelastic angular rate at any given point

MDASE of NGC Processing

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Chap. 2

on the rigid-body angular rates is valid. The transfer function from the generalized acceleration f to the normalized body-bending curve can be written as

where 5 1 is a structural damping rate, w l is the natural frequency, and Z is the reference displacement to which the mode shape 1 shown in Table 2-3 is normalized. The location of the body-mounted rate gyro on the normalized body bending mode shape 1 shown in Table 2-3 is arbitrary, and it is desired to derive an expression for the aeroelastic body angular rate at this station. T o this end, the slope of thc normalized bending curve at the rate gyro station is determined first. Then, the time rate of change of this slope, due to acceleration, is approximately equal to the aeroelastic body angular rate at the rate gyro station. The product of the normalized displacement Z N and the reference displacement Z located at the rate gyro body station give the instantaneous reflection Z D at this station. That is, Z D = ZAjZ. The slope of the normalized bending curve h at the rate gyro station is h = ZN/XD and the actual slope at the same station is K = Z D / X D= hZ. The rate of change of the actual instantaneous slope is K = h i . For small-amplitude oscillations. the slope of the tangent to the curve at the gyro station can be approximated by the angle between this tangent and the reference x-axis. In this case, the aeroelastic angular body rates 8 , at the rate gyro station is

8,

=

K

=

/ z ~

(2-43)

Thus, the transfer function from the generalized accelerations f to the aeroelastic angular body ratcs a t the gyro station is found by substituting Equation (2-45) into Equation (2-44) giving

8,/f

=

s / ~ / ( +s ~251wls

+~

f )

(2-46)

Because the first-ordcr bending frcqucncy is typically very Iargc, thc gyro sensitivity gain to the step accelcration can be expressed to an approximation as

K, = hlw:

(2-47)

For missile applications, the gyro sensitivity gain is a very .small value since the normalized slope h is a small value and the bending frequency w l is a large value of first bcnding modc.

2.4 TARGET TRACKING STATE MODELING

2.4.1 Target N o i s e and Target Maneuver Modeling Targct noise has been presented in Section 2.3. I , and is applied hcrc to gencratc the target mancuver signal-to-noise ratio. Figure 2-21 shows thc cquivalcnt signal noisc diagram, where the signal, representing a target maneuver, is in turn represcntcd by a shaping filter. As was done in Section 2.3.1, it is possiblc to convert thc total

Target Tracklng State Modeling

Sec. 2.4

Target . Position Noise Target Maneuver Model T ~position ~ ~n ~ , Target Position Spectral Density 7 Signal yT Measurement z Shaping Filter H(s)

-

a) De~ermineManeuver: b(l) b)

Brownian Target Position: u,@ --L + J

C)

Zero-Mean White Gaussian Targel Acceleration:

,,

d) Uniformly Distributed Targel Maneuver: us e) Poisson Jinking Maneuver: u, St),,.

Note: u, = white noisc with PSD 9, Figure 2-21

Target Maneuver Model

seeker angular noise to a positlon noise by multiplying the angle noise by the range to go. In general, it is possible to generate a number of random processes by passing a white noise w through a suitable filter. If it is assumed that the target noise n in Figures 2-1% and 2-21 is a first-order Gauss-Markov process with correlation time constant T,, and white noise input w , then the corresponding equivalent white noise PSD of the position noise is where r , and 4, are given by Equation (2-25). Figure 2-21a shows a deterministic model, which is the simplest case. In most applications, target position or velocity or acceleration is assumed to be filtered white noise (see Figures 2-21b through 2-21d). Uniformly distributed target maneuver. For a target acceleration of step form with r a g o m starting time varying uniformly over the flight time, the autocorrelation function of this model is identical to that of the integrated white noise (see Figure 2-21d). The uncertainty in target position y + is described by a third-order integrator with a constant PSD 4, white-noise input us as:

where n T is the magnitude of the step maneuver, and tr is the flight time over which the maneuver is uniformly distributed. Two integrators are used in tandem in order to convert the acceleration to the target position y T , as shown in Figure 2-21d.

MDASE of NGC Processing

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Chap. 2

With regard to the uniformly distributed target maneuver, it can be seen by examining Figure 2-21d that the noise spectral density level is constant, but the signal spectral density passes through three integrators. As a consequence, the target signal is frequency dependent while the target noise is not at a particular range to go. Moose et al. [I9791 give a short exposition on the evolution of the relationship of target acceleration modeling to stochastic processes. This evolutionary development is graphically depicted in Figur62-22 [Cloutier et al., 19881. Initially, as shown in Figures 2-22a and 2-21c, target acceleration was simply accounted for with white noise (depicted by a correlated process whose spectrum is flat over the system bandwidth). Consequently, tracking could be maintained only for those maneuvers remaining within the envelope of the white noise, and even then performance suffered due to the incorrect assumption of uncorrelated acceleration.

Correlated acceleration process. A correlated acceleration process, Figure 2-22b, can be achieved using Gauss-Markov models [Fitts, 1974; Pearson

(D)

Figure 2-22 Historical Development of Maneuvering Target Acceleration Models (a) Zero-Mean White Gaussian Process (b) Zero-Mean Correlated Gaussian Process (c) White Gaussian Process with llandomly Switching Mean (d) Correlated Gaussian Process with Randomly Switching Mean (Frotn /Clourirr et a!., 19881 with prmisrion porn AACC)

Sec. 2.4

Target 'Itacking State Modeling

71

and Stcar, 1974; Singer, 1970; Vcrgez and Liefer. 19841. The well-known Singer model [Singcr, 1970) has been used with various Kalman-type filters to achieve excellent tracking characteristics over a broad class of large scale maneuvers. Its primary drawbacks arc the need to specify apriori both the acceleration time constant and power of the driving noise and the inability of the model-based filter to track target motion resulting from abrupt changes (jumps) in the acceleration process. The velocity a t which the target is moving is constant, but thc target's lateral acceleration is described as a Poisson jinking maneuver. Such a maneuver, which is one whosc magnitude B (RMS target acceleration) is constant but whosc sign is randomly switching, can be modeled as white noise through a single-polc filter. This is because the acceleration autocorrelation function (RA.,.)I(7) = ~ ~ e - ~ .' f' ' = average frequency) corresponding to this maneuver process is identical to the Poisson process. Thus, the target maneuver models are as shown in Figure 2-21e

AT, = -2fAr,

+ 2B d? w ,

= -A,AT,

+ h,u,

(2-50)

where u, is a zero-mean Gaussian noise, that is,

-

~v,

with PSD

=

-

(2-51)

w , is a white noise with unit PSL), and A, = 2f = the target maneuver bandwidth. Gholson and Moose [I9771 modeled rapid, major changes in target motion by a semi-Markov process. The mean of their process, Figure 2-22c, randomly switched to a finite number of states according to a Markov transition probability matrix, with the time duration in each state itself being a random variable. Successf~~l tracking with this method necessitates a large number of preselected states, and the erroneous assumption of uncorrelated acceleration is still retained. Moose et al. [I9791 extended this work by employing the Singer model to represent correlated acceleration within each of the states, Figure 2-22d. Larimore and Lebow [I9871 developed a similar model based on a parameterized Gauss-Markov process with the parameters changing according to a point process [Cloutier et al., 19881. Lin and haf froth [1983(a)] concluded that oneof the most versatile representations of target acceleration is that of the sum of a continuous-time Gaussian process and a finite basisljump process. The continuous-time process can be obtained by means of a classical shaping filter, while the jump process which is characterized by its basis, jump size distribution, and interjump time distribution, can be obtained by driving a shaping filter with a white sequence.

Other target acceleration modeling.

As stated in Cloutier et al. [1988, 19891, another aspect of target acceleration modeling is a consideration of the aerodynamic characteristics of the target. A winged aircraft accelerates mostly orthogonal to its velocity vector, in particular, orthogonal to the plane of its wings. The magnitude of the acceleration also has asymmetric bounds (positive or negative g) which are set by pilot and/or aircraft limitations. Modeling based solely on target point-mass motion fails to take many of these characteristics into account. Kendrick

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Chap. 2

et al. [I9811 modeled normal load acceleration with a random variable whose probability density function (PDF) was asymmetrically distributed between small negative g and large positive g. Nonnormal accelerations were modeled as first-order Gauss-Markov. It was assumed that the orientation of the three-dimensional target could be obtained by means of an EO seeker in conjunction with pattern recognition. Bullock and Sangsuk-Iam [I9841 developed a nonlinear Cartesian model of target dynamics by using polar coordinates to model planar, circular turns under the assumption of constant velocity magnitude. Similarly, Hull et al. [I9831 used polar coordinates to model both acceleration magnitude and angle as random processes. This led to a Cartesian model characterized by state-dependent noise. Finally, the parameterized Gauss-Markov model developed by Larimore and Lebow [I9871 took into account aerodynamic parameters such as bank angle, lift force, and thrust minus drag. A particular maneuver has a given set of parameters, while any abrupt change in the parameters corresponds to the initiation of a new maneuver. Merging a point process, or jump process, with a continuous-time Gaussian process, either additively, as recommended in Lin and Shafroth [1983(a)], or through parametric embedding as performed in Larimore and Lebow [1987], is an excellent way to model the acceleration of a highly maneuverable target. However, of the models of this type developed to date, only one [Larimore and Lebow, 19871 has given any consideration to the aerodynamic characteristics of the target. The natural next step in this technology area is to merge the concepts of those models based on aircraft flying characteristics with those symbolized by Figure 2-22d.

2.4.2 Two-Dimensional Target Tracking State Modeling Consider the two-dimensional target tracking problem illustrated in Figure 2-10. For simplicity, let velocity VT be in the x-direction while V,, = 0 and y = 0 for a fixed tracking station. From Equations (2-2),

R =

v7-cos u,

6

=

- (VTIR) sin a

(2-52)

Differentiating these equations yields the nonlinear state equations

R =

VT

cos u - l,/6 sin g ,ii = - [ ( v ~ R- v ~ R ) / R sin ~ ] u - (VT/R)6 cos cr (2-53)

Following the treatment of Lewis [1986], assume that the target is not accelerating, that is, VT = 0. Then, assuming also very small changes in R and u during each 1 to 10-scc radar scan, one has that R ;= 0 and & = 0. Hencc, random disturbances O R ( [ ) and ow(() are introduced to account for the small changes in R and b during the scan time T, giving

Sec. 2.4

Target nacklng State Modellng

73

The processes w R ( t )and w,(t) are known as maneuver noise. Because the Kalman filter is robust in the presence of process noise, R and & can be tracked. The addition of process noise is done to offset unmodeled dynamics introduced as a result of excluding nonlinear terms. The state equations then become linear as

The measurement model, given that the radar measures range R and azimuth a is defined in Equation (2-3) and is given by

Suppose independent disturbance accelerations are uniformly distributed between t n T,then w R ( t )is a radial acceleration with variance n$/3. Since w,(t) is an angular acceleration, w,(t)R is distributed according to the maneuver noise PDF and w,(t) has variance n$/(3R2). The covariances of process noise w ( t ) and the measurement noise v ( t ) are given respectively by

where :u and a : denore the variances of the range and azimuth measurements, and a :
where

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Chap. 2

2.4.3 Three-Dimensional Intercept State Modeling

Twelve-state guidance law design modeling equation.

The linear state model, consisting of relative position, relative velocity, target acceleration, and the FCS and the airframe and propulsion dynamics, is given by Equation (3) of Table 2-1 with N = I and x, A, B, u, w defined as follows: x = [x,, y,, z,, xr, jl?, i,,AT,, AT^, A T = ,Am,, A m , , ,Am,I T

where subscripts T and m denote target and pursuer, respectively; x,, y,, and z , are, respectively, the x , y, and z components of the relative position in inertial axes; ir,jl,, and 2 , are, respectively, the x, y, and z components of the relative velocity in inertial axes; and AT,, ATl, AT, are target accelerations which are modeled as a Poisson jinking maneuver shown in Figure 2-21d with target maneuver bandwidth A and with w T as the white process noise with covariance matrix Q. Definitions of A and Q are as follows:

where A , and 4, are the same parameters as those in the horizontal plane case defined in Equation (2-51). The states A,,, A,,, A,,, represent the pursuer accelerations, and the first-order model defined in Example 2-2 is assumed here with its bandwidth o defined as (2-62)

o =

0

0

0,

The nonlinear measurement equations for a sensor measuring range, range rate, and two LOS angles can be represented as the true values corrupted by additive noise as when /I is a nonlinear function of the states and v is a white measurement noise with covariance r. The target tracking sensor is assumed to provide measurements of LOS angles u,,and a, slant range R, and range rate R (see Figure 2-9). Thus, the z vector is given by

[;][ =

I

R,,,, = 11, = (xf + yt + z ,2) 112 R,,,,. = /I, = (x,i, + y , j , + z,i,)l(x: + y t + xf)'12 2 112 u,,,, = h, = tan-'[z,l(x? + y,) ] urmr= h4 = tan-' (y,lx,)

+

v

(2-63b)

Sec. 2.4

Target nacklng State Madeling

75

where the subscript true denotes the geometric relationship, and the subscript k denotes time index. The measurement noise v is defined as R m

+ Rf + R, + R,r,,,

ut, urn

+ u , + a,, + + Of + u, +

(2-63~)

0,,,, ~gl,,,#

where, as in the case of two-dimensional engagement, the subscript rn denotes the receiver noise, f the fading noise, c the clutter noise, and glint the glint noise. Specifically, the LOS angle measurement noise v , is the same as that in the horizontal plane case defined in Equation (2-22).

Nine-state guidance filtering equation. The nine-element state vector of the linear kinematic target-pursuer engagement model is simplified from the twelve-state one, and is written as

where the measurement equations are the same as Equations (2-63), and x, u, A. B, and w are defined as follows:

where the target bandwidth A and the process noise covariance matrix Q are defined in Equation (2-61). The achieved pursuer acceleration u is assumed to be measured very accurately by accelerometers so that an estimate of u is not needed. For a sensor with a constant data rate, that is, sampling state every T sec, the discrete form of the system equation in Equation (2-64) is x ( k ) = O ( T ,A)x(k - 1 )

+ W u ( k - 1) + w ( k - 1 )

(2-65a)

where the state matrix @ is

I

D(A, A) A-'(13 - e K A A ) e -AA

(2-65b)

where D ( s , h ) = ( e - ' h - I + s h ) h K 2 ,the deterministic forcing term. T u ( -~ 1) depends on pursuer acceleration, which is assumed to be known exactly and is given by

MDASE of NGC frocesslng

76

If U(T)is assumed to be u(k

-

V( k-1

1) between the time intervals (k =

( -I , T2

Chap. 2

- 1)T and kT, then

T

-T I 0

and the discrete form of the process noise w(k

-

u(k - 1)

(2-65c)

1) is a white-noise sequence

where D ( k T - T, A) - e-(kT-~)A] -(kT-,)A

E[w(k

- I)] = 0,

E[w(k

-

and the discrete process noise coGariance matrix

where, as shown in Singer [1970],

l)w(k

a is

-

=

a

(2-65e)

Modern Multivariable Control Analysis Modern multivariable control analysis, which is the second ofthe five primary topics as shown in Figure 1 of the Preface, is mainly concerned with the analyses of robustness, performance, dynamics, and stability. This chapter deals specifically with the first two analyses, while stability and dynamics analysis is covered in Book 1 of the series. Robustness analysis (Sections 3.1 to 3.6) is particularly important for examining multivariable control systems design. Performance analysis (Sections 3.7 to 3.12) includes the various statistical methods used in the analysis and synthesis of any modern NGC system. The material of this chapter is based in large part on AGNC [1986(a), (b)].

Robustness analysis.

The background material for this discussion on robustness analysis, such as the use of matrix theory and the property of singular value, is presented in Book 1 of the series. Robustness means that the closed-loop stability and performance are insensitive to uncertainties which include modeling errors, nonlinearities, unmodeled dynamics, measurement errors, and unknown external disturbances. The more the system can preserve its performance and stability under uncertainties, the more robust it is. Robustness analysis is essential since all models of physical processes are to some extent approximations of the true plant or state. The existence of uncertainties due to incomplete system knowledge, sim-

78

Modem MultluarfableControl Analysis

Chap. 3

plified assumptions in the model, external disturbances such as air turbulence around a flight vehicle, or measurement noises are the primary reason for employing feedback. Feedback aims to compensate for the uncertainty, thus preserving system stability and performance. T6 characterize how robust a system is, a reliable and nonconservative measure of the robustness margin is essential. For a single-inputsingle-output (SISO) system, this is generally relegated to the gain and phase margins. For a multivariable system, the singular value, structured singular value, and other robustness margins have to be used. All these robustness measures will be discussed in Sections 3.1 to 3.6.

Performance analysis. The rest ofthis chapter is devoted to a comparison of various computerized statistical analysis methods employed in the synthesis and analysis of guidance systems. Through examples, the advantages and computer costs of each method are examined in detail. The design that emerges in the synthesis of a guidance system must specify allowable measurement noise, determine subsystem bandwidths that guarantee adequate stability margins, prevent component saturation, and evaluate total system performance (see Books 1 and 4 of the series and Yueh and Lin [1985(a) & (b)]). Several statistical analysis techniques can be applied to both linear and nonlinear noise-driven systems for the purpose of studying systcm performance. Linear analysis has as exact methods the adjoint technique and covariance analysis. Approximate methods that exist for nonlinear analysis includc Monte Carlo techniques, CADET, and SLAM. These performance analysis rncthods are all examincd with the help of a unifying example involving missile homing guidance and digital simulation. The unif>;ing example from inissilc homing guidance prcscntcd in Example 3-2 is uscd to demonstrate thc utility of each of thc pcrformancc analysis tcchniqucs via digital simulation IZarchan, 1979(a), 19881. When trcating thc complctc nonlinear dynamics of thc vchiclc's n ~ o t i o n anal, ysis must be pcrformcd on thc systcm of equations govcrning thc complctc N G C loop. Thc analysis n ~ u s tconsider both detcrininistic inputs such as impulses and steps, and random or stochastic inputs such as white noise and turbulence. In principle, any type of problcm involving N G C systems can be solvcd using the mcthods of pcrformance analysis. However, theoretical analyses of N G C loops almost always involve the use of cornputcrs owing to the prcscncc of nonlincar differential cquations with variablc cocfficicnts, and thc stochastic nature of thc actual N G C proccsscs. Whilc thc importance of numerical solutions obtaincd from thc application of computers cannot bc ovcrcn~phasizcd,analytic analyscs of N G C loops arc cqunlly i ~ n p o r t a for ~ ~ tpurposes of estimating thc gcncral laws governing the N G C proccss.

3.1 SINGULAR VALUE APVU,YSlS The idca of gain and phase margins has bccn uscd in spccifying and evaluating the robustness of an SISO systcm. Consider thc fccdback systcln dcpictcd in Fig~lrc 3-1. Thc transfcr function C(s) incorporatcs both opcn-loop plant dynamics and

Sec. 3.1

Singular Value Analysis

??3-+ Figure 3-1

Feedback System

any con~pensationemployed. The Nyquist stability theorem states that the closedloop system is stable if and only if the graph of Cfjo), for all o,encircles the point - 1 + j O as many times counterclockwise as C(s) has right half-plane poles. The gain and phasc margins are derived by perturbing C by KG and eimC,respectively. More precisely, the system has a gain margin k,,if PC, for all p < I*(,, satisfies the Nyquist stability theorem. Likewise, the system has a phasc margin 40 if the perturbation P'+, for all 4 < 40, does not destabilize the system. This concept of gain and phase margins has been extended to multivariable systems by Rosenbrock 119741 and MacFarlane and Postlethwaite [I9741 in which the locus of the determinant of the G(s)], is evaluated. return difference matrix, that is det[l It is, however, not possible to identify the design robustness under plant uncertainties by examining solely the Nyquist plot of det[l + G(s)]. As demonstrated in Dovle and Stein 119781, the effects of cross coupling in the internal system model are not dircctlv rcflcctcd in the stability margins obtained from the polar plots of det[! + C(s)]. Generally, one finds that multivariable generalization of the Nyquist stability criterion does not directly relate the conditions of multivariable robustness to tolerance of open-loop modeling errors except in cases where nominal systems and perturbations thereof are diagonally dominant. M3thematically, this can be artributed to the inadequacy of the eigenvalues as indicators of the singularity of the matrix (there is a pole crossing the jo-axis if I + G ( j w ) is singular). Any measure of robustness or relative stability should predict how far I + G ( j o ) is from being singular or what is the magnitude of the perturbation that can be tolerated without destabilizing the system. This observation leads to the singular value analysis in feedback system designs. The singular values of a complex p x m matrix M, denoted by ui(M), are the nonnegative square roots of the eigenvalues of M*M, that is,

+

. . . , k) where k = min ( p , m ) 2 a 3 ( M ) 2 .., 2 a k ( M ) ,the maximum and

a i ( M ) = ~ A ; ( M * M()i = 1, 2,

Ordering u i such that u , ( M ) 2 u,(M) minimum singular values can be alternatively defined by a ( M ) = U I (M) = max [ 11 MX 112/11 X llr] = II M I .Y 11220

112

and (3-1)

In the casep = m and M is nonsingular, the smallest singular value can be computed by g(M) = uk(M) = \\ ML1\\ZJ-' which measures how near the matrix M is t o being singular or rank deficient. As is well known, the matrix M + E is singular if there exists a nonzero vector .Y sach that ( M + E ) x = 0. In robustness analysis,

Modem Multiuariable Control Analysis

80

Chap. 3

at each frequency w, E represents a certain stable perturbation while M stands for some nominal closed-loop transfer matrix. If the minimum singular value of M is E is greater than the maximum singular value of E for all frequencies, then M nonsingular, which implies that the system remains stable if it is nominally stable. In fact, the singular value is the exact measure of robustness if E is unstructured (norm-bounded only). In the case that E is structured (subject to certain constraints), a singular value may be conservative. Consider the following two possible representations of uncertainties.

+

1. Additive. The plant is perturbed as G + E for some stable E. The return difference matrix becomes (I + G + E)-I. If the system is nominally stable, (I + G ) is nonsingular for all frequencies, then it remains stable as long as I G + E is nonsingular. A sufficient condition for this to be true is z ( I + C) > II E 112. 2. Multiplicative. The plant is perturbed as C(I + E) for some stable E. The system is stable if [ I + C(I + E)] or [ I + C + GE] is nonsingular or if [I + G + E ] is nonsingular or IT(I+ G -') > (1 E ((7.

+

-'

Singular value analysis is a powerful tool for checking whether the system is robustly stable. This notion leads directly to the generalization of gain and phase margins for multiloop systems. The multiloop gain margin is the pair of real numbers c1 and c2 defining the largest interval ( c l , c2) such that for all Ci(s) = C i E R in Figure 3-2 satisfying the inequalities, ci < C , < c 2 (i= 1, 2, . . . , m), the closed-loop system remains stable. On the other hand, the multiloop phase margin is the number c defining the largest interval (-c, c) such that for all C,(s) = e-'"('") in Figure 3-2 satisfying the inequalities - c < &(o)< c ( i = 1 , . . . , I N ) , the closed-loop systcm remains stable. The following arc obtained using the preceding definitions and the concept of singular values: (1) If the original system is stable and ~ ( +l C) > a , then c l 5 111 (1 - a)], c2 2 [1/(1 a)], and c r 2 sin-'[a/2]; and (2) If the nominal system is stable and ~ ( +1 G - ' ) > a , then 5 1 - a , ( 2 2 1 + a, and c 2 2 sin-'[a/2]. Clearly, the singular value plays a crucial role in determining the robustness of the system. Depending on thc typcs and characteristics of the uncertainty, therc are diffcrcnt robustness mcasures for diffcrcnt sources of unccrtaintics such as plant uncertainty or cxternal disturbanccs that have to bc considcrcd. In the next scctiotl,

+

Figure 3-2 Configuration for Multiloop C;aiti/Phasc Margin l)cfi~iiti~)~l

Sec. 3.1

Slngular Value Analysb

81

suitable robustness measures are identified and more general robustness analyses are discussed. T o show the necessity of using singular values instead of eigenvalucs, consider the following 2 x 2 matrix:

While the determinant of this matrix is always unity for any bation

E,

an additive permr-

produces a singular matrix which cannot be predicted by the eigenvalues. The singular values of M 6 defined by Equation (3-2) are the square roots of the eigenvalues [(E' + 2) t E c2 + 4112 of the matrix M*(E)M(E).Note that the singular values depend on E, while the eigenvalues and determinant do not. Note also that

s-

clearly shows that ~ ( M ( E ), ) 0 as c -t m. A more interesting example is discussed in Laub [1985]. The Ostrowski matrix is an n x n matrix whose upper triangle elements are all + 1, whose diagonal elements are all - I , and whose lower triangle elements are all zero. That is,

This matrix is nonsingular and its eigenvalues are all equal to - 1. Its determinant is ( - 1)" for any n. However, the Ostrowski matrix becomes increasingly nearer to a singular matrix as the dimension n gets larger [Laub, 19833.

In fact, the smallest singular value of the matrix P is 2-"+I and approaches zero as n + cc. Let AP be the perturbation matrix

Modem Multfuariable Control Analysls

82

Chap. 3

that is, P - h P is a singular matrix. g ( P ) is in the order of a(AP). Looking at Equation (3-j), the disturbance A D has therefore caused the closed-loop system P - AP to have zero eigenvalues even though the eigenvalues of the nominal system P are all equal to - 1 [Hewer et al., 1988(a)].

3.2 SENSITMTY AND COMPLEMENTARY SENSITMTY FUNCTIONS Consider the system depicted in Figure 3-3. In control system design, typical requirements are (1) closed-loop stability, (2) disturbance rejection, (3) tracking capability, (4) noise insensitivity, (5) robustness against plant uncertainty, and (6) functionality under component failures. It will be shown in this section that, under mild assumptions, the sensitivity and the complementary sensitivity function play crucial roles in characterizing the feedback performance. That is, the singular values of the sensitivity and the complementary sensitivity function can be measured in order to assess the robustness. First, consider the disturbance rejection property; the transmission from the disturbance source d to the output y is the sensitivity function So:

The smaller the sensitivity function, the better the disturbance rejection property. Similarly, the tracking error as a function of the reference input r is e = (1 + GK)-' r. Again, the smaller the value of (I + GK)-' is, the smaller the tracking error will be. Next, consider that the plant is perturbed as (I A)-'G for some uncertainty A. The uncertainty block matrix A can be used to represent low-frequency parameter errors or the changing number of right half-plane poles of the plant [Doyle et al., 19821. After some manipulations, it can be shown that the system remains stable if A is a stable transfer function and [I (I GK) - A] is nonsingular for all w. This implies that the sensitivity matrix (1 GK)-' must be small such that g (1 + GK) > 5 (A), which in turn means that I + G K + A is nonsingular. Since I + GK + A is nonsingular and det(1 + (1 + GK)-'A) = det(I + GK)-' det(1 + GK + A), the matrix I + (I + GK)-'A will be nonsingular. Another function that is useful in characterizing the robust stability is the complementary sensitivity function To(s) = G K ( l + GK)-'. Let the plant transfer function be disturbed as ( I + A(s))C(s) for some stable A(s) representing sensor error or ne-

+

+ + +

'

Controller K(s) Figure 3-3

Feedback System

Sec. 3.2

Sensitivity and Complementary Sensitivity Functions

83

glected high-frequency dynamics. Assume that the maximum singular value of A(s) is bounded by the inequality -d[A(jw)] < e m ( j w ) for all w 5 0. Then the robust stability in the high-frequency region is guaranteed if the nominal system is asymptotically stable and the following condition is satisfied [Doyle and Stein, 19811: -d[GK(;o){l

+ GK(;~)}-'1 < l/ern(;w)

for a l l o such that e m ( o ) B 1

(3-7)

The system remains robustly stable if Equation (3-7) is true. That is, the smaller u [ G K ( I + GK)-'(jw)] is, or the smaller the maximum singular value of the complementary sensitivity function is, the better the robust stability will be. Also, the noise effect at the output node y is y = - GK(1 GK)-'r. Hence, the smaller GK(1 + G K ) - ' is, the better the noise immunity will be. Similar arguments can be made for the sensitivity function (I KG)-' and the complementary sensitivity function K G ( 1 K G ) - ' at the plant input node. More interpretations of the sensitivity and complementary sensitivity functions can be found in Cruz and Perkins [I9641 and Kwakernaak [1985]. Hence, many feedback properties can be represented in terms of these two functions, and the robustness analysis can be done by investigating the size (singular values) of the sensitivity and the complementary sensitivity functions. For a particular class of problems, the control design technique that reduces the size of these two functions or their combinations directly leads to a robust design. There are, however, important fundamental limitations that restrict the achievable To = (I + performance. The most important one is due to the fact that So GK)-' + G K ( 1 GK)-' = I and (I KG)-' KG(1 KG)-' = I. Thus, it is impossible to make both functions small at the same time, and trade-offs have to be made. In general one requires the sensitivity function to be small in the lowfrequency range (to reject disturbance) and the complementary sensitivity function to be small in the high-frequency range (to be robust against high-frequency uncertainties). Since at low frequencies the loop gain is high, the sensitivity function in the low-frequency range can be approximated as S o ( j o ) = (I + GK)-' == ( G K ) - ' ( j o ) . Also, since at high frequencies the loop gain is low, the complementary sensitivity function in the high-frequency range can be approximated as To(jo) = GK(1 GK)-' = GK(jw). These two requirements simply show that the loop gain G K must be large at low frequencies for good tracking performance and small at high frequencies for robustness. o n e ' drawback of using the singular value in measuring robustness is its conservativeness. Conservativeness results from the structure and the nature of the uncertainty. For a repeated, block-diagonal, or real uncertainty, a less conservative measure has to be adopted. The structured singular value (to be discussed in Section 3.4) represents one attempt in reducing the conservativeness of the singular value. In the following section, another feature of singular value will be discussed, that is, how the singular value analysis of the complementary sensitivity and sensitivity functions is translated into feedback design requirements.

+

+

+

+

+

+

+

+

+

Modem Plultivariable Control Analusis

84

Chap. 3

3.3 DESIGN REQUIREMENTS Design requirements which the control system must satisfy in general are: (1) low sensitivity property, a low-frequency requirement; (2) fast response characteristic, an intermediate-frequency requirement; and (3) large stability margins, a high-frequency requirement. One of the design methods which can handle these requirements explicitly is the frequency response method of classical control theory. However, these classical techniques cannot be directly extended to multivariable control systems in general. Performance specifications and stability robustness requirements for multivariable control systems can be examined by singular value plots in the frequency domain. The plots corresponding to the maximum and minimum singular values over a given frequency range can be established as the design criteria for a multivariable control system design.

Low-frequency requirement. From Equation (3-6), the largest singular value of the sensitivity matrix can be written as

where, as defined in Equation (3-I), E and g represent the maximum and minimum singular values, respectively. In general, the upper bound p ( o ) must satisfy 0 < p ( o ) 1 to obtain a sensitivity reduction property. Using the well-known inequality o[l - + A] 2 g [ A ] - 1 in Equation (3-8), a sufficient condition of Equation (3-8) can be given as 1 dcK(jw)l> 1 f o r o < o,

dw)+

Since g [ G K ( j o ) ]rolls off with the gradient of -20 dB1dec above the maximum frequency of the open-loop poles, the upper-bound frequency in Equation (39) has to be lower than the maximum open-loop pole frequency. Following Ohta and Fujimori [1988],the bound p ( o ) is studied more carefully as follows. T o restrict p ( o ) to values less than unity means that the sensitivity and the response to disturbances are reduced to at least k ( ~x ) 100%. This also means that the maximum steady-state error will be decreased by a factor of p(0) as follows. When r(s) in Figure 3-3 is a step function given by r(s) = [I, . . . , 1IT/s = Hls, following Figure 3-3, the steady-statc crror e, can be written as 1 e , = lim SSO H = So(0)H P--=

-5

(3- 10)

The maximum singular value of e, becomes E[e,] = if[So(O)H]S F [ S o ( O ) ] ] E [ H ] . Therefore, E[e,] 5 p(O)if[H].

High-frequency requirement. The rcquircmcnt in thc high-fr~qucnc)~ region is to make a [ G K ( j w ) ]as small as possiblc. As shown in Ohta and Fujimori [1988],the upper bound of a multiplicative uncertainty of the model is assumcd to

be denoted as t',,(w). Then, the condition of stability robustness can be expressed as F[GK(jw)] < l/t,.(w) for t',,,(w) > 1. Thus, the singular value of the complementary sensitivity matrix has to be less than a given upper - - bound. This condition is effective for suppressing the effect of measurement noises as well.

Intermediate-frequency requirement or crossover frequency. The crossover frequency w, is related to the response speed of the system. The larger w, is, the faster the achievable response speed can be. However, a trade-off between the requirement in the high-frequency region and the response speed must be taken into acount. This consideration restricts the maximum crossover frequency w,,,, below the crossover frequency of t',,(w), and the minimum crossover frequency w,,,, will be determined from the settling time T,,which is an alternative measure of the response speed. Both measures ofw, and T, are closely related with the closedloop poles. As shown in Ohta and Fujimori [1988], a unity feedback system is considered with the transfer function G(s) = o:/[s(s 25w,)]. When 0 < 5 < 1, the crossover frequency and the settling time within 5 percent error of the command input are given as w, = w,,[(4i4 + 1)'12 - 2(2]1'2 and

+

T , is inversely proportional to the real part of the closed-loop pole. The smaller 5 is and the larger o,, is, the larger w, becomes. In addition, the value of w,,,, that gives a desirable settling time will be determined according to Equation (3-11). A typical requirement is shown in Figure 3-4. Design engineers then shape the loop transfer function to meet the desired requirements. There exists a fundamental limitation here due to the right half-plane poles and zeros. For the system to be stable, when the plant (or controller) has right half-plane zeros, the sensitivity function

dB

Roll-Off C.baracIcristiu

Robust Stability Requirement

Figure 3-4 GK(jw)

Singular Value Plots and Constraints for the Loop Transfer Function

86

Modem Multluariable Control Analysis

Chap. 3

must be greater than 1 over some frequency range. O n the other hand, for a plant having right half-plane poles, its complementary sensitivity function must be greater than 1 over some frequency range. These fundamental limitations set up the bottom line for design engineers in judging the achievable performance.

3.4 STRUCTURED SINGULAR VALUE It is a fact in engineering practice that any model used in the design process is at best an approximation tb the actual system. The difference between the physical svstem and its mathematical model is called plant uncertaintv. This inevitable modeling error will cause performance degradation and stability problems if plant uncertainties are not accounted for during the design process. For single-input singleoutput (SISO) systems, the standard frequency domain techniques such as Nyquist diagram, Bode plot, root locus, inverse Nyquist, and Nichol's chart can all be used effectively to ensure robust stability and certain desired performance characteristics. However, these techniques do not readily extend to multivariable systems in general. Singular value analysis discussed in Section 3.1 was introduced as a candidate for a multivariable robustness measure in the late 1970s. In principle, the stability margin of a multivariable system can be measured in terms of the maximum singular value of a certain transfer function matrix if the plant uncertainty is unstructured [Chen and Desoer, 1981; Doyle and Stein, 1981; Kimura, 1984; Lehtomaki, 19811. Unfortunately, using singular value analysis for testing stability robustness is usually too conservative because the actual plant uncertainties are more or less structured in some way. This motivates the definition of the structured singular value. In Doyle 119821, it is shown that any norm-bounded perturbation problem, regardless of structure, can be rewritten as a structured block diagonal perturbation problem. Consider, for example, a linear multivariablc plant G(3) with two mu]tiplicative perturbations appearing simultaneously a t the inputs and outputs as shown in Figure 3-5. Hcre, K(s) is a stabilizing controller for the nominal plant G(s). In a straightforward way Figure 3-5 can be equivalently redrawn as shown in Figure 3-6. In Figure 3-6, the combination of G(s) and K(s) inside the square box can be denoted by a nominally stable closed-loop system M(s), a transfer function matrix from [ d r d' vT] to [,I, T r12T e T 1. It is clear that the perturbations can then be rcprescntcd by a block diagonal feedback matrix, that is.

.

Note that the pcrturbed systcln in Figurc 3-6 is just a special casc of the gcncral structured block diagonal perturbation problem as shown in Figurc 3-7 with 11 = 2, A = block diag { A , , A,), 2 = [ d l , d2IT. Z = [n,, n2IT, 1,. = r, and = E . 11, Figure 3-7, M(s) has the following form:

Sec. 3.4

Structured Singular Value

I+A,(s)

G(s)

t-+l I+Az(s)

87

Figure 3-5 T w o Multiplicative Perturbations Appear Simultaneously a t the Plant Input and Output

The issue of robust stability becomes one of characterizing the set of allowable perturbations such that the closed-loop system remains stable. Note also the appearances of the sensitivity and the complementary sensitivity functions in the matrix M(s). It is well known that the systcm in Figure 3-7 is stable if and only if the det [I - ibl(s)A(s)]has roots with strictly negative real polynomial 'PIlnr(s)'PA(s) parts, where 'Picl(s)and 'PA($)are the characteristic polynomials of M ( s ) and A(s), respectively. Since M ( s ) is stable, all the roots of 9 m ( s ) have strictly negative real parts. Thus, if A(s) is also stable, the system is stable if and only if det [I - M ( j w ) A ( j o ) ] 0 for all A and o E R. A set of structured block diagonal perturbations is defined as

+

A(s) = block diag [ A ,( s ) , Az(s), . . . Ak(s) stable, E I A k ( j o ) ]5 6(w), for all k

(3- 13)

Then the structured singular value (SSV) [Doyle, 19821 is defined as

The following theorem is just a direct result of the definition (3-14) Theorem 3-1. an only if

The system in Figure 3-7 is stable for all A(s) E X ( 6 ( o ) ) if

p [ M ( j w ) ] < 1/6(w) for all u E R

(3- 15)

Note that the perturbation 6(w) and the structured singular value p [ M ( j w ) ] are frequency dependent rather than constants. From the preceding theorem it can be seen that the stability margin of the system M(s) can be defined by the structured singular value p [ M ( j o ) ] . While the p function is conceptually simple, its computation poses a challenge.

e

+

Figure 3-6 Equivalent System Diagram of the System Shown in Figure

Modem Multluarlable Control Analusis

88

a = [dl, 4- .... %?

n = [nl,

.

T

nz, .... bl

I

Chap. 3

Figure 3-7

Structured Block Diagonal Perturbation with A = Block Diag { A , , . . . , Am)

When complex perturbations are allowed, some earlier results by Doyle [I9821prove to be very useful:

where p(M) is the maximum modulus of the eigenvalue of M [Horn and Johnson, 19851. Moreover, let it = { U E Q I U*U = I ) and CT) = { D I D A = A D for A E Q), then p(M) = k(UM) = p(DMD-')

for all U E

and D E 9

It follows that max p(MU) I k(M) 5 inf ~ ( D M D - ' ) UEU

(3-1 7 )

DEm

It turns out that the left part of the preceding inequality is actually an equality, and this fact provides an approach to compute the p function involving multiextremum optimization. Under some restrictive conditions, namely the number of blocks m a 3, the right part of the inequality in Equation (3-17) also becomes an equality [Doyle, 19821. It should be noted, however, that when only real perturbations are allowed, the preceding relation no longer holds, and the other methods hare to be pursued. At present, there are several algorithms to compute structured singular values [Doyle, 1982; Fan and Tits, 19861. However, no single algorithm is univcrsally superior. The situation is worse where there are mixed (real and dynamic), repeated uncertainties. It must also be noted that the structured singular value is merely an analytical tool, offering no assistance for controller design if the robust stability test (3-15) fails. Recent progress in robust controller synthesis using a combination of the H" optimization technique and structured singular value has been reported in Doyle 11983, 1984). Although this so-called synthesis approach does not provide an exact solution for the robust perforn~anceproblem, it often yields a satisfactory (suboptimal) solution. Further research is thus necdcd in this area. Auothcr alternative measure of stability margin is called the block-limiting norm, proposed by Fan and Fu [1989], who extcndcd thc notion of a limiting norm [Pokrovskii, 19791 by associating with it the block structure that forms thc basis of the structurcd singular value. It is shown that the block-limiting norm is no ICSS than the structured singular value and no grcatcr than the infimum of the scalcd maximum singular value. More interestingly, where thcrc are no morc than thrcc blocks, the block-limiting norm is cqual to thc structurcd singular value. Unfortunatcly, a computational mcthod for the block-limiting norm is also unavailable to date.

See. 3.5

General Robustness Analysis

8Q

3.5 GENERAL ROBUSTNESS ANALYSIS As described in Section 3.4, the general framework of robustness analysis can be represented in Figure 3-7. Any system may be rearranged to fit in this general framework, although the interconnection structure can become quite complicated for complex systems. The uncertainty considered here may be modeled in two ways, either as external inputs or as perturbations to the nominal model. The performance of a system is measured in terms of the behavior of the outputs or errors. Robustness analysis is heavily related to the assumptions made concerning the input signal, the uncertainty, and the performance requirement. Assume that the transfer matrix from I, to can be represented in the linear fractional map

where M , , are real, rational, proper, stable transfer functions. This assumption can be made without loss of generality because nonlinear or nonrational parts can be regarded as uncertainties, and the system has to be stable before robustness analysis is carried out. Depending on the assumptions made on the input, uncertainty, and performance, diffcrent robustness measures are used as shown in Table 3-1, which is similar to Table 1 of Doyle [1983]. Thus, robustness analysis boils down to the computation of a certain "norm" of a particular transfer function. In some cases, this transfer function is a combination of the sensitivity and the complementary sensitivity functions. Also, the weighting functions used in design process are absorbed into the interconnection structure. The definitions of each "norm" follow.

where u i is the ith singular value, E is the maximum singular value, and p,(M), defined in Equation (3-14), is a measure of the amount of structured disturbance that can be tolerated without destabilizing the system. T o conclude this subsection, two important theorems from Table 3-1 are stated. Let

BA = -

{A E Q

1 ll A 112 5 1 )

This leads to the following theorems.

Theorem R S (Robust Stability). F,,(M, A) is stable for all A E only if II M22 1Iw = SUP ~ [ M 2 2 ( j w )< I 1. 0

a if and

Modern Multivarlable Control Analysk

90

Chap. 3

TABLE 3-1 VARIOUS ROBUSTNESS MEASURES

w

Perturbatiqn

P e r f o m

YIc(t) I ~ ) =I I b(t-II)

A =0

weT@)e(t)l < 1

1 2z 11%<12

Ie(t) = Ieo6(t) T qIcoIco) = I

A=O

w T ( t ) e(t)l < 1

1 2x ~~~~~2

A=O

Ilel12 < 1

lly7J=< 1

IIIcllz 5 1

IlAllm < 1

Ilel12 <

11M1111= < 1

II1cll.c 5 1

A=O

llella < 1

11%111

~ ~ I 5e 1~ ~ 2

llAll.c < 1 A = Diag (A, ... A$

JlelJ2c

l l ~ ~ l
llAll= < 1 A = a g ( A A

Ilel12<1

!\lc\12

1

-I I I c 5~ ~1 ~

oc

Robustness Measure

5

<1

1

-

1

IIMllr < 1

Theorem R P (Robust Peuformance). F,(M, A) is stable and sup II F.(jo) 1 for all A E &$ if and only if 1) M 1, < 1.

112

W

5

3.6 ROBUSTNESS OF REAL PERTURBATIONS The general robustness analysis provides a framework in robustness margin computation. However, for all practical purposes, there are other definitions or alternative algorithms that achieve the same goal. Robustness measures using graphic metric [Vidyasagar, 19841, hypersphere radius [Bhattacharyya, 19871, and Kharitonov's theorem [Barmish and DeMarco, 19871 have been proposed. A particularly important problem in feedback analysis is to compute the multivariable real-perturbed stability margin. Using p. computation to evaluate the real-perturbed stability margin tends to be conservative. In the following, several techniques for assessing the real-perturbed stability margin will be discussed based on Hewer et al. [1988(a)(c), 19891. A state-space model for real additive perturbation errors that is algebraically and dynamically equivalent to Doyle's SSV is introduced here. Based on this theory the problem is cast as a spectral assignment problem employing output feed-

Set. 3.6

Robustness of Real Perturbattons

91

back, where the variable feedback matrix is restricted to the block diagonal elements represented in Figure 3-7. Next, a link between root locus and SSV is established. T o bring out the utility of this theory a Monte Carlo search algorithm is used in conjunction with some graphical convergence strategies to compute for thirdorder systems. The Monte Carlo algorithm was used by Hewer et al. [1988(a)] to compute the SSV for the autopilot example first studied by De Gaston and Safonov [I9881 with good numerical agreement. For this example, the bounds resulting from the Monte Carlo algorithm are sharper than those obtained with the algorithm used in Fan and Tits [I9861 and Doyle's estimates [Doyle, 19821, which is in line with the well-known fact that their estimates are valid for both complex and real perturbations. The Monte Carlo algorithm is a "brute force" approach and should only be used in a concurrent manner with other structured singular value algorithms and inequalities. 3.6.1 State-Space Model for Additive Uncertainty Similar to a frequency-domain formulation, in state space formulation the system can also be arranged such that it consists of a nominal forward block

k = Ax

+ [Bl, Bz]

[:I. [:I

=

[S:]

(3-19)

with uncertainty expressed as feedback perturbation

The matrices A, B1, B2, C1, C2 are of dimension n x n, n x [, n x k, p x n, and k x n, respectively. The state vector x, the input vectors rl and r2, and the output vectors yl and y2 all have compatible dimensions. Taking the Laplace transform of Equation (3-19) yields the following two-port (p k) X ([ k) block transfer matrix [Hewer et al., 1988(b)]:

+

+

with the transfer matrices G;,(s) = C,(sl - A)-'B, (i = 1, 2; j = 1,2). The triples (A, B1, C1)and (A, B2, C2)are both minimal (that is, controllable and observable) realizations of the respective transfer matrix G ~(s)I and G22(5). The conditions under which the transfer matrix Gll(s) is exponentially stable (exp. st.) [Desoer and Chan, 19731 are that it is strictly proper (that is, Gll(m) = 0) and that all of its poles are contained in the open left half-plane. The p x 4 loop transfer matrix Gll(s) incorporates both open-loop plant dynamics and any compensation employed. The advantage of the state-space formulation is that it is convenient to treat parametric uncertainty, where the perturbations are necessarily real. In effect, the matrix A is perturbed to A AA, where AA = BzSACz. The matrix S is a k x k real-valued

+

Modem Multiuariable Control Analusis

92

Chap. 3

diagonal scaling matrix, and A is a k x k member of Q. The p function is defined as (where S has been absorbed into B2 or CZ)

A E Q and det(jw1 - A

- B2AC2) =

0 for some 0 5 w

I

Ia:

Introducing the set 48) = {A E Qs 1 det(jw1 - A - B2AC2) = 0 for some 0 The k function can be written as p,,(A, B2, C2) =

SUP A€e(-)

I w 5 m)

11 A 11'

3.6.2 State-SpaceSingular Values The following theorem, which is proved in Hewer et al. [1988(c)] is a state-space robust stability theorem. Theorem. Let (A, B1, C1) be an exp. st. system, assume e(8) is nonempty, and assume that ( A + AA, B1, C,) is a minimal system for every A E Qs. The system in (3-19) and (3-20) is exp. st. with respect to the real perturbation set Qs if and only if p,(A, BZ, C2).6 < 1. The following corollary, which establishes the basic link between the frequency-derived SSV and the eigenvalue-based SSV, depends on the following wellknown identity. Let Wand Z be n x k and k x n dimensional matrices, respectively; then det(lk

+

ZW) = det(1,

+

WZ)

(3-21)

Corollary. For every s with real part 2 0 and every A E e(8), p.,(A, B2, CZ) = p(Cz(s1 - A)-'Bz). The proof of this corollary is given in Hewer et al. [1988(b)]. These results make it evident that the additive perturbation structure laid out previously produces a workable and rigorous model that yields a nonconservative state-space test for real perturbations. With Equation (3-21), it is possible to construct a Monte Carlobased algorithm that reduces the computation of p., to an eigenvalue search. The corollary forms the basis for the Monte Carlo search algorithm outlined in Section 3.6.4 and in the appendix of Hewer et al. [1988(b)], reducing the computation of p. to an eigenvalue problem. The Monte Carlo eigenvalue search is proposed as a heuristic tool to supplement other algorithms.

3.6.3 Root Locus This section focuses on the link between root locus and SSV and is based on Hewer et al. [1988(b)]. A comparison is made between the stability margins predicted by root locus for second- and third-order polynomials and those predicted by SSV

For second-order polynomials they are identical, but for third-order systems they are distinct. Let g ( s ) denote a strictly proper nth-degree transfer function, and let ( A , b2, c2) be the minimal control-canonical form realization such that g ( s ) = c2 ($1 - A ) - ' b 2 . Using the basic matrix identity Equation (3-21) results in the following equation which relates root locus and the spectral set e(6), for some parameter A: for every s such that s l , - A is nonsingular. Any polynomial subject to additive perturbations can be represented as a dynamical control system such that ( A , b ) is in controller canonical form, and the nonzero coefficients in the row vector c r e p resent the additive perturbations of A . Thus, as an additive robustness measure, root locus represents a uniform affine shift of all the perturbed coefficients. Consider the third-order stable polynomial p ( s ) = s3 + a2s2 + a l s + a. and the canonical form

+

Let the perturbed polynomial be p p ( s ) = s3 + a2s2 a l ( l + A l ) s + ao(l + Ao). The Hermite matrix [Lancaster and Tismenetsky, 19851for the third-order perturbed polynomial is

Given that the original polynomial is stable, it can be concluded that the perturbed polynomial is stable if and only if H p ( A ) is positive definite, or, equivalently, its leading principle minors are all positive. Applying the latter test to H p ( 0 ) produces the well-known result that p ( s ) is stable if and only if a0 > 0 , at > 0, a2 > 0, and ala2 > ao. The perturbation matrix A A for root locus is Ab2c2, and the perturbation matrix A A for the structured singular value is A A = B2AC2

where the diagonal matrix is A = diag(A0, A l ) . By Hermite's test p p ( s ) is unstable when the first leading principle minor in H , ( A ) is zero. The gain margin predicted by root locus would be unity at w = 0 . The structured singular value spectral set

e(6) = { A E Q s

I det(jw1 - A ' -

B 2 A C 2 ) = 0 for some 0

Iw

< m)

would contain the matrix A = diag(1, I ) , which is not necessarily the minimum element. Now, consider the second leading principle minor of H p ( A ) , which can

Modem Multivarlable Control Analysis

94

vanish whenever the linear equation A . = m A l positive real variable m = alazlao. The matrix

+m

-

Chap. 3

1 is satisfied, where the

(

A = diag 0, l i m ) is always a member of e(S). Moreover, the maximum singular value of A is always less than 1 inasmuch as in is always greater than 1. Thus, the results of the root locus and the structured singular value tests are not the same even for this simple problem. Retracing these steps for the second-order polynomial a(s) = s2 + a l s a,,, it is easy to see that p,(A, b, c) is always equal to the root-locus value as long as the 2-norm is used for comparison.

+

3.6.4 A Monte Carlo Computation As an application of this theory, the Monte Carlo search algorithm is used t o compute a pe(A, Bz, CZ) for a third-order system, following Hewer et al. [1988(b)]. The system matrices are structured as in Equation (3-22) with a" = 1, a, = 2, a 2 = 2. T o estimate the 1.1. function, Equation (3-17) is used. Standard eigenvalue routines were employed in the computation of the spectral radius. Its upper bound is computed using a balancing algorithm. These inequalities hold true for real and complex perturbations, so they will be generally conservative for real perturbations. However, they do establish the following range where the robustness margins are smallest. Also, they serve as a useful guide to the variation in the robustness margins as functions of frequency. The Monte Carlo robustness margins predict 11 A l l = ~ ,7268 at the frequency w = ,7678. Doyle's inequality predicts p = 2.3789 at the frequency o = .8214. To transform this to a comparable delta, first take the reciprocal of p. which is ,42036 and then compute 11 diag(.42036, .42036) = 0.5945. The Monte Carlo search uses an F-norm, which represents a 2-sphere for the Qa set that describes the robustness region. The Monte Carlo algorithm is described as follows:

1. Generate A randomly with uniform probabilistic distribution such that k

Z A ? = 1. %-I

2. Start with known upper bound a for e(6). 3. Set a C 2 a until either (a) or (b) as follows happens: a. A + aB2AC2has an eigenvalue with a positive real part. Go to Step 4. -a. Start Step 3 again. b. a is greater than some cutoff value. Let a 4. Use bisection to obtain an a such that A + aBzAC2 has an cigenvalue sufficiently close to the irnaginary axis.

Robustness of Real Perturbations

Sec. 3.6

95

5. Save a. Repeat Steps 1 to 4 a sufficiently large number of cirncs. The minimum a is an estimation of robust margin I( A 1, ,. These same Montc Carlo techniques arc used to compute the robustness margin for the example of a typical SISO autopilot problem. These examples provide firm support for using the Monte Carlo scarch algorithm to find y for interesting control problems. However, as mentioned previously, the Monte Carlo scarch is only a heuristic tool that is best utilized in conjunction with other algorithms. It must always be kept in mind that the Montc Carlo scarch may deliver a reasonable result but not ncccssarily the supremum.

3.6.5 A Monte Carlo Analysis for Second-Order Systems As a second example, consider the following second-order system:

A =

[-:

-1, ='[:I, BI

CI = [ I , 0,.

B2

=

[:

c2=

[i:]

Let 0 be a random variable that is uniformly distributed in the interval ( - a , a ) , and let A = diag(r sin 0 , r cos 0) where r is a scaling parameter such that A E e(8). The characteristic equation for this problem is the random polynomial A'

+ (3 - r cos @)A + (2

- r sin 0) = 0

Now A E e(6) if and only if the following two cases occur:

These equations define a random variable r ( 0 ) which is a member of P(8) on the interval [2, 3/cos(arc tan(2/3))]. The cumulative distribution function for the random variable can be derived using standard transformation formulas in Papoulis [1984]. Upon integrating,

P(r(0)

5

d}

=

[2/a arc cos(3Id)

+ f(2/d)]/2

3

5

d 5 3.6056

where f(a) = 1 - 2 / a arc sin a. The density function corresponding to this random variable has a discontinuity a t d = 2 and d = 3. In the absence of a numerically robust and efficient algorithm to compute y, the Monte Carlo search algorithm combined with graphical convergence tools make for a practical, working method that can be used to augment the Doyle bounds, and the De Gaston-Safonov and Fan-Tits algorithms.

Sec 3.6

Robwtnes of Real Pemubatlons

1

At

Frequency

A2

A3

Monte Carlo

7.753

.I473

e.2645

-.1933

Penalty Function

7.697

,1229

-.3 164

-.?391

. 102 t

5% ESTIYATES VERSbS FRESUENCY

.

-

L

S S

. .. .. .....

1.

*

......

,,...'

V

,,,,,..,.. .""

.,

. .'

..I.

...

.:..-. ..'1.

-

4

'..'..

7

1 1

'1

(,,,..,.,.......l,.#-.....'~~

E

I' 10' T

M A T

E

S

q

9

.. .

*.- --.---

-

10" 1oO

---*--C--*r---

FREQUENCY

- RAD/SEC

('3

Figure 3-8 Robustness of Real Perturbations (From [Hewer et at., 1988(a)] with permission from S C S )

-'0'

Modern Multiuarlable Control Analysis

98

Chap. 3

3.6.6 Stability Margin Computation For the example problem considered here, the original block diagram is shown in Figure 3-8a. This example is found in De Gaston [I9851 and includes a number of essential features of an NGC system. The purpose of this example is to demonstrate state-space modeling and to illustrate Doyle's claim that a linear system can be rearranged to match Figure 3-7 and include some compound bounds for IJ.. The nominal parameter values are: K = 800, z = 2, p = 10, o2 = 6, w-, = 4, o,,= 4, o,,= 6. The block diagram in Figure 3-8a is transformed into one with scaled perturbations, and results in the diagram shown in Figure 3-8b. The scaling is selected such that a value of + 1 for the perturbation represents 100 percent of the nominal parameter. The Ai (i = 1, 2, 3) represent real perturbations and o,((i = 1, 2) are the nominal poles and scaling parameters such that the perturbation represents a percentage of the nominal parameter [Hewer et al., 1988(a)]. Transforming the block diagram in Figure 3-8a to the desired form is accomplished by rearranging the loops at the inputs to the perturbation blocks, which is illustrated in Figure 38b. Treating the inputs and outputs of these blocks as separate system inputs and outputs, respectively, conventional block algebra can then be used with Figure 38c to obtain the desired M matrix by evaluating all nine transfer functions in Figure 3-8d. The original system inputs and outputs are no longer represented in Figure 3-8c. The exact entries for M are found in Figure 3-8e. The values are identical with De Gaston except for sign differences. Following Hewer et al. [1989], the controllable and observable linear dvnamical svstem for Figure 3-8a can be derived " using the following state variables x l , x2, x-,, and x4 that are identified in the same figure. -

These equations yield the following dynamical system ( A , B1, C1):

with perturbation matrices:

[ -; !] 0

B2

=

0

-1

c 2 =

[. : :] 0 0 0 1

A = diag(Al, A2, A;)

Sec. 3.6

Robustness of Real Perturbatlons

B9

and scaling matrix S = diag(K, w2, ~ 3 ) The . perturbation matrix 6A with 6K = KAl, 6w2 = -oS,A2, 603 = - w,,A3

factors into the product 6A = BzSACz. Before the state-space equivalence of M(s) in Figure 3-8e can be determined, the inverse of the matrix (sl - A) must be determined. Let (sl - A);, denote the ( n - 1) x (n - 1) matrix formed by deleting row i and column j from A, and define the cofactor matrix Then s l - A has an inverse equal to det-'(sl - A), multiplied by the transpose of the matrix obtained from (sl - A) by replacing each element by its cofactor whenever r 65 R ( A ) where R(,-l) is the spectrum of A. Now for the De Gaston example, it is easily shown that 6 = det(s1 - A ) where 6 is defined by the equation in Figure 3-8e. The transpose of the cofactor matrix for ( s l - A ) is:

It is not difficult to verify that the matrix M(s) = SC2(sl - A ) - ' B 2 is identical to

the transfer matrix in Figure 3-8e. The transfer matrix with input scaling Ml(s) = Cz(:l - A)-'BZS is not equal to ,W(S) with output scaling. However, by the fundamental determinant identity the polynomials in the three variables A 1 , A2, and 113 defined by the equations det(1 - Ml(s)A) and det(1 - M(s)A) are identical. This observation leads to the following theorem about input-output scaling. Theorem. If a ( A ) < 0 and e(A, B2, CZ. A) is nonempty, then for any nonsingular transformation T that coinmutes with the members of A p(A, B2T. C r , A) = k(A, Br, TC2, A) Using standard root-locus techniques on the nominal system in Figure 3-Sa, the following stability margins can be obtained [Hewer et al., 1988(a)]: A1 = ,931, A2 = - ,481, A, = - ,652. However, the simultaneous use of these marginal values results in an unstable perturbed system. The scaling bound in Figure 3-8g predicts instability for A i > .15 ( i = 1, 2, 3) at the frequency 7.326. Note that the margin of stability occurs at the reciprocal of the peak. Using the Monte Carlo algorithm and one called the penalty function technique, Figure 3-8f lists two possible nearminimum values for the joint variation in A. Figure 3-8g shows some bounds for CL. Curve n is a plot of the minimum singular value of M, while curve b is a plot

100

Modem Multluariable Controf Analysis

Chap. 3

of the spectral norm of DMD-' using a balancing algorithm in Garbow et al. [1977]. A balancing algorithm is discussed in Osborne [1960]. An algorithm due t o Fran and Tits [I9861 was used to generate curve c, and curve d is a plot of the spectra] radius of M. Recall that the svectral radius is the maximum absolute value over the eigenvalues of M. Curves a an2 d can be computed using standard routines in arbo ow et al. [1977].

3.7 MONTE CARLO ANALYSIS

The Monte Carlo method [Hammersley and Handscomb, 19641 is the most general method available for estimating the performance of noise-driven time-varying systems, being applicable to linear as well as nonlinear systems. The method is founded on direct and repeated simulation,'involving a large number of random error factors. Because of this, much of the computational work revolves around looking a t a sufficient number of test cases plus postprocessing of the resultant data in order to determine the average system behavior. If there is a minor drawback to this method, it is the often large number of simulations needed to achieve an adequate level of confidence in the accuracy of the results. For this reason, it is primarily used as an evaluation tool. Owing to its wide range of applicability and the ease with which it can be used, however, it remains perhaps the most widely used tool for performing nonlinear statistical analysis. In the case of systems that are driven by white-noise disturbances, these disturbances are approximated by Gaussian random number generators. The standard deviation of these generators, which are used to approximate white noisc, relate to thc spccial density of the white noise being simulated by:

where R k is the covariancc of the Gaussian N(0, Rk)noise gencrator, @ is the power spectral dcnsity of the white noise in Hertz, and A t is the interval of simulation integration (time spacing bctwccn random nunlbcr calls). Equation (3-23) allows for the approximation of white noise since elements in the rcal covariance matrix Rb(t) take on infinite valucs. If the bandwidth of thc noise is much greater than the systcm bandwidth, then Equation (3-23) is accurate to thc cxtcnt that the noisc appcars white to thc systcm. Bccausc thc integration intcrval A t is usually taken much sn~allerthan the sn~allcstsystcm time constant, the noisc in fact looks white as far as thc systcnl is concerned. Statistical cstialation by the Montc Carlo tccl~niquc begins by computing thc standard dcviation of the rcsults according to

whcre y (1,) is the mcan valuc of final rcsults 2nd Aidenotes thc numbcr of simulations in the Montc Carlo proccdurc. Bccausc only a finite numbcr of simulation runs arc performed, thc standard dcviation dcrived in this mcthod is only approximatc and

Sec 3.7

Monte Carlo AnalysLs

NORMALIZED CONFIDENCE INTERVAL

I

I

I

I

I

I

0

50

100

150

200

250

NUMBER OF SAMPLES

X.200 RUN MONTE CARL0 SAMPLE SIZE

Figure 3-9 (a) Theoretical Confidence Intervals for Gaussian Distributed Random Variable (b) Acceleration Limit Study Utilizing Monte Carlo Approach (From [Zarchan, 19881 with permission from AGARD)

102

Modem Multluariable Control Analysls

Chap. 3

must itself be treated in a statistical fashion. This gives rise to expressing confidence in a particular estimate of the standard deviation. The confidence level rises as the number of simulations is made larger. As an example, for the case in which these statistics are Gaussian distributed, confidence intervals can be calculated like those shown in Figure 3-9a. In the case of 50 simulations, the confidence level can be shown to be around 95 percent, that is, the standard deviation is found to be between 0.85 u and 1.28 a. The uncertainty is reduced even more by increasing the number of simulations to 200, a sample size that would give rise to a 95 percent confidence that the standard deviation in fact lies between .91 u and 1.12 u [Zarchan, 19881. For this reason, confidence levels are usually presented in conjunction with Monte Carlo data.

Example 3-1. Following Equation (3-24), the miss-distance standard deviation is calculated from:

where jj(tf) is the miss distance from the ith run. Following Example 2-2, a 200run Monte Carlo simulation was performed of the homing loop of Figure 2-l5a(ii) and 2-15b, corresponding to the nonlinear mode. In the study conducted by Zarchan [1988], one of the parameters was the missile acceleration limit. The results of the study are displayed in Figure 3-9b, which includes the 95 percent confidencc intervals. This study highhghts the effect of the acceleration limit on miss distance. It can be seen that results approach those of the linear analysis only in the case of large acceleration limit. It is interesting to note that, even with the large number of simulation runs, the 95 percent confidence limits point out the high degree of uncertainty in the answers.

3.8 COVARIANCE AFWYSlS It is important to be able to calculate the vchicle responsc to random process modcls such as atmospheric turbulence. Calculating the response of dynamic systems to random processcs can be achievcd through the use ofthe covariance matrix equation, which is applicable to both deterministic and random excitation functions. Using the computer-implemented method of covariance analysis, only one computcr run is required to do an exact analysis of linear and nonlinear, time-varying noise-driven systems. Uy directly intcgrating a linear or nonlinear matrix differential equation, the covariance matrix of thc system state vector is propagated in time to generate exact time-dependent statistical performance projections of any state. Extensive use

Sec. 3.8

Covariance Analysls

103

of this nlcthod is found with problems involving optimal cstimators, as is shown in Chapter 4. The linear differential equations rcprcsenting the vehicle's equations of motion are first written as linear timc-invariant statc equations. The statc equation includes a term representing the random process input function. The statc vector is then augmented to include the components of the original statc vectors, the original systcnl input, plus any required auxiliary variables, that is, thc auxiliary disturbance statc variables, as shown in Equation (3) ofTable 2-1 where, for simplicity, D and E are assunled zero from this point on. and where w is a white-noise vector having PSI1 matrix Q ( t ) which gives the 1'SD function of the white-noise process a constant value of ( l I 2 n ) Q over all possible values of the circular frequency w. For example, the properties of the turbulence field are usually specified as PSD functions making the preceding representation suitable for aeronautical applications. Equation (3) of Table 2-1 can be used to analyze whole-body motion, flexible motions, and control system dynamics. Handling unsteady aerodynamic effects can be achieved through a quasi-steady assumption, through the use of approximate lift-growth functions, or by the numerical matching of unsteady aerodynamic matrices. Bias and drift, which are both types of nonwhite-noise inputs, can be handled by white noise passing through an appropriate shaping filter. It is also interesting to take note of the covariance matrix of the augmented state and output as follows: P(t) = E[(x(t) - x)(x(t) - x ) ~ ] ,

x = E[x(t)]

(3-26a)

~,(~)=E[(Y(~)-Y)(Y(~)-Y Y =) E~ []Y, ( t ) l (3-26b) For the augmented state, the covariance takes on the following matrix form:

u

u:.>

... ... ...

1

u;,,,

u;,

u?.~~,~

is used for the variances of the individual states .u,. is used to represent the covariances between different states xi and xj.

T o take it one step further, the diagonal elements of P(t) denote state variable variations when the expectations of the disturbance responses are zero. Furthermore, the off-diagonal elements denote the degree of correlation between the state variables. In the proceeding discussion, both the system states x and the forcing functions w are vectors, the elements of which are random variables. For simplicity, the deterministic inputs u are neglected here. There are two situations (continuous and discrete cases) to be analyzed. In the first situation, the state vector obeys the linear relationship Equation (3) of Table 2-1 without the input term Bu. The covariance matrix is defined as in Equation (3-26a) and the covariance matrix P satisfies

This equation is used to study the parameters that characterize the state during the vehicle's flight. Since Equation (3-27) can be integrated, the statistical properties

Modem Multluarlable Control Analysis

104

Chap. 3

of P(t) can be analyzed directly in only one computer run. The solution of Equation (3-27) gives the augmented state covariance matrix, which can be written as

Stability of the augmented dynamic matrix A guarantees that P(W) = 0.SO that, if PAr + NQNr. Using Equation (3-26a), A is stable, at steady state 0 = AP the covariance matrix Equation (3-26b) is given by P, = CPCr. In the discretestate model given by Equation (5) of Table 2-1, assuming 4' ' = 0, the equation for projecting the error covariance is given by

+

Furthermore, for a continuous nonlinear dynamic system described by Equation (1) of Table 2-1, the nonlinear variance equation is approximated by ~ ( t= ) A(x(t), t)P(t)

+ P(t)AT(x(t), t ) + N(t)Q(t)Nr(t)

(3-30)

where A(x(t), t) is the matrix with its ijth element defined by Aci(x(t),t) = [ a f , ( x ( t ) , t)]l[dxj]. In the nonlinear discrete case, similar results can be obtained.

Example 3-2.

This example, following Zarchan [1979(a)] and Example 2-2, again uses the linear homing guidance loop of Figure 2-15a(ii) and 2-15b to illustrate here the usefulness of covariance analysis. The system equation expressed in matrix form is

The state covariance P(t) is obtained by intcgrating the covariance propagation equation, Equation (3-27), with N equal to an identy matrix. With respect to Equation (3-27), the preceding equations define A ( t ) , and can be uscd to determine Q ( t ) as follows:

Fi ure 3-10a plots the RMS relative separation between the missile and target, as a function of time, which war found by intcgrating Equation (3-27). This value becomes the RMS miss distance at if, or the end of flight. That is, RMS

h,

TlME lrecl

a)

corariance Ana1y.i~ Provides RI Trajectory Profile TlME lucl

b)

cwariuc.

Malyml.

Provldoa

mm Accoloretion Protile Figure 3-10 Covariance Analysis Provides RMS Trajectory Profile (From [Zarthan, 1979(a)], 0 1979 AIAA)

Modem MultIuarlable Control Analysis

106

Chap. 3

miss distance = v ~ ( 3 3) , (,,,,. Because covariance analysis is capable of generating statistical information corresponding to each state, like the RMS acceleration of Figures 3-lob, it is possible with this technique to establish the correctness of assumptions concerning system linearity. That is, such an assumption would be validated if there were no acceleration saturation.

3.9 ADJOINT METHOD The method of adjoints [Besner and Shinar, 1979; Derusso et al., 1970; EASAM, 1969; Howe, 1965; Laning and Battin, 1956; Peterson, 1961; Zarchan, 1979(a)] represents a widely used technique in guidance system design. This adjoint technique is founded on the system impulse response, and is capable of analyzing exactly linear, time-varying noise-driven systems such as a guidance loop in one computer run. The exact performance projections of any quantity at any instant can be estimated using this method. Additionally, the information that measures the contribution of all disturbances (inputs) to the total performance projection (output) can be extracted. Although it has been used mainly in the past in missile guidance system design and analysis, the adjoint technique can certainly be applied to many other problems.

3.9.1 A4joint Philosophy The impulse response of the adjoint system I* is related to the impulse response of the original system 1 in the following manner:

I*(tf - ti, tf - to) = I(to, ti)

(3-32)

where t , and to are the impulse application and observation times of the original system, respectively. If it is desired to observe the impulse response of the original system at the final time tf after several impulses are applied at times t, ( i = 1, . . . , n), then the system response must be simulated for each of the impulse application times to generate [(I/, t,), as demonstrated in Figure 3-lla. For the adjoint system, however, if the observation time is the final time (to = t,), then only one adjoint response must be generatcd. Equation (3-32) reduces to I*(tf - I,, 0) = I((/, 1,). The impulse response of the adjoint system, except for being generatcd backwards, is identical in evcry aspect to that of the original system. Figures 3-1 la and 3-llb illustrate the way in which the two responscs are relatcd [Zarchan, 1979(a)]. The system rcsponse y(t) at time t to any input u is given by

T o dctcrminc how the method of adjoints becomes handy in this situation, Equation (3-32) is substituted into Equation (3-33) to obtain [Pctcrson, 19611

Aaolnt Method

~ c 3c.9

I

Uto, t , )

crvation Time 0

-------Impulse Application Time.t i

tO=ti

Figure 3-11 (a) Generation of I(tf, ti) in Original System (b) Impulse Responses of Original and Adjoint Systems are Related (From [Zarchan, 1979(a)], 6 1979 AIAA)

using y(t) =

ul

t

0

I*(tf

- T . tf -

Z = t - T

t)

U(T)

di

3

y(t)

=lI*(tf-t+z,y-t)u(r-z)dz

(3-34)

108

Modem Multiuariable Control Analysis

Chap. 3

If the final time t k is the time at which the response is desired, then Equation (3-34) becomes

The concept of adjoint system can be extended to account for stochastic inputs. A linear time-varying system that is driven by white noise has a mean square response [Zarchan, 1979(a)]

where 4, is the spectral density of the white-noise input in units of Hertz, and is assumed double-sided and stationary. Unfortunately, the many computer runs required to generate I(tf, t i ) render the direct simulation of Equation (3-36) impractical, as mentioned previously. Substitution of Equation (3-32) into Equation (3-36) yields

For the case in which the final time is of interest ( t = tf), Equation (3-37) can be expressed as

Therefore, the integral of the square of the output of the impulsively driven adjoint system from t = O to t = tf is equixralent to the mean-square response at time t, of a linear time-varying system driven by white noise. The RMS value of the output of the terminal time is the square root of Equation (3-38). The RMS value of the output of the original system in the presence of a white-noise input is then computed by first squaring, then integrating, and finally taking the square root of the adjoint system's impulse response.

3.9.2 Applications

From any linear system, an adjoint system can be constructed by the following stcps [Besncr and Shinar, 1979; Pctcrson, 1961; Zarchan, 1979(a)]:

1. Substitute tf - t for t in the arguments of all variable coefficients, that is, gains. 2. Reverse all signal flow, redefining branch points as summing junctions and vice versa. 3. The input to the adjoint system is a unit i~npulse6(t), whose application point corresponds to the output of interest in the original system.

Sec. 3.9

AdJolnt Method

10g

Using the systcnl adjoint together with the outputs described previously, it is possible to plot llMS miss at interception versus range a t the start o f hotning for each computer run. Because the RMS miss caused by each of the input quantities can likewise be obtained, it is also possible to isolate which factors have the greatest influence on RMS miss. Using the adjoint system technique is therefore much less time consuming than using Monte Carlo techniques and applying noise generators to the r c g ~ ~ l analog ar circuit. By going to acceleration adjoints, one can also study the RMS lateral acceleration requirements for the missile. A balanced design can then be achieved by comparing miss and acceleration statistics. It is easily seen that the adjoint technique is well suited to the problem of statistical atlalysis and optimization of linear time-varying systems. Using this technique, quick estimates of the required gains, natural frequencies and damping, acceleration limits, and ailowable radomc tolerances can be obtained for several initial geometry and other input conditions. As a preliminary design tool, the method of adjoints can significantly lower the amount of work involved in the ensuing analysis of a refined nonlinear system model [Peterson, 19611. The adjoint solution resulting from only one simulation Example 3-3. supplies a tremendous amount of information concerning system performance and behavior as shown in this esa~llplefrom Zarchan [1979(a)]. What has been presented can now be used to study thc homing loop shown in Figures 2-15a(iii) and 2-15b of Examplc 2-2. In the deterministic input, the homing loop has a step target maneuver xvhich has been converted to an impulsive input through integration. The ) to the step target maneuver. output to be considered is the miss distance y ~ ( t f due Figure 3 - 1 3 shows an adjoint model of the homing guidance loop of Figures 213a(ii) and 2-15b, constructed according to the rules laid down previously. The idea is to apply an in~pulseto the adjoint system at the location X4, corresponding to xvhich the output ofinterest in the original system is denoted by y,. In the actual simulation, however, an initial condition of unity is specified at X4 as opposed to a unit impulse o n the derivative. In the process of constructing the adjoint model, the three inputs o f the original system, target maneuver, glint noise, and fading noise, all become outputs. That is, the outputs in the adjoint system are now miss sensitivities due to target maneuver (y=(tf) for deterministic, y.ar,(tf) for stochastic), miss sensitivity due to glint noise y,, and miss sensitivity due to fading noise yf(tf).

Stochastic inputs. Because there is no dependence of the sensitivity coefficients of the adjoint system on the spectral density levels of the error sources, changes in the latter do not necessitate additional simulation runs. The total RMS miss distance, [E(yz(tf))]1'2 applying the principle of superposition, is given by

I Tgt Mvr + u:,(tf) I Glint + uf,(tf) I RIN + u;,(tf) I Range Dep.)Ii2

uy,(tf) = {u;,(tf)

yT(Q Step Target Maneuve~ Miss Distance Sensitivity

4 GLINT NOISE

4r ('t) FAD. NOISE

=m

M

= =

11s 1 I (1)

~2 AT,

2

q1( I f ) + FAD. NOISE

1' I +

9,( 1 0

GLINT NOISE

-,/+ =~ ~ Y L ? + ~ E) IIY($)I ~

+

1

I'

YI: (141

ADJOINT TIME (wcl (8)

ADJOINT TlME l u )

(C)

Figure 3-12 (a) Adjoint Model ofLinear Homing Loop (b) RMS Miss Distance Error Budgct Is Automatically Generated hy Adjoint Method (c) Adjoint IJrovides information on Performance Sensitivity l>ue to Target Mat~cuver(From /Zarchan, 197Y(a)l. Q 1979 AIAA; and /Zarrhat~,19881 ulirh prJnnissiot#.jam ACARDJ

Sec. 3.10

Statistical Linearization

111

Figurc 3-12b plots the individual RMS contributions to miss distance together with the total RMS miss distancc as functions of adjoint time. Hence, adjoint time implies cithcr timc of flight or time to go at which disturbances occur. The figure shows that the biggest contributor to miss distancc in this homing guidance loop is glint noise.

Deterministic inputs. It is, howcvcr, neither a difficult nor expensive task to compute othcr disturbance sensitivities as well. For example, thc miss distance scnsitivity due to a step in target acceleration y T(tf) was also calculated and plotted against timc in Figure 3-12c, which suggests an optimal time of 0.6 sec for the targct to maneuver in ordcr to rnaximizc miss distance before intercept. The figure also indicates that miss distance will bc small if the target maneuvcrs too soon, that is, if the adjoint timc is too large. As the adjoint time approaches infinity, y T(tf) will always approach zero if the guidance system is well designed. The actual adjoint timc required for the miss-distance scnsitivity curve to settle down is a function of the overall guidance system time constant.

3.10 STATISTlCAL LINEARIZATION When it is desired to treat nonlinearities in a stochastic guidance system, that is, acceleration saturation, Monte Carlo techniques must be employed to compute mean-square miss distance. The problem lies with the large number of trials that must be run to achieve even moderate results. Most often, system nonlinearities are dealt with through the construction of a linear model. Using suitable conditions (for example, assuming the actual trajectory of the vehicle is close to its nominal trajectory), a nonlinear function can be expanded in a Taylor series about some operating point (nominal state vector) of the nominal trajectory and only the firstorder or linear terms kept. This approximates a nonlinear function for small perturbations about the operating point and can be used to obtain small signal linearization. When an input is no longer accurately represented by a particular linearization, the input is relinearized about a new operating point. The entire trajectory can be segmented so that the small linearization approximation is valid over each segment to reduce the analysis errors. The perturbations of the system states around the nominal values can be studied using the covariance method. If changes in the variable N G C loop parameters are very slow in comparison with the rate of the associated processes, then the errors introduced by treating the time-dependent coefficients in the linearized equation as constants will be correspondingly small. Moreover, significant analysis errors are not introduced by the process of linearizing the N G C loop equations if: (1) the quantity in question is deterministic; or (2) if the 3u value of the random signal at the input to each of the elements does not exceed the

Modem Multiuarlable Control Analysis

112

Chap. 3

linear region of the element; or (3) the linearized function is differentiated. An even more subtle restriction is posed by the fact that the linearization procedure involves the system state vector itself. Thus, any results obtained will require a lot of computation. These types of restrictions therefore lead to Monte Carlo approaches. However, discontinuities in the system (for example, jamming and sudden target guidance) such as a limiter and two-level switches restrict maneuver in interce~t" these techniques and render linearization in the ordinary sense impossible [Gelb and Warren, 19731. When Monte Carlo techniques are required in the linearization procedure, or when the conditions for ordinary linearization are lacking, statistical linearization may be used. A statistical linearization that approximates a nonlinear operation, but depends on some properties of the input signal, is often referred to as a quasilinearization. This technique can produce different linear approximations to the same nonlinear function applied to different input signal forms. Quasi-linear approximation, unlike linear approximation, can be used for any range of signal magnitudes, and is dependent on the input signal amplitude [Gelb and Vander Velde, 19681. What has emerged as an immensely useful tool for the analysis of nonlinear systems having random inputs is the method of statistical linearization. When statistical linearization is used to analyze nonlinear systems with random inputs, the nonlinear element is replaced by an equivalent gain which depends on the form of the input signal. A method for determining the equivalent gain was developed by Booton [I9531 and Gelb (19741. Consider an approximation of the nonlinear function f(s) by a linear function, in the sense suggested by Figure 3-13, using statistical linearization. T h e input x in the nonlinear system can be decomposed into a mean component m plus a zero-mean illdependent random process I., giving x = PI + r. For this systcm, it is desired to replace the nonlinear element by an equivalent gain

.

Figure 3-13 Statistical Linearizatiori Approxinlation (Frorn /Grlh, 1Y80/ufirlr pcrrrriuiorl ,fnorn Tltc Arrillyti( Sricriic~ Corp. )

K,,. The error signal e(t) is defined as the difference bctwcen y = f ( x ) and the equivalent gain output, and has a mean-square value defined by

Using standard calculus, the minimum value of E[e2] can be found by differentiating this quantity with respect to K,, and setting the derivative equal to zero. When this is done, the result is

It is assumed that the signal r(t) is a zero-mean Gaussian random process whose u) where u is the RMS value probability density function is p(r) = e-'2'('"2) /(6 of r(t). The equivalent gains can therefore be written as

K,, =

1 d3

dr,

-7 utn

K, =

,.f(,,,

1 d 2 7 u~

+ r)e-r21(zo2)

dr

Now, when it is known that the signal x itself is a zero-mean random Gaussian process, m is zero, that is, x = r and the equivalent gain reduces to

In Gelb and Vander Velde [I9681 input-sensitive gains of the preceding type which approximate the transfer characteristics of the nonlinearity are called random input describing functions. These functions are tabulated for several important nonlinearities.

Example 3-4. This example, following the work of Zarchan [1979(a)] and Example 2-2, centers around the limiter shown in Figure 3-14a. The describing function given by Equation (3-43) becomes xe - ~

lim +

Lim

2 / ( 2 ~ 3

dx

m

xe -s1(2u:)

dx

+

-* 1

lim

J-lim

x2e-x21(2u:) dx (3-44)

Modem Multlvariable Control Analysls

Chap. 3

LINEAR (INFINITE LIMIT) PERFORMANCE 0

h

lb ;O do 50 MISSILE ACCELERATION LIMIT (g'sl (C)

Figure 3-14 (a) Input-Output Characteristics of a Limiter (b) Randoni Input Describing Function Approximation to Limiter (c) Acceleration Limit Study Utilizing CADET Approach (d) SLAM Model of Nonlinear Homing Loop (e) Describing Functions for Various Acceleration Limits vs. Forward Time (0 Acceleration Limit Study Utilizing SLAM Approach (g) SLAM Provides Information on Performance Sensitivity Due t o Target Maneuver (From [ Z a r than, 1979fa)). 0 1979 AIAA; and /Zarchan, 19881 u~irhprnnirsion.fiom ACARD)

60

REVERSED GAIN FROM CADET PORTION OF PROORAM

RMS

=

FAD. NOISE

A ~ Y

FORWARD TIME lkal

MISSILE ACCELERATION LIMIT Ig'sl

(E)

ADJOINT TIME lwcl

('3

Modem Multluarhble Control Analysis

116

Chap. 3

which can be evaluated to give 1

lim -x2/(20f)

dx

Finally, rewriting this last equation in terms of the probability integral yields as the describing function -.

Abramowitz and Stegun [I9641 give the following approximation to the preceding integral, which can also be found by table lookup, that is accurate to five decimal places

K,, = 1 - (21fi)e-""'~'(~":)

(0.43618360 - 0. 1201676w2

+ 0.937298w3) (3-47)

where

Figure 3-14b plots the describing function for the limiter against values of lim/u,, the ratio of the limit to the RMS value ofthe input signal. As expected, the describing function is determined by these two values only.

3.10.1 Techniques and Tools for Statistical Linearization Recent techniques involving statistical linearization of nonlinear system elements with covariance analysis allow for the direct statistical analysis of nonlinear systems and have been used on nonlinear missile guidance systems [Pricc and Warren, 19731. An important tool for analyzing and evaluating the statistical behavior of nonlinear stochastic systems is an approximate computer-implemented technique known as the covariance analysis describing function technique (CADET). It involvcs the direct statistical linearization of nonlinear systems, combining in the process clcments of covariance analysis with those of random input describing function analysis [Booton, 1953; Gelb and Vandcr Velde, 19681 to yield statistical pcrformancc projections in one cornputcr run. Thc dcrivation of the describing functions makcs usc of a Gaussian assumption, which may at first glance seem unduly restrictive. However, because lincar elements outnunlbcr their nonlinear countcrparts in the majority of dynamical systems, this turns out not to be the case. Morcover, nonlinear outputs that are not Gaussian in nature are easily converted to nearly Gaussian inputs through the use of low-pass filtering. What has been shown is that the large nurnbcr of Monte Carlo runs can be replaced by one computer run with this ncw method, and that the mean-square miss distance can be computed about as accurately. For 111any

types of nonlinear systems the CADET method [Gelb and warren, 19731 can often be used as a less expensive alternative to the Monte Carlo approach in order to obtain approximate performance projections. CADET has proved itself to be a useful and efficient tool in the preliminary evaluation of nonlinear missile guidance system performance. T o apply CADET to guidance systems, the following principal steps must be implemented. 1. Substitute the appropriate random input describing function gain for each nonlinear element in the original system, assuming a Gaussian probability density function for the input to the nonlinearity. 2. With the linear system model that results from the preceding substitution, use conventional covariance analysis techniques (Section 3.8) to propagate the statistics of the system state vector. Note that the describing function gains are themselves functions of these statistics. 3. Use the elements of the system covariance matrix to calculate the mean-square output at t,, for example, mean-square miss distance at the intercept time.

Example 3-5.

This example follows the work of Zarchan [1979(a)] and Example 2-2. It illustrates how CADET can be applied to the system of Figure 2lja(ii) and 2-15b when the latter is operating in the nonlinear mode with saturation effects. Following the previous discussion, the first step is to replace the acceleration saturation nonlinearity by a random input describing function Kl,,. The linearized system equation is then given by

118

Modem Multluariable Control Analysls

Chap. 3

At this point, P(t) is found by integrating Equation (3-27), in which A is obtained from the preceding linearized system equation and Q is still given by Equation (331). The previously derived describing function gain for the limiter is a function of the statistics of the unlimited commanded acceleration u, and the limit level nli,, and can be computed from Equation (3-47). T o compute the RMS level of input signal to the nonlinearity u:,, u, is first expressed as a function of the states. The resulting mean-square value can be written as 02, = (AV,)'P(5, 5). Example 2-2 provides the input values that were used to perform C A D E T analysis of the homing guidance loop. As in the case of the previous Monte Carlo analysis, the parameter that was selected for the simulation runs was the missile acceleration limit MI,,,. Figure 3-14c displays the results of this study, which were generated from six C A D E T runs. This can be compared with the 200 runs used to generate the Monte Carlo results, which are superimposed on the results of this study in Figure 3-14c. T h e results shown in the figure point to the high degree of accuracy that can be obtained using CADET. A study undertaken by Price and Warren [I9731 showed that CADET was capable of producing results comparable in accuracy to those obtained by Monte Carlo analysis involving several hundred simulation runs.

3.10.2 Statistical Linearization with A4joint Method

Statistical linearization has also been used in connection with adjoint techniques to produce successful results. The combination of the two methods allows for the assessment of primary factors in the overall measure of performance as well as the ability to cvaluatc the stability of the guidance system. SLAM [Zarchan, 1979(a)] in particular, is one of the approaches that have combined statistical linearization with an adjoint method, and is another example of an approximate, computerized technique that is available for the complete statistical analysis of nonlinear noisedriven systems. It basically combines CADET with an adjoint tcchnique, the result of which is capable of generating accurate statistical performance projections and producing an approximate error budget that quantifies the influence of each distrubance on the total system performance. In addition, SLAM has been successfully applied to preliminary analyses of guidance system performance. The following discussions on SLAM are based on [Zarchan, 19881. The principal steps to be followed for using SLAM are: 1. This stcp is identical to stcp 1 of the C A D E T method. 2. This step is identical to stcp 2 of thc C A D E T mcthod. 3. Store thc resulting dcscribing function gains for cach nonlinearity as a function of time.

Sec 3.10

Statistical Linearization

llD

4. Convert the linearized system modcl to an adjoint model in the following way: (a) substitute tr - t for t i n the arguments of all variable coefficients including the described function gains; and (b) reverse signal flow, causing the original systcm inputs to become adjoint system outputs. 5. Propagate the systcm in adjoint time. Steps 4 and 5 together replace the linearized system model of the original system by its adjoint modcl. As pointed out previously, the RMS miss distance derived from SLAM is identical to that derived from CADET. However, SLAM can additionally generate an approximate error budget that shows how each individual error disturbance contributes to the total RMS miss distance. The approximate nature of this error budget, which is valid only so long as the time history of the of the describing function is valid, renders suspect any estimates of miss gain K,,,,, distance for different error source input levels based on extrapolations. Such an error budget is still useful inasmuch as it serves to put in relief those error sources that have the greatest impact on the total RMS miss distance in the nonlinear system. It is also much less expensive to generate than its counterpart derived from the use of Montc Carlo or C A D E T techniques. These methods, in order to examine individual contributors to the total RMS miss distance, must rely on running simulations with one error source a t a time, the results of which are combined in some fashion to compute the total RAMSmiss distance. For nonlinear systems, this method of combining individual error sources does not always produce the same total RMS miss distance that would have resulted from running all error inputs at once. In addition to the aforementioned error budget, another product that arises from the use of statistical linearization methods are sensitivity functions. Although they cannot be used to compute miss distance, being more qualitative in nature, these functions relate system sensitivity to a step-target maneuver, and provide insight into the relative stability of the system. Finally, another meritorious feature of the SLAM concept lies in its self-checking nature. Should the adjoint and CADET portions of the program fail to yield identical RMS miss distances, then it is certain that the program is flawed in some manner, conceptually or otherwise. While it is not possible for any program to totally eliminate the possibility of undetected errors, the SLAM concept makes great strides toward this goal.

Example 3-6. Following Zarchan [1979(a)], the example here - presented . looks again at the nonlinear stochastic guidance system of Figures 2-15a(ii) and 2l j b to illustrate the usefulness of SLAM. Using the inputs of Example 2-2, the time history of the describing function Kli, is first generated from the CADET module of the program. Substituting tf - t for t as described earlier, the SLAM program then generates K*,,,, the gain of the reversed describing function, which

120

Modem Multiuarlable Control Analysls

Chap. 3

is entered into a linearized adjoint model of the original nonlinear system. This is shown in Figure 3-14d. An approximate error budget for the nonlinear system is then generated by running the adjoint module of the SLAM program. The parameter selected in this study was, as in the case of the previous examples in this chapter, the acceleration limit nli,. Figure 3-14e plots the describing function gains resulting from the C A D E T analysis as a function of forward time for different values of nl,,. It can be seen from the figure that, as long as no saturation occurs, the gains are unity, and that these decrease with increased saturation. This is, of course, as the theory predicted. Figure 3-14f plots the total RMS miss-distance error budget obtained from SLAM as a function of the missile acceleration limit. As expected, the total computed RMS miss distance corresponding to each of the acceleration limits is identical to that obtained from running the CADET program alone in Example 3-5 (see Figure 3-14c). When the system is slightly saturated, that is, ittin, = 60 g, glint noise can be seen to be the greatest contributor to the total RMS miss distance, which can also be confirmed by Figure 3-12b. This is quickly superseded by random target maneuver as saturation increases, or as nl,, decreases. Conversely, the contributions from fading and glint noise slightly decrease. This is understood by realizing that saturation acts to provide additional filtering. The sensitivity function due to a step-target maneuver is also plotted in Figure 3-149 as a function of adjoint time for various values of titi,,,. These functions represent mathematically the impulse response of the quasi-linearized system, but cannot, as mentioned before, be used to compute the miss distance due to steptarget maneuver. The sensitivity curve of Figure 3-14g corresponding to t~l,,,,= 5 , that is, the linear casc, first peaks then rapidly decreases to an asyn~ptoticvalue of zero. This situation, which is also depicted in Figure 3-12c, would be found to occur in a well-designed missile guidance system en~ployingI'NG. I n this casc. a target maneuver initiated a t a time to go of not less than ten guidance time constants should have no bearing 011 the total miss distance. Figure 3-14g demonstrates that the miss-distance sensitivity asymptotically approaches values other than zero as soon as an acceleration limit is introduced. Such a situation is typical of PNG systems whose effective navigation ratio is not sufficiently large. In the case of acceleration limits smaller than those indicated in the figure, there would be a monotonic increase in the sensitivity function, representing a system that is stable yet incapable of guiding effectively on maneuvering targets. Such information, while not of a purely quantitative nature, noncthclcss provides the guidance system designer with insight into thc details of system behavior.

3.10.3 Applications It has been shown [Zarchan, 1979(a)]that the preceding C A D E T and SLAM tncth-

ods produce the same degree of accuracy demonstrated in Monte Carlo methods applied to missile guidance system evaluation. In gcncral, though, thc missile ac-

Five to Seven Times the Acceleration Capability of the Tar et Is Needed by the Missile in Order to Intercept the Target sing PNG.

8

Two to Four Times the Acceleration Ca ability of the Tar et Is Needed b the Missile in Order to Intercept the arget Using APr$or Advanced duidance Law.

.F

Linear Analysis

I

I 10

Figure 3-15

I

I

I

I

20 30 40 50 Missile Acceleration Limit (g's)

I

60

I

70

Missile Acceleration for Target Intercept: Nonlinear System Analysis

celeration limit greatly influences the RMS miss distance. Figure 3-15 shows that, for 7-g maneuvering targets an acceleration capability five to seven times greater than that of the target is required by the missile to be able to intercept the target close to the accepted level if PNG law is used. However, the use of augmented proportional navigation (APN) guidance or modern guidance laws requires the missile to have an acceleration capability two to four times that of the target to be able to intercept the target. For further details, see Chapters 6 and 8.

3.11 QUALITATIVE COMPARISON The qualitative comparison presented in this section is based on material presented in Zarchan [1988]. The selection of a particular computerized method for performing statistical analysis is not always a straightforward matter. Depending on the system being analyzed and the type of information required, there are occasions that call

Modem Mu(tiuariab1e Control Analysis

122

Chap. 3

for the use of more than simply one method to gain a complete system understanding. Because it requires basically a random number generator and a postprocessing routine, in addition to a simulation program for the system equations, the Monte Carlo method is the most general and most easily applied o f these techniques. Inspection of a block diagram of the system is often easily translated into the system equations. When it is desired to use the adjoint technique, the modified diagram can be obtained from the original block diagram of the system, and these are also easily translated into system equations. For this reason, the adjoint technique is almost as easy to utilize as the Monte Carlo method. The same is not true of covariance analysis, which requires the system equations to first be expressed in statespace form. Covariance analysis is therefore more difficult to implement, and systems composed of both analog and digital sections must be handled very carefully. Problems that are basically linear except for the inclusion of specific nonlinearities such as saturations or dead zone lend themselves well to analysis by application of the CADET and SLAM techniques, both of which are more difficult than the Monte Carlo method t o utilize yet no more so than covariance analysis. Besides difficulty of implementation, another way in which the methods differ from one another is their C P U usage, or computer running time. The greatcr the number of differential equations needed to describe the system and the number of computer runs required to perform a statistical analysis, the higher is the cost of doing the analysis. The Monte Carlo method requires many more computer runs than any of the other methods to obtain accurate results; whereas, in the casc of linear systems, the adjoint and covariance methods only require one run because they are both exact methods. For an 11-state system, n differential cquations must be integrated for the adjoint technique compared to ti' differential equations for a covariance method. Thcrc arc no exact methods of statistical analysis for nonlinear system analysis. The accuracy obtained by using CADET and SLAM is close to that obtained from several hundred Monte Carlo runs [Gelb and Warren, 1973; Price and Warren, 1973; Zarchan, 1979(a)]. For an 11-stable system, 17 differential equations must be integrated with CADET, while ,I' + rr cquations must be integrated with SLAM. A comparison of the different statistical analysis techniques can therefore be made on the basis of cost, which is defined hcre as cost

A

number of equations x number of runs

(3-48)

Figurcs 3-16a and 3-16b comparc the costs of using these methods. The): s l ~ o \ ~ ~ that, in spite of the usually large number of computer runs required, thc Montc Carlo mcthod can be chcapcr to i~nplcmcntthan covariance analysis, CADET. or SLAM. In the previous discussions of the quantitative and qualitative aspects of thc various computerized statistical analysis mcthods, it has been dcmonstratcd for thc case of linear systems that the mcthods of adjoints and covariance analysis yield exact pcrformancc projections. In the casc of nonlinear system analysis, the Montc Carlo mcthod is easily rivalcd by the CADET and SLAM techniques.

COS NUMBER OF RUNS

COVARIANCE ANALYSIS 100 10

100 NUMBER OF STATES

a)

1000

coat Cmparlmn lor Linear Syatema

COST

NUMBER OF STATES

b)

Comt C-11-

lor Nonlinear Syatema

Figure 3-16 (a) Cost Comparison for Linear Systems (b) Cost Comparison for Nonlinear Systems (From [Zarchat~,19881 with permission from ACARD)

124

Modem Muitiuarhble Control AnalysIs

Chap. 3

3.12 OTHER PERFORMANCE ANALYSES Stochastic disturbances that greatly affect NGC performance must be identified and accounted for in the NGC system synthesis. The turbulence and gust velocities produce incremental aerodynamic forces and moments acting on the vehicle. The turbulence and gust disturbances are applied to the aerodynamic terms in the equations of motion. but not to the inertial or acceleration terms. Similarly. . . , their effects are included for the aerodynamic sensors but not for the inertial sensors. For example, a 30-ftlsec horizontal crosswind hitting a flight vehicle with a 300-ftlsec forward speed will create a 5.7-deg aerodynamic sideslip while the inertial sideslip still remains zero. Turbulence and gust are responsible for random variations in airspeed VtOt,r,in the angle of attack a, and in the sideslip angle P. The model accounts for turbulence and gust through gust components of a, P, Vt,,,l as a = ai + a,, Q = Q i + and Vt,,,l = V i+ V,, respectively, where If,, ol,, and p, are the increments of VtOtal, a, and p, respectively, due to the turbulence and gust. The dynamic model includes three channels of independently generated Dryden or von Karman model turbulence. Dynamic loads due to atmospheric turbulence, discrete gusts, and discrete control inputs must be calculated in order to analyze vehicle's flexibility effects. In the dynamic load analysis, the following data are needed: (1) externally produced structural data composed of mode shapes, lumped masses, generalized masses, and generalized stiffness; (2) vehicle geometry; (3) control law; and (4) information of flight condition, for example, velocity, altitude, Mach number, and mass distribution (center-of-gravity location). The maximum expected load varies as a function of the atmospheric turbulencc environment and the maneuver level of the vehicle. It is the sum of the steady-state load and the predicted peak incremental value. Results of a dynamic load analysis are rcpresentkd in terms of statistical quantities (for example, RMS responses), PSD response, time response of the dynamic loads (for instance, torsion, bending moment, acceleration, and both deflection and rate control surface activity), and singular value response. Recall that spectral (PSD) analysis is used to obtain vehicle responses to random disturbances. ~ h k r e s u l t of s PSD analysis can then be used to estimate fatigue damage rates, maximum expected loads, stability information, and crew compartment acceleration environmcnts. In gcncral, vehicle response is constraincd by aerodynamic and structural restrictions. Control surfacc activity is limited by actuator and hinge momcnts. As an cxamplc, the closed-loop system RMS control surface rate must be significantly below lld2 times the rate limit, sincc the boundary value (maximum and minimum valuc) is about times the RMS valuc. Generalized harmonic analysis is an cxtcnsion of thc spectral analysis in obtaining a more comprehensivc vehicle's dynamic response to atmospheric turbulence. In generalized harmonic analysis, the theoretical frequency response functions are derived from the nonlincar ,

f&,

~

-

Sec. 3.12

Other Performance Analyses

125

flexible dynamic equations of motion. Letting the gust input be a unit impulse and taking the Fourier transform of the flexible dynamic equations of motion yield the frequency response equations. The response equations are then solved algebraically to obtain complex motion frequency response functions. Summing the resulting airload and inertial load responses allows the load and stress frequency responses to be obtained. Consequently, the complex load and motion frequency responses are computed.

Modem Filtering and Estimation Techniques As the caption of Figure 4-1 suggests, this chapter is concerned with modern filtering and estimation techniques. As can be seen from the figure, the fundamental (purely classical) aspects of filtering and estimation are covered extensively in Book 1 of the series, which also includes discussions on signal reconstruction and observers. Filtering and estimation techniques can be divided into two approaches, deterministic and stochastic, the latter of which is applied to systems with noisy inputs. The deterministic approach leads to the design of the Luenberger observer, treated in Book 1 of the series. Both deterministic and stochastic methods are related to the classical elements of complementary filtering, also covered in Book 1 of the serics. The bulk of the current chapter deals with optimal estimation techniques, which make up Sections 4.1 to 4.5. These are stochastic methods including linear minimum variance estimation, nonlinear filtering, prediction and smoothing, Kalman filtcr analysis, and operational considerations. Finally, in Section 4.6, a combined complementary1Kalman filter approach is presented to close the gap between the hcuristic classical dcsign and the analytical modern estimator dcsign. Many dynamic modeling and control applications require at some time during operation estimates ofunknown variables in the equations and/or state of the system. Ultimately, the values of these quantities depend on measurements and the assumcd model structure, and on statistics related to measurements, the model, and inputs to the system. This section develops techniques designed to produce such estimates,

Chap. 4

Modem Fllterlng and Btlmatlon Techniques

Filtering & Estimation (Book 1)

Deterministic Luenberger Observer ( B o ~ k1)

Stochastic Complementary Filtering (Bopk 1)

+

Optimal Estimation (Kalman Filter) (Secs. $1-4.5)

Combined ComplementaryiKalman Filter Approach to Estimator Desig (Sec. 4.6)

4

Adaptive and Digital Applications (Books 4 & 5 )

Figure 4-1 Modern Filtering and Estirnatlon Techniques

which can be applied in either a batch or a recursive mode. In a batch mode, the estimate is determined after all of the measurements are recorded, whereas in a recursive mode, an updated estimate is obtained after each measurement is recorded. Recursive techniques take on special importance in the case of on-line applications involving digital computers. Modeling of a time-varying dynamic system for which there are uncertainties associated with inputs, outputs, or the system itself, is formulated through state vectors. Modern estimation methods make use of known relationships to compute the desired information from measured information, taking into account measurement errors. the effects of disturbances and control actions on the system, and prior knowledge of these variables for which the new information is desired. Filters designed from modern estimation methods for noisy data reduce the effects of drift with regard to the desired information. State estimates of a dynamic system are not confined solely to being estimates of the current state. In fact, there are occasions where it is desirable to obtain an estimate of the state at a specified future time. Such a problem arises in the application of intercept guidance. This and other types of similar problems are known collectively as the prediction problem, and estimation techniques can easily be modified to treat these. The smoothing problem represents yet another extension of basic estimation techniques. It is desired in this problem to determine the state at some previous time based on measurements recorded up to the current time. Both the prediction and smoothing problems are covered in subsequent sections. Kalman filtering is a method for estimating the state variables of a dynamic

Modem Nlterlng and Estimation Techniques

128

Chap. 4

system recursively from noise-contaminated measurements. The filter determines the system's present state by optimally combining theoretical estimates with measurement noise characteristics, based on a knowledge of the system model. In contrast to methods that -process measured data simultaneously and require substantial data-storage capability, such as least-squares methods, the Kalman filter processes measurements sesuentiallv and thus reauires much less data to be stored. Each new measurement results in a new state estimate. In addition to the estimate itself, a covariance matrix of estimation errors representing uncertainties in the estimate is also generated by the Kalman filter. For linear systems, the actual, measured data are not required to generate the covariance matrix, so that the latter can be calculated in advance based o n different models or measurement arrangements. Consequently, estimation errors are known before the dynamic system is actually implemented, allowing a determination to be made of the effect of additional state variables on estimation accuracy. This in turn permits a prediction of improvements in accuracy that can be achieved by using additional sensors. The Apollo space program is generally accepted as being the first known application of the Kalman filter. Since then, Kalman filtering has proved eminently useful in such applications as the determination of spacecraft orbits, satellite tracking and navigation, digital image processing, economic forecasting, industrial processes, and nuclear power-plant instrumentations. Some particular applications of the use of Kalman filter theory include ascent guidance at Cape Kennedy and orbital determinations for Voyager 1 and Voyager 2. In this last problem, spacecraft-based optical observations were combined with earth-based radiometric observations to accurately determine the orbit during the Jupiter encounter approach phase. In the case of nuclear powcrplant instrumentation, on-line instrument failures are detected and fault-tolerant computer control systems are designed, each by making use of Kalman filter theory [Baheti, 19871. As powerful and useful as Kalman filter theory is in so many diverse areas of application, it still cannot circumvent unacceptable results arising from poor system modeling and improper conditioning of the covariance equations. Modeling is primarily concerned with the definition of state variables, selection of the coordinate system to be used, the method of linearization, and the effects of bias, disturbance, and modeling errors. Even in the same area of application, the dynamic model developed for a particular problem may no longer be suitable if changes or new restrictions are introduced into the problem. The following sections provide a revicw of approaches and advances in Kalman filtering. The reader who is also intcrcsted in a chronological account of the development of this mathematical filtering thcory and its widespread engineering acceptance should consult Schmidt [I9811

.

- -

4.1 LINEAR MINIMUM V A M C E ESTIMATION: W M A I Y FILTER The Kalman filter technique is summarized in this section and is shown in Figure 4-2.

Sec. 4.1

Linear Minimum Variance Estimation: Kalman Filter

a) Syucm Model and Discrete Kalrnan Filter w(t.1)

i(k,+

b) Linear Continuas Kalrnan Filter

, Figure 4-2 Filter

Optimal Stale Gnimalion

(a) System Model and Discrete Kalman Filter (b) Linear Continuous Kalman

4.1.1 Discrete Kalman Nlter Table 4-1 defines the well-known Kalman filter [Kalman, 19601. A historical treatment of this subject is documented [Gelb, 19741. The Kalman filter is the optimal state estimator that provides a systematic scheme for computing the estimator gain matrix Kk.A flow chart is drawn in Figure 4-2a. Table 4-2 defines the multiplerate Kalman filter.

4.1.2 Continuous Kalman Nlter In Gelb [I9741 and numerous other places the Kalman filter is derived for a linear system with a continuous measurement equation given in Equations (3) of Table 2-1 by performing a limiting process involving Atk = tk+, - tk on the discrete filter solution. The result is a natural generalization of the discrete case and only the I result will be presented here [Powers, 19781.

1. Given the state equations (3) of Table 2-1 and the initial values E[x(to)]= 20 and E[(x(to)- io)(x(to)- 2 0 ) ~=] Po. The matrix R ( t )must exist at each t. 2. Integrate the matrix differential equation using P(to) = Po as follows

-'

Modem Nltering and Estfmation Techniques

130

Chap. 4

TABLE 4-1 DISCRETE KALMAN FILTER STEP-BY-STEPPROCEDURE

Step I : Problem Formulation

Given Eq. (5) of Table 2-1, no measurement is made at time to, and k(0)' represent the estimate of x at to with known covariance

P: = E[x(O) - i(0)') (x(0) - k ( ~ ) + ) ~ ]

is used to (1)

Application of the exbectation operator to Eq. (1) produces an apriori estimate of x(1) which is denoted by %(I)'. The associated covariance P; is obtained in a similar manner. The notation [ ](k) - represents the value of [ ] at 1, before the measurement z(k) is performed, while I ](k) + represents the value of [ ] at t, after the measurement. The latter is called the apmeriori estimate. The equations for fr(k)+ and P: are functions of %(k)-PI, and estimate of i(k)'. In z(k), and represent the linear unbiased minimum variance this recursive approach, each time a new measurement is performed, immedialely afterwards an updated estimate of x is obtained. Step 2: Time Update i Since E[w(k-l)]=O, then given %+(k-I),and fj , the expectation operation on Eq. (Sa) of k-1 Table 2- 1 gives i(k)- = Q(k-1) 8(k-1)' + Y(k-1) u(k-I) (2) Based on the definition. P; = E{[x(k) - i(k).] [x(k) - i(k)-lT) (3) The following equafon is derived: Pi = @(k-l)~;., QT(k-1) + N(k-1) Q,., liT(k-I) (4)

(LUMV)

..

Step 3: Meastrrenrerlr Update Given $,(k)- and P: ,calculate

P; = [I - K, H(k)]PL i(kf = %(k)-+ K,[z(k)

(6)

- H(k) k(ky]

(7)

Step 4: If measurements are exhausted, stop; otherwise, set k=k+l and return to step 2. -

-

-

It should be noted that in many cases, Eq. (6) is replaced - by P; = [I - K, H(k)] P i [I - K, ~ ( k ) ] '+ K, R, K: (8) because of its superior numerical properties. That is, Eq. (6) may result in bad subtraction, whereas Eq. (8) involves the sum of two positive definite matrices (which will keep P: positive definite).

-

Linear MInlmwn Varknce Estlmatlom Kalman Fflter

Sec 4.1 TABLE 4-2

Dl

MULTIPLE-RATE MLMAN rlLTER

S~epI : Problem Formulation Given Eq.(5) of Table 2-1,the problem formulation is the same as that in step 1 of Table 4-1. Let the measurement vector be defined as zf = fast measurement available every Tf sec. = q = slow measurement available every T sec. L

J

and zfis available 1times as often as q, i.e.. T, = Tf. Time is defined by the index (k, i), where k refers to the basic time period T, and i refers to the short time period Tf. At time index (k, 0) both q and z, are available giving the multi-rate measurement equation. Hp (k, 0) + v, (k, 0) v, (k,0) N(0, r,) & - H,x (k, 0) t v, (k, 0) v, (k, 0) N(0, r,) At the remainder of the time instants only q is available and the corresponding measurement equation is H?(k,i)tv,(k,i) i = 1 , 2 , ..., 11-1 =[ 0 vf (k, i) N(0, rf)

[-I'[=.

-

I

.=[a\

-

Step 2: Time Updare

Repeat for i = 0, 1, 2,

..., 1-1.

i(k, i+l)' = @(k, i) k(k, i) + Y(k, i) u(k, i)

Step 3: Measurement Update For i = 0. P(k, 0)'

= k(k,

For i = 1, 2,

.

0)-+ Kf[zAk, U) - H&(k, O).]

-1,

S(k, i)' = i(k, i)'

+ &[q(k, 0) - H,k(k, O r ]

+ K,[z&k, i) - ~ $ ( k ,i)']

Step 4:

If the measurements are exhausted, stop; otherwise, set k = k + l and return to step 2. The above results can also be applied to multiple sampling rates. Following [Lewis, 19861, let the continuous system be discretized using a period T = Ti, where Tf is much less than the data sampling period T,,where T, = k Tt. No data arrive between kT,, so during these intervals only the time update (step 2) should be performed. For this update, use the system dynamics discretized with period TI. When data are received, at times kT,, the measurement update portion (step 3 but treat K, = 0) should be performed. For this update, the measurement noise covariance R' = R,T, is used.

and define rhe Kalman gain ~ ( t =) ~ ( t ) ~ ( t ) ~ ~ - l ( t )

(4- 1 b)

The state estimate is defined by integrating

i ( t ) = A(t)?(t) + B ( t ) u ( t )+ K ( t ) [ r ( t )- H ( t ) i ( t ) ] ,

$(to) = i o

(4-lc)

132

Modem Filtering and Estimation Techniques

Chap. q

The estimation is corrected by the residual term x - H2 through K. Figure 4-2b presents the linear continuous Kalman filter. The Bu(t) term includes all the known inputs to the model dynamics so that the filter can use this information to improve its estimate. If E [ w ( t ) v ( ~ ) ~=] S(t)8(t - T) is a white-noise process, then only Equation (4-lb) is modified to give

'

K(t) = [ p ( ~ ) H ( t+) ~N(t)S(t)]R- (t)

(4- 1d)

In many cases the filter solution is studied in two phases: the transient phase and the steady-state phase, with the steady-state characteristics defined by the solution as t approaches infinity. In classical linear differential equation terminology, these correspond exactly to the homogeneous (steady-state) and particular (transient) solutions. There exists a steady-state Kalman filter gain which is the most useful for control problem applications. Supposing a t this point that the matrices A, B, N, H, Q, R, and S in Equations (4-1) are all constants, it can be shown that the limit of the covariance matrix P(t) as t approaches infinity, exists, for example, P,, and furthermore that it is defined by the algebraic Riccati equation

This last equation is nothing more than Equation (4-la) with Kalman filter gain is the constant matrix

P = 0. The resultant

and thus the steady-state estimate by Equation (4-lc) is

Given that these arc the statc equations o i thc stcady-state Kalnlan estimator, thc stability of this last differential cquation is strictly a function of the matrix A K,H. A great deal of information about the bchavior of Equation (4-4) can be obtaincd by analyzing the roots of the characteristic cquation corresponding to det[sl - (A - K,H)] = 0. A sufficient condition for the steady-state filter estimate to be asymptotically stable is that thc system is observable. Assuming Q 2 0 (positive sernidcfinite) and R > 0 (positive definite'), the system is observable if and only if the 11 x r ~ pmatrix [ H T I A ~ i (AT)'HT H ~ i ... i ( A ~ ) " - ' H ~has ] a rank 11.

Example 4-1: First-order altitude rate Kalman filter. As in Book 1 of thc scrics, thc altitudc rate h can bc infcrrcd from two mcasurcmcnts, a bar-

omctric tncasurctncnt iln and an accclcromctcr mcasurcmcnt /;:. It is worth\vhilc here to apply the stcady-statc Kal~nanfiltcr to thc samc problem. First, is trcntcd as a stochastic input with known noisc charactcristic w(t), that is, ?(I) = 1 ) = /;,(I) t u f ( t ) , whcrc ul(r) is a zero-mean whitc-noise process with variance q. Sccondly, /IB is trcatcd as a nlcasurclncnt z = = 11 + v = x v where v(r) is a

I;,

+

zcro-mean white-noise proccss with variancc r , and r is a constant. With rcgard to Equations (4-1). A([) = 0, B(r) = 0, N(r) = 1 , H(r) = 1, Q = q, and R = r. Applying the steady-state Kalman filter produces the algebraic Riccati cquation - ~ ' l+ r q = 0 whose solution is P, = d :.i The stcady-state Kalman filter gain is K I = d& and the estimate equation is h = I;:, + g& ( i l H - i). Comparing i this solution with the complementary filter solution of Book 1 of thc series, h = ;1

+ -T1 ( I.l l ,

-

jl),

shows that

T =

VG. If T E 12, 61, then, physically, the h B ( t )

measurement is about two to six times noisier than thc I;:, measurement, with respect to variance [Powers, 19781.

4.2 NONLINEAR FILTERING The linear filtering problems of the previous sections are basically solved by a single approach. In many applications, the system dynamics and the measurement models are nonlinear. Nonlinear problems in general require more problem-oriented tcchniques, along with considerably more knowledge of stochastic process theory. Fortunately, one of the most successful techniques for treating nonlinear problems developed to date is a natural extension of the linear-oriented Kalman filter known as the extended Kalman filter (EKF).

4.2.1 Extended Kalman Filter (Em) The Kalman filter is a standard filter for state estimation in linear systems. For systems with nonlinear dynamics, a natural extension is to linearize the system around the current state estimate 2(t) and apply the Kalman filter to the resulting linear, time-varying system. This is the EKF which has been described in detail in Jazwinski [1970]. This filter has also been widely used for the combined (nonlinear) problem of estimating th'e state and the parameters of a linear system. Discussions of the latter application are presented in Book 4 of the series. The EKF is presented in this section along with the procedure for treating the continuous dynamic model, discrete measurement case. The continuous nonlinear system is described by Equation (1) of Table 2-1. The main idea is to use expansions of the nonlinear functions f and h about the estimate 2 whenever linear forms are necessary in the development. There is not a great deal of difference in the structures of the covariance and gain computation equations for the Kalman filter and for the EKF, and these are summarized as follows. The initial condition is also assumed to have a normal distribution, E[x(to)] = 20, E[(x(to) - 20)(x(t0) - 20)T] = PO.

Modem Filtering and Estimation Techniques

134

1. Define A[i(t), t]

=

-

H[?(t), t] =

Chap. q

(4-5a)

2. Integrate the coupled (in general) differential equations

where K[2(t), t]

= p(t)HT[2(t), t ] ~ - ' ( t )

(4-5d)

The EKF yields very nearly optimal estimates if +e linearization is accurate. The EKF involves the integration of n + [n(n + 1)/2] '(since P is symmetric) coupled differential equations in the variables 2 and P with the initial conditions given by Equations (4-5b and 4-5c). As mentioned previously, the idea behind the EKF is that of linearizing the nonlinear equations about the current state estimate 2(t), a technique which is termed relinearization. In utilizing such a procedure, the approximation remains linear because the estimated state is as often as not closer to the actual state. The structure of the algorithm for the EKF is illustrated in Figure 4-3. Because of the linearization procedure, the EKF covariance (and hence the gains) depends on the current state estimate and the measurement data. Consequently, it is not possible to compute the EKF covariance and gain matrix as a function of time off line as in the linear case of Section 4.1 since the state estimates of the EKF are no longer optimal. O n the other hand, if an a priori nominal trajectory, for example, x*(t) replaces 2(t) as the base trajectory in the expansion procedure, then Equations (4-5a to 4-5d) will involve known functions of time (for example, A[x*(t), t], H[x*(t), t ] , and so on) and the Kalman gain can be computed a priori. However, this procedure has not met with the same success as the EKF in actual applications. Note that the standard KF guarantees stability while the EKF does not.

E~s. (4-k&d) Eqs. (4-5a) Calculation of

Initial Estimation Linearization of Estimation Error Error Covariance -b Covariance p(t) and 4- Model About State Estimate P(t,) Kalman Gain K(t) State Variable Model A K(t) of Nonlinear System Measurement Data Initial State Variable Estimate

+Computation of

Figure 4-3

State Estimate Eq.(4-Sb)

*

State +Variable Estimates

Structure of an Extended Kalman Filter

Sec 4.2

Nonlinear Flltertng

135

TABLE 4-3 CONTINUOUS DYNAMICS, DISCRETE MEASUREMENTS: EXTENDED KALMAN PILTER

Given Eqs. (2a&b) of Table 2-1, and E[x(to)]= Step I: Defme

4,

Step 2: Given a(!;), Pi, integrate forward to ti+, = qa(t),u(I), ti, = ~ i ( t11)P. + PA[$(^). 1 Set k = k+ 1and go to step 3. Step 3: Given $(ti), Pi, z(k), deternine = $(ti) + K,{z(k) h[!!(t;),

8

P

&ti)

-

E[(x(to) $,)

-

1+~Q(0.

(~(1,)

- to)T] = PO.

P(!,) = P(~;)

kl)

P: = [I - KrHk[%(t;)]]P; where Go to step 2.

= Pi ~:[ft(ti)l{Hk[a(ti)lPi H:[s(~~)]+ %I-'

Note that if to < I,, then the definitions %(t:) = ko, P; = Po are made, and step 2 is implemented as the first step. If a measurement is given at to, i.e., I, = to, then the definitions $(ti) = 9,. P; = Po are made and step 3 is implemented as the first step.

Several modified versions of the EKF have been proposed and implemented to deal with nonlinearities in the model. Some of the more successful variations of the EKF are represented by the second-order filter, the iterated EKF, and the Jadaptive filter Uazwinski, 19701. The algorithms developed are capable of yielding significant benefits only in certain applications and are not of a very general nature. For this reason, the application of any of these algorithms must be carefully examined and acccomvanied bv a substantial amount of simulation studies. The EKF filter for the continuous-time nonlinear system with discrete-time nonlinear measurements (see Equations (2a & b) of Table 2-1) is presented in Table 4-3, from which the linear case of continuous dynamics, discrete measurements can be easily deduced.

4.2.2 Statistical Linearization Technique A linear approximation is desired for a vector function f(x, t) of a vector random variable x, having probability density function p(x). Following the statistical approximation technique, a statistically linearized filter is given in Table 4-4 in which it should be observed that the structu2e is similar to the EKF in Table 4-3; however, now K, and K h have replaced A and\ H k . The computational requirements of the statistically linearized filter may be greater than for EKF derived by taking Taylor series expansions of the nonlinearities because the expectations f , h[x(tk), k], and so on, must be performed over the assumed Gaussian density for x. However, the A

..

Modem Filtering and Estimation Techniques

136

Chap. q

CONTINUOUS DYNAMICS, DISCRETE MEASUREMENTS: STATISTICALLY LINEARIZED FILTER

TABLE 4-4

Given Eqs. (2aBrb) of Bble 2-1, and vx(to)] = kO, E[(x(to) It is reasonable to approximate f(x,t) by the linear expression

- 9,) (~(t,)- 9,) T] = Po.

f(x, t) m a(t) + x(t) (1) where a(t) and K,(t) are a vector and a matrix to be determined. Defining the error e f(x, I) - a(t) - %(I) x(t) (2) it is desired to choose a(t) and K, such that the quantity J = ~ [We] e ~ (3) is minimized for some symmetric positive semi-definite matrix W. Substituting Eq. (2) into Eq.(3)and setting the partial derivative of J with respect to the elements of a(t) and K, to zero, gives a({) = ?(XI - yk, K, = {[fxT],,, - ?kT} P-I (4) To derive approximate minimum variance filtering algorithms, Eq. (4) is substituted into Eq. (1) yielding f(x, 1) m kx) + &(x - 8). h[x(tk). k] h[k(tk). k] + I$,(k)[~(t,) 9(ti)l

-

-

(5)

This formulation necessitates computing f, hi, &, and Kh, each of which depends on the probability density function for x, a quantity that is generally not readily available. Consequently, an approximation is needed for p(x) that permits the above quantities to be calculated. For this purpose, it is frequently assumed that x is Gaussian. Since the probability density function for a Gaussian random variable is completely defined by its mean k and its covariance matrix P, both of which are already computed in any filtering algorithm, it is possible to compute all the averages in Eqs. (5). Assuming that the right sides of Eqs. (5) are calculated by making the Gaussian approximation for x, the statistical linearization for f and h can be used for the nonlinear filter. The step by step procedure of the statistically linearized filter is summarized as follows: Srep 1: Definitions [fxT],,, ?, Et are expectations calculated assuming x -N($, P). k, k(ti) and [h xT(t;)],,, are

-

expectations calculated assuming ~ ( t k ) N [$(ti), pi].

Step 2: Given kcti),

P;, integrate forward to ti+, Et = REt(t), u(t), I], P = K$P + Q I ) , P(1,) = PO;)

PC+

= {[fxqC,,- ?kT} p-l(r) Set k = k+l and no to step 3. Step 3: Given a(&). Pi, a k ) , detlrmine %(ti) = k(tk) + Kk { ~ ( k )h[k(tk), k]} &(I)

-

-

-

= [I $K,,(k)] PC where

Kk= Pi K L ( ~() q ( k ) Pi ~ l ( k+) RI,)-'

From [Gclb. 19801 with prrmission from The Analytic Scie~lccsCorp.

Sec. 4.2

Nonlinear Mitering

W7

performance advantages offered by statistical linearization may make the additional computation worthwhile. A detailed derivation can be found in Gelb [1974].

4.2.3 Multiple Model Estimation A multiple model approach considered here is to assume two or more levels of process noise. A filter is set up for each model and, based on their likelihood functions, the probability for each model being correct is obtained [Bar-Shalom and Luh, 19861. Let Ae ( e = 1, . . . , m) bc the event that model e is correct with prior probabilities P{Ae} = pe. The likelihood function of measurements up to time k under the assumption of model Ae is Le(k) = P ( z k 1 Ae) =

n P [ v e ( i ) ]where k

zk

,=I

= { z ( l )... z ( k ) }and, under the Gaussian assumption, the probability density func-

tion of the innovation from model

e is

Using Bayesian rule, the a posteriori probability that model E is correct at time k is

PIAe I z k } = P { A ~ } P { ( zAe} ~ j=l

P{A,}P{zk I A,} = L e ( k ) p e

j= 1

(4-6) It is assumed in Equation (4-6) that the same model has been in effect from the

initial time. However, this is obviously not true if the model changes at some time during the interval [ I , k ] . In view of this, a "sliding-window" likelihood function has to be used. It is now assumed that the model changes at time k - N, so that one considers only the likelihood functions of the measurements from k - N + 1 to k. Each filter starts from the same initial condition at k - N . The likelihood function corresponding to model e is then

Le(k - N , k) = P [ z ( k - N

+ I), . . . , ~ ( kI Ae, ) ~

( -k N ) , . . . z ( l ) I

probability of model P{Ae I z k } and t at time k:

Modem N1terin.q and Estimation Techniques

138

Chap. q

The state estimate is a weighted average of the model-conditioned estimates with the preceding probabilities as weights: 2,

E{x(k)I zk} =

E { x ( k ) 1 A,, z k } P { A j 1 z k )

(4-8)

j=1

A "fading memory" approach to the computation of probabilities in a multiple model situation is now considered. The discounted (fading-memory) average norm of innovations with discount factor 0 5 a-,5 1 ( j = 1, . . . , m ) is k-e

- 1)

=

P.;(K)

+ ~ ; r ( k ) s ; l ( k ) ~ , ( k=) 2 a ~ - C ~ ~ ( t ) ~ . ; l ( e ) (4-9) ~j(e) E=O

where v j ( k ) is the innovation from the filter based on model j. The "effective" memory length is (1 - a j ) - I . Let S j ( k ) = S, be the steady-state covariance of the innovations from filter j. Then, it is possible to replace Equation (4-7) with 1

P_I ( k )

=

~ , e - ' ~ " ~P;(O) '

1 ,P<(k)

Cce-1

--

Pe(O), C , = I S j

I

?(I

I

- a,)

(4- 10)

/:I

where Pc ( 0 )are the prior model probabilities. If these prior probabilities are equated to the posterior probabilties, then Equation (4-10) simplifies to Pi = I

/

1

- A typical multiple model estimation is drawn in Figure c,~-$"~) cce -7,:') 4-4 in which the combination of the estimates is done according to Equation (4-8).

4.3 PREDICTION AND SMOOTHING The methods of the previous sections are concerned with the problem of estimating the state of a system at time t based on the measurements from time io to i. This problem is called the filtering problem. In this section, formulae for the predicted state, that is, ? ( t ) based on nleasurements fro111to to I.,- < I , and the smoothed state, that is, 2 ( t ) based on measurements from to to I,,., t E (to, tn) are presented. Since prediction and s t ~ ~ o o t h i ninvolvc g state estimates at times othcr than the measurement time, the follo\ving notation is employed in this section. 2(f I t ~ ) expected : value of x ( t ) based on measurements from to t o t N

P(t I I,.):

covariance matrix at time t based on measurements from to to

I~

(4-11)

Ssc. 4.3

Predlctlon and Smoothing

System Dynamics

+

Combination of Estimates

Measurement System

+% I

Calculation of Probabilities

Figure 4-4

$ p

I

Multiple Model Estimation

In a discrete-time system, Equation (4-11) is rewritten as .+(k I N):cxpected value of x(k) based on measurements from k = 0 to k = N P(k I h'):covariance matrix at time k based on measurements from k = 0 to k = N (4- 12) I t should be noted that many intermediate and advanced books on state estimation employ this notation for all developments including the filtering problem. Equations (4-11) are simply the estimates at t conditioned under the measurements up to t , ~ , and thus are called the conditional expectations of the state and covariance.

4.3.1 Prediction Actually the prediction problem has already been presented as part of the filtering problem. Optimal prediction can be thought of in terms of optimal filtering in the absence of measurements, that is, R = 0, and hence the filter gain is zero. In particular, for the continuous dynamics, discrete measurement case, the state and covariance propagation equations between measurements represent the best estimates of the state and covariance. A similar situation is true for the discrete dynamics case, and thus the prediction equations are as follows: Linear, discrete Kalman filter-Equations

(2) and (4) of Table 4-1

i(kIN)=@(k-l)i(k-lIhq+Y(k-l)u(k-1) ~ ( ( khq = @(k - i ) ~ ( k- 1 I iv)aT(k > I )

+~

fork>N

( -k i ) ~ ~ - , ~ 1) ~ ( k (4-1 3)

Linear, continuous Kalman filter-Equations

(4-la and 4-lc) with R-'

= 0

Modem Filtering and Estimation Technlques

140

Chap. q

and K = 0, that is

B(r 1 t , )

=

A(t)i(t 1 tN)

~ ( 1 t,t)

=

A(t)P(t 1 t,)

+ B(t)~i(t) for t > tN (4- 14) + P(r 1 tN)A T(t) + N ( t ) ~ ( t ) ~ ~ ( t )

Nonlinear, continuous EKF-Equations K=O i(t

P

It 1t

~ =) f[t, I ( t I tN), i*(t)]

(4-5b and 4-5c) with R-' = O and

for t > t~

) = A[t, q t ( tN)lP(t ( t ~ +) P(t 1 t ~ ) A ~ [ t , 1 t ~ ) 1+

(4- 15) Q(t)

With R-' = O implying infinite variance, that is, no measurement or worthless measurement, K = O gives no weighting to a measurement aspect in propagating 2.

4.3.2 Smoothing The primary advantage of Kalman's solution is that the equations that specify the optimum filter are recursive in nature, so that they can be adapted easily to the digital computer. However, Kalman filtering does not consider the important problem of smoothing. AGNC [1986(f)] presents a simple and elegant derivation of the principal equations for smoothing. The derivation is based on the method of maximum likelihood and depends primarily on the simple manipulation of the probability density functions. The resulting equations of the linear discrete smoother are summarized in Table 4-Sa, while the linear continuous smoother can be derived and is summarized in Table 4-5b. Finally, the stnoother of this section is called a fixed-interval smoother since the interval [0, N] is fixed and 2(k 1 is obtained from k E [0, S ] .T w o other types of smoothers are the fixed point (fixing k and r = constant, dedetermining 2(k 1 N) as N increases) and the fixed lag (given h termine 2(k - A1 I k) as k increases). Equations for each of these smoothers for continuous and discrete time are developed in Gelb [1974].

4.4 KALMAFl FILTER DESIGN AND PERFORMANCE ANALYSIS The design of a well-tuned Kalman filter is achieved only after successfully iterating a process that combines system modeling with performance analysis and trade-off studies. While system modeling is based mainly on physical insight with respect to the problem a t hand, performance capabilities arc often subject to computational constraints and requirements imposed by the actual hardware involved. Therefore, although Kalman filter theory embodies a good deal of mathematical formalism, much effort is still spent arriving at an acceptable design. When the Kalrnan filter state estimates arc used as the state feedback, that is, when the Kalman filter is inserted into the control loop, a precise design methodology may be as follows [Maybeck, 19791. The first step consists of developillg a

Sec. 4 . 4

Kalman Filter Design and Performance Analysls

TABLE 4-5

(A) LINEAR DISCRETE SMOOTHER (6)LINtAR CONTINUOUS SMOOTHER

141

a) Linear Discrete Smoother Given Eq. (5) of Table 2-1, Step 1: Apply the linear discrete time Kalman filter (Table 4-1) for k E[O, N], and denote % (k I k) = P(k I k) = P;

he)+,

P(k I k-1) = P; and %(k1 k-1) = %(ky Step 2: Smoother State Estimate %(kI N) = %(k1 k) + Ck[4(k+l I N) - k(k+l 1 k)] where Ck = P(k I k) oT(k) P(k+ll k).' and k(N I N) = &N)+ Step 3: Error Covorinnce Matrix Propagation P(k I N) = P(k I k) + C,[P(k+ll N) - P(k+ll k)] C: where

P(NIN)=P;

b) Linear Continuous Smoother

Given Eq. (3) of Table 2-1 and k(to) = %,, P(to) = Po Step I: Apply the linear continuous Kalman filter (Sec. 4.1.2) for resultant state and covariance by iF(t), PF(t).

t E [to, i], and denote the

Step 2: The smoothed state and covariance at t E [to, i] are denoted by %(tI i) and P(t I i), and are

determined by integrating backward the coupled differential equations from i to I, d -1 = - P ~ A - A T -1 1 T -1 -P P ,-P;NQN P, + HT R -1 H, p B 6 j1 = O dt B

T -1

~ = - ( A ~ + P ~ N Q N ~ ) ~ + Px, ~yB( i ~) =+o H R and then substituting the resultant y(t), pB(tjl into:

p(t1ij1 = pB(tjl + p,(tj1

-

i(t Ii) = &,(I) + ~ (1 i)t [y(t) ~ i ( tQt)] )

dynamic model that ideally resembles system behavior and validating the model with experimental data. While it may not be of any benefit to develop an overly complex physically based model, it is absolutely necessary to develop one which lends itself well to Kalman filter design. The second, and perhaps most significant, step is to design and tune the filter against performance requirements, comparing estimation errors with the specifications and taking into account computational constraints. The complexity of the filter may be influenced by the definition of state variables and the choice of coordinate system for certain applications. In the next step, an attempt is made to combine and remove states where possible, and to delete

142

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weak cross couplings, in order to obtain reduced-order filters. At this stage, possible approximations to the optimal gain are also evaluated. Following this, a covariance sensitivity analysis and Monte Carlo simulations-are conducted, after which it is then possible to select a final design based o n a trade-off between the performance and the computational requirements. Finally, the last step consists of performing a checkout, final tuning, and an operational test of the filter. Further details of Kalman state feedback design are presented in Book 3 of the series.

4.4.1 Kalman Filter Design Process This section highlights in more detail the second step listed in the design procedure outlined previously. Regarding Section 4.1.2, to reiterate, the design parameters that can be selected to design the Kalman filter are z(r), w ( t ) (some of which may be fictitious), Q, and R. The estimator dynamics gain and bandwidth are completely determined by the measurement signal selected and the designer's specification of Q and R for the design. Q and R are design parameters selected by the designer to create the filter gain K, via the Riccati equation (4-la), and thus determine the dynamics characteristics of the estimator. Actual physical process noise and measurement noise will in general differ from these design values. For good estimator performance, these specified noise intensities should roughly correspond to disturbances or uncertainties that will actually be encountered, since the estimator will be tuned to these conditions. Sometimes, inexact inte~lsitiesare provided to the Riccati equation in ordcr to achieve certain desirable filter characteristics, but the engineer should be aware of the costs and benefits. Thc error covariancc P i s useful in terms of certain limiting cases where constraints are established by the open-loop model dynamics itself to bc placed on the effectiveness of the cstimator. Some very rough Kalman filter design rules of thumb are given as follows. For small Q and large R , the design tends to be such that the filter poles are close to the model dynamic poles, the filter is a low-bandwidth filter, and there is slow state estimation and good measurement noise rejection. For large Q and small R , the design tcnds to be such that the filter poles approach the nlodel dynamic transmission zeros or stable reflections thcreof (some actually move to infinity in the left half-plane), the filter is a high-bandwidth filter, and there is fast state estimation and poor measurement noise rejection. Some useful transfer functions for design follow. From I A ) (to ~ ) ~ ( r )H(s1 : - A ) - I .Y From x(r) to i ( r ) : ($1 - ('4 - K H ) ]- ' KH From z ( t ) and rr(t) to i ( f )[ s:l - (A - KHJ- 'IBK]

(4- 16)

(4- 1 7) (4-1 8)

4.4.2 Selection of Noise Intensity Matrices For given Q and R, specificd for a Kalman filtcr design, some propertics of the filter follow. 111 general, the magnitude of estimator gain matrix K is proportional to the magnitude of the Q and R - ' matrices. Multiplying Q and R by the same positive

Sec. 4.4

Kalman Filter Design and Performance Analysls

143

scalar k yields the same estimator as Q and R. The initial guess of Q and R can easily be made using thc following rule of thumb. Since K is lowest for Q R - ' = 0 and increases as Q R - ' increases, the estimator break frequency is lowest for Q R - I = 0 and increases as Q R - ' increases. The estimator bandwidth is high for low measurement noisc and low for high measurement noisc. It is assumcd that Q = diag(ql, qz, . . . q,,) and R = diag(rl, rz. . . . , r,,,). A nominal process with noisc intensities 71, . . . , q,, is applied, and the response intensity ?', . . . , ?,, that comes through at the sensors due to the nominal process noisc Q (as the nominal sensor response intensity) is recorded. Then R = diag(arl, a?,, . . . , a?,,,) is chosen as the measurement noise for the filter design wherc a is a scalar that will be allowed to vary, and will serve as the scanning parameter for the filter design.

.

0. In the case of stable, open-loop model dynamics, as R goes to infinity or as Q approaches zero, analysis shows that the estimator operates open loop. Moreover, P will be zero, K approaches zero, and the Kalman filter poles move to the positions of the open-loop poles. Q9X-l-

Example 4-2: Stable, first-order filter. The model dynamics cquations are $(I) = ax(t) + b ~ r ( t )+ w(t), E[w(t)] = 0, E[IU(~)LV(T)] = q8(t - T), z(t) = hx(t) + ~ ( t )E[v(t)] , = 0, and E [ v ( ~ ) v ( T = ) ] r8(t - T), whcre a < 0, q 2 0, and r > 0. The model dynamics transfer function is G(s) = hl(s - a). The Riccati equation as specified by Equation (4-la) is h'P?ru - 2aP - q = 0 and yields P = r[a + q a ' + h2(qlr)]/ll'. The Kalman filter gain is k = [a + a + h'(q/r)]/h and the state estimate equation is

The preceding equations s h o ~ vthat the estimator dynamics are a function of the ratio qlr, and not of either q or r alone. T h e root locus of the Kalman filter is shown in Figure 4-5a and is discussed here. Using Equation (4-17). the transfer function from x(t) to 2(t) is kh k(3) X(r) s - (a - kh)

k(0) kh where the DC gain is --- X(0) -a+kh

(4-20)

The magnitude of x(s)/x(~) as a function of frequency is plotted for increasing values of qlr in Figure 4-5b. For a given value of r, increasing the value of q in the design results in increased values of the gain k. From Equation (4-20), this means that X(O)/X(O)will tend to unity. Also, for a constant v, increasing q will increase the estimator bandwidth, while for a constant q, decreasing r will have the same effect. N o w , if the input u is zero, and w is a constant but unknown disturbance, then at steady-state conditions the model dynamics equations imply that x = w l a = constant, while, from the state estimate equation, i= 0 and 2 = (khx

+ kv)l(-a + kh)

(4-21)

Modem Filtering and Estimation Tecftnlques

Chap. 4

Imaginary

Real

a

b\

Imaginary qir = 0 k =0

qhk- =

1 Real -a a Open-Loop

Model ~ ~ n a m iPole cs Figure 4-5

Selection of Noise Intensity Parameters q and r

Assuming n o measurement noisc, that is, v = 0, and using Equations (4-20), it is true that 2 = kkhxl(-a

+

kh) = X(O)X/X(O) = k ( O ) u ) / [ ~ ( O ) a ]

As noted previously from Equation (4-20). specifying a large q means that ~ ( 0 ) l X(0) will be close to unity, and that x wrill be close to 2 at stcady state. Under thcsc conditions, the state estimate is a good disturbancc cstin~atc.If, on the other hand, w = 0 but v is a constant but unknown disturbancc (for example, a sensor bias), then Equation (4-21) at stcady state becomes 2 = kvl( - a + kh). When q > 0 and r > 0 , then Equation (4-19) bccorncs 2(t) =

-da2 +

h2(q/r) :(I)

+

b11(t)

+ la

+

qa2

+ k2(qlr)]~(t)lh

and thc transfer function fro111 the measurement inputs z ( t ) to the state estimatc

Sec. 4.4

Kaiman Filter Design and Performance Analysls

i ( t ) , given by Equation (4-18) is x ( s ) / z ( s =)[a + g a 2 V a * + }t*(q/r))].T w o limiting cases are now considered. Case I : q

145

+ h2(q/r)]l[h(s +

0. This case corresponds to zero process noise, implying no uncertainty about the model dynamic statc, including x(O), exists. For this case, P = 0, k = 0, and the statc estimate equation is ;(I) = o.?(t) + bu(r) and i ( 0 ) = x ( 0 ) . The filter therefore generates open loop. For a constant step input at steady state, 2 = x = hula = constant. -r

0. In this case, the filter bandwidth, or break frequency, approaches infinity, k also approaches infinity, but P approaches zero. Also, as shown in Figure 4-5c, x(s)/z(s) = Ilh. Case 2: r

-r

Unstable open-loopmodel dynamics. When the open-loop model dynamics are unstable, the estimator gains are nonzero, and the error covariance will not go to zero even when there is little or no process noise. In addition, the estimator poles move toward the positions of the stable poles which are the left half-plane reflections (about thejw-axis) of the unstable open-loop poles. This is termed the minimum measurement effort estimator, and sets a lower limit on thc estimator gains. Even in thc Lvorst case, therefore, the Kalman filter always guarantees c l o d loop system stability by providing the minimum measurement effort estimator as long as the model dynamics are completely observable. However, in classical design approaches, when the model dynamics have right half-plane poles, the closed-loop system is unstable for low filter gains. Example 4-3: Unstable first-order filter. The open-loop model dynamics for this example are the same as the previous example, except that now a > 0. Similarly, the model dynamics transfer function, the Kalman filter gain, and the state estimate equation are also the same, respectively. Finally, the root locus of the Kalman filter is sholvn in Figure 4-5d in which the estimator locus begins at the stable reflection of the open-loop model dynamics pole. If q > 0 and r > 0, then not only is the state estimate equation identical to that in the previous example, ) also identical. Again, it is possible to consider two limiting cases. but A ( s ) / ~ ( s is Here, the variance and filter gain are nonzero and are given k = 2 a / h , and the filter equation becomes i ( t ) = a$(t) + bu(t)

Case 1: q + 0 .

by P = 2aih"and + 2az(t)lh.

Case 2: r -+ 0.

The results are the same as those in the previous example.

Example 4-4: Neutrally stable first-orderfilter. In this example, the open-loop model dynamics are again the same as those in Example 4-2, but this time a = 0. Setting a = 0 in Equation (3) of Example 4-2 also gives the model dynamics transfer function, while the state estimate equation becomes i ( t ) = bu(t) + k(z(t) - h?(t)) and k = g&. Hence, as in the previous two examples, the estimator dynamics are a function of the ratio q l r . The important point to highlight

146

Modem Filtering and Estimation Techniques

Chap. 4

in this example, however, is that, if q = 0, then k = 0 and i ( t ) = bu(t), and the estimator is neutrally stable, which is unacceptable. If q > 0, the estimator is stable with a pole at - kh. Therefore, if the open-loop model dynamics have a pole on the jw-axis, it must be disturbed with a process noise in order to have a stable estimator. For a zero input, u = 0, and a constant initial state x(O), at steady state, x = x(0). From the state estimate equation, 2 = x + v l h . If v = 0, then 2 = x = k(0). QR-'-, m. In this case, all measurement covariances R go to zero (that is, ideal sensors with excellent measurements) or the process covariance Q goes to infinity. As R goes to zero or as Q approaches infinity, the estimator gains increase drastically, and the bandwidth of the resulting Kalman filter increases very rapidly. When all the transmission zeros in the model dynamics are in the left half-plane, that is, minimum phase, the closed-loop poles asymptotically approach the transmission zeros, and the residues of the slow model which is created by the poles approaching zero have small values in 2 ( t ) = x ( t ) - 2(f). However, the residues of these modes in K[z(t) - H<(t)] are not small. If the number of sensorslmeasurements is greater than or equal to the number of disturbances, the error covariance P approaches zero under the condition of ideal sensors or as Q approaches infinity. However, in both minimum and nonminimum phase systems, if there are more disturbances than measurements, the minimum value P can achieve for any cornbination of Q and R is always strictly positive. This is true even if the product QR-' becomes infinite. Therefore, even in the event of having all ideal sensors, perfect estimation cannot be achieved as long as there are not enough sensors. As shown in Examples 4-2 to 4-4, as qlr approaches infinity, the estimator poles move toward the transfer function zeros located at - x (see Figures 4-ja and 4-5b). Nonminimum phase systems impose dcfinitc restrictions on the accuracy which the filter can achieve. When the trans~nissionzeros of the model dynamics phasc), as the measurcrnents approach are in the right half-plane (no~~minimum perfection the cstimator has high gains, and the closed-loop Kalman filter eigenvalucs move towards the positions of the stable poles which are the left half-plane reflections (about thejw-axis) of the unstable transmission zeros of G(s). Thus, unstable transmission zeros are not desirable because they lead to an inefficient estimator. The residues of these modes remain large as the estimator gains increase for the closcdloop system. Even if the number of measurcrnents is greatcr than or equal to the numbcr of disturbances, the resulting error covariance docs not approach zero and is much larger than that for minimum phasc systems. Optin~alestimation always guarantces closcd-loop system stability evcn in thc evcnt of high cstimator gains. Howcver, in classical design, for a nonminimuln phase systcn~,thc closed-loop system is unstablc for high controller gains.

4.4.3 Error Analysis ,Examining thc state cstimatc equations (4-I), the structural csscnce of thc Kaln1a11 filter is that which is also used in the general case. Also, the definition for K ( t ) is that which defines the optimal gain, and thus the optimal filter, and is shown to bc

Sec 4.4

Kalman Filter Design and Performance Anaiysls

147

a particular case of the general filter gain matrix developed in this section. For the case of a general filter gain matrix, the state estimate error is defined as f(t) = ?(I) - x(t) and E[.f(r)] = 0, which yields the error dynamics of the estimator

The linear error dynamic system is therefore being driven by white noise, as shown in Figure 4-6a. As shown in Figure 4-6b, the Kalman filter is thought of as a regulator loop in error estimate .i. coordinates. The equation defining the error covariance for the filter with a general filter gain matrix K(t) is given by

If the eigenvalues of A - K H are in the left half-plane, the estimator is stable and, as r approaches infinity, the state estimate i ( t ) converges to the actual state x(t) and the estimator error 2(r) approaches zero. The design procedure consists of selecting the filter gain matrix K(t) that places the estimator eigenvalues in the desired locations. A wide bandwidth gives fast rate estimates but lets measurement noise corrupt the estimates. Using the Kalman filter gain Equation (4-lb), the error dynamics Equation (4-22) is a stable linear system driven by white noise. Substituting in Equation (4-23) the Kalman filter gain gives the same equation for Pas in Equation (4-la). As t becomes large, the error covariance P(t) in Equation (4-23) satisfies P(;) = 0 and the steady-state covariance matrix is obtained by solving the linear Lyapunov matrix equation,

The Kalman filter gain minimizes P(t). That is, P ( t ) from Equation (4-24) using a

+

b

I

A-KH

'f+@

I-------

I _I

Figure 4-6 Error Dynamic Model for Kalman Filter

Mdern Filtering and Estimation Techniques

148

Chap. q

general filter gain matrix is always greater than P(t) from Equation (4-2) using the Kalman filter gain Equation (4--3). minimizing (by Kalman gain) P =

+

minimizing trace u:, u s + ... u:,,

+

+

For the particular case in which all the states are measured, H = I and R = diag(rl, r2, . . . , r,), and the Kalman filter gain then becomes

K

=

PH~R-'=

(4-25)

The ith component of the state estimate is

2

2

u*--;pi + ... + ((t)

- 2,))

u*;,, + ... + @,,(I) - i

t

+ ...

r,, If the noise intensity ri is large, the gains uZ,+;lrj will be small, and the signal zi(t) will not be weighted heavily in computing 2i(t). If, however, the error covariance crzjr, is large, the gains ~ : , ~ , l will r , be large, and the signals will be weighted heavily in computing i , ( t ) . rj

4.5 OPERATIONAL CONSIDERATIONS Problem-dependent judgments must often be made to enhance filter performailce (for example, reduce the transient time in the estimation procedure at the expense of a greater variance), reduce computational requirements, furnish missing information (for example, the Kaln~anfilter needs complete a priori covariance information on the initial state, process noise, and il1easuremcnt noise), and/or establish procedures for stabilizing diverging filter estimates. In the Kalman filter in~plcmentation, the computation of the transition matrix together with the propagation of the error covariance matrix represent a substantial fraction of the overall COIIIputation effort necessary. Fortunately, because the accuracy of these quantities nccd not be as great as that required by the state vector, the former may be computed at a slower rate than the state vector update. In numerous applications, essential

Set 4.5

Operational Considerations

149

statistics information concerning the process and measurement noise, in the form of standard deviations and correlation time constants, may cither be missing altogether or be poorly defined. Moreover, the analytical formulation ofthis information rarely reflects the behavior of the actual nlcasurements, resulting in reliable data for an extended period of time followed by a series of poor data points. Filter operation is further affected by modeling errors, linearization approximations, and the reduced-state nature of practical filters. Because of all this, the covariance matrix computed from the Riccati equation of the Kalman filter may not resemble the true covariance matrix of errors in the cstin~atedstate. For this reason, many simulations need to be performed to adequately examine filter performance [Baheti, 19871.

4.5.1 Filter Performance The Kalman filter is capable of solving a certain optimization problem (minimum variance), which is an indicator of filter accuracy. For numerous problems, however, additional considerations in the filter design (for example, settling time to a steadystate value, sensitivity to parameter variations) may be more significant than determining the estimation with the least variance. For instance, it may be necessary to obtain a constant gain filter that has the minimum settling time subject to an upper-bound constraint on the trace of the covariance. Figure 4-7a shows hypothetical state estimates and variances for the minimum variance and minimum settling time filters [Powers, 19781. It is also possible to design a filter that minimizes a function of both the variance and the sensitivity of the filter to modifications in a system parameter. Figure 4-7b illustrates a hypothetical case where the Kalman filter is the minimum variance filter for a = a" (that is, the design value); however, it is not necessarily the minimum variance filter for off-design values. In this particular case, the minimum sensitivity filter, being less sensitive to parameter changes, would probably be the best filter for an application in which a was known to vary over the range [a,, az]. In defense of the ~ a l m a nor minimum variance filter, it is possible to compute the Kalman gain without iteration. In the two problems noted previously (minimum settling time and sensitivity), iterative parameter optimization schemes are required to determine the filter gains; thus, they would be employed only when the Kalman filter is not adequate for the given application. An appropriate reduction of the number of states modeled will cause a suboptimal filter to become a noticeably smaller computer. Figure 4-7c illustrates that this suboptimal filter may be employed to reduce the sensitivity to uncertain parameters.

4.5.2 Computational Requirements Kalman filtering fundamentally represents a digital computer-oriented approach to state estimation. Thus, the analysis of a Kalman filter would consider all problems inherent to the digital computer.

Modem FIlterlng and Estimation Techniques

150

Chap. q

Minimum Variance Estimate

Minimum Settling Time Estimate

Trace P(t)

r For Minimum Settling Time Filter

I

For Minimum Variance Filter

Trace P

Figure 4-7

(a) Minimum Variance vs. Minimum Settling Time Filter Performance (b) Covariance as a Function o f Parameter Variation for Minimum Sensitivity and Variance Filters (c) Conceptual Example of Minimum Sensitivity Filter

I and sensitivity trade-off) I

I

I I

I

I

I I

I I I I

I I al

C

Expected Range of Uncertainty

Figure 4-7

n>m>P

i '

(Continued)

Square-root filter. The reliability of time-varying Kalman filters has been significantly improved by the so-called square-root or (uDu') algorithms [Bierman, 1977; Kaminski et al., 19711. They propagate the square root of the symmetric covariance matrix instead of the covariance matrix; this provides almost twice the accuracy for a certain computer word-length as Kalman's original algorithm. In addition, this process also prevents numerical truncation from generating a covariance matrix that has negative eigenvalues [Bryson, 19851. Filtering problems can result from the finite word length of the digital computer. For instance, after numerous measurements have been processed, round-off error build-up can produce theoretically impossible filter behavior. In other words, the Kalman filter is enhanced assuming infinite precision, and accumulated round-off errors can cause the covariance matrix update to result in a nonpositive, semidefinite matrix. The main way of dealing with this problem is to utilize the square-root filter. This filter was initially developed [Potter, 19641 to rectify Kalman filter implementation problems in the Apollo program. The computer employed in the Apollo spacecraft program was a fixed-point digital computer. The square-root filter involves a reformulation of the propagation and update equations in terms of a matrix S , where P = S S T (that is, S represents the square root of the covariance matrix). Utilizing S instead of P causes the precision of the computation to essentially double. In addition, it also prevents P from being nonpositive and semidefinite. Unfortunately, the fact that the matrix S is not unique results in a proliferation of square-root algorithms [Andrews, 1968; Bellantoni and

152

Modem Filtering and Estimation Techniques

Chap. 4

Dodge, 1967; Kaminski et al., 39711. The adverse effect of employing the squareroot filter is that it requires more computer time. However, recent developments with more efficient implementation have substantially decreased this effect. Bierman [I9771 discusses these efficiencies and the computer program listing of the Jet Propulsion Laboratory orbit determination program square-root filter.

Suboptimal filter. Another computer-oriented problem that relates to the Kalman filter is the potentially numerous differential equations that must be integrated (that is, state and covariance differential equations). Because the major portion of computer burden in Kalman filtering is typically represented by the solution of the error covariance equations, an effective way to make the filter satisfy computer hardware limitations is to precompute the error covariance, and thus, the filter gain. As a result, the suboptimal filters for a simplified system model are usually implemented instead of the optimal Kalman filter. Suboptimal filters may deal with fewer state variables (for example, states that produce negligible effects in the Kalman filter are ignored), fewer covariance equations, and in many cases, simplified filter gains which approximate the optimal Kalman gains. T o avoid on-line covariance calculations, the error covariance matrix can be computed at an earlier stage and the filtcr gain histories are obtained. While a prerecorded set of precise gain histories could be stored and utilized, a more effective method is to approximate the optimal gain behavior by analytical functions of time, which can be easily computed in real time. A piecewise constant suboptimal gain and a decaying exponential function gain can be employed. Steady-state gains referring to the limiting case of a set of pieccwise constant gains can also be employed by choosing each gain to be constant over all time. A Kalman filter that utilizes steady-state gains is idcntical to a Wiener filtcr. In numerous practical systcms, the constant-gain Wiener filtcr offers estimating accuracy that is equivalent to that of the more sophisticated Kalman filter, but a t one third of the computational cost [Powers, 19781. Kalman filter design software. A parameter estimation subroutine package for the construction of efficient multipurpose estimation algorithms has becn developed [Bicrnlan and Nead, 19781. This software was developed under NASA contract of thc Jet Propulsion Laboratory, and has been applied to several Kalman filter designs and simulations. More rcccnt works [Pappas et al., 1980; Verhaegcn and Van Dooren, 19861 have rcsultcd in significant improvements in the computational reliability obtainable in the nu111cricalsolution of the Riccati equation. The algorithms arc applicable to problems with singular transition matrices \vhich arisc in thc case of sampling a continuous-time systcm with time dclays. A softwarc package has also becn devcloped [Arnold and Laub, 19831 that solves the gcncraIizcd algcbraic Iiiccati cquation. Many research organizations have already dcvelopcd an interactive Kalman filtcr analysis and design package. By making the softwarc user friendly and easily adaptable, providing a broad range of graphics capabilitics, 2nd supplying numerically cfficicnt algorithms, thcy have produccd a rathcr elegant 2nd powerful tool for the dcsigncr's use. Such tools will invariably si~nplifythe Kalmnn

Sec. 4.5

Operational Considerations

153

filter dcsign proccss in the future and may at the same time substantially improve performance (Bahcti, 19871.

4.5.3 Absence of A Priori Information Much a priori information, that is, .?(I)), P, Q(r), and R ( r ) , is required by the Kalman filter. If this information is not available, then straightforward least-squares techniques (recursive or batch) should be applied. In many cases, the Kal~nanfilter approach can still be elnployed successfully even though only a portion of the required a priori statistics is known. This occurs because of several reasons. First, the order of magnitude estimates or relative magnitudes of the statistics can be guessed, and then a Kalman filter can be established. Often, the resultant filter performs adequately. Second, if the first reason given prcviously produces undesirable performance, then an analysis of the simulations should indicate which missing a priori statistics are most critical. An acceptable filter design can be achieved through a parameter study involving the critical statistical parameters. Finally, in numerous cases, .?,, and Po are the absent values. In some systems, these parameters mainly affect initially stable, transient effects and have little impact on the long-term operation of the filter. Note that the steady-state Kalman gain is independent of ko and Po. Another technique of dealing with absent statistics is to develop an adaptive filter (that is, the statistics are generated as the proccss proceeds by analyzing differences between the model predictions and the measurements) [Powers, 19781. Adaptive filter techniques can be found in several textbooks. Details of adaptive filters are presented in Book 4 of the series. An identification problem occurs when @(k) and q ( k ) in a discrete case, or A and B in a continuous case, are unknown. This problem is the topic of Book 4 of the series. If process noise w ( k ) in a discrete case, or w ( t ) in a continuous case, is known, then the on-line identification of the process covariance Q h in a discrete case, o r Q ( t ) in a continuous case, is required. A problem will occur if the initial value is not known and thus, P is unknown.

4.5.4 Filter Divergence T w o sources of error which may cause the measured performance of a Kalman filter to differ from what was analytically predicted are modeling errors and computer round-off errors. As a result of these, errors in the estimates may become intolerably large at some point during the operation of the filter. Filter divergence is often explained in terms of the calculated covariance matrix, the assertion being that this matrix becomes unrealistically small and results in placing unreasonable confidence in the estimates. Consequently, the filter gain is reduced and subsequent measurements are effectively ignored. In the study of Kalman filters, divergence refers to the situation where the estimated filter error converges to a value less than the true filter error. In some cases of divergence, the true error converges to a finite value; this is known as an apparent divergence. This is typically caused by errors in the mathematical model of the process (for example, neglected forces which generate

154

Modem Filtering and Estimation Techniques

Chap. q

a bounded-state response). The resultant filter may perform satisfactorily for the particular application, with the only deficiency being the estimate of the error by / the Kalman filter approach [Powers, 19781. True divergence represents the most critical type of divergence in filter design. This type of divergence means that the true error becomes arbitrarily large as time increases while the estimated error converges to a finite value. True divergence is usually the result of unmodeled effects which can cause an unbounded-state variable response, or in some cases, the type of instability presented in Section 4.1.2. Several methods. of which the use of larger vrocess noise covariance and es~. ponential data weighting have been particularly successful, have been applied t o the problem of avoiding filter divergence and improving the estimation accuracy. The increased noise covariance offsets modeling errors and improves filter stability, while exponentially weighting the measurements prohibits obsolete data from saturating the filter and sets a lower bound on the filter gain [Baheti, 19871. A primary factor in the onset of filter divergence that surfaced after many Kalman filter implementations is computer round-off errors in the covariance propagation equations. To maintain acceptable numerical accuracy, these equations generally require doubleprecision computations. As shown in Section 4.5.2, algorithms have been developed based on matrix square-root propagation, which are capable of producing an effective precision that is double that of the conventional Kalman filter. A procedure similar to the square-root formulation and called U-D covariance factorization, has also been developed and tested in many applications for consistency and stability. Studies emphasizing number of operations, actual C P U time, and data storage requirements have demonstrated that optimally coded U-D factorization and the conventional Kalman filter implementations are very similar. However, for processing large sets of data with infrequent estimates, an even more efficient algorithm has been developed called a square-root information filter [Bierman, 19771. Most of the procedures that deal with true divergence are conservative, straightforward methods that increase stability, and thus, degrade the performance. Although one could obviously attempt to estimate unmodeled states, the resultant filter size would most likely be This is especially significant in practical applications where the process models are not always completely validated. Increasing the process noise in the model (for example, making Q(t) a positive, definite matrix) represents a simplified way to approach the divergence problem. This approach implies that one does not know the model exactly, and will not allow the state estimate propagation equation to predict 2(t) with absolute certainty (that is, P(f) = 0). As noted in previous examples, if Q(r) = 0, then the Kalman gain may approach zero and thus not use any additional measurements in the state propagation equation. The effect of Q(t) > O causes P(r) > 0. In othcr words, a nonzero lower bound implies a nonzcro Kalman gain. An increase in the error estimate and poor performance will be thc effects of an arbitrary selection of Q(t) > 0. More sophisticated techniques that produce less pcrformancc loss are discussed in Book 4 of series. The use of finite memory filtering represents an alternate solution for preventing divergence.

" .

Kalman filter divergence and suboptimal design. If the eigenvalues of Equation (4-24) arc in the right half-plane, the estimator is unstable and, as t becomes large, the state estimate i ( fdiverges ) from the actual state x ( t ) and the estimate error 2 approaches infinity. One of the principal causes of this failure is that the systcm modcl contains states or models that are undisturbed by the model process noise (modcl uncertainty). One cure for such problems is the periodic restarting of a time-varying Kalman filtcr. Other cures include minimum variance observers with eigenvalues constraints, added noise, pole shifting, and destabilization. Thc constrained minimum variance problem may be stated as follows [Bryson, 19781. Find a set of observer gains K that minimize J = trace (2hlP) where P is found from Equation (4-23). ,\.I is a positive semidefinite matrix, and the gains are C ( K ) > 0. This constraint ensures that the eigenvalues ( s i ) of A - K H always lie in the left half-plane of the original point at - a , where u is a specified positive number, that is, R,(si) 5 -0. The nccessary condition for minimizing J with constraints Equation (4-23) and C ( K ) > 0 is

The following example of estimating an undisturbed constant state demonstrates this technique. The simplest model of a Kalman filter for a neutrally stable plant is given in Example 4-4 where b = 0, q = 0, and h = 1 have been assumed here. The initial state x ( 0 ) has zero-mean value and variance Po. The Kalman filter for estimating x ( t ) is = k ( z - i )where k ( t ) = ( P o / r ) l [ l (Polr)t]instead of zero. Clearly for t (rlPo), k ( t ) -+ 0 , and the estimator ignores the measurement z. In practice, such a filter will almost certainly diverge for (Polr)t % 1 with drift errors. From C ( K ) > 0, k = u where a is the eigenvalue constraint level. The variance 06 the estimator is predicted to be P ( t ) = Poe-2"' + ar(1 - e-2"')/2 so that P approaches ur12 in steady state, but the estimator will not diverge. Schmidt [I9701 presents a method for modifying unsatisfactory Kalman filters by introducing additional noise into the system model Equation (4-la). T h e spectral densities of added noise may be varied until the filter has the desired stability. The reason that this technique works can be seen by examining the steady-state error covariance matrix equation in Equation (4-2). If certain elements of Q are zero, corresponding elements in the steady-state value of P, and consequently K , are zero. However, if these elements of Q are assumed to be nonzero, then the corresponding elements of K will be nonzero, and the filter will always try to track the true system. The choice of the appropriate level of the elements of Q is mainly heuristic, and depends largely on what is known about the unmodeled states. Breza and Bryson [I9741 present a pole-shifting method in which the first step is to design the steady-state Kalman filter. If the stability is unsatisfactory, the slow eigenvalues are shifted to specified, faster eigenvalues by changing the gains so as to leave the other eigenvalues unchanged. The method does not give a unique solution if there are two or more measurements.

*

+

Modem Filtering and Estimation Techniques

156

Chap. 4

4.5.5 Modeling Process Noise and Biases Fundamental to Kalman filter theory is the use of white Gaussian state and measurement noise models. In actual practice, random inputs are likely to be time correlated and represented by first-order or second-order Gauss-Markov process. Moreover, the dynamical system may be subject to certain parameters whose values are not well known. Augmenting the original state vector to include the exponentially correlated process noise and biases is a method commonly employed, but this can greatly increase the amount of computations required for many applications [Baheti, 19871. An algorithm for partitioning the estimation of the bias variables, or unknown constants, from the dynamic variables was published by Friedland [1969]. One advantage realized by the use of this algorithm is reduced data storage requirements in the computer. A second advantage of the partitioning is that it creates additional analytical insight to the filter behavior based on special properties of the variables. The following reviews the basic theory behind separate bias estimation. It begins by considering a discrete-time process

x(k

+

1 ) = A ( k ) x ( k ) + B(k)b + w ( k ) and z ( k ) = H ( k ) x ( k ) + C ( k ) b + v ( k )

where w ( k ) and v ( k ) are white Gaussian noise processes and b is a constant but unknown bias vector. The optimum estimates ? ( k ) and b^(k)are given by ? ( k ) = Z(k) ~ 6 ( k where ) Z ( k ) is computed as if no bias errors were present. That is, -x ( k ) is the optimal "bias-free" estimate, which is then corrected to account for the effect of the bias. The matrix V represents the ratio of the cross covariance matrix of the bias-free state cstimate and the covariance matrix of the bias. Such a decomposition becomes very practical in the case of a large number of bias parameters. The dynamical modcl parameters, noise statistics, and initial estimates of a particular system to which the Kalman filter is applied generally do not agrce with the real system that generates the observations. In the case of using a model of fewer dimensions than the original model to reduce the number of computations, such differences are imposed deliberately. The price of using a reduced-order model may, however, be degraded filter performance. Algorithms for analyzing the mean-square error performance when the system model is not the model used in the filter dcsign have been developed Uazwinski, 19701. The error analysis generated is particularly useful for conducting sensitivity studies. That is, one can study the effect of some known variation about a norllinal value in a ccrtain system paramctcr on the filter pcrformancc, if the filter is dcsigncd using the nominal valuc. Timc-corrclatcd noisc is occasionally rcfcrred to as colored noise, as opposed to white noise. In thc discrctc modcl case, it is callcd sequentially correlated noise. The utilization of this type of information should cnhancc filtcr performance. Following Powers [1978],procedures for dealing with timc-correlated noise with diffcrential equation models arc discusscd here. Suppose that the measurcmerlt r~oise is white, but the proccss noisc is colored with thc corresponding discrctc-time niodcl cases,

+

where wl ( t ) represents zero-mean white noise with variance G, and @ ( t o ) and E{[w(t,,) - @ ( t o ) ] [w(to) - @(to)lT}are assumed known. If the random process [Equation (4-26)) is a Gauss-Markov process, then it can always be expressed by a linear dynamic system with white noise forcing function only

where li, is zero-mean white noise with variance Q. Equation (4-27) and the measurement equation can then be treated by all of the results developed in the previous sections. Note that Q - is never required in these equations, which allows the coloredprocess noise case to be treated by simply defining the augmented state vector of Equation (4-27). The colored-measurement noise case is more complicated because it invariably leads to a singular measurement covariance matrix, which is not allowable in the standard Kalman filter equations since they involve R - ' . Supposing

'

.G = A x

+ Nw,

Z

=

Hx

+ v,

b = Fv

+ Gvl

(4-28)

where w and v l are uncorrelated zero-mean white noise with variance Q and R , respectively, and Equation (4-28) defines the correlation properties of the mea~ ] Define surement noise v, in which E[v(to)]= 0 and ~ [ v ( t ~ ) v ( t =o )Vo.

which implies, upon differentiation of z , i(t) = (H

+ H A - FH)x + H N w + Go,,

+

i t ) = x

+ I7

(4-30)

+

where H = H H A - FH, I7 = HNw C v l , and E[i;] = H h l E [ w ] + G E [ v l ] = 0, E[i;ijT] = H N E [ w w T ] N T H T G E [ v l v T ] G T = [ H N Q N ~ H + ~GRGT]r x 8(t - 7 ) = ~ 8 ( -t 7 ) . Here, use has been made of the fact that E [ w ( ~ ) v I ( T ) ~ ] = 0. It should be noted that in the new problem variables ( x , i ) , the process and measurement white noises w and 17 are now correlated as

+

However, this case is easily handled by employing Equation (4-ld) instead of EquaT. tion (4-lb) with S(t) = Q ( t ) N ( t ) T ~ ( t )Thus

As shown in Bryson and H o [1975],one need not perform the differentiation of the measurement signal z in Equation (4-29) by introducing

x*

9

i ( t ) - K ( t ) z ( t )J i*

= (A

-

KH)(X* + K z ) - ( K

.

+ KF)z,

xF = i0- K(to)z(to)

(4-32)

Modem filtering and Estlmation Techniques

158

Chap. 4

which would then replace Equations (4-31) in the filter calculations. Finally, to determine 2(t) and P(t) from Equations (4-31), care must be taken with respect to the initial conditions for the integration. Note that the derivation of the Kalman filter assumes that 2o and S(to) are not conditioned upon any other information. However, in the x, 2 system, ,?(to) is conditioned upon knowledge of z(to) and 4(to), that is, Equation (4-29). Thus, for a proper application of the Kalman filter equations, i ( t ) and P(to) should also be conditioned upon this information. One and Po corresponding to 2(tc) and P ( t f ) , respectively, can view this as the given io with a measurement 2(to)at to. Thus, the initial conditions for integrating Equations (4-31) are

i ( t o ' ) = 20

+ K(t0)[4(to)- ~ ( t o ) i ~ ] ,P ( t 2 ) = [ I - K(to)H(to)]Po

where K(to) is defined by Equation (4-31) with P = Po.

4.6 COMBINED COMPLEMENTARYIKALMAFI FILTER APPROACH TO ESTIMATOR DESIGN This section presents, derives, and verifies the continuous form of first-, second-, and third-order complementary/Kalman filters. As discussed in Book 1 of the series, conventional complementary filter architectures have been used with the filter gains derived from Kalman filter techniques. These low-order filters are shown in various NGC applications in Book 1 of the series. It was noted previously that complementary filters are really simple Kalman filters, but this section further highlights this fact, especially after a thorough comparison of complementary and Kalman filters.

4.6.1 First-Order Complementary/Kalman Filter The generalized continuous mechanization forms of the first-order complementary/ Kalman filter are illustrated in Figure 4-8. A comparison of the complementary filter and the Kalman filter is also shown in this figure. As in the standard firstorder filters, rate limiting is achieved by inserting a limit function after the Kalman gain correction block. The limit function ensures that the rate of change of the innovation sequence v = z - i is limited. When the rate of change of v is less than the imposed limit, the filter functions identically to the linear first-order filter. Ho\vever, when the rate of change of 11 is equal to or exceeds the limit, the changc of the estimated output 4 must be due to a pure integration of the input signal M. Consider the model for the first-order dynamic system in which x = position and i = rate, and the equation of motion in statc-space form is where w and v arc uncorrclatcd and zero means with variances q and r, respectively An illustration of the preceding model dynamics is shown in Figure 4-8a ~vhicll

Sec. 4.6

Complementa~~lKalman Fllter Approach to Estimator Deslgn

159

Kalman Filter

I

+Rate Limit

Complementary Filter

Low-Pass Comparison

k, t bu, Z=

4

x t v = xl+bu1+v

z = xtv. v = high frequency noise 1

U $U

X

dt

y = xtn, n = low frequency noise

k=fi

lh

Figure 4-8'

First-Order Complementary/Kalman Filter

includes the optimal state estimator. With regard to Kalman filter formulation,

+

- b u l l where k is the Kalman and the state estimate equation is & I = u k [ z filter gain. The estimate outputs are 2 = 2 , + bul. Substituting Equation (4-36) into Equation (4-18) yields

A , ( s ) / ~ ( s=) l / ( s +

k),

X I (S)IUI (3) = - bkl(s

+ k),

.k,(s)/Z(s) = k l ( s

+ k)

(4-35)

Therefore The characteristics e uation is \s + k = 0.The algebraic Riccati equation is -P2/ r + q = 0 or P = &r and hence the Kalman gain is k = q r . The Kalman filter gain k is the inverse of tkcomplementary time constant and the Kalman filter provides more insight into the filter gain characteristics.

k

Case 1: 40~itiot1 bias. the actual measurement z from Equation (433) also contains a bias error xb beside the measurement noise v , that is, z = x -

Modem Filtering and Estimation Techniques

160 xb

Chap. q

+ v , then the first approcch in implementation is that the bias error xb must be \

removed by adding a ul = $ x b correction after the i~tegrationand b = k . The second approach is to set b = 0 and treat xb as an extra input equal to Au = kx, to be added to the input u .

Example 4-5: Estimation of center-of-gravity (CG) lateral accel. eration. As shown in Book 1 of the'series, the inertial lateral acceleration is measured at the IRU, which is a t a distance L , from the center of gravity (CG), as opposed to being measured at the actual CG location. Consequently, A, measured at the IRU contains an error due to the yaw acceleration R, that is, xb = L , R / ~ . Hence x = AYcC and z = A = x - xb v . The first approach is to set b = k and ul = $ xb = Rt = LxRIg. The acceleration rate measurement jc = A,,, is not available, and hence u = 0. The second approach is to set b = 0 and u = kxb (requiring R to be available). Therefore, the first approach is selected because it is simpler.

,,,"

+

Example 4-6: Estimation of CG n o d a l acceleration.

Similar to

the estimation of the CG lateral acceleration, from Book 1 of the series x = Azcc,

UI

= J x = ~

L,QIg, z=Azlnu=x-xh+lv,

and

u = O

Exampie 4-7: Turbulencelgust velocity estimator. Let z = Vl = inertial acceleration, u l = VTAs= air data velocity, b = - k , and u = 0. Substituting this equation into Equation (4-36) and using the definition VTAs = V, - VC yiclds A

Z(s) = V c =

k ks k . ks sk Z(s) - -U , ( s ) = -v, - -VTAS= s+k s+k s+k s+k s+k 1

VTASand k are operated on by the filter to produce a gust rate estimate Vc. That is, as can be seen from the preceding equation, the rate of change of the inertial component of VTASexactly cancels V I , resulting in the production of the aforementioned gust rate estimate.

Example 4-8: Turbulencelgust angle-of-attackrate estimate.

Let

z = & I = inertial angle of attack rate, u , = a* = air mass angle of attack, 11 = k , and u = 0. Substituting this equation into Equation (4-36) and using a* = ar ac yields

+

a~ and a, are operated on by the filter to produce a turbulence/gust a rate estimate The rate of change of the inertial component of a A exactly cancels the resulting in the production of the aforementioned turbulencelgust a rate esti~nntc,

&.

Sec 4.6

ComplementarylKalman Filter Approach to Estimator Desfgn

161

Example 4-9: Altitude rate estimator. The problem of estimating altitude ratc is fornlulated as z = l>k = s k k , r r l = All + I/c;ycsls = hls, rr = 0, and b = k. Applying Equation (4-36) yiclds This equation verifies that the problem formulation is corrcct. If thcrc cxists an acceleration bias A h in z , then z = hlk + Alak = 1;ik + sAj11k. Substituting the new z into Equation (4-36) yiclds

As mentioned previously, the acceleration bias Al;?k can be implemented in the position bias term by adding A hlk to u r as ul = hls + A hlk. Substituting thc new u l into Equation (4-36) also yields Equation (4-38). This verifies that an alternative implementation can be done.

Case 2: Rate bias, b = 1. If the rate input contains a bias error jCb, that is, u = j: + i b , then the rate bias error jCb can be removed by a position correction input u l = xh after the integrator as in the position bias case. Since the rate input u has a unit forward loop gain, u l should also have a unit forward loop gain, that is, b = 1. Example 4-10: Mrst-ordera estimator using inertial a rate at IRU. In this case, the C G normal acceleration estimate is not used in the inertial a rate computation, and the problem is formulated as follows: x = a = angle of attack, z = x + v = a~ = a1 + a~ = measured or derived air mass a, a r = inertial a, X D = ac = turbulencelgust a, and u = &I,,L, = inertial a rate at IRU. Also, aIIRU contains an error due to pitch acceleration, Q, from the inertial a rate at the actual C G location iurC,, as discussed in Book 1 of the series. The correct terms are &rcc = + dlb and & b = LxQ/LTAs One way to obtain arc, information at and then integrate the sum of the CG would be to add the error term & b to iYI, , these two terms. Unfortunately, Q, and hence & b , are not available. A better scheme and then add a pitch rate correction term u1 = a b = L,QI is to integrate &I,R,, VrAS with b = 1 after the integrator to obtain accurate arcG. Example 4-11: First-order sideslip estimator using inertial sideslip rate at IRU. The problem is formulated the same manner as in Example 4-10 as follows: x = P '= sideslip, z = x + v = PA = PI + PC = measured or derived air mass P, pr = inertial P, x~ = PC = turbulence/gust P, u = pl,,, = inertial sideslip rate at IRU, U I = P b = LxRIVTAs, and b = 1. Case 3: NO bias, b = 01. In this case, b = 0. Hence, u l = 0 and the estimate is 2 = u + k(z - 2 ) . If there exists a disturbance x D in the x measurement,

Modem Filtering and Estimation Techniques

162

Chap. q

+

then z ( t ) = x ( t ) x D ( t ) and the rate input is u = i.hn the Laplace domain, Ul(s) = 0, U ( s ) = s X ( s ) , and Z(s) = X ( s ) +I X D ( S ) .Substituting the preceding result into Equation (4-36),

Example 4-12: Rate estimator. The problem is formulated such that x = x = rate, z = x + v = measured/der,ived X , u = ji = measuredlderived ul = 0, and 6 = 0. The estimate equation is k ( t ) = ~ ( t +) k ( X ( t ) - ~ ( t ) )Applying . Equation (4-36), ~ ( s )= [ X ( s ) + k ~ ( s ) ] l (+s k ) = ~ ( s )This . verifies that the formulation is correct. The first-order altitude rate estimator in Example 4-1 can be solved by this formulation with x = h , and x = ha. The estimate equation is the same as in Example 4-1.

x,

Example 4-13: First-order a estimator using inertial a rate at CG. Following Example 4-10, if the CG normal acceleration estimator is used in the inertial a rate computation, then b = 0, 1 4 1 = 0, and ir = &I,, = inertial rate at CG. Here, & tracks mainly the air mass a.4 a t low frequencies so that the complementary a rate A & = k ( a A - &) is zero a t low frequencies. Applying Equation (4-39) yields &(s) = a l ( s ) + k a c ( s ) l ( s + k ) . Thus, a~ and &I,, are operated on the filter to produce an a that contains pure c r l plus filtered &c. Example 4-lk First-order sideslip estimator using inertial side. slip rate at CG. Following Example 4-1 1, if the C G lateral acceleration cstimator is used in the inertial sidesiip rate computation, then 6 = 0, i d l = 0, and PI,, = inertial sideslip rate at CG. fi tracks mainly the air mass sideslip P A a t low frequencies so that complcn~entarysideslip rate A @ = k ( P A is zero a t low frequency. Applying Equation (4-39) yields fi(s) = Pl(s) + kPc(s)l(s + k ) . Thus P A and OI,, are operated on the filter to produce a fi that contains pure inertial sideslip P I plus filtered fit.

p)

Example 4-15: First-order velocity estimator. mulated as follows:

x = I~T.AS 2

= x

+

=

air data vclocity, x ~

=

Thc problem is for-

Vc = turbulence/gust vclocity

v = V I - V c = air data velocity measurement, V I = inertial velocity

u = V I = inertial acceleration,

ul = 0

Applying Equation (4-39) yields ~ T A S ( S = ) c l ( s ) - K V C ( S ) I ( Sf k ) . Thus, V r ~ s and 01are operated on the filter to produce a velocity estimate 6 ' ~ that ~ s contains pure V I plus filtered 6 ' ~ .

ComplementarylKalman Nlter Approach to Estimator Design

Sec. 4.6

Example 4-16: First-order altitude estimator.

la3

The problc~nis for-

mulated as follows: x = It = altitudc,

z = ItR

= x

+

u

= /I = altitude ratc via INS and FCS

v = barometric altitudc,

The resulting altitude estimator is = it + k(k important in INS and FCS altitudc tracking.

11,

=

0

- I;). The altitudc estimate is very

Example 4-17: First-order Euler attitude estimator.

The problem

is formulated as follows: x =

4

z =

IJJ,

=

attitude,

tr =

r/r

= attitude ratc transformed from body rate

rr, = 0

= INS attitude,

The resulting Euler attitude estimator is shown in Figure 4-9.

4.6.2 Second-Order ComplementarylKalman Filter Figure 4-10 shows a generalized second-order complcmentarytKalman filter. Consider the model for the second-order dynamic system. Applications for this filter can be classified into three major categories: (Formtrlntiorr 1) Position and Rate Estimators with Position and Acceleration

Measurements: Given measuredlderived position x and measuredtderived acceleration 2 , it is desired to obtain the best estimate of the position and rate. (Forrrrrrlntion 2) Position and Rate Estimators with Position and Rate Measurements: Given measuredlderived position x and measuredlderived velocity k, it is desired to obtain the best estimate of the position and rate. (Formulation 3) Position and Rate Estimators with Rate and Acceleration Measurements: Given rneasuredlderived rate k and measuredtderived acceleration x, it is desired to obtain the best estimate of the position and rate.

Second-order complementarylKalman filter: formulation 1. This formulation is used to design position and rate estimators with position and accel-

Roll Rate Pitch Rate Yaw Rate

Body to

Euler Axis Transformation

Figure 4-9

Euler Attitude Estimator

Modern Filtering and Estimation Techniques

164

Chap. 4

Kalman a l t e r

L.

Complementary Filter

High Pass

4 =Fl -nT Low Pass

n-v

z

s*

+ 2 j wn s +a);;

Low Pass

G(s)

Comparison

el + bu2

L

z = r + v = n l + b%+v

z = s+v. v = high frequency noise

JJu &dl + Jul

y = x+n, n = low frequency noise

k/kI Figure 4-10

d

-

2j!w,,

-

T

-

= complementary filter time constant

Second-Order ComplementaryiKalman Filter

eration measurements. Consider the model for the second-order dynamics system in which x l = x = position, xz = i = rate, and 14 = x = acceleration. T h e equation o f motion in state-space form is i l ( t ) = ~ ~ ( and 1 ) i , ( t ) = . ( I ) , with position x as an output, and z ( t ) = x, ( I ) . If the acceleration measurement ii contains a bias crror A x , that is, ii = x + A x , then the acceleration bias error A x can be removed by a rate correction input u l = A i after the rate integrator so that i l ( t ) = x 2 ( t ) + I I I ( ~ ) + w l ( t ) where w l is a zero-mean white process with variance q l . ril can be a rate correction equal to the rate difference between the actual rate measurement and the sensor rate measurement. The measured/derived acceleration ji is treated as a control w Z ( t ) ,wherc w z input u with known noise characteristics such that i 2 ( 1 ) = U ( I )

+

Sec. 4.6

ComplementaryIKaiman Filter Approach to Estimator Deslgn

165

is a zero-mean white process with variance q 2 . The preceding model dynamics are illustrated in Figure 4 - 1 0 which includes the optimal state estimator. A comparison of a second-order complementary filter and Kalman filter is also shown in the figure. As in the first-order filtcr formulation, the measured/dcrived position x is treated as a measurement z with bias error bu2 and a measurement noise v , and z = xl + bir? + v , where v is a zero-mean white process with variance r. With regard to Kalman filter formulation,

where K is the Kalman gain matrix, and the state estimate equations are

where i = .?

+ u l ( t ) + k [ z ( t ) - i l ( t ) - bu2(t)] (4-40) i2(t) = ~ ( t+ ) k I [ z ( t ) - . t l ( t ) - 6112(t)] = .GI + bit2 = position estimate, .t2(()= rate estimate, and i ( t ) =

.??(I)

=

il(t)

+ irl(t)

=

22(t)

calibrated rate. Applying Equation ( 4 - 1 8 ) yields

where gains k and k l have been expressed in terms of the second-order parameters. Therefore,

i(s)

=

2(s)

=

kl(s)

+ bU2(s)

+ s U ~ ( S +) b s 2 U r ( s ) + ( k s + k J z ( s ) ] l D ( ~ ) k ( s ) = k2(s) + UI(S) = [(s + k ) U ( s ) + (s2 + k s ) U l ( s ) - bkIsU2(s) + s k I Z ( s ) ] l D ( ~ )

(4-42a)

= [U(S)

(4-42b)

All the transfer functions from either the control inputs u , u l , and 242, or the measurement z to the state estimates are low-pass filters. The estimator gains k and kr

Modem Filtering and Estimation Techniques

166

Chap. q

can be related to the estimator poles sl and s2 by the following formulae:

The location of the estimator poles sl and s2 are chosen based on applications and are different for different applications. The transfer functions from the measured z ( t ) t o k and 4 are simply the last column of Equation (4-41a), k ( s ) = (ks

+ kl)Z(s)lD(s),

~ ( s =) klsZ(s)lD(s)

(4-44)

The steady-state values for the elements of P, and K , are found to be

L

If ql = q2 = 0, then in the steady state the elements of the error covariance matrices all vanish, that is, P, -+ 0, and the Kalman filter gain vanishes. Since the perfect model and the filter need only to find the missing initial condition of the state, the processing being observed is nonstationary and the estimate converges on the true value of the states which is purely the double integral of the illput Case (a).

U.

Case ( b ) .

If q, = 0, qz # 0, then in the steady state k

=

~ ' Z ( ~ ~ l r ) ' / k1 ~ ,=

Using Equation (4-41 b), w,, =

(4-46)

k 6= (q2/r)114and 5 = - 0.707. Thus, the 2%

damping ratio is a constant and the undamped natural frequency is determined to be a function of only the ratio of errorlnoise in the rneasuredlderived acceleration x to the measuredlderived position x . Furthermore, w,, depends only on the relative noise between q2 and r, not on the absolute noise levels. and kl If q 2 = 0 and ql # 0, then in the steady state k = = 0. From Equations (4-40) k1 = i 2+ U , + k [ -~ 2 , - bu2] and i2= u. Since qz = 0, a perfect accelerometer model is used and the filter needs only to find the missing initial condition on the rate state, the process being observed is nonstationary, and the rate converges on the true value of the rate states that are Case (c).

Sec. 4.6

Complementaryllblman Filter Approach to Btlmator Deslgn

167

the integral of the accelerometer input. The position estimate is the first-order completnentary/Kaltria~~ filter position estitr~ate. If the Kalman gain K is assumed zero where the forcing function in the filter design is not considered, the covariance equations (4-23) are employed here with K = 0 since they hold for an arbitrary equation K-matrix.

The error covariance will grow according to Equation (4-47) cvcn if there is only a small amount of white noise of spectral dcnsity q l forcing the .ul state or a small amount of white noise of spectral dcnsity q- forcing the xz state.

Example 4-18: Navigation filtering for position and velocity estimate. The navigation filtering for position and velocity estimate presented in Chapter 7 falls cxactly into the previous category, with the following definitions: x =

X, Y, Z position; z = X, Y, Z position measurements;

i= I!

=

A, Y , z velocity X , V , z acceleration;

til

=

A X , A Y , A Z velocity corrcction; u2

=

0

Following Example 4-1 and defining x l = h = altitude and x2 = h = altitude rate, this example can then be applied to the design of an altitude and altitude rate estiftlator where z = ha.. = h + v = X I + v = Z vosition measurements. The .. acceleration bias error sail is assumed to exist so that h = I;, s h h + w = u j t r l + w , and the state equation is obtained as $1 = x? + ul and i z = u + w. Thus, this is the same problem as that in Case (a) wherein q l = 0. q, = q, and r = r. The Kalman gain is given by Equation ( 4 - 4 3 , and the state estimate eqyations are given by Equation (4-40) where i = i1 = h = altitude estimate and h = 22 = altitude rate estimate. The calibrated altitude rate estimate is h , = 22 + u l . The transfer function from z ( t ) = h R ( t ) to h^ is h^lhR = ( k s k,)lD(s) = s + V & ) l ( s 2 + V'2Vqlr s + d&)and it is a low-pass filter. In the Laplace domain, from Equation (4-42) the altitude estimate and the altitude rate estimate in terms of u = hlRS, u 1 = A h , and z = h are h^(s) = ( U ( s ) + s U l ( s ) f (o: 2<w,s)Z(s)]/ D ( s ) = [ h ( s ) s h h ( s ) + (w: + 2<w,,s)h(s)]lD(s)and = [ ( s + 2<w,,)h(s) + (s2 + ~ < w , , s ) A / L (+x )SW: h ( s ) ] / D ( s ) The . problem of estimating gear altitude and altitude rate is formulated as follows:

+

+

A

-

+

x = gear altitude,

(a

+

i(j)

+

z = gear altitude measurement,

u = inertial altitude acceleration measurement, u2 = 0 u l = inertial altitude rate correction,

Example 4-19: Position error and rate error estimator with POsition error and acceleration measurements. Trajectory errors in terms

Modem ~iiteringand Estimation Techniques

168

Chap. 4

of position error and rate error are important parameters for NGC and hence need to be estimated. The problem of estimating trajectory error is formulated as follows: x = AX = trajectory position error, z = x

u = ul

x

+v

= trajectory position error measurement

= accelerometer measurement

= A X = trajectory rate correction

Applying EAquation(4-42) yields A ~ ( S )= [ ~ ( s + ) SAX(S) + (ks + kl)AX(s)]/ D(s) and AX(^) = [(s + k ) ~ ( s + ) (s2 ks) AX(S)+ sk~AX(s)]/D(s).In estimating the glideslope deviation, the problem is formulated as AX = A h = altitude error, A X = ~h = altitude rate error, and x = h = vertical acceleration. In estimating the localizer deviation, the problem is formulated as AX = A y = lateral position error, A X = A j = lateral velocity error, and x = j = lateral acceleration.

+

Second-order complementaryIKalman filter: Formulation 2. This formulation is used to design position and rate estimators with position and rate measurements. The following are defined: x = position (same as before),

z

=

x

+v

=

measured position

u(t) = Ax(t) = acceleration bias error (if it exists) instead of x in Formulation 1

rate measurement (regarded as a control input) instead of A i in Formulation 1

ul(t) = i ( t ) =

xl(r)

=

position (same as before), x2(t) = A i = rate measurcment disturbance instead of i in Formulation 1

2 , ( t ) = position estimate (same as before), turbance estimate i(t) =

?(t) =

i(t) = i2(t)

.G2(t) =

rate measurement dis-

+ bu*(t) = position estimate (same as before)

+ u l ( f ) = rate estimate (same as before)

The state and measurcment equations are the same as those in Formulation 1 , that is, i 1 = x2 + U I + U ' , . i 2 = u + ul2, and z = x, buz o. The results of the second-order filter (Formulation 1) can all be applied here with a slight revision as follows. If there exists a disturbance X D in the x measurement, then z(t) = >-(I) + x ~ ( t )In . the Laplace domain, U(s) = 0, Ul(s) = sX(s), and Z(s) = X(s) + X D ( ~ ) . Applying Equation (4-42) yields

+

+

Sec. 4.6

CompiernentaryIKalman Niter Approech to Estimator Design

169

Thus, z ( r ) and r r , ( 1 ) arc operated on by the filter to produce: ( 1 ) a position estimate i ( t ) that consists of pure position .u(t) and a filtcrcd i I > ( t )arid ; (2) a rate estimate ; ( I ) that consists of a pure ratc i ( r ) and a filtered io(r). The preceding results assume the acceleration bias rr = A x to be zero. If it is nonzero, then a third state needs to be modeled. Therefore, a third-order complcmcntaryIKaln~anfilter such as the one presented in Section 4.6.3 is required.

Example 4-20: Second-order a estimator. This example follows the first-order a estimator problem formulation in Example 4-10. Case 1: Using inertial a rate at ZRU. In this case, u l = k = &I = &, = inertial a ratc at IRU, b = 1, and u = 0. A means of removing offsets in the accelerometer used to compute the inertial a rate is to bias the inertial a rate by the integral of the error between a.+and ti. This also removes computational errors. As in Example 4-10, a better scheme of correcting ah is to integrate &I,,,, and then add a pitch rate correction u 2 = a b = L,Ql VTAS after the position integrator (see Figure 4-10) to obtain an accurate a at C G information. The estimates are given bv & = i t + 6rr2 and & = .?? + 1 4 , . Note that & has the same formulation as that i n the first-order estimator, while & is not provided by the first-order estimator but is provided here by this second-order estimator. Case 2: Using inertial a rate at C G . In this case, u l = jr = Cul = &I,, = inertial a ratc at C G , 6 = 0, tr = 0, and 1.12 = 0.The solution of this case is simpler than that in previous Case I . Applying Equation (4-48) yields & ( s ) = ads)

+

(ks

+

kl)ac(s)/D(s),

&(s) =

&l,,(s)

+ klsaG(s)lD(s)

Thus, a.4 and &l,:,; are operated on by the filter to produce: (1) an & that consists of a pure a 1and a filtered tic; and (2) an & that consists of a pure &I,, and a filtered s

kc. Example 4-21: Second-order sideslip estimator. This example follows the first-order sideslip estimator problem formulation in Example 4-11. Case 1: Using inertial sideslip rate at IRU. In this case, U I = 2 = PI,,, = inertial sideslip rate a t IRU, b = 1 , u = 0 , and u2 = Pb 7 LxRIVTAS.The state estimate is given by = 2 1 + bu2 = sideslip estimate and P = i2 ul = sideslip rate estimate. *Note that p for the same formulation as that in the first-order estimator, while 6 is not provided by the first-order estimator but is provided here by this second-order estimator.

+

p

Case 2: Using inertial sideslip rate of C G . In this case, u l = jr = P I = Dl,, = inertial p rate at CG, b = 0, u = -0, and u 2 = 0. The solution of this case is simpler than that ip previous Case 1. The state estimates are now P = 21 = sideslip estimate and 6 = 322 u1 = sideslip rate estimate. Applying Equation (440) yields

+

fi = ?2 + 0lcC + k(PA -~fi),

i 2

= k l ( P ~-

Cj)

Modem Filtering

170

Applying Equations (4-42) and using yield

and Estimatton Techniques

@I($)

=

s P ~ ( s and )

PA(s)

=

Chap. g

PI(s) + PC(S)

+ ( k s + k r ) P ~ ( s ) ] f D ( =s ) PI(^) + (ks + k ~ ) P c ( s ) l D ( s ) b ( 3 ) = [(s2 + k s ) b ~ ( s + ) sk~P~(s)]lD= ( s )b l ( s ) + k ~ s P c ( s ) l D ( s ) Applying Equations (4-48) also yields the same equation. Thus, P A and PI,, are = [X@I(~)

fi that consists qf pure P I and a filtered fiG; and (2)

oppated on to produce: (1) a a 3( that consists of a pure @I,,

and a filtered

bc.

4.6.3 Third-Order Complementary/Kalman Filter Figure 4-11 shows a generalized third-order complementary/Kalman filter.

Third-order complementary/Kalman filter: Formulation 1. Consider the model for the third-order dynamic system in which x = position, i =

U

4

a) Formulation 1

U

-

,A

Wi3

b) Formulation 2 Figure 4-11

Third-Order Complementary/Kalrnan Filter

Comp(ementarylKa1manNlter Approach to Estimator Design

Sec. 4.6

171

rate, and x = acceleration, and the equations of motion in state-space form are as follows: XI

=

where x = is

X?

s l

+

111

+

LVI,

E \ W I ( ~ )= ] 0.

E [ I v ~ ( ~ ) w ~ =( Tqlb(t )] -

T)

+ 6 1 1 2 ,i = x 2 + l r l , and .t = x.3 + rr, and the measurement equation

The preceding model dynamics are illustrated in Figure 4-lla which includes the optimal state estimator. With regard to Kalman filter formulation,

where K is the Kalman gain matrix, and the state estimate equations are

r

and the estimator outputs are i = 2 = it + b112, x = i 2 + U I , and x = 23+,,. The solution of the second-order filters is identical to that of the third-order filter presented in this section, with the exception of k, being set to zero. This results in the elimination of the last state x 3 in the third-order filter. The transfer function from the control inputs u ( t ) , i r l ( t ) , and u 2 ( t ) , and the measurements z ( t )to the state ) estimate vector ~ ( t are

-

A

Modem Nlterlng and Estimation Techniques

Chap. q

where d ( s ) = O is the characteristic equation. Therefore, Z(S) =

A($)= kl(s)+ bU2(s) =

[sU(s) + s 2 U l ( s )

+ bs3U2(s) + (ks2 + kls + k l ) Z ( s ) ] l d ( s )

The transfer functions from z ( t ) to ~ ( trepresent ) the last column of Equation (451) as

As shown in Equation (4-52), all the transfer functions from either the control inputs u, u l , and u2. or the measurement z , to the state estimates and the estimator outputs, are low-pass filters. The algebraic Riccati equation is

and the steady-state Kalman gain is

In this case, 43 = O, q l # 0, and q2 # 0, and Equation (4-55) with kl = O has been found to be the same as the second-order filter gain Equation (445). Therefore, when q3 = 0, that is, the process noise intensity on the third-state is zero, the second-order filters employed in Section 4.6.2 are identical to the thirdorder filter presented. Case 1.

Case 2.

In this case, ql = 0, q2 = 0, q3 # 0, and Equation (4-55) has been

found to be k = 2(q3/r)I1",

kl = 2(q3/r)'I3,

k l = (q3/r)1'2

(4-56)

Sec. 4.6

ComplementaryIKaImanFilter Approach to Estimator Design

173.

Case 3. If there exists a disturbance x n in the x measurement, then r = x XI,. If ~ c ( t )= 0, u l ( t ) = jc = rate, and u2(t) = 0, then in the Laplace domain, U ( s ) = 0, U l ( s ) = s X ( s ) , and Z ( s ) = X ( s ) X D ( S ) .Applying Equation (4-52)

+

+

yields

?(s) = X ( s ) A

X(S)

= X(S)

X(5) =

X(S)

+ (ks2 + k r s + k l ) X n ( s ) l d ( s ) , + (kls2 + k l s ) X ~ ( s ) l d ( s ) + kls2X~(s)/d(s)

(4-57)

By setting k , = 0, Equation (4-37) is reduced to the second-order filter Equation (4-48). If a gain h is added to the measurement input, then z' = h r = h x , + hbuz + hv and v' = hv is the new measurement noise with zero mean and variance r' = h2r. The new state estimate equations and Kalman gains are Case 4.

21 = 22

+

141

+ k ' ( ~ -' h.?] - hbL42)

j3= k ; ( z ' - hG1 - hbuz),

k'

=

klh,

k ; = krllt,

ki = kllh

The error covariance P is the same as that in the previous formulation except that r is replaced by r'lhZ.

Example 4-22: Trajectoly estimation.

In a three-dimensional flight, the trajectory estimation problem involves finding the best estimate of position, velocity, and acceleration in the X , Y, Z inertial coordinates, given the position, velocity, andlor acceleration measurements. The problem is defined by

x = X , Y , Z position,

u = X, Y , z acceleration

z

=

X, Y, Z position measurement,

ul

2

=

X, Y,

z velocity,

=

AX, AY, A Z velocity correction

u2 = AX, A Y , A Z position correction

Note that the problem formulation is similar to the navigation filtering for position and velocity estimate presented in Example 4-18. However, uz is nonzero here and the acceleration estimates are provided as the estimator outputs which were not available in that section. In some cases, X, Y , Z acceleration measurements may not be available and hence need to be estimated. Then, u = AX, A Y , A Z acceleration correction and x j = unknown acceleration to be estimated. In altitude, altitude rate, and altitude acceleration estimator design, the problem is formulated as

x = h = altitude

+

v =

+ bu2 = altitude.measurement

z =

x

u

h = altitude acceleration

=

XI

Modem Filtering and Estimation Techniques

Chap. 4

ul = ~ h = altitude rate correction u2 = A h = altitude correction Terminal guidance tracking filter (presented in Chapter 7) is another example of the trajectory estimation problem discussed in this section.

Third-order wmplementary1Kalman filter: Formulation 2. The problem is defined the same way as in the third-order complementary/Kalman filter (Formulation I), except x 3 here is assumed to be filtered white noise, that is, the state Equations (4-49a) now become and the state estimate equations are

i1= 22 + u1 + k(z - 21 43

= -2317

- bu2)

+ k l ( z - 21 - bu2)

Applying Equation (4-la),

Equations (4-61) are solved nun~ericallyfor P ( t ) to generate gains k, kl, and k ~ , which mechanize the third-order filter depicted in Figure 4-11b. The steady-state Kalman gain is

k = Plllr,

k,

= P12/r,

k, = P l J r

(4-62)

If a gain h is added to the nIcasurcment input, then z' = Irz = h x l + I~brrz + Ifv and v ' = hv is the new measurement with zero mean and variance r' = h'r. The new state estimate equations and Kalman filter gains are

ComplementaryIKalman Filter Approach to Estimator Deslgn

Sec. 4.6

175

The error covariance P i s the same as that in the previous case except that r is replaced by r'lh2. Assuming measurement noise v in Equation (4-49b) to be a colored noite, then following Equations (4-28) to (4-31) yields the new measurement equations z = x v, ir = F v G v l , = ~ H HA - FH = [ - F , 1,0], E [ f i ( t ) ~ ( ~=) ~ ] [ H N Q N T H T GrGT]6(t - 7 ) = r,S(t - 7), andS(t) = Q ( I ) N ( ~ ) ~ HApplying (~)~. Equations (4-31) yields the new equations for state estimates, covariance matrix, and Kalman filter gain matrix. If F and C are scalar parameters, then Equations (431) yield the following simple form of new covariance and Kalman gain equations:

+

-~ J ' I-I dt

+

- 2P12

+

+ q l - kfr,,

+

-PI-2 dr

-

P22 +

PI3

- k,krcr,

Inertial Navigation' The block diagram of Figure 5-1 is provided to illustrate how the different parts o f the subject matter presented in this chapter are related t o one another. For instance, the discussions of common requirements, navigation con~putation,and coordinate systems together form the framework for the design of a self-contained INS. The t w o types o f self-contained INS are the gimballed INS and the strapdown INS. Following discussions of each of these two types o f INS, a comparison o f the two is presented. Combining INS with other navigation aids introduces the subject of filtering, and finally extending INS uses to provide flight control outputs leads to an integrated INS dcsign.

5.1 INTRODUCTION Inertial navigation is thc process of measuring accclcration onboard a vchiclr 2nd then integrating that accclcration to dctcrminc the vehicle's vclocity and position relative to a known starting point. Accclcrotnctcrs arc uscd to sense thc magnitude of thc accclcration, but acceleration is a vector quantity having direction as wcll as

'

This chapter has becn technically cditcd and improvcd by I l r . J. Stanley Ausrnan of Litton Guidance and Control Systcms.

176

Sec 5.1

lntroductlon

Common Requiremenu Coordinate Systems (Sec. 5.3)

Self-contained INS (Secs. 5.4 & 5.5)

Navigation Computation (Sec. 5.3)

Comparison of Gimballed INS & Strapdown INS I (Sec. 5.6)

I External Navigation Aids (Sec. 5.7)

Figure 5-1

Inertial Navigation

magnitude. For this reason a set ofgyroscopes, or simply gyros, are used to maintain the accelerometers in a known orientation with respect to a fixed, nonrotating coordinate system, commonly referred to as inertial space. This does not mean that the accelerometers themselves are kept parallel to the axes of this nonrotating coordinate system, although some systems are mechanized this way. Instead, the accelerometers may be kept parallel to a set of earth fixed axes by causing the gyroscopes to precess at earth rate. O r they may be maintained locally level by precessing the gyros at earth rate plus the angular rate of the vehicle over the earth. In most of the modern aircraft inertial navigation systems today, the gyros are precessed to rotate with the aircraft itself, resulting in what is known as the strapdown system. One characteristic inherent in accelerometers is that they do not measure gravitational acceleration. For example, an accelerometer in free fall will measure zero acceleration (0 g), but it is really accelerating downward at the acceleration of gravity. For another case in point, consider an accelerometer sitting on a stationary table. It will measure 1 g due to the gravitational reaction force of the table pushing upward on the accelerometer, but the accelerometer is not really going anywhere, at least with respect to the earth. In order to overcome this deficiency of accelerometers, it is necessary to add in the effect of gravity based on a gravity model. A computer uses this model to calculate gravity as a function of position, which in turn is determined from the double integration of the measured acceleration. The most common gravity model employed is one which accounts for both the latitudinal and altitudinal variations in gravity. This model is symmetric about the earth's polar axis and also about the equator. For most navigational purposes, this ellipsoidal model is sufficient, but highly accurate systems must in addition account for gravity anomalies which are deviations from this simpler model.

178

Inertlal Haulgation

Chap. 5

It is difficult to navigate a flight vehicle at night or in bad weather, even with the availability of radio and radar navigational aids. Similarly, smaller, faster, more maneuverable surface vehicles armed with guided missiles, torpedos, and cannons meet with the same difficulty. Consequently, the tremendous need for improved navigation gave rise to the necessity for onboard systems that would supply the position, velocity, and attitude of the vehicle on which they are mounted. The first inertial navigation systems (INS) in operation were those in the German V-2 rockets of World War 11. Since then, INS have become standard equipment in ballistic missiles and in all modern fighter aircraft, airliners, and naval ships. what makes INS particularly suitable for ballistic missile guidance is their autonomous mode of operation during the boost phase. This feature yields a jam-proof guidance system [Bar-Itzhack and Berman, 19881. Much of the progress in the design and use of inertial systems can be traced to the strategic missile programs of the 1950s and, specifically, to the work done at the MIT Instrumentation Laboratory (currently the Draper Laboratory). It was during this time that the need for high accuracy at ranges exceeding 1,000 k m using autonomous systems became more and more apparent. As a result, the U.S. Air Force requested the laboratory to develop inertial systems for the Thor and Titan missiles, while the U.S. Navy requested the same laboratory to develop an inertial system for the Polaris missile. The practicability of a self-contained all-inertial navigation system (INS) for aircraft had just been demonstrated by the laboratory in a series of flight tests using a system known as Spire [Draper, 19811. With the MIT Instrumentation Lab providing the spark, several privately-owned companies took up the challenge and developed and produced their own INS for long-range cruise missiles (North American Aviation for the Navaho missile and Northrop Aircraft for the Snark missile), for strategic bombers (Sperry Gyro Corp. for the B-38), and for ballistic missiles (GE for Polaris and American Bosch Arnla for the ATLAS missile). The remarkable succcss of those early missile and aircraft programs paved the way for applications in aircraft, ships, missiles, and spacecraft. Today, inertial systems are standard equipment in both commercial and military applications. The types of gyros currently found in inertial navigation systems (INS) include the floated rate integrating gyro, the electrostatically supported gyro (ESG), the tuned rotor gyro (TRG), and the ring laser gyro (RLG). The first three are mechanical in nature, relying on a spinning mass to produce a large angular momentum which tends to stay fixcd in inertial space. The floated and tuned rotor gyro suspend the spinning mass in mechanical gimbals, while the ESG suspends the spinning mass, generally a spinning ball, with electrostatic forces so as to prevent any mcchanical contact between the spinning ball and the outside case [Schmidt, 19871. The RLG is a closed optical cavity containing two beams of single-frequency light traveling in opposite directions around the closed cavity. The two beams interfere with one another to form a standing wave pattern around the cavity. This pattern of bright and dark spots tends to stay fixed in inertial space, so that an observer or light detector fixcd to the cavity can sense rotation of the cavity as he (or it) rotates relative to this fixed pattern. A more complete dcscription of the four

types of gyros listed earlier, together with accelerometers and error models for each can be found in Savage 11978, 1984(a)]. Inertial navigation systems are either gimballed or strapdown. In a gimballed system the gyros and accelerometers are isolated from the rotations of the vehicle so that they can be maintained in a specific orientation relative to the earth or inertial space. This greatly simplifies the computations of velocity and position and also greatly reduces the requirements on the gyros which would otherwise have to measure very high angular rates. In a strapdown INS the gyros measure the full rotation rates of the aircraft and keep trace of the instantaneous orientation of the accelerometers in order to properly integrate the accelerations into velocity and position. I t was the advent of small, high-speed digital computers that made strapdown INS possible. In general, the three basic functions of an INS are sensing, computing, and outputting. The accelerometers and gyros perform the sensing function and send their measurements of acceleration and angle rate to the computer. T h e computer uses these data to generate velocity, position, attitude, attitude rate, heading, altitude, range and bearing to destination. If true air speed is available from an air data system, the INS can also compute wind speed and direction as xvell as drift angle. The outputting function simply means sending the appropriate data to thc flight control system, fire control system, reconnaissance sensors, or the control and display unit (CDU) as required for the particular mission. Accordingly, the significance of the navigation system to the flight mission becomes obvious, as the former provides dynamic parameters for flight and path control, assists in terrain-following flight, allows avoidance of enemy defenses via corridor flight, and reduces operator load. The ability of a navigation system to accomplish these tasks greatly enhances mission success. Particularly desirable features in a navigation system include: self-containment; security; worldwide capability; bounded navigation errors for position, speed, attitude and heading; no attitude, weather, or terrain limitations; minimum airframe structural impact; minimum size, weight, power, and cost; and high availability. If aided by other external navigation devices, the INS is called an aided INS. The so-called external navigation devices can be position aids or velocity aids. The former consists of GPS, MLS, MLS-GPS, LORAN, OMEGA, T A C A N and barometer (see the following for definition). The latter consists of Doppler, odometer, and laser velocity meter. The following list covers several important navigation devices and the corresponding section in which each one is covered in some detail: global positioning system (GPS), Section 5.7.2; tactical air navigation (TACAN), Section 5.7.3; long-range navigation (LORAN), Section 5.7.4; terrain contour matching (TERCOM), Section 5.7.5; Doppler radar, Section 5.7.6; and star trackers, Section 5.7.7. An aided INS composed of the INS and external navigation devices is depicted in Figure 5-2a. The evolution of these flight vehicle navigation devices is given in Figure 5-2b. It is seen that a GPS is the most up-to-date navigation device, offering the highest accuracy for flight vehicle navigation. The operational characteristics of

Inertial Nauigation

180

1950

1960

1970

INERTIAUDOPPLER

1 MILE ABSOLUTE

198)

Chap. 5

Sec. 5.2

181

Common Requirements for lnertlal Navigators

CHARACTERISTIC

L REOUlRES REMOTE COOPERATING TRANSMIllER

8. ANTENNA LOCATION I S IMPORTAN1

2.

POLAR REGION LIMITATIONS

9.

TIME FRAME DfPENMNT

3.

HEAVILY SMOOTHED SPEED I S AVAllABLE

10.

DOPPLER SUBJECT TO D m C l l O N

- POST 198)

4 WIND VELOCITY VECTOR REQUIED

11 STORMS AND SUN SPOT ACTIVITY

5.

U. U.

CERTAIN WEATHR LIMITATIONS

6 LIFE OF S I C H I LIMITED

7.

LIMITED OVER WATER AND HOMOGENEOUS LAND I.E.

MANEWERS M G R A M ACCURACY ACCURACY LIMITED TO MAG COMPASS

DESERT (C)

Figure 5-2 (a) Navigation Parameters Available from Sensors (b) Navigation Technology Evaluation (c) Operational Characteristics of Navigation Sensors ( F r o ~ n[.21izrri11, 19861 w i t h pennissio~zfrom Litton)

these navigation devices are given in Figure 5-2c. Generally speaking, the accuracy required of a navigation system for most applications is on the order of 1 nautical mph of flight. More stringency may be placed on the required accuracy in certain phases of flight, such as ground attack for example, in which the accuracy must be in the neighborhood of a few meters. In these cases, an external reference is necessary to update the INS in order to achieve the required accuracy.

5.2 COMMON REQUIREMENTS FOR INERTIAL NAVIGATORS

Vehicle and weapon systems. The requirements of a navigation system for weapon systems can be categorized as follows. In the case of piloted vehicles, medium accuracy (1 nautical mph) is mainly what is required of the navigation

Inertial Navigation

182

Chap. 5

system for tactical aircraft, whereas landfall position fix, velocity, and nuclear hardness are all required of the navigation system for long-range strategic bombers. Furthermore, low accuracy or simple attitude reference is the basic navigation requirement for observation aircrafts, while size and weight are critical t o helicopteri. In the case of short-range missiles and tanks, size, weight, and low cost figure predominantly in the requirements, whereas medium-range missiles require somewhat better navigation. Reliability is a key requirement for intercontinental ballistic missiles and tanks. Submarines and ships require extremely high accuracy over long time periods. For military use, the ideal navigation system should be fully autonomous, undetectable, unjammable, usable worldwide in all weathers, and of vanishingly low weight and infinitesimal cost. These aims have driven development for the past thirty years. Systems are now sufficiently reliable and accurate for virtually all modern fixed-wing military aircraft to be equipped with an inertial navigator [Barnes, 19871.

Navigation requirements for weapon delivery.

Navigation accuracy is a key factor for successfully carrying out low-level attacks against targets at known locations. First, it gets the attacking aircraft to the right position to find the target. Once the target is identified, the INS must provide an attitude reference for the targeting sensors (optical sight, laser range finder, radar, IR sensor, or TV sight). After the target position is located with respect to the aircraft, that position is continually updated by the INS as the aircraft moves into its weapon launch position. For the weapon delivery itself, the INS provides initial position, velocity, and attitude data for the weapon. Stand-off weapons require the same set of funct~ons,but the missile itself takes over the final target acquisition and terminal homing. For such weapons, the INS in the launch aircraft simply has to provide initial conditions for the missile, which may also have a small INS for midcourse guidance, good enough to steer the n~issile into a target acquisition basket.

Stand-off weapon systems.

The following discussion on navigation requirements for precision-guided stand-off weapon (PGSOW) delivery are based on detailed results shown in Shapiro (19861. The successful delivery of a PGSOW depends, in general, on three key elements: knowledge of launch position parameters, knowledge of target position, and accuracy of navigation between these two positions. The basic PGSOW mission scenario is shown in Figure 5-3 which depicts the flight legs from aircraft takeoff to weapon launch point, then the weapon flight to target arca window, and finally to the impact point. The focus hcrc is on the choices for providing accurate navigation to the lowcost, short-range (20-mile) RPV or powered glide bomb which cannot sustairl the size, weight, or cost penalties of a high-precision navigation system. In order to ascertain the accuracy requirement for the PGSOW, it is necessary to estimate the capability of the terminal seeker which, in the final analysis, determines the sizc of the target window or search arca. The burden placed on the terminal seeker incrcnscs

Sec. 5.2

Common Requirements for Inerthi Naulgators

183 TARGET

/SEARCH - ~

TARGET

LAUNCHING AIRCRAFT

FLIGHT ORIGINATION POINT Figure 5-3

PGSOW Mission Scenario ( F r o ~ n[Shapiro, 19861 w~idtpennisrionjom ACARD)

with the size of the required search area imposed by uncertainty in relative weapon and target positions. In the case of an 1R seeker, for example, the window may be relatively narrow for a given angular field of view because of the limited range due to atmospheric attenuation. An active millimeter wave radar can provide longerrange operation in bad weather (several kilometers); however, the limited aperture available for the antenna and the consequent limited angular resolution creates problems with false alarms and poor detection probability as the search area is increased. Therefore, the PGSOW must navigate to the target zone with sufficient crosstrack accuracy to insure that the target lies within the swath described by its forward motion and the scanning limits of the terminal seeker. Based on the foregoing types of considerations. it can be concluded that a vractical search area width is about 1 km. The corresponding navigation cross-track error budget should be limited to one half of this, or 0.5 km. The along-track error is dependent on the allowable false alarm and desired detection probability criteria. Obviously, as the along-track extent of the search decreases, these probabilities become more favorable. The alongtrack navigation error can be specified, however, as being the same as the crosstrack error, namely 0.5 km. T o achieve the preceding performance objectives, the navigation system for the PGSOW must have a heading error over the 20 miles of better than 1 deg and an along-track error ofless than 1.5 percent of distance traveled. The future will see an increased demand for more agile missiles for both airand ground-launched applications. Gun-launched missiles involving high accelerations at launch will contain guidance systems. Inasmuch as strapdown guidance systems have been included in missiles that are subject to high roll rates and high

184

Inertial Navigation

Chap. 5

acceleration, the need for more rugged navigation instruments having a dynamic range greater than what conventional gyros offer is brought to the forefront. A fiber-optic gyro is described [Kay, 19873 that, although it is still in the laboratory stage, has demonstrated good performance and, at the same time, ruggedness, quick starting, and low cost.

Visual attack systems. Following Barnes [1987], in an effort to expose the aircraft to as little ground-based radar as possible, much of the development subsequent to World War 11 has been focused on low-altitude flight. Another goal related to this effort was, until the development of look-down Doppler radars, to make it more difficult for a defensive fighter aircraft to identify an incoming raid. Considering that the total time of an attack pass may be just a few seconds, target acquisition is rendered more difficult by low-altitude attacks. In these few seconds, the pilot of an attacking aircraft armed with unguided weapons must maneuver on to an attack heading, stabilize the aircraft path, and release the weapon. Precise navigation and an accurate calculation of weapon release are two elements that are absolutely essential for a successful attack. The former ensures that target acquisition occurs as early as possible, while the latter minimizes delivery error. Regarding attacks that can be conducted under visual conditions, the requirements for precise navigation with an acceptable workload, especially at high speed and low altitude, have been met by the introduction of inertial platforms and moving map displays. Although such developn~entssimplify the task of acquiring preplanned, fixed-position targets early, they do little for acquisition of targets of opportunity. Similarly, two cle~nentshave combined to meet thc rcquirements of accurate calculation of weapon release. The first involves the ability to obtain data on aircraft altitude, heading, and speed from the inertial platform. Thc second rclates to the ability to digitally process in the aircraft range to target obtained from a ranging systenl. The ability to carry out high-speed, low-altitude attacks a t night and in poor visibility has also bcen enhanced owing to other dcvclopmcnts. One of the first of thcsc was ground-mapping radar, which enabled pilots ro attack targets giving discrete radar returns and targets whose position was known relative to some observable and recognizable feature on the radar screen. However, radar resolution is poor relative to visual wavclcngths and forward-looking infrared (FLIR) scnsors greatly cnhancc the ability of the aircrew to identify both targets at known Iocatiol~s as wcll as targets of opportunity. 5.3 NAVIGATION COMPUTATION AND ERROR MODELING

5.3.1 Coordinate S y s t e m s I n navigation computations, thrcc colnponct~tsof accclcration scr~scdby a n orthogonal triad of accclcro~nctcrshave to be transfornlcd onto a rcfcrencc coorditlntc systcm. Thc vclocity and position of thc vchiclc arc then computcd with rcspcct to

Sec. 5.3

Naufgatlon Computation and Error Modeling

Figure 5-4

(a) Earth-Fixed Coordinate (b) Geographic Coordinate

this reference coordinate system (RCS). Because the reference coordinate system affects both the computation time and accuracy, a suitably selected RCS can minimize the processing errors. Beginning with an earth-fixed set of coordinates (Figure 5-4), we can, by a series of two rotations first about the z-axis through the longitude angle and then about the negative y-axis through the latitude angle plus 90 deg, transform from the earth-fixed coordinates into a navigation coordinate system. In this case the navigation coordinate system is defined as one in which the x-axis is north, the yaxis is east, and the z-axis is down. The transformation is depicted graphically in Figure 5-5. Figure 5-6 derives the resulting direction cosine matrix. Such a transformation is known as an Euler angle transformation. A similar transformation will take us from the navigation coordinates into the aircraft bodyaxis coordinate system. The first rotation is about z (down) through the heading

Figure 5-5 mation

(a) [Z] Transformation (b)

[YI] Transformation (c) [Yz] Transfor-

Inertial Navigation

186

X Y Z

NAV

X

X Y

w s h sin h 0 -sinhwsh 0 0 0 1

-sin @ 0 - a s @ 0 1 0 ms @ 0 -sin cP

Z

Chap. 5

Z

FAR'IH

FARlH

- sin@ms h - sin@sin h - cosQ <=

-sin h cos@msh

msh 0 cos@sin h -sin@

Figure 5-6

where h = longituic (East) Q = latitude (North)

Earth-to-NAV Direction Cosine Matrix

angle +. The second rotation is about the new y (right-wing) axis through the pitch angle 0. The third and final rotation is about the new x (roll) axis through the roll angle 4. The direction cosine matrix for this transformation is derived in Figure 57

1.

In a gimballed system, the accelerometers would be maintained parallel to the navigation system axes. In a strapdown system, they would be parallel to the aircraft body axes (or in some fixed relationship to the aircraft axes) and their outputs have to be transformed into navigation axes in order to calculate velocity and position with respect to the earth. The inverse of the direction cosine matrix shown in Figure 5-7 could be used for the body-to-NAV transformation, but this involves computing nine quantities a t a high iteration rate. More often this transformation is done with quaternions, a four-parameter representation of the same thing. Conceptually quaternions describe a coordinate transformation by a single rotation. Three of the parameters are related to the direction cosines of the axis of rotation, and the fourth parameter is related to the magnitude of the rotation about that axis.

5.3.2 Position and Velocity Generation

A simple block diagram of space-fixed navigation computation is given in Figure

-

-

-

5-8. In the figure and xSare inertial position and velocity vectors, respectively, in a space-fixed coordinate system. Navigation and guidance of the Saturn launch vehicles using this navigation computation has been implemented onboard with negligible error contributions from the computations. For propellant economy, the

X Y Z

BODY

1 0 0 0 w s + sin+ 0 -sin+ a s +

w s 8 0 -sin0 0 1 0 sin0 0 w s 8

ws 0 sin v

cosems~,

%-- - cos 4, sin yl + sin + sin 0 cos 9

sinIV 0 - s i n v cosv 0 0 0 1 aSIV

+ - +

+

X

X Y 2

NAV

-sin 0

+

cos cos + sin sin 8 sin I) sin cos 0 + sin 4 sin Q + cos 4, sin 0 cos 9 sin cos 9 + ms 4 sin 0 sin I$ cos cos 8 Figure 5-7

+

NAV-to-Body Direction Cosine Matrix

Z

SAV

where Q = heading

0 = pitch = roll

Sec 5.3

Naulgatlon Computatlon and Error Modellng

MEASURED INERTIAL VELDCITY

A ~ I

- - INCREMENTAL

&AI

XI

-TOTAL

. x l -TOTAL

Ap i p

-

-

-

I:

;I

-

I

-

I

INERTIAL VELOCITY VECTOR

-

- ... is * (X*,Y.,Zs)

SPACE-FIXED VELOCITY VECTOR

INERTIAL VELOCITY VECTOR INERTIAL ACCELERATION VECTOR

-TOTAL

GRAVITATDNAL VELOCITY VECTOR

-TOTAL

GRAVITATIONAL ACCELERATION VECTOR

NOTE: SUBSCRIPT i DENOTES

INITIAL VALUE.

I

-

1

GRAVITY CotAFWfATuN SPACE-FIXED POSITION VECTOR

Figure 5-8

Iqavigation Computations (Frorn [Haeusserrnar~n, 19811, O 1981 AIAA)

iterative, path adaptive guidance mode was selected. The flight vehicle is guided along a minimum propellant trajectory to obtain the required end condition, such as the required velocity vector and vehicle position in orbit or the velocity vector for translunar injection. Figure 5-8 shows how the vehicle state is derived from the velocity, measured by the accelerometer integrator, in the form of position and velocity vectors in a space-fixed coordinate system which has its origin at the center of the earth. The diagram further shows the calculation of the acceleration in the space-fixed coordinate system as thrust-to-mass ratio, Flm. The iterative guidance mode for flight path adaptive guidance along a mini. mum propellant trajectory determines the necessary vehicle attitude corrections and the urouulsion termination command to obtain the reauired end conditions for the flight vehicle. The guidance functions for the thrust direction and for its termination were expressed by a series development of the reference attitude and of the propulsion cutoff time in terms of position and velocity components, time, and acceleration; the coefficients of the terms in the series were determined using calculus of variations and stored in the guidance computer. Forover-the-earth flight vehicles, the required position vector and velocity vector x are not expressed in spacefixed coordinates but in earth-fixed coordinates, as seen in Section 5.3.1 [Haeussermann, 19811.

.

i

x

Inertial Navigation

Chap. 5

5.3.3 Data Processing

A block diagram of the processing of a generalized navigation computer is given in Figure 5-9. In this figure, it is seen that: (1) the outputs A 8 c and A V A of the inertial sensors, that is, gyros and accelerometers, are incremental due to digital processing; (2) coordinate transformations are required in order to transfer the body coordinates to earth coordinates ,(see Section 5.3.1); and (3) filtering is sometimes required for data smoothing.

5.3.4 INS Error Analysis and Modeling Following Bar-Itzhack and Berman [1988], the outputs of an ideal INS are the exact position, velocity, and attitude of the host vehicle. The error analysis of INS or, in particular, terrestrial INS brings to light many interesting features typical of INS. It is no wonder that much of the literature on INS is concerned with its error analysis and error propagation. In analyzing INS errors, questions arise immediately concerning the behavior of the errors as a function of time, the ability to even measure all error components, the ability to estimate those errors that cannot be directly measured, and whether or not it is possible to control the errors. Rather than focusing, therefore, on new features of the INS, emphasis is placed on casting the known INS characteristics in terms of linear systems and control theory concepts and exploring hidden relationships between system parameters and states that affect system performance. In the examination of system observabiljty, a straightforward transformation is used into observable and unobservable subsystems that, in turn, exposes the states that hamper the estimation of INS errors during the initial alignment and calibration

.-

1) A0 Proossing: (Angle Inawmt)

Coordinate Earth Caord. -Thnsforma~on - - - +-lian~form~ti~n sq - - - - -

-We,

Navigation Procasing

Ve

-& 6 6

+e"E -h

Be 2) AV Ptocssing: (Vekxity Inaancnt) &Ordinatc -%>~O,-AO~-

Ffltering

V~ --+ Naviption

&

P'occssing

---4 ,

+,q: EulaAnglcs 0 ~ : Body Rate in Body Coord. 6 ~ :Body Rate in Earth Chord. r:

VB: Vclodty in Body Cmrd. VE: Vdm'ty in Earth Coad. Xt YB & Pcsitions in Earth M.

T m Lag

Figure 5-9

Processing of a Generalizrd Navigation Computer

+

xD YES%

+k +

--b

AE r

Sec. 5.3

Naulgatlon Computatlon and Error Modellng

189

phase o f operation. This approach was adopted successfully in the past by Kortum [I9761 who considered the problem of INS platform alignment, in which the measurements were the horizontal accelerometer outputs. A comparison of this approach with the classical one that is presented here, as well as the discussion of uniqueness and thc relationship between observability and quality of estimation, provide additional insight into the observability issue. The examination of INS as a unified system from a control theory point of view sheds more light on the systcm and contributes additional insight into the analysis of INS [Bar-Itzhack and Bcrman, 19881.

I N S error models. There arc two approaches to the derivation of INS error models. O n e of them is known as the perturbation (or trrre frame) approach, and the other is known as the psi-angle (or cotnputer frame) approach [Benson, 1973). When deriving the perturbation error model, the nomlnal nonlinear navigation equations are perturbed in the local-level north-pointing Cartesian coordinate system location of the INS. The psi-angle error that corresponds to the true geographic . model, on the other hand, is obtained when the nominal equations are perturbed in the local-level north-pointing coordinate system that corresponds to the geographic location indicated by the INS. The development of the first model can be found in Broxmeyer [I9641 and Britting [1971], whereas the development of the psi-angle model can be found in Pitman [I9621 and Leondes [1963]. Benson [I9751 showed that both models are equivalent and yield, therefore, identical results. The differential eauations that describe the error behavior of the INS are divided into equations describing the propagation of the translatory errors and equations describing the propagation o f the attitude errors. Bar-Itzhack and Berman [I9881 pointed out that most of the published works on INS errors adopt the psi-angle approach and use the velocity error version of the translatory error equation. This model is also used in the present analysis. It possesses the advantage that the translatory error is not coupled into the attitude error equations [Pitman, 19621. The physical attitude difference between the platform and the local-level north-pointing coordinate system is calculable using the position and attitude errors obtained from the solution of these INS error equations. The choices made yield a complex terrestrial INS error model expressed by the following equations [D'Appolito, 1971; Nash, 19721:

where v , r, and I) are, respectively, the velocity, position, and attitude error vectors; 0 is the earth-rate vector; w is the ang~~lar-rate vector of the true coordinate system V is the accelerometer-error vector; f is the specificwith respect to an inertial frame; force vector; Ag is the error in the computed gravity vector; p is the vector of the rate of rotation o f the true frame with respect to earth; and E is the gyro-drift vector.

Inertial Naulgatlon

190

Chap. 5

It can be shown [Leondes, 19631 that in the local north, east, and down coordinate system,

a=

R cos L [ 0 - R sin L

]

(5-2)

where L is the local latitude. The vector w is computed as follows:

o = n + p

(5-3)

where

.=[

icos L

.-i

1

(5-4)

- A sin L

and ); is the longitude rate. When Equations (5-1) are resolved in the true frame, nine scalar differential equations are obtained that can be put in a state-space model. If the expressions given in Equations (5-2) to (5-4) are used, the resulting statespace model is shown in Figure 5-10. The INS error propagation is of interest when the system is at rest. In most cases, INS alignment and calibration, which are a most essential phase of the operation of any INS, take place when the system is at rest. Also, the distinct behavior of INS errors is exposed when the system is a t rest. When the INS is at rest, Equations (5-1) reduce to:

V . ~

d -

0 0 0 0 -w2

dt

vD

4E -.

-

0 O 0 0

0 1 0 0 0 0 0 0 0 -20D 202 0 O O 0 0 0 0

0 1 0 2RD 0 -2nN O 0 0

0 0 1 0 2fiN

o O

o 0

0 0 0 0 -2 0 O -RD 0

0 0 0 A?

0 0

RD

o

-ON

-

0 0 0 0 0 0 0 R,W 0 -

where 0' = g / R , R N = cos L is the north component of earth rate, and R D = -a sin L is the down component of earth rate. The latter model can be writtcn as i ' = A f x + f'

(5-5b)

Sec. 5.3

Naulgatlon Computation and Error Modellng

191

where the definitions of x ' , f ', and A' arc obvious. Initial alignment and calibration arc usually performed when the INS is resting at a location whose geographic cor~ r~ 0 [Pitman, 19621. ordinates are known almost perfectly; that is, r N For this reason, the first three states of the state vector can be eliminated. The ~ V E , but v ~ , measured signals during initial alignment and calibration are v . and the vertical velocity error, is of no interest. Moreover, the vertical channel is only very weakly coupled with the horizontal channels [Hutchinson, 1968; Yavnai, 19801. Thus, v r , does not play any role in the development of the measured signals v ~ and V E ; hence. v ~ can also be eliminated from the state vector ofthe model that describes the INS error behavior during the initial alignment and calibration stage. In modern systems, a Kalman filter is used to perform the initial alignment and calibration; however, the model given in Equations (5-5) is not suitable for use in a Kalman filter since the accelerometer error and gyro-drift rates are not white-noise processes, as required for proper use in a Kalman filter. This obstacle is very easily overcome since, fortunately, the statistical characteristics of realistic accelerometer and gyro error data can be represented quite accurately by the outputs oflinear models driven by white-noise processes [Gelb, 1974; Heller, 19751. When this is done and the models are augmented with the basic INS error model, the driving force of the augmented model does not contain correlated noise, and the model can then be used in a Kalman filter. In the analysis here, it is assumed that the accelerometer errors are basically bias errors and that the gyro errors are basically constant drifts, that is,

- - -

When r . ~ ,YE, r D , and v ~ are removed from the state of the INS error model of Equations (5-5) and the resulting model is augmented with Equations (5-6). the following is obtained:

This model can be written as

where the definitions of x and A are obvious. The latter model describes the propagation of v~ and v~ when the INS is at rest and the accelerometer and gyro errors

Inertial Navigation

192

Chap. 5

sL and CLdenote, rcspcaively, the sine and &e

of L,and the subscripts N, E, and D denote the n o d , cast, and down mmponents, rcspsctively. Figure 5-10 State Equation of Terrestrial INS Error Model (From [Bar-ltzhatk et al., 19881, 0 1988 AIAA)

are constants. Although it seems that this model is not general enough, it turns out that the conclusions derived from it are quite useful. An estimate o f the state vector yields the estimation of $, as well as the estimate of the accelerometer and gyro constant errors. Obtaining the former is known as alignment, and obtaining the latter is known as calibration. An examination of the models given in Equations (5-5) and (5-7) from the point of view of a control analyst is presented in BarItzhack and Berman [1988].

5.4 GIMBALLED INERTIAL NAVIGATION SYSTEM (INS) In inertial platform gimbal mechanizations, the gyroscopes mounted on a stable element in a gimbal1 system measure angular rates, and the gimbal drive system uses the angular rate information in a feedback manner to rotate the gimbals and null the angular motion sensed by the gyroscopes. In this manner, the gyroscopes and accelerometers mounted on the stable element are inertially stabilized from the vehicle motion and the stable member physically represents an inertial reference frame. Alternatively, control torques to the gyros can cause them to precess a t precisely the same rate as the combination ofthe earth's rate of rotation plus the angular ratc of thc vehicle's position with respect to the earth. In this way two of the accelerometers can be maintained in a locally level plane, the third being parallel to the vertical axis a t all times. Most aircraft and marine gimballed inertial navigation systems are mechanized in this manner.

5.4.1 Gimbal Mechanism In aircraft applications, a four-gimbal mechanism as shown in Figure 5-1 1 for the platform unit of a gimballed INS is required to avoid gimbal lock. In surface vchicle applications, a three-gimbal mechanism for the platform unit of a gimballed INS suffices as shown in Figure 5-12.

Sec 5.4

Glmbalied Inertlal Navigation System (INS)

Outer

Accelerometers)

Figure 5-11

Four-Gimbal Mrchanisn~

5.4.2 Inertial Sensors on the Stable Platform Inertial sensors on the stable platform are gyros and accelerometers. If SDF (singledegree-of-freedom) gyros are to be used, three gyros (mutually orthogonal) are required for the north, east, and vertical (NEV) coordinates. If 2DF (two-degreeof-freedom) gyros are to be used, two gyros are required and installed. In Figure 5-13 three gyrolaccelerometer pairs are installed so as to measure the angular rate and acceleration along each of three axes.

5.4.3 Platform Alignment Modes There are two alignment modes, coarse alignment and fine alignment. In the coarse alignment mode, accelerometers are used for leveling control of the platform, while

Figure 5-12

Three-Gimbal Mechanism

Inertial Navigation

Chap. 5

Figure 5-13 Inertial Sensors on the Stable Platform: Three-GyroIAcceleromet~ Pairs

the gyros sense the direction of zero earth rate (east-west). In the fine alignment mode, A Z gyro slew gives a more accurate aiignment of the platform. During the fine alignment or gyro compass phase, the objective is to keep the pitch, roll, and AZ gimbal aligned with true north and the horizontal. Any AZ misalignment causes the east gyro to sense a component of the eaith rate and thus tilts the platforn~about the east-axis. The north accelerometer then generates a pseudo velocity due to the platform tilting about the east-axis. The pseudo velocity is then used to drive a r gyro torquer and its associated gimbal servo to achieve the proper alignment.

5.4.4 Navigation Mode Gimbal systems provide a good dynamic environment for inertial instruments, particularly in vchidcs that are to perform at high maneuver Icvels, since the gimbals from the rotational environment. In fact, the state of the art is such isolate the gvros ", that navigation performance of gimbal systcms can approach alnlost error-free instrument operation, to the point where uncertainties in the knowledge ofthe gravity field become the dominant sources of navigational error. A simplified diagram to show the relationship between the platform and the navigation computer is given in Figure 5-14. The two mechanizations for the platform attitude reference are the space stable and local vertical. The platform space-stable inertial mechanization is given in Figure 5-l5a. Figure 5-l5b illustrates the case in which the geographicalnorth-pointing local-level navigation frame is the instrumented stable member frame for a gimbal system. The local~levelframe is the most common for gimbal systems. The local-vertical frame is usuallv a form of a .,fiec-arimirth reference frame in which no attempt is made in a gimbal system to point one of the stable-member axes north, as illustrated in Figure 5-15c. This mechanization eliminates difficulties in flying over the earth's poles. T w o of the axes are maintained level but are free to rotate in azimuth aboutthe third axis, which is maintained along the local vertical [Schmidt, 19871. Figure 5-16 shows a block diagram of the x channel in a navigation computer. The operations of a navigation computer include: (1) The correction rate o, is used

Sec. 5.4

Gimballed Inertial Navigation System (INS)

I

Acaltr~metm 1

-.

Navigation

Gym 'bquing Rates mpuution P l a t f m Heading

&Attitude L - - - - J

Figure 5-14

R e l ~ r ~ o n s h bcr\vecn lp rhe Plarfi)rni a t ~ t~l i ~N L Y I ~ L ~CIoO~ Ii i~p i ~ t e r

to torque the y gyro, and thus causes gyro precession. The gyro output is then used to drive the platform servo so as to maintain the x-axis horizontal. (2) A s , Ay, and Az are the measured accelerations from accelerometers in the navigation coordinates. (3) The correction at the Ax summing point is composed of coriolis, centrifugal, and transport accelerations.

5.4.5 System Internal and External Interfaces It~rc'rtrr~l Itlterfilce: The gimballed INS typically consists of eight main functional

blocks. For each function block, the interface infor~nationis given as shown in Figure 5-17. E.urrnrnl 111teuf;lce:The I10 (inputloutput) interface between the gimballed INS and vehicles is given as follows: (1) Inputs-In order to permit operation in aided-inertial configurations, a standard INS accepts the following digital inputs in M U X serial format (MIL-STD-1553): position update (latitude and longitude), velocity update (velocities in INS coordinates), angular update (angles about INS axes), and gyro torquing update (torquing rate to INS gyro axes); and (2) Outputs-The system provides pitch, roll, and heading in both analog (synchro) and digital form. In addition, the following outputs are provided on a serial M U X channel (MIL-STD-1533): present position (latitude, longitude, altitude), vehicle attitude (pitch, roll, true and magnetic heading), vehicle velocity (horizontal and vertical), and steering information (track angle error).

5.4.6 Navigation Mechanization and Error Model

Mechanization.

In the example of Figure 5-18 Uoos and Krogmann, 19811, the diagram consists of two portions: I M U and navigation computer. O n the I M U platform, there are three SDF gyros and three accelerometers with their sensitive axes pointing in (N,E,V) coordinates. In the navigation computer, the vertical channel is aided by the baro-height. The navigation computer includes compensations for earth rate, vehicle rate with respect to the earth, coriolis acceleration, and gravity. Figure 5-19 depicts Doppler aiding where the aiding velocity reference V.YDcomes from a Doppler radar [Streeter; 19721. With the errob e,, the original

U kcelermeter trtod m o n inertally stabilized platform

!

Pccekrometermeasured spccif ic force in lnertml CQardiwtes

t

Alt~tudeInformation

Init~olc o n d ~ t m s

I

Posttlon ~n lnert~olcwrdtnotes

Nav~got~on computer

( L) * Lotitude Lonpitude ( I I

t

Alt~tude( h )

itotiliied platform

(B)

Indial Altitude Wndtt~otn mfamoton Velocity lv7 ws~tlon(L,I,&

Figure 5-15 (a) I'latforn~ Space-Stable Inertial Mcchanization (b) I'latform Geographic Local-Vertical Mechanization (c) Platform Free-Azimuth Mechanization (Rrprinfed with prnnissio~tjotn (Srh~nidf,Sysfons fi C o ~ ~ f r61cytlopedio), ol Copyright 11987), Per.qafnon Press PLC)

Sec. 5.4

GImballed lnertlal Navlgatlon System (INS)

Figure 5-16 x Channel Mechanization o i a Navigation Computer

undamped loop can be damped as follows: (1) With free inertia ( k = 0, n o Doppler glR = 0 with natural frequency w, = aiding), the characteristic equation is "s (glR)'12 = 2 n l T where the period T is the so-called Schuler period; and (2) With Doppler aiding, the characteristic equation is s' + ks + glR = 0 with damping ratio 5 = kl(2w,,). Thus, with Doppler aiding, the original undamped system will be damped with damping ratio 5. With altimeter nidirlg (see Figure 3-20), the altitude reference h~ is given by an altimeter and thus generates the flight height error E,, to damp the loop. In Figure 5-20, thc feedback gains K I and K 2 can be selected so as to nicely damp the vertical 100~. *15V DC

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Error models. In the Doppler-aided INS error model shown in Figure 521, there arc nine states in the loop, five statcs in the A.y loop and four statcs in the A loop (not shown). For altitude aiding, the error model is depicted in Figure 520. A self-contained navigation system without aiding has the error model shown in Figure 5-22a. Figure 5-22b illustrates a block diagram of a generalized mechanization for a local-level free-azimuth INS. Performance trade-off analysis. The errors 6x in any inertial system mechanization (for example, nine states as in Figure 5-21) can be cast in the usual state-variable form 6x = F6x + Gq where the coefficient matrices are functions of the system mechanization and vehicle dynamics [Schmidt, 19781. This allows the standard covariance matrix propagation equations to be used in predicting inertial system performance. If external aids (see Section 5.7) are to be considered, the Kalman filter update equations can be applied to study requirements on aiding accuracy, frequency of updates and so on, to predict accuracies in the estimates 6 i that could be achieved in the actual implementation [Schmidt, 19871. 5.5 STRAPDOWN INERTW NAVIGATION SYSTEM (INS)

A generalized strapdown INS consists of IMU, electronic unit (EU), and computer unit (CU). The IMU is a sensor package with a gyro triad and an accelerometer triad without a gimbal device. This is the main difference between strapdown navigation systems and gimbal navigation systems. Since there are no gimbals to keep the sensor package in a prescribed orientation, there is an extraordinary computational load imposed on the CU. The basic concepts of the strapdown inertial navigation are that the body-to-NAV direction cosine matrix is computed using strapdown body-mounted gyro outputs. Then the horizontal and vertical accelerations are computed analytically using this direction cosine matrix to transform accelerometer outputs into local-level navigation coordinates.

5.5.1 Generalized Strapdown Mechanization A simplified block diagram showing the IMU and navigation computer is given in Figure 5-23. An overview of the strapdown mechanization is given in Figure 524. From this figure, it is seen that the acceleration Ab (in body coordinates) is

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Strapdown lnertlal Naulgatlon System (INS)

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transformed onto local-level axes to compute the NAV axes acceleration components A,, A,, and A,. The internal structure of the strapdown navigation system is given in Figure 5-23. Note that in this figure, the angle and velocity output of the inertial sensor components are small increments to be integrated in the body attitude and velocity computations. Thc small increments in A 0 and A V are processed as shown in Figure 5-26. Figure 5-27 Uoos and Krogmann, 19811 is another view of the same system depicting the gyros and accelerometers. A detailed block diagram of a rate gyro strapdown inertial navigation system is presented in Boggess [1986].

5.5.2 Strapdown Navigation Computer The strapdown navigation computer may transform the body-axis sensor data into either inertial coordinates or into geographic local-vertical coordinates. The strapdown system computing in inertial coordinates is given in Figure 5-28a. The strapdown system computing in geographic local-vertical coordinates is given in Figure 3-28b. Figure 5-29 shows a generalized block diagram for strapdown inertial computations. This diagram shows that the computations are partitioned into two parts: a high-speed section that transforms the body-axis measurements of acceleration and attitude rates into navigation axes, and a slower-speed section that performs the navigation routines just as they would be computed in a gimballed system.

5.5.3 Strapdown Computer Requirements Typical computer requirements for a strapdown NAV computer call for a 16-bit computer whose properties include a speed of 250-350 KOPS (the computer throughput

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Inertial Navigation

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IMUICOMPUTER ELECTRONICS

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may rcach 3.0 MIPS to support a wide bandwidth autopilot for high dynamic applications), a memory sizc of 12-15 K, 1 p S for addition, 8 p S for multiplication, and 10 FS for division. Thesc con~putationrequircmcnts are for 1 NMIIhr navigation systclns. Thc eri~~irotirtictitalreqrrirctr~etitsinclude an accclcration of 13-30 g, 500°-10000/scc maximum angular ratcs, a 100 g (20 MS) shock, a MIL-E-5400 (curve IV) vibration, and a MIL-E-5400 (Class 2) thcrmal.

5.5.4 Strapdown Error Model and Analysis

Error sources.

l3asically, thc strapdown crror sourccs can bc grouped illto thc two following catcgorics.

Instrument Errors. Thcrc arc significant diffcrcnccs in the navigation performances bctwccn gimbal and strapdown systclns bccausc o f calibration limitations, sensor inaccuracics, colnputational errors, and sensor crror propagatioll cf-

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Inertial Navigation

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fects. O f these, the instrument errors can be caused by the following error sources [Schmidt, 19871. (1) Calibration limitations on strapdown systems arise because the inertial sensors, which are rigidly attached to the vehicle, cannot be arbitrarily oriented at different angles to provide known inputs from the earth's gravity and angular velocity vectors for a stationary vehicle. (2a) In single-degrce-of-freedom gyros, a gyro drift, called anisoinertia-induced drift, will be prescllt whenever the gyro experiences accelerations such as vibration concurrently about its spin and input axes. To eliminate such an error source requires complicated compensation, so the system designer must understand the interplay between vehicle environment, sensor BODY

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Sec 5.5

Strapdown Inertial Navigation System (INS)

Figure 5-27 Gyro and Accelerometer Components of Strapdown Navigation Systcrn (Frorrr Uooi nrrd Kroyrnarm, 19811 wirl~purnrrisio~rjorrr AGARD)

design, and the resulting navigation system performance when selecting a particular type of gyroscope for a strapdown system. (2b) Gyro-torquing scale factor error, particularly asymmetrical scale factor, causes errors in a vibratory environment. (2c) Output axis rotation error causes the strapdown gyro to have a drift error proportional to the angular acceleration along its output axis. This error source usually dictates how the gyros are mounted in the vehicle; the output axes of the gyros can, for example, be placed along the yaw andlor pitch axes in an aircraft application. (3) Alignment errors incurred by initializing the transformation matrix using the accelerometers for leveling information result in tilt errors which produce navigation errors. (4) In strapdown navigation systems that have external position or velocityaiding sensors one can calibrate accelerometer biases in flight, since they produce observable errors in a strapdown system, but not in a gimballed system. Computational Errors. Computation-induced errors in strapdown systems refer to those navigation errors introduced by the incorrect transformation of the body-measured specific forces to the computational frame. The factors causing the incorrect transformation matrix include: the algorithm used to generate the transformation matrix, the speed at which the transformation occurs, the roundoff due to computer word length used, quantization of gyro outputs, truncation due to integration scheme, and renormalization of the quaternion vector. They all play a part in contributing to computational errors. Since this type of error also depends on the vehicle's dynamic environment, most designers evaluate it by computer simulation [Savage, 1984(a)].

-

Inertial Naulgation

Chap. 3

s p c ~ f ~cone c m

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Error model.

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5.5.5 Operation Flow Diagram The functional flow of strapdown navigation systems is dividcd into five blocks as shown in Figure 5-31.

Sec. 5.6

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5.6 COMPARISON AND ANALYSIS OF GIMBAL VERSUS STRAPDOWN

5.6.1 Feature Comparison Not only is a gimbal system inherently more accurate than a strapdown system, but calibration is also easier with a gimbal system. In addition, there is less computation involved and less demand placed on the gyro. O n the other hand, a strapdown system is smaller in size and more reliable than a gimbal system, eliminating AID conversion of synchro signals as well as electromechanical components. Attitude and heading data in a strapdown system are in particular most accurate. Furthermore, angular rates are inherently available in body-referenced axes, making strapdown INS suitable as a flight control sensor. Multiple redundant systems of strapdown systems are required when used for flight control because in that role they become critical to flight safety.

5.6.2 Navigation Errors Comparison This section presents a comparison of navigation errors based on a detailed analysis appearing in Schmidt [1987]. While the strapdown system errors listed in the discussion that follows appear in Section 5.5.4 (Instrument Errors), they are again mentioned here for the purpose. of a direct comparison between strapdown and

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gimbal systems. There are significant differences in the navigation performance between gimbal and strapdown systems because of calibration limitations, sensor inaccuracies, computational errors, and sensor error propagation effects. The inability to arbitarily orient the rigidly attached inertial sensors of strapdown systems to provide known inputs from the earth's gravity and angular velocity vectors for a stationary vehicle results in calibration limitations. Conversely, a gimbal system is mechanized so that the inertial component outputs at different angular orientations can be compared with the known input components of gravity or earth rate and the sensor errors calibrated in a series of test positions that expose the error sources. Since only a few of the strapdown sensor errors (compared with a gimbal system) can be calibrated, the system designer must depend on stability of the instruments between removals of the system from the vehicle for calibration. The errors that cannot be calibrated in a strapdown system propagate into navigation errors when the system begins to navigate. Sensor inaccuracies that arise from motion of the vehicle are even more difficult to calibrate in a strapdown system. The dynamic response characteristics of the inertial sensor under consideration for use in the navigator must be well understood and the limitations on navigation performance must be evaluated early in the system design. Gyro-torquing scale factor errors such as asymmetrical scale factor errors in vibratory environments represent another significant source of strapdown system

212

Inertial Navigation

Chap. 5

sensor error. In a local-vertical gimbal system, gyro scale factor errors usually have a small effect on navigation system performance, since the torquing angular rates are small, being nearly equivalent to earth rate plus the angular velocity of the vehicle over the earth's surface. In a strapdown system, the gyros must, in addition, be torqued for the attitude rate of the vehicle; consequently, gyro scale-factor-induced navigation errors can be quite large. Misalignments of gyro input axes in strapdown systems are also important, for similar reasons. Another major sensor error source in strapdown single-degree-of-freedom gyros is output axis inertia. In this case, the strapdown gyro has a drift error proportional to the angular acceleration about its output axis. Computation-induced errors in strapdown systems are those navigation errors introduced by the incorrect transformation of the body-measured specific forces to the computational frame. The transformation algorithm, its iteration rate, the computer word length and the inertial sensor quantization all play a part in determiningcomputational errors. In addition, this type of error depends on the vehicle's dynamic environment [Savage, 1984(b)]. Sensor-induced navigation errors propagate differently in a strapdown system. As noted in Schmidt [1987], in a gimbal system during self-alignment, the accelerometer outputs are used to level the stable member and the misleveling of the stable member is due primarily to accelerometer bias. When the system starts to navigate, no significant position or velocity errors occur due to this error source, since the platform tilt-error contribution exactly cancels the accelerometer bias-error contribution. In a strapdown system, where self-alignment occurs by initializing the transformation matrix using the accelerometers for level information, the identical type of tilt error results. However, when the system starts to navigate and the vehicle heading changes, the orientation of the inertial sensors changes with respect to the alignment (or navigation) axcs and the net rcsult is to introduce a step of acceleration error, equal to the value of the accelerometer bias if the heading changes by 90 dcg. Subsequently, velocity and position errors arise both from accelerometer bias and from the initial tilt error that bias introduced during alignment. These error effects in strapdown navigation systems that have external position or velocity-aiding sensors means that one can calibrate accelerometer bias in flight, since it produces an observable error in a strapdown system. Detailed computerbased evaluation of performance, including algorithm errors, in a strapdown systcm design requires the use of a whole-number type of simulator rather than a statistical (or covariance) approach. This simulator can also evaluate the effects of nonlinear sensor errors and quantization errors. Such computer programs are used widcly in the inertial systcnl manufacturing industry. Generally, designers will use the rule of thumb that for gimbal systems 0.0l0h-' of gyro drift will produce a position error growth rate of 1.85 km h-' and a velocity error of approximately 1 ms-' for a terrestrial navigation system; for a strapdown system, the errors would be 20 percent to 50 percent worse. However, as mentioned in Schmidt [1987], thc difference in errors can be small, depending on the vchiclc environ~nentand thc use of other aiding sensors (Doppler velocity, position-fixing radar, satcllitc navigatiol1

Sec. 5.7

External Navlgatfon AIds

213

receivers, and so on). Consequently, the process of preliminary system selection involves the choice not only of system conlponcnts and mechanization but also of aiding devices and frequency of updates. These trade-offs are usually conducted with a general-purpose covariance analysis program [Schmidt, 19781.

6.7 EXTERPUU, NAVIGATION AIDS An unaided INS has the advantage of possessing all-wcather, all-attitude, and nonradiating characteristics; high-dynamic response; and self-contained navigation. Disadvantages include unbounded position errors and oscillatory velocity errors, absence of in-flight alignment, long reaction from cold start, and poor recovery from power interruption. Among the disadvantages, unbounded position error and oscillatory velocity error are of most concern to the designer. Thus, it is often necessary for navigation systems to be aided by some other external navigation sensors.

5.7.1 Aided Inertial Navigation Mechanization Referring to the nine-state error propagation equation shown in Figure 5-10, for example, there are generally three types of aiding available to inertial navigation systems. These are position aiding, velocity aiding, and attitude aiding. Each of these augmentations directly observe one or more of the error states in the INS. In . addition, various combinations of these three aiding techniques may be used. GPS, T E R C O M , LORAN, and T A C A N are position-aiding devices. Doppler radar and GPS can both provide velocity updates. Another very important velocity update is available on stationary vehicles, namely a zero-velocity update. Stellar observations made by a telescope mounted directly on the stable element of an INS provide attitude updates, actually psi-angle updates. Other examples of attitude updating are stored heading alignment and attitude matching transfer alignment from a master INS to a slave INS.

Effect of augmented inertial mechanization Error Improvement. From Figure 5-32, it is seen that the augmented inertial mechanization may keep acceleration, velocity, and position errors in a fixed region. Figures 5-32a through 5-32c represent self-contained navigation systems. Among them, the first figure represents the worst case involving free-inertial navigation, and the last figure represents the best case involving position navigation. However, augmented inertial mechanization as shown in Figure 5-32d, provides acceleration, velocity, and position errors which are much better than those of the three selfcontained systems. Cost and Performance Improvement. From Figure 5-33, it is seen that, in the region of drift rate O.OOlO/hr,and 0.02"/hr, the position- or velocity-aided inertial

Inertial Navigation

214

Chap. 5

TIME --+ AUGMENTED INERTIAL NAVIGATION

Figure 5-32

Augmented Inertial Rationale (From [Martin, 19861 with permission from Litton)

REQUl RES OCl TY

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Sec. 5.7

btemal Naulgation Aids

215

mechanization has a lower cost than the unaided one. This is not true for stellaraided systems which cost considerably more than an unaided INS.

Advantage of Augmented Position-Velocity Inertial. The advantages of augmented position-velocity inertial mechanization include: multibackup modes; protection against spoofing o r position; rate-aid for position sensor; good dynamic response; in-flight startup and alignment; and bounded position, velocity, and heading errors. When an augmented inertial mechanization is no longer self-contained. that is, it requires external cooperation, its dominant disadvantage is its susceptibility to RF noise and propagation errors. 5.7.2 Global Positioning System (GPS) Among the current generation of positioning and navigation aids, it is perhaps GPS coupled with newer and better-designed field-qualified receivers that provide unprecedented positioning and navigation capability.

Requirements. The objective of a GPS is to develop a worldwide precision navigation system based on satellites so that it can be applied to all classes of military and possibly commercial operations with an accuracy of 25 ft in three dimensions. In conjunction with other systems, a GPS, with its precise position, velocity, and time is very versatile and is applicable to a variety of air, land, and sea services and operations. GPS will be an integral part of future navigation systems. The applications of a GPS include: time transfer; range instrumentation; geodesy and survey; antisubmarine warfare; launch vehicle guidance improvement; mine and sensor delivery; missile inertial system updating; field artillery and shore bombardment; close support and common grid coordination; riverine and small-craft operations; land vehicle navigation; enroute nautical and aeronautical navigation; VTOL, STOL, and helicopter takeoff, landing and cruise operations; and photomapping, phototargeting coordinate bombing.

GPS overview. GPS is a space-based radionavigation system which is functionally divided into the space, control, and user segments. The space segment consists of a constellation of navigation signal time and range (NAVSTAR) satellites. A full constellation will have 21 satellites plus 3 active spares. They will be uniformly distributed in 6 orbit planes, providing 4 to 7 visible satellites at any time anywhere on earth. The planes are inclined 55 deg with respect to the equatorial plane. The orbital altitude is 10,890 nautical mi. NAVSTAR measures range to a set of four satellites in 12-hour orbits, by timing the arrival of radio signals transmitted from the satellites at precisely known times. Theoretically, a minimum of only three satellites would allow a fix to be obtained but since three satellites may not always be in suitable positions, and because timing errors in the receiving system have to be eliminated, a fourth satellite is necessary [Barnes, 19871. Each satellite trarismits specially coded signals that allow individual satellites to be distinguished, and the

216

Inertial Navigation

Chap. 5

range and rate of range change to the user to be measured. The signals are pseudorandom binary noise codes (PRN). T w o different codes are transmitted in phase cluadrature, providing a standard positioning service (SPS) and a precise positioning service (PPS). A low-rate data message is also transmitted. T h e control segment has five monitor stations that track all satellites in view of their antennas. Data are transmitted to a master control station where processing takes place to determine orbital and clock modeling parameters for each satellite. The information is then uploaded to the satellites by one of three upload stations. The satellites incorporate this information into the data message. The user segment consists of equipment designed to receive and process the satellite signals. The unique codes transmitted by each satellite allow the use of common RF carrier frequencies throughout the constellation, a process known as code division multiplexing. Measurements from four satellites are required in the general case. The United States Department of Defense has developed two classes of receiver, continuous tracking and sequential tracking. The continuous tracking receiver provides a dedicated channel for each satellite being tracked and a fifth channel that performs ancillary functions (for example, ionosphere correction, interchannel bias measurement). The sequential receiver has one or two channels, for low- and medium-dynamic applications, respectively. In these receivers the channels are time shared among satellites and housekeeping chores [McGowan, 19871.

Measurement principles. The following discussions on measurement principles are based on McGowan [I9871 as follows. Range Measurement. The PRN code in each satellite is reset to its initial state precisely at midnight, Saturday (Universal Coordinated Time). The codes have sufficient period so that no state will be repeated bctween the weekly resets. The receivers generate a replica of the code and phaselock on it using correlation tcchniques. The receivcd code phase provides a dircct indication of the time of transmission of the code and allows the transit time of the signal to be calculated against a local clock. Scaling this measurement by the speed of light (c) determines the range betwecn the user and satellite antennas. This quantity is known as pseudo-range (PR) because it contains errors. T w o dominant error sources are the error in the local clock (its quality is at least an order of magnitude worse than that of the satellite clock) and atmospheric dclays suffcred by the signal. T i m e Measurement. The mcasurcn~cntof transit time by the rcceivcr contains thc bias bctwecn the satellite and uscr clocks. Since it affccts all mcasurcmcnts cqually, it can bc cstimatcd as part of thc solution. Whcrc three 111easurcmcnts arc nccdcd to dctcrminc thrcc position coordinatcs, a fourth allows clock bias to bc determined. Range Rate Measurement. During pcriods of sufficient rcccived signal quality (CINo > 29 dB), thc rcccivcr can phaselock onto the carricr. Thc Doppler shift observed 1s uscd to dctcrm~ncthe linc-of-sight (LOS) velocity betwccn the satcllltc and uscr. This mcasurcn~cntis implcmcnted as an integrated Doppler known as the

Sec. 5.7

External Naulgatlon AIds

217

pseudo-delta-range (I'DR), which is a measure o f the range change during the integration interval.

Performance.

Table 5-1 lists the error budget for the PR measurement, assuming that the dual-frequency ionospheric co~npensationtechnique is used. As shown, a measurement error of 5 m is predicted. The test results indicate that the high-frequency fluctuation of the error is less than 1 m. The error was dominated by a bias component that ranged from 7-10 m [McGowan, 19871.

INS updating.

Figure 5-34 illustrates a block diagram of a typical implementation using the GPS system as an update aid for an inertial system. Note that the filter estimates receiver errors as well as inertial navigation errors. However, realistic modeling of all error sources usually results in a state vector dimension that prohibits full implementation in the vehicle computer. Trade-offs to determine the appropriate suboptimal filter to implement are then required [Schmidt, 1976; Setterlund, 19861. From these performance trade-off analyses, a set of system performance specifications can be developed prior to implementation and integration [Schmidt, 19871.

5.7.3 Tactical Air Navigation (TACAN) T A C A N is a two-dimensional navigation system used for piloted aircraft and is based on generating azimuth and distance to a fixed ground station for the aircraft. The mechanization of TACAN centers around an airborne-interrogator and ground transponder. The flight vehicle is provided with range and bearing lines of position TABLE 5-1 GPS PSEUDORANGE ERROR BUDGET Segment Space Segment

Control Segment

User Segment

Error mechanism Clock Stability Track Perturbations Other Segment RSS Ephermeris Prediction and Modeling Other Segment RSS Ionospheric Delay Tropospheric Delay Receiver Noise/Resolution Multipath Interference Other Segment RSS System RSS

From [McGowan, 19871 with permission from AGARD

System budget (1) meters

Inertial Navlgatlon

Chap. 3

Uncorrected

nOviVion vorlo ICS

~ovigationcorrections

pseudoronge and pseudo delta range

Figure 5-34 Aided Inertial-GPS Mechanization (Reprinted with permission jot11 (Schmidt, Systems fi Control Enrydopedio), Copyrishr (1987), Pergemort Press P L C )

by a ground transponder which uses a nearly pulse type of transmission for range1 bearing loci measurements. It operates on a frequency in the L-band (1.0-GHz) range, and has an operational range of 150 N M (LOS) at an altitude of 15,000 ft. It has an accuracy that is on the order of 200-1,500 ft in range/(bounded) radial position error and 0.5 deg-1.0 deg (100 ft/NM) in bearing/(bounded) angular position error. The resolution is approximately 100 ft in range and 0.5 deg in heading. It is the bandwidth that limits the range resolution to 100 ft, while the L-band frequency and the beamwidth of the transponder antenna limits the bearing resolution. In Figure 5-35a, it is seen that the angular position error is 10,000 ft at 100 NM, corresponding to the bearing accuracy mentioned earlier. O n e way to overcome the large angular position error is to go to a dual-TACAN mechanization. By implementing a ~ U ~ ~ - T A Csystem A N or range-range mechanization, the position error is reduced to 500 ft (CEP), as illustrated in Figure 5-35b [Martin, 19861.

TACAN advantages and limitations. Advantages of T A C A N include excellent enroutelhoming navigation aid, and accurate range and bearing over short distances. Furthermore, it is amenable to fixed and mobile (that is, ships) ground stations. One of the limitations of TACAN is that it is not a navigation system and thus requires a heading device. In addition, it is not sclf-contained because it requires a radiating ground station. At long ranges, its cross-track errors are large. There are also limitations on its LOS, which require aircraft to fly high in mountainous regions and for long-range measurements. There is a disadvantage in the fact that radiation from the aircraft can be detected during military OPS. T A C A N is also subject to natural and enemy environments as well as to ground maintenance ell-

Sez. 5.7

External Navigation Aids

218

OPERATIONAL RANGE

150 NM (LO9 AlRCRAFl AT 15,000 FT ALT.

RADIAL POSITION ERROR (BOUNDED)

200

ANGULAR POSITION ERROR (BOUNDED)

LO0 ( - 100 FTINM)

- 1500 FT

I

TACAN STATION

ACAN STATION R

Figure 5-35 (a) TACAN Characteristics (b) Dual TACAN Capability Reduces Large TACAN Cross Track Position Errors (From [Martin, 19861 with permission from Litton)

220

Inertial Navigation

Chap. 5

vironment, Due to the earth's curvature, as the range to the T A C A N station in, creases, so must the altitude of the flight vehicle increase. A relationship between the TACAN range and the altitude of the flight vehicle is given in Figure 5-36.

5.7.4 Long-Range Navigation (LORAN) LORAN is a time-difference of arrival system that requires one master and at least two slave ground stations that transmit radiation to the flight vehicle. Like TACAN it is a two-dimensional system. The transmission of the ground station is approximately in the form of 8 or 16 pulsed (groups) at 10 KHz PRF, entailing measurements of hyperbolic loci of position provided by pulse envelope and phase difference. It has an operating frequency of nearly 100 KHz and an accuracy that is on the order of 50-500 ft or more, contingent upon the geometric dilution of precision (GDOP). The resolution of the transmission is approximately 0.01 psec X GDOP.

LOBAR characteristics. LORAN characteristics are depicted in Figure 5-37a, which include an operational range of 600-1,000 N M on land and 1,0001,500 N M over water and a radial position error of approximately 300-500 ft in the primary zone. The radial position error is the region of interception of the M-S#l line of position (LOP) (see Figure 5-37b), and the M-S#2 LOP as given in Figure 5-37a. M-S#1 and M-S#2 LOP are hyperbolic curves between M-S#l and M-S#2, respectively. The hyperbolic LOP is constructed based on the receiving time (that is, range) difference measurements between the master station and the slave station. LORAN advantages and limitations. Advantages of LORAN include simplicity and reliability of airborne units, accurate medium-range navigation aid, and nonradiation of passive receiver. There are some limitations and disadvantages associated with LORAN, one of which is that radiating ground stations are required for LORAN. In addition. LORAN, like TACAN, is subiect to natural and enemy " environments as well as to ground maintenance environments. Also, like TACAN, LORAN is not a full navigation system and needs to be augmented with heading, velocity, and altitude. Its performance is reduced by the GDOP. The long smoothing circuits of LORAN require low dynamics during measurement or rate aid to maintain accuracy/track.

5.7.5 Terrain Contour Matching (TERCOM) TERCOM is a form of correlation guidance based on a comparison between the measured and the prestored features ofthe profile of the ground (terrain) over which a missile or aircraft is flying. Generally, terrain height forms the basis of this comparison. Obtaining the reference data requires prior measurement of the ground contours of interest. This type of guidance is used for updating a midcourse inertial guidance system on a periodic basis, and has been applicd to the guidance of cruise

Inertial Naulgatlon

222

Chap. 5

missiles, which usually fly at subsonic speeds and fairly constant altitude [Heaston and Smoots, 19831. A TERCOM system uses an airborne altimeter and a data processor to correlate the measured terrain contours with the prestored contours to obtain the best estimate of position. The transmission characteristics of the airborne altimeter include an operating frequency of approximately 4.4 GHz (incidental to operation) and a transmission type that is pulsed or CW. As the missile flies, the radar altimeter first measures the variations in the ground's ~rofile.These measured variations are then digitized and processed for input to a correlator for comparison with stored data. High-density digital storage permits large quantities of data to be stored, and modern microprocessors allow the comparison to be done sufficiently and quickly [Barnes, 19871. Through this process, the missile can determine its position and correct any errors that have developed since the previous update. Accuracy is 100-1,000 ft and resolution is 100-200 ft. Figure 5-38 illustrates the basic concept employed by TERCOM. The terminal guidance stage may bebased on the final TERCOM update and a preprogrammed course relying on the inertial system, or a separate terminal homing seeker may be employed that can recognize the target and provide the final guidance commands [Heaston and Smoots, 19831. The block diagram shown in Figure 5-39 shows the mechanization of the TERCOM system. OPERATIONAL RANGE RADIAL POSITION ERROR SLAVE I1, STAT ION

@XI- 1OOO N M (LAND) lOCKl - 1500 NM (WATER) -300 - 500 FT (PRIMARY ZONE)

SLAVE X2 STATION

MASTER STAT l ON (A)

Figure 5-37 (a) LORAN Characteristics (b) LORAN Lines o f Position ( a ) j o m [Martin, 19861 with permission j o m Liftot~( b ) courfesy 0jF.U'. Hardy, 1986)

Sec. 5.7

external Navlgatlon Alds

(B)

Figure 5-37

(Continued)

TERCOM advantages and limitations.

The advantages of TERCOM include total automation of its operation, accurate position location with a good heading device, and self-contained operation requiring no external support. The

Inertial Navigation

224

Cruise

Cruise Missile

Terrain

Chap. 5

Elevation Map Stored In Onboard Computer

Terrain Mapped Previously

Guidance

Update

T A R G E T AREA

OPERATING RANGE RADIAL POSITION ERROR

- UNLIMITED (COMPUTfR - ZW - % FEnI

CAPACITY1

Figure 5-38 (a) Terrain Corltour Matching (b) T E R C O M ~haracteristics ((aJjom /Heaston and Smoors, 198.71 lr'irh prrmissiot~j o m GACIAC (b)jottr /hlnr.ritlt 19861 with prrmission j o m Lirrorr)

Sez. 5.7

External Navigation Aids

Data Cornlation

4

Recommended Flight Path

I

h

Figure 5-39

AutopiSol

TERCOM Mechanization

limitations of TERCOM are as follows. TERCOM requires varying-altitudc tcrrain and a radiating altimeter. Moreover, thc prestored maps must be obtained, generated, and programmed in advance. In order for the updating tcchnique described earlier to perform properly, there must be sufficient variation in the terrain to generate a usable signal. Obviously, such a system will not work over water. It may have difficulty in the midwestern region of the United States where the ground is very flat. Therefore, consideration must be given to the terrain surrounding the target when employing this type of guidance. The data collected for the reference map will provide the information necessary to predetermine if the system will work in the areas of interest [Heaston and Smoots, 19831. 5.7.6 Doppler Radar

Doppler radar on an aircraft provides a measure of the aircraft's velocity relative to the earth. It usually consists of four beams, one forward, one aft, one to port, and one to starboard. They are all angled downward at a 45-deg to 60-deg angle. The fore and aft beams measure velocity along the track heading of the aircraft, while the left and right beams measure cross-heading velocity. All four beams measure vertical velocity. The Doppler set measures velocity in aircraft axes. The INS, or sometimes an AHRS, provides the reference coordinate system for resolving the Doppler velocity components onto along-track and cross-track axes or onto earthfixed axes.

Doppler advantages and limitations. Doppler radar has the advantage of being autonomous; it does not require any ground-based or space-based facilities. Because it does not have its own attitude reference, it must always be used as an adjunct to an INS or an AHRS. The Doppler return signal over water is poor due to more specular and less diffuse reflection off the surface. In addition, the scattering cells on the water surface may be moving due to currents or wind-blown spray. For this reason Doppler radars usually have two modes, one for land and one for over water. This landlsea switch can be used to reduce the amount of feedback gains to the INS during over-water operations. 5.7.7 Star Trackers

Astro-inertial systems consist of a telescope mounted in two gimbals, azimuth1 elevation, which in turn are mounted directly onto an inertial system. The systems in operational use today are integrated into a !gimballed inertial system. This means

Inertial Navigation

226

Chap. 5

that the whole system must include at least five gimbals: the pitch, roll, and azimuth gimbals of the inertial system plus the azimuth and elevation gimbals of the stellar telescope. Consequently, these systems tend to be quite large and expensive. Conceptually, the gimballed inertial system could be replaced by a strapdown inertial. thereby reducing the size of the overall system. T o date, however, this development is still in the early stages of design. The basic idea of the astro-inertial system is as follows. Given the position of the aircraft from the inertial system, the computer looks up the ephemeris data for a particular star from the star catalog contained in its memory, and calculates the azimuth and elevation angles for that star at that time. It then drives the telescope to those angles and looks for the star. If the star is off the axis of the telescope, the star tracker provides correction signals to center the star on the telescope axis. These same azimuth and elevation correction signals go the inertial system as psi-angle corrections (see Figure 5-10). One such star fix can update two attitude error components. For a full threeaxis update, the telescope is driven to another star, widely separated in angle from the first, and the process is repeated.

Star tracker advantages and limitations.

Like inertial and Doppler systems, the star tracker is autonomous, requiring no outside facilities, except of course for the stars whlch are always there. Cloud cover is an obvious limitation for star trackers, and this relegates their usefulness to high-altitude flight. Because it is a psi-angle measuring device, it basically updates the gyros and not the accelerometers. (Recall that the psi equations are independent of the acceleration equations.) This means that acceleration errors can grow without bound. The astro-inertial system will simply apply a tilt error to compensate for the position error; it cannot distinguish between the two. However, with an occasional position fix or with Doppler radar updating, the integrated system can produce a navigation system with bounded errors over very long periods of time. 5.7.8 Kalman Filtering

In order to combine the INS data with one or more of the foregoing augmentations, most modern avionics systems use a Kalman filter (KF). The KF is a linear feedback system with time-varying gains. The gains are varied in an optimal or nearly optilnal fashion so as to take account of the relative accuracy of the INS and the updating measurements as well as the geometry of the measurements. Because the accuracy of the INS varies with time, the KF must include an error model of the INS. This error model includes a number of error states, the dynamic coupling betwccn these states, and the noise which drives these states. A similar but simpler error model for the measurement noise allows the KF to properly weight the feedback gains to the INS. The feedback corrections themselves may be applied in two different ways, One is called open-loop correction, and the other is called closed-loop correctioll,

External Naulgatlon Aids

Sec. 5.7

227

In the first, the corrcctions are added to the INS outputs and do not affect the operation of the INS itself. Figure 5-40 illustrates such a mechanization. The advantage of this type of mechanization is that spurious measurements affect only the corrected INS outputs and not the INS itself. If the KF diverges as a result o f these spurious updates, the INS is still operational, and the system can recover by reinitializing the KF. The disadvantage of the open-loop KF is that the state errors in the error model arc not driven to zcro and can become large enough for nonlinearity effects to disrupt the system operation. In thc closed-loop feedback system, the feedback corrcctions actually correct the INS itself so as to maintain the INS state crrors near zcro. This eliminates the nonlinearity concerns. O n the other hand, spurious updates can drive the INS so far off that it may not be able to recover.

Kalman filtering approach. A standard Kalman filter minimizes the state-vector variance and estimates the state errors. The Kalrnan filter is constructed based on an error model, as mentioned previously. First, an INS model and a measurement model arc given as jc = Z =

.4.u

Hs

+

+

E[~(~)Iu(T = ) Q6(t ] - 7) E[v(I)v(T)] = R6(t - T)

(INS) E[ru(t)] = 0, v (measurement) E[v(t)] = 0,

lu

INCREMENTAL ERROR ESTIMATES

I

I

SENSOR SUBSYSTEM

SENSOR SUBSYSTEM

-c

tl

SUBSYSTEM ERROR ESTIMATES

ERROR

-

SUBSYSTEM CORRECTED OUTPUT

1 MEASUREMENT ERROR VARIABLE DIFFERENCES

SENSOR SUBSYSTEM

MATRIX

SUBSYSTEM OUTPUT RATIONAL SUBSYSTEM ERROR

SUBSYSTEM CORRECTED OUTPUT

ERROR

Figure 5-40 Generalized Mechanization of Filtering Open-Loop Error Control System (From ['Martin, 19861 with permissionfrom Lirton)

Inertial Navigation

228

Chap. 5

where x is the error state with X E R ~ ,and w and v are white noises. The measurement z can be a combination of attitude, velocity, o r position measurements. If position measurement is available, position error 6, must be included in the state vector x, An example application of error-covariance analysis to inertial platform errors is constructed as illustrated in Figure 5-41. As presented in Chapter 4 (Sections 4.1 and 4.2), the fiow diagram in Figure 5-41 [Tucker and Stern, 19861 can be con.. structed, with v = a as the input. From this figure, it is seen that there are nine error state variables: three position errors, one along each of the coordinate axes; three corresponding velocity errors; and three misalignment errors, one about each of the coordinate axes. The velocity errors are obtained by integrating the acceleration error inputs, and the position errors are obtained by a second integration. In addition, the misalignment error about any axis leads to an additional acceleration input along each of the other two axes. The resulting dynamics matrix (A) from Figure 5-41 is as follows:

I

0 -k? 0 0 0 0 0 0 0

-

1 0 0 0 0 0 0 k 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 0

kz 0 0 0

0 0

0 0 1 0 0 0 0

0 0 0 a47

0 aj7

0 0 0

0

0

a28

a29

0 0 0 ass 0 0 0

0 049

0 0 0 0 0 *

Examination of the A matrix and the flov,. diagram in Figurc 5-41, howcver, indicates that the systcm can bc decoupled into three indcpcndcnt subsystems, cach of dimension four. For example, states 1, 2, 8, and 9 form a subsystem. It IS noted that cach of thc misalignment error states 7, 8, and 9 feed into two of the three subsystems, but this does not cause any direct coupling among the subsystcms, since there are no feedback inputs into the misalignrncnt states. T h e four-dimensional problem for states 1, 2, 8, and 9 is of the form

5.7.9 Kalman Filtering Performance

For conventional filtering, the position error (in ft) ncvcr converges to zcro, as shown in Figure 5-42a. For Kalman filtcring (optimal filtcring), on the other hand, the position error (in ft) convcrgcs to, but ncvcr rcachcs, zcro, as shown in Figure 5-4%. Figure 5-42c displays a table showing thc relationship of mechanization and performance.

Sec 5.8

Integrated Inertial Navigation System (IINS)

9 n3 n5

n2

-

x velocity

n7 ' misalignment about x-axis

= y Position

n4 = y velocity

nB = misalignment about y-axis

= z position

n6

ng

x Position

Figure 5-41 AACC)

= z velocity

-

misalignment about z-axis

Flow Diagram (From [Tucker and Stern, 19861 with permission from

INTEGRATED INERTIAL NAVIGATION SYSTEM (IINS) When inertial systems were first included as part of the avionics package on aircraft circa 1960, they performed only the navigation function. Typically, there would also be an AHRS for driving the flight instruments, and there would be rate gyros

Inertial Naulgatlon

230

CONVENT lONAL FlLTER

OPTlMAL (KALMAN) FILTER

(A)

'HARS

- BOUNDED loHCADING,

Chap. 3

(B)

DG

PIHRFRfE.

VG 1 8 1 ~ ~

Navigation PerFigure 5-42 (a) & (b) Integrated Performance (c) Augme~~red formance Summary (From /Manin, 19861 ulith pcrmir~ionJiom Lirrotr)

Sec. 5.8

Integrated Inertlal Naulgatlon System (IINS)

231

for flight control sensors. Gradually, as the INS became more reliable they began to take over the AHRS functions. Then as the strapdown INS came into being, their use for flight control became a real possibility. In order for the INS to be part of the flight control system (FCS), it must be made ultra reliable. This leads to the design of multiplcly redundant INS. Such systems, properly updatcd, can providc reliable and accurate position, velocity, and attitudc for the navigation function, as well as body-axis accelerations and attitudc rates for the FCS. Such systems are called intcgratcd inertial navigation systems (IINS). Only strapdown INS arc candidates for IINS, because the gimballcd INS cannot providc the attitude rates for the FCS with a wide enough bandwith.

5.8.1 Integrated SensinglFIight Control Reference System (ISFCRS) One way to improve the reliability of the INS is to employ redundant sensors, as in the case of an integrated sensinglflight control reference system (ISFCRS). The ISFCRS uses six ring laser gyros and accelerometers in an arrangement of two orthogonal triads, onc of which has its axes aligned with the flight vehicle's body axis. The geometry is depicted in Figure 5-43. The sensors are numbered from 1 to 6 with 1, 2, and 3 corresponding to the x-, y-, and z-axes of the body aligned unit, and 4, 5, and 6 corresponding to the x-, y-, and z-axes of the skewed unit. A functional block diagram of an ISFCRS is presented in Figure 5-44. The ISFCRS contains signal selection, lever arm compensation, and parity equations. The ISFCRS was proposed for the MFCRS and MIRA systems that provide all required incrtial reference and air data in advanced transport and fighter aircraft for flight control, navigation, weapon delivery and targeting, terminal area functions, automatic TFITA, sensorltracker stabilization, flight instruments, and displays, as illustrated in Figure 5-45. Figure 5-46 shows a conventional approach of comparison with MIRA [Boggess, 1986; Young et al., 19831.

5.8.2 Integrated Sensory Subsystem (ISS) Detailed discussions of the integrated sensory subsystem (ISS) presented in this example can be found in Toolan and Grobert [1983]. The ISS is composed of three major elements. The first element consists of the various sensors/sensory data and preprocessing electronics. The second element is a redundant I 1 0 system which provides data transfer between the sensors/transducers and redundant computers. The third ISS element is the data handling system (DHS) contained within the redundant digital computers in which redundancy data management and output parameter calculations are performed. The ISS sensor set consists of redundant skewed arrays of strapdown integrating rate gyros and accelerometers, redundant air data probes and transducers, magnetic heading reference sensors, radio aids, and command inputlsurface position transducers. A system-level block diagram of the

Ineriial Navigation

232

Figure 5-43

Chap. 5

ISFCRS Geometry

ISS DHS is shown in Figure 3-47a. The skewed and dispersed gyros and accelerometers supply raw input data to the DHS. The DHS removes the vehicle bending and kinematic effects and provides orthogonal rate and acceleration data to the SAHRS (strapdown AHRS) algorithms, the FCS, and the user systems. The six gyros and six accelerometers are configured in a symmetrical conical array and provide a 2-fail operational sensor capability. In addition, the sensors are dispersed to assure a survivable system. The ISS design philosophy for inertial sensor packaging and dispersion for survivability permits installation o f gyros and accclerometers in less than ideal locations with respect to bending nodeslantinodes and the aircraft center of gravity. Without appropriate compensation generated by the use o f a state estimator, the sensor locations can result in FCSIbending-mode coupling andlor nuisance trips of the RDM failure detection routines. The configuration of the gyro and accclerometer RDM routines and the state estimator is shown in Figure 5-47b.

5.8.3 Integrated Inertial Sensing Assembly (IISA) Another type o f IINS is the IISA. This is an integrated inertial sensing assct1lbJ~ (IISA) that uses six ring-laser gyros and six inertial-grade accelcromcters in two, separated clustcrs. It is a system that is well suited to the combined navigatioll and flight control sensing needs of modern high-performance fly-by-wire vchiclcs. The background of integrated navigationlflight control sensors is given by Ebncr and Wei [1984]. Krasnjanski and Ebncr [1983], and Jar~kovitzct al. 119871 includillg a spectrum of possible system configurations as a function of avionics redundancy

Inertial Navigation

234

?-={

I

I

I

I

iI

MILSTD.1553A

I I I

L

---

Chap. 5

USERS Right Control Navigation Weapon Fire Control Delivery & Trajectory Sensorhcker Stabilization NAV Axis Cockpit Display

I

-l

Figure 5-45

MlRA System

requirements. Within each inertial navigation assembly (INA), sensor axes are orthogonal but skewed relative to the vehicle yaw axis, as shown in Figure 5-482, One accelerometer and one gyro in an INA are oriented along each skewed axis. Each of the three channels is largely independent, as shown in Figure 5-48b. Figure 5-48a also depicts the orientation of axes when the INA is installed into the left equipment bay of the vehicle. When an identical INA is installed into the right equipment bay, with 180-deg rotation about yaw relative to the left INA, the six sensor axes are then distributed uniformly about a 54.7-deg half-angle cone. No two axes are coincident, nor are three in the same plane. Thus, any three sensors may be used to derive three-axis outputs in vehicle axes after suitable computer transformation. The IISA development uses high-reliability laser gyro and accelerometer sensors packaged in a strapdown system configuration to provide a common, efficient source of aircraft body rates, attitude, and accelerations. These measurements provide the essential inertial data inputs for all core and mission avionic functions including stability control and stability augmentation system (CSAS). The IISA (Figure 5-48c) consists of five assemblies: two identical INAs containing the inertial sensors and navigation computers (shown in Figure 5-48d); two identical digital computer assemblies (DCAs) each containing dual redundant flight control redundancy management and sensor selection logic computers; and a multifunction

L

k Air Data

User

Figure 5-46 Conventional ~ p p r o a c ~ to Sensing and Flight Control Referellce System

ORTHWWAL RAT1b ACCIL DATA

OW111 UtR FIST#MS PITCH L ROLL

IU OAT A HANDLINO SYSTEM

DELIVERY

INTEORATID N A V SVSTEM CONTROL SYSTEM

-- -

SKEWLD OVROS ACCEL8

I

IEXTRAPOLATED EENDlNG ACCELERATIONS1

I

I

USE EXTRAPOLATED ESTIMATES TO REMOVE BENDINO ACCELERATIONS

SKEWED ACCELERWETER

AIRCRAFT DYNAMICS

USE GYRO RDMS OUTPUT 6 EXTRAPOLATED ROTATIONAL ACCELERATION ESTIMATES TO REMOVE KINEMATIC ACCELERATIONS

p-r-4 ACCELEROMETER

7

ESTIMATOR -TE

t+

TO SAHRS 6 CONTROL L A M PITCH 6 ROLL ATTITUDE FROM SAHRS

ELBP USE EXTRAPOLATED ESTIMATES TO REMOVE BENDINO RATES

I

1

1

IEXTRAPOLATED BENDING RATE ESTIMATES1

(B)

Figure 5-47 (a) Inertial System Block Diagram (b) Inertial Sensor RDM (From [Toolan and Groberf, 19831, 0 1983 IEEE)

Inertial Navigation

Chap. 5

control display unit (CDU) for displaying IISA data and providing the operator interface for initializatior~and mode selection.

Configuration of sensor locations.

Table 5-2 is a comparison of configuration and attitude/velocity redundancy for the IISA. T h e four different system configurations with variable levels of attitude/velocity redundancy are described as follows [Ebncr and Wci, 19841. (1) Configuration 1 assumes that neither attitude nor velocity is required for flight safety. The INS can be nonskewed relative to aircraft pitchlrolllyaw axes for minimum size. A companion unit would be required

Integrated Inertial Naulgatfon System (IINS)

Sec 5.8 INA

--

MILStD-IS538

NAY ILICIRONIC~

1014 Hx

3 RLOS 3 ACCtU 3 HVPS 3 fC ILICTRONICS

N A V TO * I C

- -=-

rcro AIC(MH.,

-

d,

PRoClssoR

+-(I

PC T O A I C (80 Ha)

DCA INA

A

---

-

3 RLOS 3 ACCIU 3 HVPS 3 PC I L I C I R O N I C S

-

-- -

PROCESSOR

NAV tLtCIRONIU

DCA

a

-

-

-

N

-

-,I

-

rc TO AIC

(80 HZ)

-I,

c rcc TO AIC POHZ) A

V

TO AIC

(C)

INSTALL

Figure 5-48 (a) IISA Sensor Orientation (b) INA Functional Block Diagram (c) IISA Functional Block Diagram (d) INA Installation Configuration (a) G. ( c ) j o m [Krasnjanski and Ebner, 19831, O 1983 IEEE (b) G. ( d ) j o t n uankovitz yr al., 19871 with permissionjotn AGARD)

238

Inertial Navigation

TABLE 5-2

CONP'IGURATION VS. ATTITUDEWELOCITY REDUNDANCY

Config

Attitude redundancy

Velocity redundancy

None Dual Dual Quad

None None Dual Quad

1 2 3 4

-

Unit 1 INS INS INS INS

(nonskew) (nonskew) (skew) (3NS+lS)

Chap, 5

Unit 2

-

Sensors only (skew) SD AHRS (skew)

-

INS (3 N S + IS)

From [Ebner and Wei, 19841, O 1984 AlAA

containing three gyros and three accelerometers, skewed relative to the INS axes s o that n o t w o are coincident and three are not in the same plane. (2) Configuration 2 would be virtually identical to configuration 1 except that a computer is added to unit 2 to solve the strapdown equations in order t o derive the aircraft attitude and heading. (3) Configuration 3 consists basically o f t w o identical INS (skew) to provide the required fail-operationlfail-operationifail-safe (FO-FO-FS) angular rate and acceleration outputs, that is, dual attitude and navigation outputs. (4) FO-FO-FS navigation can be achieved by adding one gyro and one accelerometer to each INS. Since the four sensors within an INS are bolted together rigidly, with proper geometry good navigation can be achieved using any three of thc four. Among the four configurations, configuration 3 is commonly used for redundancy. For this configuration, thc sensors are contained in t w o INA, each of which provides full, independent incrtial navigation outputs.

Effect of gyro dither.

Basically, IISA utilizes ring laser gyros that dcpcnd on a small amount of angular mechanical dither motion to avoid lock-in between C W and C C W lascrs to achieve full navigation accuracy. This mcchanical dither produces accelerations sensed by the acccleronlcters. If nonlinearitics occur somewhere in the proccss, diffcrent frcqucncics bctwectl gyro dithers o r aliasing with the sampling frequcncy can causc low-frequency beats to occur on acccleration outputs that could enter the FCS and causc wander or actuator flutter. The acccleration output from thc actual operational IlSA hardware with gyro dithcring is random with a noise arnplitudc of 0.05 ftlsec' (less than 2 milli - g) [Ebner and Wei, 19841.

Redundancy management.

Redundancy managcmcnt providcs FOFO-FS opcration through the usc of parity cquations which lincarly conlbinc the sensor outputs to dctcrminc thc residual crrors. Thc opcration scqucncc of rcdundancy managcmcnt is givcn in Figure 5-49a. Thc six axcs of skcwcd angular rate and accclcration arc sent to an cxtcrnal computcr for redundancy managcmcnt pro~

-

Figure 5-49 (a) Itcdundancy Management Operation (b) Trajrrtory Flow Chart and System Response Sinlulator (c) I'arity Flow Chart and llesign Equation Sinlulator (a) .font Oankovirz rr a/.. 1Y87/ luirh prn~~issiotl . f i ~ n A C A R D ( h ) G (0fro111 /Ebttrr and Wri, 19841, 0 I984 A I A A )

SlWSql DATA

OYRO AND A C C U R O U T S B SELF-TfST

0

o I/O A V A I L U I L R I ,o D Y X L X C E I 0 N A B L I : I E S S

I

1

i s taurrrons

o

o ISOLATE 2 S x n a t A N E o u s FAILORES

mon UNFILTERED omms FILTERED OUTPUTS MR P E R W N I FAILSELF-ADJOSTING TRIP E E L S

o RAPID DETECTION 0 O

( SENSOR SELECTION

+I

1

o SENSOR PERCORCUNCL INDEX RESULTS USE BEST 3 OR 4 SENSORS

o -0

o 29 COI4BINATIOll WJATIONS 0

LEAST SQULRE CSTItiATE FOR I) SEXSORS

1

I

h I G 8 T CONTROL OUTPUTS (A)

1 I

INA-1 TRAJECTORY

I

SYSTEM DYNAMICS MODEL

I

u G EFFECT

a GYRO DESIGN EQUATION

I I ? I l I I I LEVER ARM COMPENSATION

ERRORMODEL

ACCELEROMETER PARITY

QUANTIZATION FILTER ANDSAMPLER

I

ACCELEROMETER DESIGN EQUATION

I

Inertial Navigation

240

Chap.

cessing to derive body-axis rates and acceleration, free from the effects of any tx~, hard or soft sensor failures. The redundancy management consists of parity equations, sensor selection logic, and design equations. The parity equations ensure detection of up to three sensor failures by comparison of redundant sensor data. Sensor errors above a predetermined level (threshold) are defined as failures. Because of information limitations, a third sensor failure of the same type can be detected but not compensated. Sensor selection logic considers the state of all the parity equations and determines which sensors are to be used to derive angular rate and acceleration outputs. Based on selected sensors, design equations are used in Ebner and Wei [I9841 to derive the required outputs, removing skew angles and accounting for redundant data where applicable. Citing Jankovitz et al. [1987], realistic simulations have been performed, to evaluate the effects of factors such as vibration isolators, antialiasing filters, and misalignments on the redundancy management process. The redundancy management simulator consists of the following three programs: trajectory simulator, system response simulation, and parity and design equation simulator. The trajectory and system response simulator for trajectory and system response simulation are illustrated in Figure 5-49b. The system dynamic model in this figure is based on the sensor assembly model that results from rigid body dynamics, including effects of vibration isolators and gyro dither mechanisms. The input excitation is obtained from the trajectory sin~ulator[Ebner and Wei, 19841. The parity and design equation simulator is flowcharted in Figure 5-49c.

IlSA performance. requiremcnts arc:

Following Jankovitz et al. [1987], the performance

Radial position error rate Velocity errors, per axis Reaction time

1 nmiihr (1.852 kmlhr) (CEP) 3 ftlsec (91.44 cmlsec) (rms) 5 min

The specified accuracy of outputs to the FCS follows. Actual accuracy will be significantly better since the outputs are derived from inertial navigation grade sensors:

Scale factor Bias Alignn~etit llcsolutioi~ Range

Error sources.

Angular rate

Accclcration

4(X) drciscc

20 g

The primary sources of error associated with llSA follow. (1) Axis alignt~~cr~r emor: Achieving the 1-2 arc scc axis alignment stability needed, if a significant portion of flights is to contain tcrrain avoidance and cvasivc n~ancuvcring, rcquircs very careful design. (2) Atcelcratint~scale factor stability: Skewing 0f

Sec 5.8

Integrated Inertlal Naulgatlon System (IINS)

241

accelerometer axes requires that accelerometer scale factor stability be signifcantly better than for a nonskewcd configuration. (3) It~putaxis bending: Input axis bending is the major error source for strapdown navigators in a vibration environment. (4) Data synchronization: 1mportant.considerations for flight control are the time delays and synchronization of data from the IISA when used as part of a digital FCS controlling the states of an aircraft in real time. Data sampling and processing time delays in the sensor clement have a destabilizing effect in an aircraft control system and must be carefully selected. (3) Noise andfilterittg: Modern FCSs are digital, and sensor data are sampled at some fixed frequency. for example, 80 Hz for modern fighter aircraft. Sensor noise or vibration inputs at high frequencies can be aliased by the sampling process to a frequency within the flight-control bandwidth, causing control surface flutter or pilot discomfort. In light of the preceding error sources, extensive laboratory testing of the system must be undertaken to insure that IISA is suitable for installation and flight test in a flight vehicle. These tests are described in detail Uankovitz et a]., 19871, which examines IISA system performance for navigation and flight control.

5.8.4 Helicopter Integrated Inertial Navigation System (HIINS) Fingerote et al. [I9871 present the subject of HIINS, including the following discussions on accuracy requirements as they relate to configuration 1 shown in Figure 5-50. The roles of the military maritime helicopter include search and rescue, antisurface surveillance and targeting (ASST), antisubmarine warfare (ASW), and antiship missile defense (ASMD). Many of the missions must be carried out at ultralow altitudes under all weather and visibility conditions. The increased range, speed, and accuracy of modern weapon systems impose stringent accuracy and reliability requirements on the aircraft navigation system.

Accuracy requirements. In view of the often dangerous missions that must be carried out, the small crew of the helicopter must not be burdened with monitoring the functioning of, or updating, the navigation system. Consideration of these factors has led to the following accuracy requirements. 1. Radial position error (95 percent), with external aids = 2.0 nautical mi (nm), and without external aids = 1.5 nmlhr. 2. Radial velocity error (95 percent), with external aids = 3.0 ftlsec, and without external aids = 4.0 ftlsec. 3. Attitude error (95 percent), with or without external aids = 0.5 deg. 4. Heading error (95 percent), with o r without external aids = 0.5 deg. Another similar requirement proposed by Hassenpflug and Baumker [I9871 is listed in Table 5-3. Comparing with the preceding requirements, it is seen that the ac-

Inertial Nauigation

Chap. 5

Inertial Navigation

+

MIL-STD-1553B Data Bus

Reference

(h4ILSTD-1750A)

Figure 5-50 HIINS Configuration 1 (Front [Fingerore et a l . , 19871 with permission jom ACARD)

TABLE 5-3

PERFORMANCE PARAMETERS

Parameter

Range

f90°

Refreshrate [HZ 1 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 b.25 6.25 6.25

Accuracy (95

.5* .So .5%+.25kt .5X+ .25kt .6%+.2 kt .6%+.2 kt .5%+.25kt .Olg .Olg .Olg .25*ls .25@/s .25*ls 2% 300m/l/b h 1.

O++15Okm/h f 90. -25++100mls f l4mls f l5mls -45tr70aC 480t1100mb OIZSOOf t f90*/f180. 0 + 360' t5Okmlh flOO* t30' 10.1s

6.25 6.25 12.5 12.5 12.5 6.25 6.25 50 12.5 6.25 6.25 6.25 6.25 12.5

1.2m1s 1 Zmls 2mIs Imls 2*C*:T 1100: 3mE .5m 0.5% 0.5nm 1 I km 1 0.1. 0.6'16

INS kquircmcnts ........................................................................................ Pitch 8 -30 + 45. .So Roll 6 f 90. .5 * Heading True Heading Velocity along Velocity across Velocity vertical geographic vertical Ground speed Acceleration Acceleration Acceleration Angular rates Position(Enr0ute) Position(N0E) Drift Wind Direction TAS

:. Vx v Y z v v g

ax a Y *z P 9

r P.P

P. P 6 vW *W u v w

T e m p e r a t u r e static Static pressure Height above ground Target Desired T r a c k XTrack T r a c k An6le Error Roll commanded

WPT DTK XTK TKE

Turnrate

dqldt

-

*e

360. 360. -60++400kmIh f50kmlh f l5mls tl5mls -60++400km/h t. 5g f.5g -.5g++3.5g 100~1s 60n/s LOO*ls

From [Hassenpflug and Baumker. 1987) with pcrniission from AGAHI)

%)

HWS .25* .25O .5* .5'

.5%+.2kt .5%+.2kt .Z%+.lkt TED .5%+.25kt .01g .Olg .Olg .2*ls .2*/s .2*ls 1.5% 25Om/1/4h .5 1.2mIs 1'

2mIs 2mIs lm/s 2 * C + : T 11001 3mE .5m 0 .5X 0.5nm 1' I km 1 0.1' 0.6'1s

Sec. 5.8

Integrated Inertial Navlgatlon System (IINS)

243

curacy requirements given by this table have the same attitude and heading accuracy requirements and slightly different position accuracy requirements.

Configuration 1. Figure 5-50 illustrates HllNS configuration 1. The recomnlendcd system shown in this figure comprises an airborne processor to which four primary subsystem/sensors are interfaced by means of a MIL-STD-l553B serial multiplex data bus: (1) inertial navigation system (INS); ( 2 ) NAVSTARIglobal positioning system (GPS) receiver; (3) Doppler radar velocity sensor; and (4) magnetic heading reference. Also interfaced to the proccssor arc a radar altimeter, a T A C A N receiver, and an air data subsystem which supplies barometric altitude and true-air spccd information. T h e self-contained INS is a standard form-fit function (F3) inertial navigation employing ring laser gyros (RLGs).

Configuration 2.

Configuration 2 of HIINS (based on Hassenpflug and Baumker, 1987) and its 110 parameter are shown respectively in Figures 5-51. The HIINS is a heading- and velocity-augmented INS system, providing three-dimensional navigation information in conjunction with a radar altimeter and calculating the wind vector by means of a TAS system for the entire speed regime of the helicopter. The latitude range is ?80 deg ( U T M range). Angular rates and linear acceleration in the body-frame coordinate system for flight control and weapon delivery purposes are supplied by the INS. The autopilot functions are supported by the following signals: Radar altitude h~ Doppler vertical velocity v , ~ Magnetic heading +,L, Attitude +, 0 Velocities in the navigation frame

Inertial altitude h, Inertial vertical velocity v , True heading Body velocities v,, v,, v,

+

VE, VN,

bv

Besides calculating the present position coordinates, the following navigation functions are available: (1) bearing and distance to the selected waypoint; (2) time to go to this waypoint based on the momentary speed; (3) optimal steering information to the selected waypoint; (4) targets of opportunity; (5) position update by flying over known landmarks whereby the position coordinates of these landmarks are: (a) already stored, (b) read from the map and manually inserted after freezing the position flown over, and (c) gathered and inserted by means of a map display after freezing the position flown over. Figure 5-52a is another view of an integrated NAVJFCS for helicopters, a detailed discussion of which appears in Osder [1986]. It is a double fail-operative (fail-op2) system in which the navigation function achieves the same level of failop2 performance as the flight control computers. Each flight control computer (FCC) contains its o w n internal redundancy structures and techniques (hardware and software) needed to detect its own faults, and achieve the required reconfiguration to support the system's fail-op2 specifications. The emphasis here is on the

Inertial Navigation

------------

r----------

1 HUNS 1I

7I

I I I I

I

I

II I

I L

I

DVS RDN 808

I

II I

Chap. 5

t

Q-

mu RADM ALTII(E1ER

-

I I

I

-

1I

--

-

I I

I I

I

,

INS

I

I

, ucs

I

I

*

I

L

---------- -------

fl

-----_I

II

I

DUAL MIL-BUS lSS3B

Figure 5-51 (a) HIINS Configuration 2 (b) Configuration 2 110 Parameter (Fr*m [HasscnpJug and Baumker, 19871 with permission j o m AGARD)

Sec. 5.8

integrated Inertial Nauigatlon System (IINS)

245

system architecture advantages obtained by making the navigationlflight control sensor function an autonomous navigation subsystem by virtue of the consolidation of sensor measurements, state estimation algorithms and self-contained reversion1 recollfiguration stratcgics. Figure 5-52c illustrates the data flow interfaces between the integrated navigationlsensor assembly and the flight control computers and external sensors. The broadcast bus transmissions of flight critical data to the FCCs are summarized in Figure 5-52d. The tight coupling of data from various measurements is implied in Figure 5-32b by the position vector output which contains the four states of latitude, longitude, altitude, and GI's clock bias ( C R ) For . accurate navigation, the GI's clock bias as well as its first and second derivatives are included in the navigation filter.

5.8.5 Integrated Missile Guidance Systems This section discusses the design of integrated navigation, guidance, and control (NGC) systems for tactical missiles using optimum control and estimation theory based on Williams et al. [1983]. Sensors used in this design consist of strapdown accelerometers and rate gyros and a strapdown homing seeker. Guidance and control algorithms, in addition to performance studies, is presented in Book 3 of the series. The overall system is shown in Figure 3-53a. The design uses a passive strapdown seeker and a set of strapdown inertial sensors (gyros and accelerometers). These sensors provide the two LOS angles between the missile and target as measured in the missile pitch and yaw planes, and the missile body rates (P, Q, R) and accelerations (A,, A,, A,) along the three principal body axes of the missile. T h e LOS angles, plus information about the missile body rates and accelerations as provided by the inertial sensors, are inputs to the navigation and FCS. The integrated N G C system is shown schematically in Figure 5-33b. Since the guidance algorithm requires knowledge of the inertial LOS rate, which cannot be directly obtained from the output of a strapdown seeker, an algorithm by which these rates can be estimated given the angular inputs is essential in order to achieve good performance. In the design of Figure 5-53b, these rates are not obtained by a finite difference algorithm which could produce serious errors in the presence of noisy o r infrequent data, but instead an extended Kalman filter (EKF) includes these rates as state variables. The EKFs are part of the navigation filters as shown in the figure. Since both channels have identical dynamics, the navigation filters for each channel are also identical and have the structure shown in Figure 5-53c. The block diagram is intended to show that the output of the filter u, as well as the state estimates 8 and 4 are continuous-time variables, whereas the inputs to the filter are discrete-time variables. (In actual implementation, the continuous-time variables are also discrete-time variables, but the sampling interval is much smaller.) The navigation filter automatically extrapolates the state estimates from one observation until the next in accordance with the missile dynamics. This is very important when the seeker-sampling interval T, is relatively long compared with the dynamic behavior of the missile. It also automatically solves the problem

Inertial Navigation

Chap. 3

Sec. 5.8

Integrated Inertlal Naulgatlon System (IINS) Parameter

DesCrIDtlon body angular rates body I l n e a r accel. P l t c h , R o l l . Iltadlng

vn. vc,

L.A.

Yz

h l m d other coordlnatc systea equivalents)

North. East. Dan Veloclty Latitude, Longitude, A l t l t v d e

Baro A l t l t u d c Calibrated Alrspeed True Airspeed A t r Dtnslty A l r Temperature ('=)

El .

Figure 5-52 (a) Integrated NavigationlFlight Control Overview Block Diagram (b) Flight ControllNavigation Interfaces (c) Summary (d) Typical Strapdown Block Diagram for Low Cost/Performance IMU with Kalman Filter Updating (From [Osder, 19861, O 1986 IEEE)

Inertial Nauigatlon

Chap. 5

Sight Angler

--

.-PI

*rr.L-

C001101NATE TIIANSFORUTIO*

YziT.la

-

A"TOPILOT

".A

b

----C

CO*I"OL SURFACE DEFLECTIONS

. 8

0

V

DIIECTIOY

UKI*E

UlC"

I O U "A.

UIlllX l*TEt"AT,O*

Q

tf'

P R

Figure 5-53 (a) Overall System (b) Integrated Guidance and Control System Schematic (c) Navigation Filter Schematic (From /Mrilliatns rt al., 19831, 0 1983 A I A A )

Sec. 5.8

Integrated inertial Naulgation System (IINS)

MEASURED INERTIAL LINE-OF-SIGHT

of seeker blinding which occurs when the target conlpletely fills the field of view of the seeker and the latter can no longer provide useful directional signals. When this happens, the switches shown in Figure 5-53c remain open and the missile continues by dead reckoning to the interception.

Another example of an integrated seeker-guidance navigation system. In a classical design, a weapon guidance system includes two distinct inertial sensor sections. Figure 3-34a showsthe separate flight control using a reference gyro for yaw and pitch corrections. Also shown in this figure is the additional seeker equipment, which in this case is an optical device including a tracker, that relies on a second reference gyro system for LOS stabilization. Figure 5-54b(i), however. shows one stra~downreference svstem made UD of the reference unit and the strapdown computer. This single unit is capable of referencing both the flight control and the seeker system. Figure 5-54b(ii) shows how future weapon system guidance and seeker-processing tasks will be integrated into one processor. Figure 5-54c illustrates a structure of the navigation software development system (NSD). Figure 5-54d depicts a block diagram for the LOS control. The gimbal control including the controlled element (gimbal and torquer) together with a high-bandwidth angle transducer are both contained in the inner loop. The nonlinear characteristics shown in the figure represent the effects of friction found with all gimbal

.

Inertial Naulgation

(B)

Figure 5-54 (a) Functional Diagram ofa Classical <;uidance& Seeker System with Separate Gyro Rcfcrenccs (b) Functional Diagram o l a Guidance & Seeker System with a Strapdown llcferencc (i) Distributed Electronics (ii) Low-Cost Version with an Advanced Processor (c) Structure of the Navigation Software Development System (NSD) (d) Block Diagram of Line-of-Sight Control (From [RahlJs rr a / . , 19871 with prmrissio!~jo!n A C A R D )

Chap. 5

Integrated Inertfal Naulgatlon System (IINS)

Sa. 5.8

REFERENCE OATA: ACCELERATION. RATE. ATTITUDE. POSITION

II

1

-

nlmAm

MIUL

m 1 m

MTA

SDGDR ElWl m 1 m

MIA RUS

REAL TIME

AafWT.

RN RDCESSIrn

PATE

CONVERSION

m

80285187 PISUL (C)

1 LOS

-

LOS..

. J ~ n eo f

M R ~ D SIW

s~gnt

CLOS

Guidance Processing Current methods used in guidance processing are examined in this chapter based on AGNC[1986(i)]. Figure 6-1 illustrates the order in which these methods are presented, including their subsections. The primary functions of the elements that make up the guidance system include sensing, information processing, and correction. The guidance algorithm is used for correction. Several different guidance system configurations are currently in use as solutions to numerous types of guidance problems. The guidance problems are mainly determined by the nature and the location of the target, the circumstances for weapon delivery, and the environmental conditions.

6.1 GUIDANCE PROCESSORS As discussed in Book 1 of the series, precision-guided munitions (PGM) depend heavily on their guidance processors. Imbedded processors are currently present on all guided vehicles in order that the entire system operates based on a particular algorithm. A general guidance system is drawn in Figure 2-2. Theguidance algorithm as part of the guidance loop represents an essential component in the design of an integrated guidance system. The information subsystem, which is needed to perform the guidance task of pursuer-target intercept, determines basically the configuration of necessary sensors and information processing, including the guidance tracking filter.

Quldance Processors

Sec a1

GuidanaRarssor ( k c . 6.1)

I Guidana Algorithms

I Guidasc Mission & Pctfonnance

G u i d m Rrfonaance (Sec. 6.2.1)

I

G u i i e Phase (Sec. 6.2.2)

Opcratiin PresetGuidancc (Sec. 6.23) (Sec. 6.4.1)

II

I

I Multiple Mode Guidance Modeling

hG&omsc (Sce. 63.1)

Dinct Guidance ( k c . 6.4.2)

I Command

I ~e-ding

Guiance Law ( k c . 6.5) I

I

I LOS An& Guidance

LOS Rate Guidance

( k c . 65.1)

( k c . 6.5.2) Pursuit

Deviated Pursuit1

Constant

Proportional Navigation Guidance (PNG) and Its Variations

Fixed-Lea

I

Sensitivity and Comparison of Guidance Laws (Sec. 6.5.3 & Chap. 8)

I

Guidance Mode Options (Sec.6.6)

Single Mode (Sec. 6.6.1)

Dual Mode ( k c . 6.6.2)

I

Multimode (Sec. 6.6.3)

I Future Guidance Processing

Air Defense

Performance Systems Parameters (Sec. 6.7.2) (Scc. 6.7.1)

I Talos

Active Semi- Passive

'Itnninal E m r Analysis Model (Scc. 6 3 2 )

Guidance Filter (Chap. 3

I ~oiing

Technological Advames (Sec. 6.8.1)

F i g u r e 6-1

Missile Guidance (Sec. 6.8.2)

Guidance Processing

254

Guidance Processing

Chap, 6

Types of targets include surface as well as subsonic, supersonic, and hypersonic aerial, and the target itself may be either friendly or enemy. A refueller aircraft is one example of a friendly target, while enemy aircraft, tanks, ships, submarines or missiles are examples of enemy targets. The pursuer, although generally an aircraft or missile, is not limited to these two types. The control process makes use of signals coming from the target (where the target may be a source or reflector). Within the guidance loop shown in Figure 2-2, individual guidance processses are characterized by a system of differential equations. Some of these describe the mathematical model used to represent the pursuer, while others describe models of the kinematics, the information subsystem, the control elements, and the ~peratorlpilo~. With such models of the guidance system it is possible to analyze the latter's accuracy and stability, and to synthesize the information subsystem and the laws governing the functioning of the control elements. In the particular case of missile applications, the guidance and control weapons and ballistic missiles embrace the methods and technologies required to correct the missile's actual trajectory in flight so that it maintains an intended trajectory to a target. A clear distinction must be drawn between guidance to a fixed geographical location, suitable for a strategic missile, and homing on to a moving target. If the retina of the eye is the sensor, and the pupil, cornea, lens, eye muscles, and movable head make up the seeker, the human brain is the processor. All control of the system falls on the processor. Signals generated by the sensor are processed for target information. If the signals do not contain what the processor is designed to call a target, the seeker is commanded by the processor to continue its prearranged surveillance pattern. It is the processor as well that decides when a received signal represents a target signature. The seeker is then commanded to acquire the target, lock on it, and track it. In the case of autonomous acquisition, or the detection and recognition of targets with military value without human intervention, the processor is again the key (See Figures 6-2a and 6-2b). The main function of the processor, after lock-on occurs, is to relate the coordinates of the missile with the coordinates of the target and to issue commands to missile components to eventually make the coordinates match. It contains the guidance laws used to direct the missile on the best course to intercept the target, and essentially commands the missile to hit the target. The demands on the missile acreleratirig capability as an important system parameter depend strongly on the kind of guidance law. In order to accomplish this objective, the processor is actually an intermediary between a data source and the missile control. The data source is the seeker which provides information 011 the targct location. The missile control is the autopilot which tells thc missile how to get to the targct [Heaston and Smoots, 19831. In thc dcfcnse scenario, the missile is responsible for pcrforn~ingthe search, acquisition, and lock-on operations, although other methods exist for accomplishing these functions. For examplc, a ~articu~arly simple approach has thc launch station performing the scarch and acquisition functions and sending target position information to the missile seeker, directing it to lock on to the targct before the missile is launched. Such a procedure is referred to as lock-on bcfore la~rrlrlt (LOBL). The

Ssc. 6.1

Guidance Processors

255

missile seeker is not always able, however, to lock 011 while still on the launcher by reason of interference from the radar associated with the launcher. because the target cannot be tracked through the high-acceleration launch phasc, because the targct is not within the field of view (FOV) of the seeker, or simply by reason of ocher circ~~rnstances relating to a particular system. In such an event, it is necessary to go to a more complex operating sequence. For instance, target position information tnay be provided to the missile to aid it in acquiring the targct after launch, yet the rnissile must cycle through a search and acquisition phasc after being launched. Such a procedure is referred to as lock-orr flfier laltttch (LOAL). Figure 62a depicts a stand-off autonomous LOAL for air-to-surface environment. A I'GM is a very complex package that has been subject to numerous tradeoff decisions in its design. The ability of a guided missile to reach its target after launch in the absence of any form of support provided by the launch vehicle or operator is known by various phrases includingfire and forget and launcl~ntrd leave. These types of systems can be LOBL or LOAL, but much research is currently being conducted in the area of autonomous acquisition, which is the most advanced type of LOAL. Figure 6-21 just presented is one such example. The con~pleteelimination of operator assistance, requiring automatic target detection. association, and tracking is a challenging goal and makes this mode of operation very difficult. Modern weapon guidance will require all-weather capability for aircraft attrition, autonomous target acquisition and long-range midcourse guidance for stand-off, passive and low RCS sensors for stealth, terrain following/avoidance for weapon . attrition, multirole weapons with increased accuracy for weapons loadout, and multisensors (dual mode) and image processing for countermeasures. At the same time, technological advances in areas such as digital processing throughput and memory, artificial intelligence technique, guidance sensor performance, and multiple/complementary guidance sensors will be exploited to a greater extent to produce smart weapons with increased accuracy. An autonomous weapon algorithm sequence is shown in Figure 6-2b [Hardy, 1986, 19901. The missile seeker uses target indication signals either to acquire and track the target without any search, or to perform a small sector search. When the missile seeker must perform such a search, the time required for it to switch from search to track is minimized by the target signals provided by the aircraft's information subsystem. Control signals. on the other hand, permit verification of the missile's operabilitv. The actions that the missile will take depend on the target and control signals, and are communicated to the computer through feedback signals [Maksimov and Gorgonov, 19881. Figures 6-3a and 6-3b look at factors that go into determining target acquisition time. Chief among these are the search volume, processing bandwidth, and signal-to-noise ratio (SIN) optimization. Figure 6-33, which deals with target acquisition with a missile seeker, shows how this requires searching through expected displacement, velocity, and angular uncertainties. Figure 6-3b, which deals with processing bandwidth and S/N optimization, shows how the time for missile-seeker target acquisition can be as long as the terminal guidance time interval. Figures 63c, 6-3d, and 6-3e look at multiple-target resolution with range gating, Doppler

-

-

WEAPON LAUNCH

TARGET SEARCH1 DETECTION

MIDCOURSE NAVIGATION

A

A PRIOR1 TARGETING KNOWLEDGE

*

TARGET 4 CLASSIFICATION/ PRlORlTUATION

+

.

a

-

TARGET lDENnFfCATloN1 AIMPOINT SELECTION

TERMINAL AIMPOINT MAINTENANCE (TRACKING) (B)

Figure 6-2 (a) Stand-off Autonomous Lock-on-after-Launch (LOAL) (b) Autonomous Weapon Algorithm Sequence (Courtesy of F. W.Hardy, 1986)

Field of view that needs to be searched

,/

Missile position at start of search

/'

Translational error due to inertial reference drift

Unpredicted Q ,tet motion

due to attitude reference drift

since last update (A)

Required homing range

T S = Time Interval for search through range velocity, angle uncertainties

I R Range to start search +

RSS = RH VC TS (VC = closing velocity magnitude) Receiver processing bandwidth, ~f will be 1 Number of looks proportional to Dwell time Ts 2

IR

Type system Signal power Noise power

Semi-active radar

Sa 'N a

1

Sa

(RH + VC %I2 1 JAT~ -

JG

(RH +

NaAfa

Maximize SIN by finding Ts for TS for max SIN

Ts =

-.

RH 3%

Ts =

Active radar

1 VC TsI2

1 -

Ts

aSIN aTs

RH Vc

1

Sa

+ VC T s ) ~ 1 NaAfa (RH

Ts

=0

Ts =

RH 3Vc

(8)

Figure 6-3 (a) Target Acquisition with Missile Seeker (Target acquisition with missile seeker requires searching through expected displacerncnt, velocity, and angular uncertainties) (b) Time for Missile-Seeker Target Acquisition Can Be as Long as the Terminal Guidance Time Interval (c) Multiple Target Resolution with Range Gating Requires Trajectory Shaping (d) Multiple Target Resolution with Doppler Gating Requires Narrow Bandwidth Doppler Tracking (e) Multiple Target Resolution with Angle Gating Requires Small Beam Widths or Fields of View (Courtesy o J K . Hiroshigr, 1984) (Figure r o ~ ~ r i ~on ~ uopposite rs pogc.)

Target resolution

(D)

Figure 6-3

(Continued)

Guidance Processing

260

Chap. 6

Radar antenna beamwldth

@B.w.

- Wave length Aperture dia

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Figure 6-3

(Continued)

gating, and angle gating. Figure 6-3c shows plots of target resolution versus crossing angle for different values of range resolution. Multiple-target resolution with range gating requires trajectory shaping. Figure 6-3d also shows plots of target resolution versus crossing angle but for different values o f Doppler-difference velocity or Doppler-difference frequency. Multiple-target resolution with Doppler gating requires narrow-bandwidth Doppler tracking. Finally, Figure 6-3e shows plots of target resolution at 10,000ft versus crossing angle for different values of angular resolution. Multiple-target resolution with angle gating requires small beamwidths or FOV [Hiroshige, 19861.

Precision guidance requirements and achievements. The factors that will necessitate the weapon guidance requirements previously listed include aircraft and weapon attrition, stealthier planes, electronic countcrmcasures (ECM), autonomous guidance and control system, weapons loadout, and stand-off. Additional challenges will be posed to effectively deal with and overcome high-clutter environments, nonstatic thermal emission signatures, and severe target shape variation. Regarding modern weapon guidance requirements, some o f the conclusions based on recent trends in weapon guidance that can be drawn include the following

Sec 6.1

Guidance Processors

261

Breakthroughs in IR focal plane arrays (FPA), providing substantial increases in sensitivity, resolution, and FOV, hold the greatest promise for autonomous acquisition of ships and high-value target (HVT). Solid-state MMW devices having affordable all-weather capabilitv will make for sol~leof the lowest-cost autonomous antitank weapons. These will also provide low-cost synthetic aperture radar (SAR) potential for H V T [Hardy, 19861. Meeting and conqucring nearly all of the challenges posed earlier will depend heavily on a continually forward-looking research and dcvclopment (R&D)effort on the part ofthe aerospace and clcctronics industries. The incrcasc of digital processing capability for netting onboard PGM components and for con~putationalpurposes will permit thc implementation of complex guidance algorithms in further smart weapons. Microproccssors will continue to be at the foundation of future system performance improvcments, as increasing efforts are made to advance beyond current state-of-the-art capabilities. The need for such capability has been recognized for many years, as evidenced by Jotrrnal of Astrot~acrtics G Aeronautics declaring in 1982 on their front cover, "Smart Missiles, Reshaping the Navy?" [AIAA, 19821. One particularly important area of advancement will be that of producing intelligent microprocessors. Major improvcments in processors and microprocessors in the 1970s and the introduction of large-scale integratcd (LSI) circuits point to another order-of-magnitude leap forward in this area for the 1980s. The continuing advancement of microelectronic technology from integrated circuits (IC) to medium-scale integrated (MSI) circuits to LSI and beyond circuits has led to a steady improvement in signal processing capability and reliability. This has resulted in improved precision and consequent reduction in circular error probability (CEP) of PGM. Technology is moving ahead so quickly that fourth-generation, very large-scale integrated (VLSI) circuits and very high-speed integrated circuits (VHSIC), and fifth-generation ultra-high-speed integrated circuit (UHSIC) coxnponents are already being tested in the laboratory. The introduction of VLSI circuits, VHSIC, and UHSIC approaching submicron in size will mean more processing capability, lower cost, lighter weight, and increased versatility, all of which will bring about greater signal processing capability in smaller PGM packages. N e w M M W sources, imaging devices, lower power requirements, and new complex signal processing techniqies will also a d d significant features to PGM. Very rapid advancements mean that developmental systems are only partially fielded before major improvements are inserted. The second generation of STINGER is already being adapted into the STINGER-POST, with production of the latter phased into making u p part of the total STINGER requirement [Hcaston and Smoots, 19831. Increased sensor perfortnance also contributes to the advancement in guidance and control technology. For example, in imaging sensing, it is predicted that 3-5 micron-scanning FPA (fourth generation) and 8-12 micron-staring FPA (fifth generation) will be ready for systems applications soon. Some unique capabilities such as gyro-on-a-chip (sixth-generation technology) are on the horizon. High-frequency electromagnetic sources and detectors in small practical packages have been developed that provide improvements in range and angle resolution, allowing warheads

Guidance Processing

Chap. 6

to now be guided to the target with a higher probability of kill. Increases of two to five orders of magnitude in the sensitivity or output of detectors and active sources have been demonstrated in the laboratory. Charge coupled devices (CCD), F ~ A (scanning and staring), integrated optics, fiber optics, and spectral filters are already a part of some breadboard and flight test-hardware. As shown later in this chapter and in Chapters 7 and 8, the use of complementary/Kalman filters and multiple/ complementary guidance sensors such as multiple sensor fusion, and the development of multimode radomes and IR domes offer unique performance capabilities, The accuracy required for missiles to cut a runway, for example, in a number of places to prevent its subsequent use without time-consuming repairs cannot be achieved over ranges of 200 kilometers or more by inertial systems alone. As presented in Chapter 5 , improved covert navigation systems are needed. One possibilit\. would be to make position fixes from a satellite-based GPS. However, if an a"tonomous missile navigation system is required, alternatives must be sought. In the United Kingdom, terrain-matching techniques have been under examination. AS with aircraft, the terrain-based navigation system is only possible because small fast microprocessors and techniques for compact data storage are now available. The aim of future work will be to achieve delivery accuracies of the order of one to five meters circular error probable so that targets such as bridges may be attacked. This will require algorithms of autonomous detection and recognition of targets. Such algorithms are presented in Section 6.8.2 and Book 4 ofthe series. A similar problem, which may arise first, is the remote attack of armor by missiles delivering terminallyguided submunitions (TGSM) (see Example 6-10) [Barnes, 19871.

Example 6-1: STANDARD missile guidance system development. This example is based on discussions in Witte and McDonald [I9811 as follows. Figure 6-4a depicts the STANDARD Missile-1 (SM-1) along with the STANDARD Missile-2 (SM-2) extensions. The use of modular packaging removed a great deal of potential difficulty in upgrading the STANDARD Missile to SM-2. A new guidance section together with additions to the autopilotlbattery section were the primary upgrades, the remaining sections involving almost n o modifications. The concept of semiactive guidance is illustrated in Figure 6-4b. Continuous wave, semiactive guidance, whereby highly pure, sinusoidal RF energy illuminates the target and makes objects distinguishable from one another on the basis of their velocity by the Doppler shift of the reflected energy, is used by both SM-I and SM-2. In this figure, since the high-power shipborne radar transmitter and the missile receiver are linked but not colocated, homing is considered semiactivc. To deal with potential jamming by the target, a passive homing capability is also provided. The onboard homing receiver together with the steerable front antenna make up the seeker. The seeker not only provides target tracking, but also supplies the guidance computer with information for computing missile steering commands. Figure 6-4c shows STANDARD Missile operational sequence. SM-I, which is supplied target acquisition data while it is on the launcher, then initiates homing soon after the unguided boost phase of flight. By contrast, the SM-2 expericnccs a

I

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8

Target detection by fuze Warhead detonation

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Homing Phase Missile homes to target 8 Semiactive or home-owjam guidance

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Midcourse Phase (SM-2 only) 8 Missile flies to vicinity of target Guidance intelligence supplied by ship

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Boost Phase

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superson~cspeed

Figure 6-4 STANDARD Missile Guidance System Development (a) STANDARD Missile-1 (b) Semiactive Guidance Concept (c) STANDARD Missile Operational Sequence (d) Typical Signals Received by the Missile in Flight (e) SM-? Midcourse Plus Semiactive Ter-

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Figure 6-4

(Continued)

266

Guidance Processing

Cbp.

midcourse phase under the ship's control. At a specified point occurring much further in flight, target acquisition takes place and homing is initiated. In the endgam, phase, a missile-borne proximity fuse detects the target, at which point the warhead is detonated. Figure 6-4d shows typical signals received by the missile in flight, Using either target-reflected energy or jamming energy from the target, it is p ~ s s i b ] ~ in most circumstances to obtain good steering information. It is essential, however, that interfering signals be rejected. These signals, which can surpass the target skin return by several orders of magnitude, can owe their existence to either enemy or friendly sources, or to natural environments. Figure 6-4e shows SM-2 midcoursc plus semiactive terminal guidance. In the AEGIS application shown in portion (i) of this figure, the ship guides SM-2 through the midcourse phase of flight. The missile inertial reference unit converts uplink acceleration commands, which are acknowledged via the downlink, into steering signals. Terminal homing which can be delayed longer than in TERRIER or TARTAR because of greater midcourse accuracy is commanded by uplink. In the TERRIER and TARTAR combat systems indicated by portion (ii) of the figure, SM-2 guides itself to a point in space using its IRU. When the trajectory of the missile must be altered, new points are sent to the missile by the ship via the uplink. Downlink information allows missile flight progress to be monitored. There is a predetermined time prior to intercept at which the missile switches to terminal homing. This time can be adjusted during the missile's flight by sending an uplink command. Figure 6-4f shows a functional block diagram that represents terminal guidance. STANDARD Missile terminal guidance centers around processing an RF signal rcceived from the target. Target angle and velocity information, as determined by the receiver, is translated by the guidance computer into steering commands. Adjustments in the aerodynamic control surfaces to alter the n~issilc'strajectory are initiated by the autopilot. Also, a solid rocket propulsion system supplies kinematic energy. Figure 6-4g illustrates the scanning receiver operation. An intentional amplitude modulation on the received signal is accomplished by a nutation of the scanning secker antcnna. The result of such modulation is the creation of sidebands about each return, whether from the target o r from clutter. From the magnitude of the first-scan sideband associated with the target signal, seeker angle error information is derived. However, scan sidebands associated with clutter can interfere with target detection and tracking. Figure 6-4h illustrates how a monopulse receiver functions. The relative magnitudes of the R and a signals, formed by summing and differencing the signals received by the four symn~etricallyoffset lobes of the inonopulsc antcnna, contain angle error information. Because this simultaneous n m surement approach bypasscs the need for anlplitudc modulation in time-sequc~ltial measurement (scanning) receivers, there is a substantial reduction in ~ o ~ ~ l c r - b a n d interference, Ensuring that pace would be kept, in the context of missile capability, with the long-range, high data-rate, target-detection characteristics of the ANISI'Y-1A radar required an evolutionary upgrade of the existing SM-1. O n e such upgrade consisted of adding a command midcourse flight phase, which would arm the A E G I ~ combat systcm with increased intercept range and greater firepower. This newer

Sec. 6.1

Guldance Processors

267

version missile with inertial midcoursc guidance. ternled SM-2, also provided upgraded TEIIRIEII and TARTAR combat systems with a concurrent increase in capability. The inclusion of a new homing receiver common to all SM-2 variants was responsible for improving terminal guidance accuracy. During the early 1970s, a t which time SM-1 was in production as the primary weapon for fleet air defense by TElZRlEIl and TARTAR ships, the dcvcloprnctit of SM-2 began. It was originally thought that the pritnary modification to SM-1 ~ o u l dconsist of adding a midcoursc guidance system involving cornmunicatioi~links, a digital guidance computcr, and an IRU. Figure 6-43 shows how these additions were included physically. Concerning terminal guidancc, the initial plan called for cniploying the existing SM1 scanning receiver. This plan was scrapped in 1972, however, to include a new monopulse terminal homing receiver. In the end, the guidance section for the final SM-2 configuration was almost entirely new. Other features of the final SM-2 configuration included a repackaged autopilot to accommodate the IRU, and the existing SM-I ordnance, propulsion, and steering. control systems. In the two years of developing the new homing receiver, during which time significant advances were mad^. in the areas of signal processing and logic, the communication links had progressed substantially, and it was not long before hardware interface tests were being conducted to demonstrate ship system compatibility. During flight tests conducted a t White Sands Missile Range in 1975, the SM-2 began to be viewed as a highly accurate missile. -

-

Example 6-2: The Talos guidance system. As a result of continued development efforts that began with the Bumblebee and Talos programs came the Talos guidance system, which was based on technology that included the first beamriding system and the first semiactive interferometer homing system. The emerging guidance system, whose development was an evolutionary process, was virtually unjammable, providing the missile with capabilities against piloted aircraft, antiship missiles, surface ships and boats, and radar targets. At the time consideration was first given to the Bumblebee program, missile guidance technology had just come into existence, and early guidance development work was based on pulse radar technology, as pulse radars were fully usable by the middle 1940s. The first guidance concept centered around using a radar beam that followed the target to guide the missile to it. It was not long, however, before it was determined that 10 nautical miles were about the maximum distance within which the maximum allowable miss distance could be achieved using such a beam-riding system. In order to increase the target intercept range for the missile, the guidance concept was altered to use beam-riding for the midcourse phase and semiactive homing for the terminal phase. The resulting guidance system could obtain very small miss distances, regardless almost of intercept range. The objective of greatest focus concerning guidance in the latter years of developing the Talos system was the homing system. As a result, a monopulse homing system was developed that was virtually unjammable together with an antiradiation missile seeker that provided the missile with the capability to home on radar targets [Gulick et al., 19821.

GufdanceProcessing

268

Chap.

Given that USS Oklahoma City, with the last Talos missile system onboard was decommissioned in 1980, a few comments are in order concerning this systeG and its components. As Dean [I9821 noted, "Though AEGIS reflects the techno, logical advancements and refinements of the past decade, it was the broadly based research and development efforts of the Talos program that supplied the underlying technology that gave birth to the present generation of surface-to-air missiles."~ Whereas missile guidance systems for many generations to come will probably take concepts employed in the Talos program and build thereon, the Talos program had no similar, existing missile guidance program on which to build. Rather, it was forced to rely on a pioneering spirit and to do things that had not been done before [Goss, 19821. Some of these "firsts" that the Talos program can boast of include the first rocket-launched supersonic ramjet engine, the first interferometric homing guidance, and the first beam-riding missile system. Moreover, the Talos system was the first to incorporate a tactical nuclear warhead. In the Vietnam conflict, Tabs distinguished itself in early engagements by its deadly accuracy at long range. According to Goss [1982], Talos will not be distinguished merely by its achievements, which alone cannot be understated, but by the developments that would emerge from the program. These range from the current and future generations of antiaircraft missile systems (hardware and concepts), to a technology that is pervading such fields as satellite systems and biomedical engineering.

6.2 GUIDANCE MISSlON AND PERFORMANCE Fundamentally, the role of the guidance processor is to ensure that a particular mission is accomplished \vith thc best possible performance. As shown in Figure 6-1, the critical guidance issues relating to this are guidance performance (Section 6.2.1). guidance phases (Section 6.2.2), and operation (Section 6.2.3). Before a decision can be made as to how to combat a threat, mission analyses must be performed. In such analyses, mission areas are designated based on functions of the troops countering the threat; however, a standard set of mission areas has yet to be adopted across all branches of the armed forces. Guidance missions can be divided into air-to-air (AA) missions, air-to-surface (AS) missions, surface-to-surface (SS) missions, and surface-to-air (SA) missions. In their underwater deployment of I'GM, the Navy enlploys a modified version of these definitions. All these missions are outlined in Figure 6-5, in each of which the missile is located in the middle between the launcher to the left and the target to the right. The lines drawn in these figures denote possible data links or sensor transmission paths employed by the PGM [Heaston and Smoots, 19831. The air launch mission for tactical missiles is considered to bc described by two different (not mutually exclusive) scenarios: AA versus AS combat and stand-off versus short-range combat. According to the phase

'

This quote is taken from an article by Frank A. Dean entitled "Guest Editor's lntroduction~'' Johns Hopkinr APL Tetlltliral Di,qerr. Vol. 3. N o . 2. April-June 1982, p. 115.

Sec. 6.2

Guidance Misslon and Performance

269

of flight, the guidance missions can also be divided into lntrt~ch/prelaurichphase missions, ttiiric.o~rust~phasc missions, Near !errriitra/ and rerrtri~ralphase missions. Figure 66 graphically depicts the midcourse and terminal guidance phases of a tactical missile. At the initialization of missile launch, kinematic parameters for each of these three phases and the criteria for the missile's decision to transition from one phase to another arc specified. Corresponding to the AS scenario, a midcourse guidancc phase is entered following launch in which the n~issilcis directed to the target area. This is followed by a search and acquisition phasc during which the missile seeker scans a region looking for a target. Upon detecting a target, the seeker locks on to it, that is, tracks its position, and acquisition is accomplished. This results in the initiation of terminal guidance, where the seeker is used to provide information concerning the relative positions of the missile and target. Such information is then used to guide the missile to impact. Here, terminal guidance could have been used interchangeably with homing guidance. 6.2.1 Guidance Performance

F i g ~ ~ r6-1 e illustrates the different guidancc mode options in these three phases. Figure 6-7 presents multiple guidance mode considerations for an AA environment, while Figure 6-8 presents that of surface-to-air missile (SAM) operation. The guidance problem for range-enhanced missiles is to obtain an optimal trajectory and guidance law for midcourse and terminal phases. These missiles generally include a command-update inertial midcourse guidance and active or a semiactive radar seeker terminal guidance. The function of the midcourse guidance is to minimize energy loss and bring the heading error to zero at handover. Handover from midcourse to terminal guidance occurs following target acquisition by the missile seeker. In general, midcourse guidance response is neither fast nor accurate enough, even at short range, to consistently achieve the desired miss distance that is within the lethal radius of the missile warhead. Therefore, a terminal guidance mode is necessary following midcourse guidance during which the missile homes on the target until intercept occurs and the missile warhead is detonated. The use of midcourse guidance followed by a relatively short period ofterminal homing offers a significant improvement in firepower and missile intercept coverage at the expense of the inclusion of the sensors and data links necessary for implementation. Several factors bear on the success or failure of the missile during the terminal guidance phase. Chief among these is seeker acquisition at the desired time of handover. Near intercept, the missile's kinematic capability, or speed and maneuverability, relative to that of the target becomes critical. Obviously, it is much more difficult to deal with targets that do a good deal of crossing and maneuvering than with directly incoming nonmaneuvering threats. Guidance accuracy is also, affected to a large degree by factors such as an angle scintillation phenomenon known as target glint and, in certain circumstances, severe fades in the target-reflected signal at a critical time prior to intercept. Missile heading error, which is a measure of the

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Figure 6-5 (a) Surface-to-Surface (Launch platforms may be tripods, ground vehicles, cannon tubes, or ships. Targets may be a t ranges of one to greater than 1000 kilometers and consist of tanks, ships, men, material, and fortifications. Primary interest is by the Army, Navy, and Marines.) (b) Surface-to-Air (Launch platforms may be shoulder-mounted. tripods, ground vehicles or ships. Targets are airborne helicopters, fighters, missiles, bombers, and supporting aircraft at ranges of one to greater than 500 kilometers. Primary interest is by the Army, Navy, and Marines.) (c) Air-to-Surface (Launch platforms are helicopters, close-air support fixed wing, fighter aircraft and bombers. Targets may be any enemy high-value assets including men, tanks, vehicles, material, air fields, SAM sites, bridges and ships. All three Services and the Marines have an interest.) (d) Air-to-Air (Launch platforms are primarily fighter aircraft and bombers. Self-defense systems for helicopters are under consideration. Generally, targets are of like kind at ranges from three to 150 kilometers. Navy. Marines and Air Force have major interest.) (e) Undersurface (Launch platforms are submarines, ships and sometimes aircraft. Targets are enemy submarines, ships, and aircraft a t one to 100 kilometers. Weapons may be mines, torpedoes or missiles. The Navy and Air Force have interests in this area.) (From [Heaston and Smoofs, 19831 with permissionfrom GACIAC)

271

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Figure 6-6 Typical Tactical Missile Trajcctorics (From [Heaston and Smoors, 198.?/ witlt per. mission from GACIAC)

Guidance Processing

2 74

1

Engagement

Chap. 6

Sea clutter

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degree to which the missile is not steering toward the actual intercept point at handover, can also have an influence on the final miss distance [Witte and McDonald, 19811. This influence depends on the missile-to-target range, missile speed, time of handover, autopilot and control loop dynamic responsetime lags, and guidance filter time constants. It is particularly important in the case o f slower missile response and severe radome coupling as the speed and intercept altitude increase. Performance evaluation requires modeling the interdependent acquisition and miss-distance parameters. A flow diagram of various acquisition and miss-distance parameters is shown in Figure 6-9a. Figure 6-9b depicts different noise sources and their RMS noise levels (see Equations (2-20) and (2-23)) as functions of range-togo R. Figure 6-9c illustrates primary sources of miss distance for an air target engagement. Figure 6-9d lists the effects of different sources on miss distance as functions of closing velocity, missile velocity, and total time constant (T = 72). Figure 6-9e illustrates the effects of the total system constant on RMS miss. A missile constant is typically selected to minimize conflicting accuracy characteristics* as depicted in Figure 6-9f (for a given A and Vc)as presented in later sections 2nd in Chapter 8.

Sec. 6.2

Ouldance Mlsslon and Performance

terminal radar Parameters

Search parameters Target khematlcs Terminal acquisition Midcourse detection i3

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heading Midcourse

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0

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Figure 6-9 (a) Performance Evaluation Requires Modeling the Interdependent Acquisition & Miss Distance Parameters (b) Noise Sources and Their Noise Levels as Functions of Range to Go (c) Major Miss Distance Contributors (d) The Miss Distance Dependence on Closing Velocity & Time Constant (T = 7,) Varies for Different Contributors & Type of Seeker (e) Effect of System Time Constant on Miss (0 Missile Constant Selected to Minimize Conflicting Accuracy Characteristics ((a), ld), (e) courtesy oJK. Hiroshige, 1984)

Guldance Processing

276

Chap. 6

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(Continued)

Example 6-3: Advanced missile guidance system against very high-speed target. In this example, the guidance law consists of a midcourse guidance phase and a homing guidance phase, as indicated in Figure 6-10. A midcourse guidance law is used to navigate the missile in the midcourse phase, minimizing the deviation angle at. lock-on from 180-deg head on. The terminal guidance law takes over as soon as the missile seeker acquires the target. While current tactical antiair missiles are designed chiefly for use against aircraft, there i s a growing concern surrounding the threat of air-to-surface missiles (ASM). Advanced missile guidallcc systems used in antiair er~gagcmcntswill have the capability o f intercepting a very

Cumulative miss distance (feet)

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Figure 6-9

(Continued)

Guidance Processing

Chap. 6

-.

Figure 6-10 Geometric Relation between a Missile and a Target (at Lock-on) a b M l r X Y (From [Kuroda and Imado, 19881, 1988 rrissiie 0 ospea AlAA)

0-

high-speed target with a relatively low-speed missile, given some constraints on the missile g performance and the aspect angle at lock-on. The closer the interception geometry is to head on, the shorter the miss distance is. Simulation results show that as the target speeds increase, the deviation angles from 180-deg head-on at lockon must correspondingly decrease in order to achieve satisfactorily small miss distances. For an extremely fast nonmaneuvering target as compared with the interceptor, a small miss distance can only be achieved if the aspect angle at lock-on 1s near 180 deg. This tendency becomes more acute as altitude is increased. The situation is almost the same against a maneuvering target, exept that the miss distance becomes small when a target maneuvers towards a missile and becomes large when it maneuvers away from a missile particularly at a high altitude (15 km). The results concerning the effectiveness of the advanced missile guidance system against a very high-speed maneuvering target and the aspect angle constraint at lock-on are as follows. Below a certain altitude (approximately 13-14 km), the advanced guidance system of a SAM is effective against a very high-speed maneuvering target. Little is gained in the way of being able to relax the aspect angle constraint at lock-on with the advanced guidance law with compensation of the target lateral acceleration (APN, see Chapter 8). The same is true of the advanced guidance law with comnpensation of the missile velocity change (velocity compensated PNG, see Section 6.5). Neither the target lateral acceleration nor the missile velocity change is primarily responsible for the aspect angle constraint. The time-to-go estimation error at lock-on in the PNG without the initial seeker gimbal pointing error has a significant effect on the aspect angle constraint. It appears that a missile heading error induced by the estimation error is the primary cause of the constraint. Because of the time lag of the missile dynamics, the missile is unable to deal with the abrupt change in the intercept geometry when the heading error (or the time-to-go estimation error) exists. A very effective way, thcreforc, to relax this constraint is to reduce the missile noise filter time constant. However, this is difficult to do by reason of filter capability degradation, autopilot instability, and other effects. If it is possible to reduce the missile noise filter time constant, the miss distances brought about by the preceding time-to-go estimation error decrease and the result is that the aspect angle constraint is relaxed. Concerning the advanced guidance system, as a practical matter, the magnitudes of the command signals are confined to lie between some maximum values owing to hardware limitations. If the closing speeds are large, saturation of the command signals is likely to occur, resulting in large

Sec. 6.2

Guldance Mlsslon and Performance

279

miss distances. However, if it happens that the missile.and targct arc very close to a head-on geometry, the LOS rotational ratcs are usually very small, so that the command signals tend not to become saturated [Kuroda and Imado, 1988, 19891. 6.2.2 Phases of Flight

Following the launch phase, a midcoursc phase serves to direct the missile to a region near the targct. In this region, accurate target information is provided by the homing scckcr. This is followed by a terminal homing phase, which is the final phase of pursuer flight prior to intcrccpting thc maneuvering target.

Launch phase. The missile launch phase can be described as follows. The missile begins its flight with the speed and altitude of its base or platform, which may be a moving vehicle such as a ship or aircraft. In progressing from these to its desired speed and altitude, the missile must use the specific values of acceleration/decelcration, and climbldivelturn rates given in its initialization message for the launch phase. The transition from launch to midcoursc takes place when the missile has reached a slant range from its point of launch that is equal to or greater than the value of the field midtransition range given in its missile initialization message. If the missile reaches the specified transition to terminal phase while still in its launch phase, the missile transitions directly to terminal phase, skipping midcourse phase entirely [Wildberger and Hunt, 19861. If the missile reaches its launchphase goal speed andlor altitude before its required transition to nlidcourse phase, the missile remains at those goals for the rest of its launch phase. If the missile is required to transition to midcourse before reaching its launch-phase goals, it switches immediately to begin seeking midcourse goals and using midcourse kinematic parameters. During launch phase, the azimuth (horizontal) heading of the missile is determined by an initial heading ordered by a weapon control system (WCS) task or else by the missile's own solution to the target intercept problem, depending on its operational state. Midcourse phase. A missile is in midcourse phase from the moment it reaches the slant range from its point of launch that is specified in the field midtraniition range of its initialization message. The missile remains in midcourse phase until it transitions to terminal phase. The transition from midcourse to terminal takes place when the missile has reached a slant range from its target that is equal to or less than the value of the field term transition range given in its missile initialization message. Regardless of whether the missile has achieved the desired speed and altitude set for launch phase, when it transitions to the midcourse phase, the missile must take as its new starting conditions whatever speed and altitude it has at the time of transition to midcourse. If the missile reaches midcourse-phase goal speed and/or altitude before its required transition to terminal phase, the missile remains at those goals for the rest of midcourse phase. If the missile is required to transition to terminal phase before reaching its midcourse-phase goals, it switches immediately to begin seeking terminal goals and using terminal kinematic param-

280

Guidance Processing

Chap. 6

eters. During midcourse phase, the azimuth (horizontal) heading of the missile is determined by the heading it was on when it transitioned to midcourse phase, and by any heading ordered directly by a WCS task or by its own solution to the target intercept problem, depending on its operational state as further explained in the following section [Wildberger and Hunt, 19861.

Terminal phase. A missile is in terminal phase from the moment it reaches the slant range from its target, which is specified in the field term transition range of its initialization message. This is true even if the missile is still in launch phase when it reaches terminal transition range from its target. The missile remains in terminal phase until it detonates or destructs. Regardless of whether the missile has achieved the desired speed and altitude set for midcourse phase, when it transitions to the terminal phase, the missile must take as its new starting conditions whatever speed and altitude it has at the time of transition to terminal phase. In progressing from these to its desired speed and altitude for this phase, the missile must use the specific values of acceleration/deceleration. and climbldivelturn rates given in its initialization message for the terminal phase. During terminal phase, the azimuth (horizontal) heading of the missile is determined by the heading it was on when it transitioned to terminal phase, and by its solution of the target intercept problem [Wildberger and Hunt, 19861. Example 6-4: Typical laser-guided weapon mission. A typical mission has three distinct phases: launch, midcourse, and terminal. In the first of these, the weapon is launched in the direction of the target. After launch, the laser sensor in the weapon begins searching for the reflection from the target, corresponding to the midcourse phase of flight. The midcourse phase may involve a ballistic flight path, or an autopilot may steer a prcprogramnled path to the target's proximity, dcpending on the tactics selected by the weapon designer. Upon detecting the target's reflection, the autopilot transitions into the terminal guidance phase and begins steering an intercept course to the target. The laser may be turned on before launch or it may be delayed. The guidance system requires very little time to correct out the launch and midcourse flight path heading errors [Christal and Mackey, 19841.

6.2.3 Operation Guided weapons. Guided miss~lesare classified into four broad categories, dcpending 011 launch and target position characteristics: (1) air-to-air r~lissiles (AAM), launched from an aircraft at an airbornc target; (2) air-to-surface missiles (ASM), launchcd from an aircraft a t a target on the carth's surface; (3) surfacc-toair missiles (SAM), launched from the carth's surface a t any airbornc target; (4) surface-to-surface missilcs (SSM), launched from the earth's surfacc at a target on the earth's surfacc. Each of the four categories of guided weapons, together with several examples of cxisting missiles, are describcd briefly in thc following subscc-

Sec. 6.2

Guidance Mission and Performance

281

tions. As shown in Book 1 of the series, the major trade-offs and decisions required ofthe N G C systcnl engineer are brought out by the mission and threat requirements.

Air-to-air weapons.

AAM arc launched from one aircraft against another aircraft-type target (see Figure 6-7). Missiles in this category include! the shortrangc, Ill-guided Sidewinder missilc with a n approximate range of 3.000 ft; the medium-range, radar-guided Sparrow missile with a rangc of about 20 nmi; and the long-range, radar-guided Phoenix missile with a range of about 80 nmi. All of thcse missiles use solid propellant rocket motors [Titus, 19851. The development of improved stand-off weapons becomes increasingly important as modern defenses make it more and more difficult for aircraft to penetrate heavily defended areas. One drawback to current weapons possessing sufficient stand-off is that thcse arc effectively limited to targets of high value and physical prominence. The limitation surrounding high-value targets is a consequence of the high cost of the weapons themselves, while that surrounding targcts of physical prominence is a matter of limitations in the navigation and target discrimination capabilities of these wcapons. T o a great extent, the guidance and control system determines both the cost and performance of these weapons. The serious limitations of short-range AAM become very pronounced when viewed against the increased maneuver capabilities of modern fighters. With longer-range AAM, the strapdown IRS has some good features that allow it to be used to decouplc the homing head from missile body motion, and compensate for radome aberration, thus improving homing accuracy [Baron, 19871. The advantage of IR-guided tnissiles is that the seeker head is self-contained, needs no connection o r equipment in the launch aircraft (except possibly to a cryogenic refrigeration system), and probably results in the lowest total missile cost. In general, an IR missile cannot be used in bad weather or at low altitude (where most targets are now likely to be found), and in many cases was limited to attack from the rear in order that the seeker should see lock-on to the hot jetpipes. Today, new types of IR seekers offer vastly improved sensitivity and better ability to distinguish between real and false targets and, in the latest designs with Cassegrain optics, focus the radiation on either one or a whole matrix of detectors, overcoming virtually all of the previous problems [Gunston, 19791. Radar-guided tnisriles are longer range and can either be semiactive, active, or both. A semiactive radar-guided missile homes in on bistatic reflections coming from the target being illuminated by the main weapon control A1 radar in the interceptor. An active radar-guided missile (that is, AMRAAM) has its own radar, and hence does not require the interceptor's A1 radar to illuminate the target while it is being guided to it. The advantage of active radar guidance is that it provides a launch-and-leave capability [Schleher, 19861. At present, AAM development is centered around two basic classes. The first class includes medium-range missiles that have a range of about 20 miles, while the other kind of AAM is the close-range weapon which must be able to be fired at a moment's notice and have such instant response and unprecedented maneuverability

DEEP TACTICAL AIRFIELDS POL. ETC

(IIR)

MOBILE AA, SAMS. TANKS, APC KNOWN TO FORWARD AIR CONTROLLER

BATTLEFIELD SUPPORT

/

REQUIRING GROUND DISCRIMINATION OF FRlENPlFOE

FRIENDLY FORCES

(A)

Figure ( a ) Mlsslotl Sccllarlo i'crspcct~vc(b) Air-to-C;round Attack ~ c c r l a r I(o( a ) co,,rte3,, of F W Hardy. 1986 (1,) botn [Ka~natid Cloutrer, 1989/, 1989 AIAA)

Sec. 6.2

Guidance Mission and Perfonnance

FEBA \

283

-

Figure 6-11

(B) (Continued)

as to kill a fighter crossing the bows of the launch aircraft. The seeker head for the dogfight missile could be either RF radar or IR, or perhaps both [Gunston, 19791. A detailed description of such a threat environment and the appropriate management to employ are discussed in Book 4 of the series. In general, missile maneuverability decreases with altitude. This case of AAM guidance and control concept against maneuvering targets is particularly challenging.

Air-tomsurface weapons.

The scenario shown in Figures 6-6 and 6-11 represents an ASM application. ASM are launched from an aircraft toward surface targets, one of the most important classes of which comprises ships. Other targets are far more challenging and demand either guidance against a point in a background scene (as in firing a rifle) or self-homing. The following describes currently available tactical AS weapons and their characteristics. Direct attack munitions (freefall) include CBU, bombs, and mines. Tactical AS weapons for arms include Shrike, Sidearm, and Harm. In a directfire an operator may be able to aim an E O or T V seeker in the nose of the missile at the target and, by the operator's own action, lock it on that target. The missile can then be released, after which it will home on the target. One of the direct-fire weapons is the TV-guided Maverick missile, which is used against small concentrated targets such as armored vehicles, gun positions, parked aircraft, and communication vans. This missile is equipped with a two-stage, solidfuel rocket motor. Other weapons used include TOW, Hellfire, Skipper, Walleye, Laser and IR Mavericks, Bullpup, GBU-15, and Laser Guided Bomb [Hardy, 19861. In an indirect fire the missile is released and either flown to the target by the operator, who sees a T V picture transmitted from the missile on the cockpit monitor, or else uses inertial or DME as midcourse guidanceso fly near the target whereupon it can be made to home by its own seeker. unfor;unately, anything calling for a missile to send back a T V picture, and an aircraft to send radio command signals,

284

Guidance P r o c e s ~ ~ g Chap. 6

is wide open to ECM interference by the enemy [Gunston, 19791. Harpoon is one example of an antiship indirect-fire ASM. The medium-range, TV-guided Condor missile may be included in the category of ASM having an approximate range of 40 nmi. This cruise missile is usually used against heavily defended, high-value surface targets. Wherever possible, the best approach to the problem of ECM interference seems to be to turn the target into an emitter of radiation. Whenever possible, an almost ideal method of turning the target into an emitter is to direct a laser at it. As seen in Figure 6-lla, a laser designator can be aimed by forward troops a t anything they can see in hostile territory, and any missile tuned to that laser's wavelength will home automatically on the light (which may not be in the visible band of wavelengths) scattered from the target. From an operational standpoint, a particularly important challenge faced by those involved in weapons development concerns how to effectively utilize in a low-cost manner stand-off airto-ground weapons. Here, stand-off air-to-ground weapons denote those weapons which allow a single launch aircraft to release weapons from secure positions hidden from threats located in close proximity to the targets (Figure 6-llb). The targets may be fixed, HVT (such as bridge, airfields, or C3 sites), or mobile force targets (such as armored fighting vehicles or air defense units). A major difference between stand-off weapons and current inventory weapons is that the latter require target overflight by the launch aircraft or lengthy periods during which LOS to the target must be maintained from the launch aircraft or a designator aircraft [Kain and Cloutier, 19891. Air-to-Surface Roles and Missions. The mission flight phases include launch over a wide range of airplane speed and altitude conditions, missile midcourse flight over a wide range of speed and altitude conditions, and terminal homing on ground targets. Four separate missions, all involving different AS applications, are shown together in a single mission scenario in Figure 6-1 la. The individual missions found in the figure can be labeled close air support, tactical interdiction, deep tactical strike, and defense suppression. Descriptions of each of these follow [Hardy, 19861. Close A i r Support. In this mission, there is a large quantity of small mobile targets including tanks, APCs, AAAs, mobile ADUs, and troops. Because of their close proximity to friendly forces, it is imperative to be able to react quickly in all types of weather conditions. The major threat facing friendly forces a t the forward edge of battle area (FEBA) is from short-range IR and RF ADUs. Weapon requirements for this mission therefore include: shorter-range lock-on before launch, smalllhard and largelsoft targets, high accuracy (hit to kill), and area saturation (submunitions). Tactical Interdiction. Here, the objective is to launch a strike against massed enemy armor moving to the front. Staging areas include vehicle parks and ammo dumps, but there is also tactical bridging, airfields, and POLS. The primary threat is from mobile and fixed RF ADUs, which can be netted. For this mission, weapon

Sec 6.2

Guidance Mission and Performance

283

requirements entail medium stand-off range, variable warheads (smalllhard, for tanks, and penetrating, for overburdened command post), and launch and leave.

Deep Tactical Strike. The focus of this mission is a preplanned stand-off attack against high-value targets, both fixed land and ship. The greatest threat facing this mission is from area and point defense ADUs such as SA-I1 and SA-12, for example. Weapon requirements for this mission therefore include longer stand-off ranges, larger warheads (against, say, industrial targets), and imaging sensors. This last requirement is for target classification and aimpoint selection. Defense Suppression. This mission is broken down into two separate missions, a defensive mission and an offensive mission. Defensive misrior~ means the quick reaction and self-protection of strike aircraft against the threat of mobile and fixed-radiating ADUs. The aim of the offensive mission is to roll back enemy air defenses prior to delivering the strike force. Possible threats here include mobile and fixed ADUs, and others listed in the Threat Definition. The Threat Definition provides graphic descriptions of SAM, antiaircraft artillery, and laser weapon systems, illustrating the depth of technology. Included in the threat are countermeasures to protect air defense systems for lethal suppression. Countersuppression techniques are expected to include decoy emitters, fake signals, jammers, emission control, cooperative engagements between adjacent sites, and passive sensors. An analysis in Pastrick et al. [I9861 compared the military value of the five baseline weapon systems (GBU-15, MAVERICK-F, MRASM, JTACMS, and HARM) and a variety (49 different types) of potential defense suppression weapon development candidates. Each weapon was evaluated against twelve threat units projected for the future, including AAA sections, EWIGCI sites, nine different enemy SAM batteries, and a tactical high-energy laseriparticle beam weapon. The enemy A D systems were given a relative worth in order to determine and quantify their value in battle. Figure 6-12a shows the systems considered in the simulation. The scenarios that were studied via the simulation model are shown in Figure 6-12b indicating the baseline. weapons, candidate seekers, sortie depth of penetration, warhead weight, and stores per aircraft. The study concluded that the required technologies to achieve a costeffective weapon system would best be directed toward a long-range, semiballistic missile with a dispenserisubmunition warhead. Important lessons that have been learned in conflicts ranging from those in Vietnam to those of the Middle East are that enemy threats, particularly aircraft, will undertake extreme maneuvers if it becomes a matter of survivability. Defense suppression, as the entire range of offense options has come to be called when applied against ground-based defenses, includes: direct attack with iron bombs, which the Israeli Air Force has used effectively; attack with stand-off weapons such as antiradiation missiles used o n a wide basis by the U.S. Air Force in Vietnam; use of stand-offjamming aircraft to mask quiet penetrators with broad-band noise, as used by the U.S. Navy in Vietnam; use of sophisticated jamming by the aircraft under attack; use of chaffto hide penetrating aircraft, used extensively in the Soviet invasion of Czechoslovakia; and penetration at very low altitudes to underfly the defenses.

Guidance Processing

286

Chap. 6

1

RELATIVE TARGET VALUE ESTIMATE

SYSTEM 1. SA.X.12 BATTERY 2. TACTICAL HEL 3. SA-10 BATTERY 4. SA.11 BATTERY 6. EWIOCI SITE 8. S A 4 BATTERY 7. SA4 BATTERY 8. SA-S BATTERY 9. SA.13 BATTERY 10. SA.3 SITE 11. SA.8 PLATOON 12. ZSU.2lH SECTION r,

42 29 21 19 16 16 7 6 6 4 3 2

EASELINE WEAPONS:

SOVIET SYSTEM COST ESTIMATE

CLOSEST U. 8. EOUIVALENT PATRIOT HEL IMPROVED HAWK LMPROVEO HAWK NOT ESTABLISHED HAWK NlKE HERCULES ROLAND NO COUNTERPART NO COUNTERPART CHAPARRAL SOT. YORK

S100M OcM 2W 2W

3W 17M 1W 42M 1W 10M SM

BM

CANDIDATE SEEKERS:

..

OBU-16 MAVERICK MRASM JTACMS HARM

1 E.0

2 IIR 3 MMW 4 RF 6 ARH

6 CO (30km CRUISE ONLY) 7 6 ARHIMMW O RFIRAC

MAIIIR

33 km

70 km

126 kg u*.lrpm

U*rontl

200 km 1 W k#

uhm

Figure 6-12 (a) Relative Worth o f Air Defense Systems (b) Candidate Weapon Concepts (From /Pastrick rr a l . , 19861 with prnnisrionfrom AGARD)

Another offense option of growing importance in this list is the use of stealth techniques to reduce the electromagnetic signature of both piloted and unpiloted vehicles. During the time of the Hawk system's design in the 195Os, several of these techniques were understood and incorporated into the design so as to counteract their use by attacking aircraft. It was not fully realized at that time, however, just how extrelne enemy aircraft pilots could be in the maneuvers they undertook in order to insure survival, when this was on the line. A good deal of effort since that time has therefore

Sec. 6.2

Guidance Mlsslon and Performance

287

been directed at counteracting whatever means of escape an attacking aircraft might employ at any time [Fossicr, 19841.

Surface-to-Air Weapons. SAM usually consist of antiaircraft and antimissile weapons, and are typically mounted on mobile or portable launchers. SAM, including land-based and shipboard versions, generally have a solid propellant, twostage rocket motor. Most of the early SAM uscd radio CLOS pridatrce. A second method of guidance with early SAM was rcldc~rcori~rrratrd,uscd by several o f t h e most widely deployed missiles in history such as the Hercules systcm. Hcrc, t\vo pcncilbeam radars are used, usually operating on diffcrcnt frequencies. One is used to lock on and track the target and the other to lock on and track the missile. 111 addition to SAM, the vulnerability of today's warships has brought about the development of ship-to-air tr~issilesablc to shoot down other missiles with great reliability and rapidity. Ship-to-air missiles that can be loaded on to their launchers by hand are often part of a weapon system that fills half a frigate and costs more than the basic ship [Gunston, 19791. T h e total height and range envelope are such that ground commands to the missile arc required between launch and acquisition by a terminal homing sensor. The target spectrum is broad, including high- and low-flying aircraft (for example, multiple and maneuvering targets) assumed to be capable of sensing the missile's presence and performing evasive maneuvers, and formation flights designed to counter a one-on-one firm lock-on and track. Simulations show that a midcourse speed loss due to an overresponsive control systcm inducing too much drag limits the range. High response is required after lock-on to cope with maneuvers. With both midcourse and-terminal phases, considerable avionics are on board, and sharing computing functions is desired. Finally, even though the threat is considered advanced, techniques and hardware proven in the past are desired to minimize program risk [Goodstein, 1972(a)].Requirements for SAM design are dictated by missions, as shown in Figure 6-13a. This figure shows altitude versus downrange of four regions: outer air battle, area defense (bombers, ASM, ballistic missiles), point defense (cruise missiles, fighters, ballistic missiles), and CWS (leakers). Area defense threat includes multiple bombers, missiles launched from bombers, multiple fighters, and stand-off jammers. Major issues in area defense threat involve firepower, target discrimination, and guidance accuracy [Hiroshige, 19861. Area Defense. An area defense analysis is depicted in Figure 6-8. As illustrated in this figure, E C M environment and target models must be included in an area defense analysis. Figure 6-13b explains the characteristics of firepower multiple illuminators. As can be seen from the figure, track-while-scan or launch-and-forget systems are required in order to increase firepower multiple illuminators. Requirements for area defense in a multiple target/ECM environment can be summarized as follows for high firepower, for target~resolution,and for high PK. First, in the case of firepower, the following requirements are essential: multiple illuminators or high data rate (more than 10 sec) track-while-scan for semiactive systems; active seeker o r passive seeker; high average velocity; and short time for search, acquisition (see Figures 6-3a and 6-3b), and terminal homing. Next, in the case of target

go 80 70 -

Area defense (Bombers, air-to-surface missiles, ballistic missiles

60 -

Altitude

50 40 30 - Point defense (Cruise missiles, fighters, 20 - ballistic missiles) 10

ClWS (leakers)\ 1 1

.

-

Initial acquisition Establish track + Threat evaluation Weapon assignment 100 Missile launch

-

1st intercept 300 Kill assessment 2nd Missile launch /

10 Downrange (miles)

100

1000

(A)

20 I

40 I

60 I

Slant range (miles) 80 100 120 I I I

140

160

180

200

Range for missile seeker target acquisition

Time between launch

U)

Ng = Number guidance channels Number of targets illuminated at > 10 hitslsecond Vt = Targel velocity Vm = Misslle velocily (B)

Figure 6-13 (a) Missions Dictate Requirements for Surface-to-Air Missile Design (b) Firepower Multiple Illuminators (To increase firepower, multiple illuminators, track-while-scan or launch-and-forget systems are required.) (c) Point Defense Against Ballistic Missiles Hequires High Interceptor Velocity (d) A Large Missile with a High Mass Fraction llocket Motor Is Ilequircd for the 13allistic Missile Threat (e) Delayed Acquisitioti 8; Track initiation for Cruise Missiles Ileducc the Numbcr of Possiblc Engagements (1) Multipath Environment Tracking (Tracking in a multipath environment is biased and erratic until after the image is resolved.) (g) High Fire Ratc (3.000 rdslmin) (Gun systems are effective at ranges less than 1.5(Io meters.) (11) Inside 2.(KKl it the Line-of-Sight Hates 8; Accelcratiotls Increase Rapidly to 3 radlscc 8; 10 radlscc2 at 2(K) ft. respectively (Courrrry o J K . Hiroshkr, 1984)

Sec. 6.2

Guidance MIssIon and Performance

Altitude (1,000 ft)

10 20 30 40 50 6,O 70 8 0 90 100 8

slant range (miles) (

I lnterceptorl missile

Seconds

2

10

12 14

16

18 20

Ballistic missile / altitude

/

,

,

I

22 r

Ballistic missile slant range

-

3Entry angle = 45 deg WlcDA 2,000 lblsq ft Intercept altitude = 30 kft Slant ranae = 4.5- .miles ....- 'z Flight time = 3.8 sec Average velocity = 6,250 ftlsec

4 -

Launch weight per 1,000 HI ol m r o c k e t motor weight Weight propellant Total webht of the rocket motw

klocity at the end of boost (fUsec) (D)

Figure 6-13

(Continued)

kaquired for average velocity a 6,250fUsec

Guidance Processing

Chap. 6

Range (miles)

24-

Time (seconds)

,-

Radar horizon equation R h = m + , r q Rh = Radar horizon (miles) Hr = Radar antenna height (ft) Ht = Target altitude (ft)

1nsufficiel:t time for guidance close-in-weaponsystem (CIWS) required (E)

+1.0 +0.8+0.6+0.4-

-

During resolution

9

+0.2- After resolution

Boresight 0 elevation angle, degrees -0.2 -

-0.4-0.6-0.8-

-------_______

gAo O -.,VVV

-1.0

Pre-resolution

6,uOO

8,000

2,OO"

Target range

- meters

10,000 L

-

Borealght equillbrlum ~CSSII~WS normel closed loop tradc C-band radar- 1 beamwldth. 60 It above water. Tsrget at 100 It allltude

-

(F) Figure 6-13

(Contmued)

Sec 6.2

Guidance Mission and Performance

1.o 0.8

At least N cumulratlve hlt

291 Cumulative hlt

30 20 +IT 10

0.6 0.4 0.2 0

0 0

500

1,000

1.500

Total miss distance

Single shot hlt probablllty

MlSSD (MI

PSS

500

VT sin o = 'i= -

R

1,000

500

1,500

1,000

1',500

VTD ~2 x2

+

acceleration

---

0.01 I I 1 I 0 1400(800 200 600

-

2,000

X(ft)

0 (H)

Figure 6-13

(Continued)

1goo X(ft)

2,000

Guidance Processing

Chap. 6

Figure 6-14 Area Defense Scenario (From [Kuroda and Imado, 19881, O 1988 AIAA)

resolution (see Figures 6-3c, 6-3d, and 6-3e), the requirements include: range, Doppler, and anglc rcsolution, trajcctory shaping to improve rcsolution (see Figure 6-3c); and multi~nodcwith additional discriminants. Finally, rcquiretncnts for high PK includc (as shown in Sections 6.5.2 and 6.7 and in Chapter 8) homing time equal to ten titnes that of guidancc time constant, maneuverability o f three titnes that of the targct, and short guidancc ttme constant [Hiroshige, 19861.

Example 6-5: Area defense.

The area dcfcnse scenario is summarized as follows [Kuroda and Imado, 19881. The object to be protectcd from enemy threats (see Figure 6-14) is surroundcd by multiple tnissilc launchers in ordcr to have a counterattack capability against such threats. If the launchers arc all spaced equidistantly from the object's center, the launcher either nearest to o r furthest from the target is used. This is done to satisfy the requirement that the target aspect be as close as possible to 180 deg, as shown in Example 6-3. Bascd on this, the missile launcher near thc vertical plane containing the object's center and the target should be uscd. It 1s thcrcforc reasonable to introduce a two-dimensional approximation restricting, for simplicity, the motion in a vertical plane. As discussed in Example 6-3, thc nlissilcs must bc as closc as possible to the head-on gconlctry against the targcts, which thcmsclvcs rnay bc dcsccnding at any incidencc anglc, and aiming at any position. It is possiblc to effcct a desirable lock-on gconlctry in somc spatial rcgions by adjusting the missile flight-path angle at lock-on to the targct incidence angle using midcoursc trajcctory shaping. The more steeply the targct dcsce11ds, the greater the flight-path anglc is required to be at lock-on. T o investigate the attainable ranges o f ~nissilcflight-path anglcs corrcsponditlg to diffcrcnt lock-on positions, the perfornlance indcx = y(~,,,~.,,,) is selected where the lock-on time

+

Sec. 6.2

Guidance Mlssfon and Performance

293

determined from the next terminal condition [x(t), h(r)l = (x, II),,,., .,,,,,,,, diacd. The performance index is to be maximized for large missilc flight-path angles, and minimized for small ones. Using optimal control, the missilc is navigated from the ground through the predicted lock-on point, where the pursuer is supposed to lock onto the target. This corresponds to the midcoursc guidance phasc. It can be seen that thc optimal control law results in a wider interceptiblc range of target incidence angles than docs Al'N guidancc law. A wider successful counterattack region is obtainable with the optimal guidancc law than with the less sophisticated APN guidancc law. Basically, n~issilescan intercept the targets successfully when nonlinear optimal control is used to guide the tnissilc in the midcoursc phase, which minimizes the deviation angle at lock-on from 180 dcg head-on, and when I'NG is used to guide the missile in the homing phasc. rloik .(.,,is

Point Defense. In a special point defense, the ballistic missile nosc cone with ballistic coefficient is equal to 2,000 Iblft? The discrimination of nosc cone from decoys and tankage is at 10OK ft altitude. The velocity of the nosc conc is 20,000 ftlsec at 30,OW ft altitude. Thc nose conc deceleration increases from 6 to 60 g's at 30,000 ft altitude. Point defense against ballistic nlissiles rcquires high pursuer velocity, as illustrated in Figure 6-13c [Hiroshige, 19861. Ballistic Missile Threat. A large missilc with a high mass fraction rocket motor is required for the ballistic missilc threat, as illustrated in Figure 6-13d. Requirements for ballistic missile threat can be summarized as follows. A rocket motor is required to achieve an average velocity of over 6,230 ftlsec over 3.8 seconds. The time to intercept is too short for extensive terminal guidance correction, as shown in Example 6-3, and hence the interceptor should be launched into an antiparallel trajectory to minimize the target deceleration component perpendicular to the LOS. Warhead, rather than guidance accuracy, is the determining factor in achieving a high PK [Hiroshige, 19861. Low-Altitude Cruise Missile Threat. In a special point defense threat of a low-altitude cruise missile, target acquisition is delayed due to late unmasking and cross-section reduction techniques. Target track in the vertical plane can be inaccurate due to multipath. Delayed acquisition and track initiation for cruise missiles reduces the number o f possible engagements, as can be seen from Figure 6-13e. Requirements for the low-altitude cruise missile threat can be summarized as follows. First, the reaction must be short. The terminal guidance requires E O o r M M W radar to resolve multipath possibilities (see Figure 6-13f). The propulsion may be traded off for more warhead, eliminating the possibility of a second shot in a shootlook-shoot policy. If the target altitude can be estimated, homing guidance in azimuth and constant altitude in elevation is a possibility. Finally, multimode guidance is also a requirement [Hiroshige, 19861.

294

Guidance Processing

Chap. g

Close-in-Weapon System (CIWS) Threat. In a close-in-weapon system (CIWS) threat situation, all targets have penetrated the area and point defense svstems. The engagement zone for CIWS is inside the inner boundary of guided missiles. Figure 6-13g demonstrates that high fire rate (3,000 rdslmin) gun systems are effective at ranges less than 1,500 meters. Inside 2,000 ft, the L O S rates and accclerations increase rapidly to 3 radlsec and 10 radlsec2 at 200 ft, as illustrated in Figure 6-13h. Requirements for CIWS can be summarized as follows. First, the reaction time must be fast. Accurate target tracking requires less than 2 mrad with angular rates of 10 rad/sec2 and angular acceleration of 10 radtsec. T i m e o f flights of up to . 2 seconds are required for an accurate intercept predictor. High projectile velocitv is employed to reduce time of flight, and high fire rate is employed to impro\,c cumulative probability of hit. Dispensing of decoys, chaff, and physical obstructions are also part of the requirements [Hiroshige, 19861. Surface-to-Surface Weapons. The scenario shown in the lower portion of Figure 6-6b [Heaston and Smoots, 19831 represents an SSM application. Launch is followed by a gathering phase in which the missile enters a wide FOV or wide beamwidth of the cont;oier. During this same phase, the command system takes control of the missile. directing" it to the LOS whcrc a narrow F O V or beamwidth is used to track the target. Commands are thcn sent to the missile causing it to follo\v this LOS, thereby achieving impact with the target. This type of command guidance is termed CLOS. SSM encompass thc whole rangc of missiles from ground- or shipboard-launched assault weapons such as antitank, antiship battlefield, and coastal support wcapons, to the largc ICBM generally known as strategic weapons [Titus, 19851. Figurc 6-15a shows a rail-launched missile, boostcd to flying speed, and then powcrcd by a cruisc cnginc on a relatively long subsonic flight. The flight is programmed so that a long, high-altitudc midcourse flight is followed by a low-lcvel dash and homing on a specified targct. Some of the major mission issues which the enginecr is faced with are shown in Figurc 6-ljb. The major trades and decisions required of the cngineer arc brought out by the mission requirements [Goodstein, 1972(a)]. Target T y p e s . Guidcd wcapons can also be grouped according to the type of targct for which thcy arc used, for cxamplc, antisubmarine warfare (ASW) missiles, antiship missilcs, and antitank missilcs. Surface targcts arc generally statiollary or slow moving, although thcy may be difficult to dctcct and track. O n the other hand, acrial targcts arc highly mancuvcring and unprcdictablc, but usually easier to acquirc. From a guidancc and control point of view, acrial targcts stress terminal guidancc thc most. Antisubmarine Warfare ( A S W ) Missiles. Virtually, all thc ASW missiles in usc at present dcrivc their guidancc information mainly or wholly from sonars. ThC narnc conlcs from SOlrrtd .Ya~,igatiotrAid Ratl
Sec. 6.2

Guidance Mlwlon and Performance

Figure 6-15 (a) Example S-S, Surface-to-Surface Mission (b) Mission Desires and Constraints, Example S-S (From [Goodrtein, 1972(a)] with permirriorr/?orn .4GARD)

noises of a submerged submarine under way at a range of many kilometers. Most sonars, however, are of the active type. Although detection by sonar is still in use, modern detection means rely on EM detection. Extremely powerful sonars are built into the underside of the hull of ASW surface ships, and into the bows of most submarines. In most warships, sonar systems can also be used as an underwaer or secure means of short-range communication. Sonar communication is also used to transmit digital data in ASW systems, and, for example, can transmit target position information from a sensor platform to a ship equipped with ASW missiles. In the

Guidance Processing

296

Chap.

same way, U.S. Navy submarines equipped with the Subroc missile nus st bc able to use sonar communications to keep the missile heading for the updated target position throughout the airborne phase of its trajectory. It remains to develop the ASW missile intelligent enough to home on its quarry by itself [Gunston, 19791,

Antiship missiles. Currently, a large group of missiles have guidance systems tailored to this role alone, and are almost useless for any other. However, the attack of maritime targets presents some similarities to and some differences from other operations. Sea-skimming techniques have been developed that reduce the time available for the ship to take defensive action, together with improved night and poor-weather capability and improved resistance of the missile guidance system to E C M . Nonetheless, saturation tactics are still required to ensure successful penetration of shipborne defenses. This means that the attacking aircraft have each t~ launch more than one missile. Moreover, because ships move a significant distance from the time of initial detection to the time of missile impact, the guidance system of each missile must be autonomous. Further, because high-value targets such as capital ships are usually surrounded by escorts, the missile must incorporate autonomous target selection capabilities. Radar seekers remain the preferred choice, capable of covering a sea area sufficiently large to ensure that, in spite of any errors in target position fed to the missile by the aircraft at the time of launch, target detection is still achieved. As defenses improve, stand-off range will have to increase, thereby making the navigation and guidance problems more difficult. Dual-mode sensors, for example, combined IR and passive radar, may be requircd [Barnes, 19871. Events in the South Atlantic during the Falklands campaign highlighted the threat a sea-skimming missile can impose on a surface ship. The basic rcquiremcnts of a sea-skilt~rtlirqtt~issilcarc that it can fly just above the sea surface at a high speed over long distanccs in cxtrcmcs of weather, and with a high probability ofachieving a kill. Such a weapon system may be launched from a land-based vehicle, a ship, ~ or an aircraft, and has a flight trajectory as shown in Figure 6-16 [ S u t and Windley, lV841. An important consideration in the development of s u c h a weapon system is an appropriate autopilot in the missile altitude control system (ACS). Book 3 deals with the dcsign of the ACS of a sea-skimming missilc niancu\~cringfrom an intermediate or midcourse height to a sea-skimming trajectory.

Antitank missiles.

An antitank missile, as notcd in Gunston 1197'11. is by far the silnplcst of all missilc classcs in most rcspccts, dealing with targets consisting of large niasscs of nictal which are rclativcly slow moving and unablc to dodge anything. Hostile armor of this type has traditionally bccn engaged a t closc range, after direct visual sighting. Compared with ships and aircraft, their ECM and IItCM capability has bccn at best low and usually nonexistent. Another aspect of tllcsc targets is that therc is no spccial problem in carrying a warhead able to picrcc their armor and causc havoc inside. As to the missiles thcnlsclvcs, most have advanced sight systems that offer the operator the best sight that optical and IR technologY can provide. Today, some of the most attractive antitank missiles are semiactive

Sec. 6.2

Guidance Mission and Performance

AIRCRAFT HEIGHT

. SEA SURFACE OR snip

TIME

LAUNCH PHASE

INITIAL CRUISE PHASE

d

Figure 6-16

FINAL APPROACH PHASE Y

Flight Trajectory of a Sea Skimming Missile in the Vertical Plane 19114/ ~uirlrpcnniiriut~fiottl SCS)

(Frum [Slrrrotl otld Windlry,

laser homers. Hayman [I9831 describes an IR imaging antitank missile that is fired and forgotten. It is a shoulder-launched weapon that lofts to an altitude of approximately 150 meters and then homes toward the targct using PNG. A gyro-stabilized seeker is precessed to the target by pointing commands. These pointing commands are developcd from an imaging sensor by a new tracking algorithm that performs well in a highly cluttered background.

Range and function.

Missiles can also be classified by range into long-, medium-, and short-range missiles. Barnes [I9871 proposes to consider long-range missiles and short-range missiles separately, proposing furthermore to subdivide the long-range category into those intended to attack high-value fixed targets on land and those intended to attack ships. T o attack targets such as airfields located well behind the FEBA, large payloads are required and the weapons must be delivered with high accuracy. Current capability requires overflight of the target by the manned aircraft. As air defenses continue to improve, stand-off weapon systems provide the only means of protecting the attacking aircraft, to the extent that missiles must have sufficiently long range to allow them to be launched from friendly airspace. Another distinction is made between tactical and strategic missiles. Broadly speaking, the terms are used here to denote the kind of target rather than its distance from missile launch. Strategic weapons are taken to be those that strike at enemy heartlands, either in counterforce attacks or in countervalue attacks. Tactical wcapons are those that influence a battle, and the battle may be by land, sea, air, or altogether. The tactical missile seldom has a nuclear warhead, while the strategic kind seldom has any other. The strategic category of missiles includes the ground-launched Minuteman series of ICBM with a range of more than 7,000 nmi and the Poseidon submarinelaunched missiles with a range of about 2,500 nmi. Both missiles are equipped with

298

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multistage solid-propellant rocket motors. These missiles achieve speeds of over 10,000 nmi per hour and altitudes of over 500 nmi. The future generation of strategic missiles may also include cruise missiles which move at subsonic speeds (about 400 nmi per hour) and at low altitudes of about 5,000 feet. Cruise missiles are powered by jet engines and resemble a small aircraft. These missiles can be manufactured at low cost and, because of their low-altitude flight trajectories, can escape detection by ground-based radars. Ballistic missiles follow a ballistic trajectory mainly determined by the gravitational field of the earth. In a short time the missile is accelerated by a single or multistage rocket propulsion system to the initial speed for the ballistic trajectory. In the propulsion phase, a guidance and control system stabilizes the missile in flight and guides it according to a preset program for the flight direction, to put the missile on its predetermined trajectory. In the terminal phase, a guidance and control system can govern the accurate delivery of warheads, correct for atmospheric or other disturbances, and even home the vehicle onto the target. Ballistic missiles are classified according to their range [Vriends, 19871.

6.3 MULTIPLE MODE GUIDANCE MODELING The information available, the maneuvers required, and the control mechanisms in use may differ from one portion of the pursuer trajectory to the next. Thus, different guidance laws may need to be used to acconlplish the mission for the entire trajectory.

6.3.1 Midcourse Guidance Midcourse guidance refcrs to the process of guiding a missile that cannot dctect its target when launched, from the launch point to the target's proximity, and is primarily an energy managerncnt and inertial instrumentation problem. The midcourse guidance law is usually a form of PNG with appropriate trajectory-shaping modifications for minimizing energy loss. The two types of midcourse guidancc commonly used diffcr only in the coord~natcsystems employed. In one missile steering signals are dircctly supplicd in an agreed-upon coordinate frame. In the other an inertial navigator is updatcd by cxtcrnally supplicd target information. Thcse tw70 types of midcoursc guidancc can further be classified as incrtial. common position grid, command incrtial, conlmand, command to L O S (bcarn rider), antiradiation homing (AIIH), and correlation matching. The first of thesc, incrtial midcourse guidancc, has a rangc-dcpcndent accuracy, high rcsistancc to C M , and a rangedepcndcnt rclativc cost. Common grid midcoursc guidance is charactcrizcd by accuracy, modcrate resistance to C M , and a moderate rclativc cost, whilc command to L O S has a rangc-dcpcndcnt accuracy and low resistance to C M , and is r c l a t i v ~ l ~ incxpensivc. Antiradiation homing (ARH) posscsscs modcratc accuracy, a low shutdown thrcshold in thc prcsencc of C M , and is ~noderatclycxpcnsivc. Finally, torrelation matching is charactcrizcd by a high accuracy, modcratc rcsistancc to CMq

Sec. 6.3

Multlple Mode Guldance Modellng

299

and a high relative cost [I-Iardy. IOXhl. As described in Exa~nplc6-1, the implerncntatioll of midcourse guidance in SM-2 was the result ofadding to SM-I an IRU together w ~ t hn~issile-shipcomniunication links. The IIIU comprises an instrument package and a s p e c i . ~ l - p ~ ~ r pconlputcr. os~ Included i l l the instrutncnts are three acccleromctcrs, two rate-integrating gyros for measuring angular motion at right angles to the missilc body, and a space-stable, single-asis platfor~ilfor monitoring missile roll. The primdry f~unctionofthe co~nputeris to solvc the equations ofmotion to keep track of missile position, velocity, and attitude in the inertial coordinate frame cstablishcd before launch. In the AEGIS application, c o n ~ m a n dniidcourse guidance is ~ ~ s c~vhile d , in TEIIRIEII and TARTAR applications, inertial midcourse guidance is used [Witte and McDonald. 19811.

Command midcourse guidance. With the AEGIS combat system, the ship uses direct control to command-guide SM-2 during midcourse flight. This is depicted in Figure 6-4e(i). Missile and target tracking are provided by the ANISPYIA radar, which also f~inctionsas the shipboard data link transceiver. The ship weapon-control computcr is used to compute steering commands, which are then transmitted to the missile on the ~iplink.The missile in turn acknowledges receipt using the dohvnlink. A form of PNG \ ~ i t hsuitable trajectory-shaping modifications serves as the nlidcoursc guidance lalv. The primary function of the IRU is that of an attitudc reference system, converting the uplink guidance commands in inertial coordinates to steering conlmands in missile body coordinates. Prior to intercept, there is a specified time at which an illuminator is assigned by the weapon system to the engagement to support the terminal phase. Following this, the missilc is given a command to search for the target based on updated seeker pointing and Doppler frequency information. Hdndovev, ~vhichis the term for the change from midcourse to terminal guidance, takes place after the missile seeker acquires the target. The use of midcourse guidance brings about a marked improvement in intercept capability relative to SM-1 in both downrange and crossrange [Witte and McDonald, 19811. Command-to-LOSmidcourse guidance. Figure 6-17 depicts a radar command to LOS guidance for air-to-surface environment [Hardy, 19861. Most of the early SAM used radio command-to-LOS guidance. The operator(s) tracked the enemy aircraft in a steerable telescope, fired the missile, and then sent steering commands by means of a radio link to keep the missile constantly aligned with the target. Even today, radio command-to-LOS guidance is very important, though all command systems using a radio link are vulnerable to countermeasures. Details of command-to-LOS guidance are presented in Section 6.4.2. Inertial midcourse guidance. Inertial guidance is a self-contained trajectory guidance method and demands an IMU, or portions thereof, to sense missile motion and generate guidance signals to control the missile to fly to a specific location. Inertial guidance is used in various simplified and relatively less costly forms to enable the missile to attack well-defined (usually long-range) stationary

CONSTANT RANGE SPHERE TRAJECTORY

NULL PLANE

Figure 6-17

Radar C o ~ l l t n a n dto LOS Guidancc (Courtesy oJF. W . Hardy, 1986)

Sec. 6.3

Multiple Mode Guidance Modeling

301

targets, the user having first translated the present or future position of the target into numerical coordinates acceptable to the guidance computer. Inertial guidance trajcctorics arc usually optimized to achieve maxinium missile range. Ballistic trajectories, which arc deter~nincdby a consideration of gravitational forces and air resistance and do not depend on aerodynamic lift for missile support, are commonly used in inertial guidance. The trajectory data arc storcd in the missile prior to launch. The missile through its internal sensors, such as the Ills and accelcron~cters,also measures its position relative to the la~~tlch point. I>cviarions from storcd and measured position data arc converted to acceleration commands which arc executed through the missile control system [Titus, 1983]. In the TERRIER and TARTAR applications, given that command guidance is not practical since these combat systems do not track the missile, SM-2 uses inertial midcourse guidance for self-navigation, A three-dimensional search radar is used to provide target data, which is in turn used by these ship systems to direct the missile to a point in space. Actually, the missile may be directed to a succession of points if the target is changing course and/or speed as shown in Figure 6-le(ii). When necessary, new guidance points, along with target position and velocity data, art communicated to the missile via the uplink. The ability to downlink missile information allows the ship system to monitor flight progress. In the midcourse guidance law, the steering equations are explicit functions of the current and desired boundary conditions (position and velocity). For this reason, it is termed explicit grridorlce. The missile's IRU supplies the missile with knowledge of its own position and velocity. As with the AEGIS system, there is a predetermined time in the flight at which an illuminator is assigned by the weapon system to support the terminal homing phase. The IRU and uplink data are used to compute the Doppler frequency and angle information required for target acquisition by the missile seeker. The greater uncertainties in missile position and heading associated with the TERRIER and TARTAR systems mean that terminal handover, which follows acquisition, cannot be delayed as long as in AEGIS. Intercept coverage of TERRIER and TARTAR ships with SM-2 is also significantly increased relative to SM-1 against both incoming and crossing targets [Witte and McDonald, 19811.

Example 6-6: Midcourse guidance for stand-off tactical weapons. Stand-off (or midcourse) guidance is required when the missile is launched at such long ranges from the target that either the missile seeker cannot "set" the target or, if it can, the available guidance information is of sufficiently poor quality that it is unusable. In such cases the guidance law usually consists of a preprogrammed strategy such as "maintain launch heading and a constant altitude" or "fly directly at where you think the target might be." In some cases tactical missiles do not have seekers, and the complete trajectory can be thought of as a type of nlidcourse guidance. A requirement exists for an autonomous, all-weather, jam-proof, quickly targeted midcourse guidance capability for use in tactical stand-off weapons. Pure inertial guidance has always been an attractive option for midcourse guidance, but the cost of high-accuracy (high-quality) gimballed inertial navigators may preclude

Guidance Processing

Chap.

NAVIGATION lNlTlALlZATlON DATA [POSITION. VELOCITY. ATTITUDE1

AIRCRAFT

- I - - - -

MISSILE VELOCITY &

I

SYSTEM

t

KALMAN FILTER ESTIMATOR

AIRCRAFT WEAPON

I

CORRECTIONS TO NAVIGATION PARAMETERS (I SENSOR ERROR ESTIMATES

Figure 6-18 Airborne AlignmentiCalibration Concept (Froin [Perliilurter aiid Fitsihm, 19801 with permission fvom AGARD)

their use in "throw-away" tactical weapons, and the inaccuracy o f low-cost strapdown inertial navigators with low-cost sensors has prohibited their use unless POsition aided by an external source. The primary cause of the inaccuracy in these strapdown navigators is gyroscope and accelerometer sensor errors. As low-cost sensors typically have large turnon-to-turnon bias shifts, through a careful in-flight alignment and calibration process the predominant sensor errors can be reduced by an order of magilitude. This performance whcn coupled with a precise position and velocity initializa;ion, projecccd to be available from the tactical weapon-carrying aircraft through hybrid LORAN or GPS inertial systems, makes unaided strapdown inertial navigation a viablc midcourse guidance candidate for tactical weapons. The in-flight initialization, alignn~ent,and calibration technique is illustrated in Figure 6-18. The strapdown navigator is initialized with currcnt position, velocity, and body attitude information from the aircraft navigation system, after which the system keeps current its own na\,igation solution. The Kalman filtcr estimator contains an crror model for the missile navigator which ~nathematicallydescribes critical strapdown gyro and accelcromcter error terms. This crror model is driven by the trajectory dynamics as scnsed by the strapdown system. Periodically during the mated flight thc Kalman filtcr samples aircraft and missile velocity and computes velocity, alig11111cnt. and scnsor corrections to thc strapdown system. Thcsc corrections arc fed back to thc navigation proccssor and incorporatcd into the navigation solution. The estimation proccss is itcratcd until weapon rclcase. To enhance obscrvability of dominant crrors, aircraft mancuvcrs can bc performed [Pcrln~utter and Fitschen, 19801.

6.3.2 Terminal Guidance

In those cascs whcrc a tcrn~inalseckcr is iockcd onto a target and providing reliable tracking data (short-range combat), the stratcgy is callcd tcn,rir~alp i d a r ~ c e .Thc dy-

Sec. 6.3

Multiple Mode Guldance Modeling

303

namic rcquircmcnts of tcrnlinal guidance are usually niore stringent because all the trajcctory errors which have accumulated must be corrcctcd in a very short time. As discussed in Chaptcr 2, typical terminal guidancc sensors include laser, TVIvidco, imaging IR (IIR), MMW, microwave, and ARH. Thcse typcs of sensors provide different typcs of tcrminal guidance tracking.

Example 6-7: Laser guidance. Lascr-guidcd wcapons homc on rctlected cncrgy. Whcn an observer directs a laser beam a t the intended target, a spot of laser light is caused to appear on thc target. As the weapon detects rcflccted laser light from the target, the autopilot stcers a course to impact on the lascr spot. The guidance system using special equipmcnt is capablc of detecting the lascr cither a t night or in bright sunlight. The laser beam is produced by a device known as a designator, which can, be man-portable and which can also be carried in an aircraft , must be a direcr, LOS between the or in a RPV (rcmotcly piloted,vehicle). There designator and the target, and the laser must operate during terminal guidan'ce. Although the target may be screened by dust or smoke, laser systems are normally designed to operate durlng marginal weather conditions and low clouds [Christal and Mackcy, 19841.

.

Example 6-8: Fiber-optics guidance (FOG). The optical-fiber guidance (see Figure 6-19a) involves a gunner locaccd in the ground station ~ v h oreceives information on the general location of enemy troops. Based on this information, a missile is launched toward the general target area from the ground station. Following a climb to cruise altitude, the operator begins a search and acquisition phase (see Figure 6-19b). The near-constant altitude and heading may be preselected. During this time, while viewing the target scene presented by the missile-borne imaging sensor, the operator has control over the sensor FOV, look angle, and the missile flight path. Having acquired a target of interest, the operator then initiates the terminal trajectory. Interesting to note is that throughout the flight, the launch vehicle may remain in a protected position and is not subjected to direct fire from enemy systems. There is considerable freedom in selecting the optimum trajectory both for launcher protection and for maximizing the total system effectiveness. The use of a human operator during missions and other unique features determine what the performance requirements are, which include trajectory ;haping directed at optihizing the operator's performance in the areas of target acquisition and terminal guidance. This in turn dictates specific aerodynamic performance of the flight vehicle as well as the missile-borne sensor. In order to take advantage of the unique qualities of optical fiber, a launcher design would have to allow engagements from defilade. This means near vertical launches necessitating unique missile control and aerodynamic capabilities. Additional requirements are placed on the missile trajectory to account for the pilot-in-the-loop and the functions of navigation and target acquisition that the pilot must perform. In addition to this, deploying a bidirectional optical-fiber data link is required. Thetocation of the target is generally known but the actual detection of the target is a function of the ground controller which uses

Guidance Processing

SEARCH/ACQUISITION

TERMINAL

LAUNCHER/CONTROL (A)

DATA COLLECTION

LAUNCH VEHICLE

OPERATOR DISPLAY

ow Figure 6-19 (a) Optical Fiber Guided Missile (b) Target Acquisition (Fro~n/Holder, 1983 & 19841 ujirh ycnnissiotz.fiot11 SCS)

Chap. 8

Sec. 6.3

Multiple Mode Guidance Modeling

305

information transmitted downlink from the missile-borne scnsor. Models used in the primary computer sin~ulationsinclude target location and composition. environmental factors, sensor parameters, and operational requirements. Outputs are target acquisition probabilities and specific sensor performance data. The TV sensor information is transmitted downlink for both human and electronic processing, and guidance commands arc then transmitted uplink to the autopilot [Holder, 1983, 19841.

Example 6-9: STANDARD Missile-2 terminal guidance. In the selfnavigation phase that follows midcourse guidance, known as terminal guidance, SM-2 homes on the target until intercept takes place and the missile warhead is detonated. The necessity for a terminal guidance mode is a consequence of the fact that midcourse guidance is not suitably accurate, even at short range, to regularly achieve miss distances less than the lethal radius of the warhead. Every SM-2 missile variant employs the same terminal guidance equations. The SM-Zterminal guidance loop is shown in block diagram format in Figure 6-4f. Characteristics, such as the angle of arrival, of the signal received from the target will be affected by the type of targct and by missile motion. M~ssilepitch and yaw motion are ren~ovedby inercially stabilizing the seeker antenna that receives the signal. The two main ourputs of the missile receiver are an LOS measurement and a closing velocity or Doppler measurement. These measurements are used by the guidance computer to generate steering commands in accordance with a PNG law. The autopilot, just as it did during the midcourse guidance phase, converts the guidance computer's electronic commands to missile tail deflections in order to vary the trajectory of the. missile. The lateral missile motion provided by this process, coupled with longitudinal motion provided by propulsion, causes the missile ro fly on an intercept course with the target. The endgame is the final phase of flight in which there is a high-speed encounter with the target. Crucial to this phase is accurate timing of the warhead detonation. When the missile is in close proximity to the target, a small radar aboard fuse, senses the target's the missile, known as a target detectiondevice or presence. The optimum time to trigger warhead detonation is then computed. The missile also contains a contact fuse to detonate the warhead on impact should the missile collide with the target before the computed optimum time [Witte and McDonald, 19811. Example 6- 10:Terminally-guidedsubmunitions (TGSM) guidance. With TGSM, there is the problem of achieving accurate guidance in very limited space and at low cost. Following Trottier [1987], this example looks at the design problem of attacking a tank from a TGSM initially in level flight. The assumption of an upper limit to seeker sightline rate resulted in a low TGSM speed and hence a large wing size, and it may therefore be preferable to go to the more usual vertical approach from parachute suspension. The last several years have witnessed increasingly sophisticated flight control and seeker technologies for guided weapons. The implementation of inertial strapdown reference systems that rely on low-cost fiber-

306

Guidance Processing

Chap. 6

optic gyros will undoubtedly contribute greatly to the solution ofthe cost and weight problems associated with more complex navigation, flight control, and seeker attitude reference systems. The requirement for ground-attack aircraft to effectively engage n~ultivehiclearmored targets in both close air support and battlefield interdiction roles places increasing demand on the intelligence and adaptability of the weapon system employed. O n e solution to the problem is the use o f a weapon which consists of a flying dispenser (bus vehicle) which is released by an aircraft some distance from the target and which subsequently delivers a number of smart, autonomous TGSM into the target area. T w o weapon concepts are investigated solutions to the problem. The first concept involves "shoot-to-kill" sensor-fused munitions which, following release from the bus vehicle, climb to provide the titude they require for target search, deploy a parachute to stabilize vertical descent, rotate an off-set detector about the vertical while scanning the ground in a decreasing spiral scan, and fire a high-velocity slug at the target upon acquisition. The other concept involves "hit-to-kill" aerodynamically controlled submunitions equipped with a seeker which guide to the target and detonate a shaped charge warhead at impact. Improved warhead effectiveness can be obtained when hitting the top armor from a near-vertical dive. This adds two constraints to the conventional guidance problem: steep impact, and minimum angle of attack at impact. Whcn TGSM arc released from a bus vehicle, the scenario is many-on-many in that several TGSM have to handle several targets against various backgrounds and levels of clutter. The targets arc armored vehicles 6 x 3 x 2 m moving in columns on roads, moving in clutter, stationary in clutter with engine running, and stationary in clutter with engine off. If one considers one TGSM only, the scenario becomes one on many. A typical TGSM attack scenario is sho~\.nin Figure 6-20a. Its flight comprises three phases: (1) the fly-out phasc during which the bus vehicle is dispensed from the aircraft and flies up to reach a fixcd altitude h at which the TGSM are released. Upon release, the TGSM initiates the deployment of its lifting and control surfaces a,nd becomes ready for target search;; (2) the search phase during which it flies level (or glides) while scanning the ground for a target; and (3) the tcrminal homing or track phase during which it guides to hit the target. During the tcrminal ho~ning guidance phase, the scenario is one on one. Figure 6-20b depicts the geometry of the scarch phasc. The TGSM is at altitude h and flies lcvel at vclocity V,,,. I t is first postulated that the TGSM is cquippcd \vith only onc scckcr which can opcratc in t w o modes: (1) a scanning modc for the scarch phasc, and (2) a tracking nlodc for thc tcrminal homing guidance phase. The seeker searchcs the ground at a look-down anglc a. The maximum slant rangc R at which the target can bc dctcctcd is lill1itcd by the detection rangc if thc scekcr uscs an M M W radar. Thc altitude / I is limited by cloud cover if thc scckcr uscs passivc 111. If the look-down anglc is fixed, both 11 and R are linked. The vclocity V,,, is also linlitcd by the characteristics of thc scarch modc. From altitude /I and level flight, any stationary object on the ground will appear as moving angularly at angular speed u = I/,,, sin' all^. If the look allg]~ is 45 deg and the maximum gimbal rate of the scekcr is 25"/s, thcn, for 11 = l j O

Multlple Mode OufdanceModellng

Sec. 6.3

RELEASE Of THE TQSM

DUAQC

DEPLOYMENT OF FINS AND WEKiS

ELEASE OF THE BUS VEHICLE

>

I 400

t

200

IMPACT ANOLE

I

I

600

800

I000

1200

1400

-

--------,----------.REFERENCE

T U

I

LINE O F SIGHT TARGET

I

- DOWNRANGE x (B)

Figure 6-20 (a) Sketch of a TGSM Attack Scenario (b) Search and Initial Conditions for Terminal Homing Guidance (From [Trottier, 19871 with permisrionjom AGARD)

Chap.

Guidance Processing

308

IPulse Rocket I

- (Chap.

Target Tracking Sensor

-

8.9)

lntegraled Sensing/Flight Reference S Control stem

J

ISFCR~ (Chap. 5)

or

Inlcgraled Inerlial Navigation System (Chap. 5 )

Figure 6-21

Midcourse Phase NGC Processing

m, the preceding equation yields V,,,lh = 0.873. Beyond this value, any attempt to eventually track a target is hopeless. Since moving targets are also considered, the relative velocity must account for an inbound component of targct velocity. For Ih = 150 m, and inbound target velocities of 20 mis, the T G S M velocity cannot exceed 110 mls. A terminal homing guidance law that will guide the TGSM to a near-vertical hit with the tank must be determined [Trottier, 19871.

6.3.3 Error Analysis Model Development

Midcourse guidance system analysis.

The missile rnancuvcrs are rclativcly small in the midcourse phase. Hence. thc resulting crror analysis is uscd to study the hcading crror of the missile to the intercept point at the handover from midcoursc phasc to terminal phase. The hcading crror at handovcr has an important effect on the miss distancc and servcs as the initial crror of thc terminal phase. Thus, to compute this heading crror, the crror sourccs need to be analyzed. Thcy include instrumental initialization error and drift, rcccivcr noise, guidancc algorithm, and flight control response. Computing the heading crror requircs a realistic mathematical model as shown in Figure 6-21. The IRU onboard provides thc flight vehicle position and velocity in inertial coordinatcs. Thc targct tracking radar uplink providcs t11c targct position and velocity. Thc trockir~,~ filter is uscd to attcnuatc noisc cffccts and thcrcforc rcducc the hcading crror at handovcr. Trajcctory shaping and optimization is applicd to causc the flight vchiclc to rnancuvcr cffcctivcly. that is, with minimal cncrgy loss and in thc shortcst timc, along tllc dcsircd trajccrory. The output of trajcctory shaping and optimization is a tnidcoursc steering command in incrtial coordinatcs. A pulse rockct firing control is uscd to i n ~ p r o v cthc propulsioll pcrforrnance o f a flight vchiclc in order to tnaxinlizc thc tcrmillal vclocity and hcllcc to cxtcnd intcrccpt rangc. The FCS can bc cithcr tl~rustcontrollcd or scrvo cot]trollcd. If thrust vcctor control is uscd, a thrust 1 ~ ~ 1 0co~tra~ocld 1. ,qccrc.rororis nccdcd to gcncratc thc thrust anlplitudc and dircction commands. A thrcrst c~ectnr(nr~tr.c~ll~'r is

Sec. 6.3

Maneuver Modeling

-

Multiple Mode Guidance Modeling

3Og

-

Homing and End Game Target Trackin Procasing (Fig. k 8 ) Actuation

Figure 6-22

Terminal IJhase NGC I'rocessing

used to follow thc thrust vector command. The thrust amplitude obtained by controlling the exhaust mass flow rate, and the thrust direction generated by controlling the thrust vector control servo are combined to construct a thrust vector control. If a servo control is used, an autopilot is used to follow the trajectory shaping and optimization commands and to stabilize the system during flight. The actuator (which consists of control surface and servomechanism) is used to change flight vehicle attitude and therefore change flight vehicle trajectory. The servomechanism may be pneumatic, hydraulic, or electric, depending on the maximum hinge moment of the control surface.

Terminal guidance system analysis. Steering of the missile during the last few seconds of flight has the most direct effect on terminal miss distance. Hence, the terminal phase trajectory of a tactical missile represents the most critical period. A homing sensor providing accurate information about the target's location relative to the missile contributes to the overall success of a mission. A steering law capable of achieving intercept in the presence of target maneuvers and measurement errors also contributes to the mission success. Environmental turbulence and gust, instrument initial bias (IRU initialization error and drift), target maneuvers, seekerltracker noise and bias, guidance law capability, flight control response and dynamic nonlinearity are all important factors that affect the miss distance in a homing-guided missile. For a radar-guided missile, on the other hand, the radome refraction is a very important factor. The error analysis of terminal phase trajectory is illustrated in Figure 6-22. Prescribed error sources and realistic mathematical models of the guided missile are included in the error analysis study to assess the miss-distance statistics. A multimodel seeker is considered in Figure 6-22. The relative motion among any of the multiple targets and the flight vehicle is detected by R F detection, IR detection, andlor T V detection. In RF detection processing, a radar search mode is needed to acquire target positions in the FOV of the seeker, and a tracking mode is then taken to track and lock on the target. Search mode is activated when the estimated target-flight vehicle range is within the detection range of the RF seeker. There are three major search patterns used in the search mode: spiral scan, raster scan, and sector scan. In the tracking mode, range-tracking servo uses the range gate technique to position the target return signal in the middle of the gate. In case of gate stealing due to the effects ofjamming or clutter, a coast mode or gate search mode is required.

310

Guidance Processing

Chap. 6

As soon as the target position is acquired in the FOV of the seeker, the target is locked on and tracked by the seeker. However, outputs of RF detection are con, taminated by disturbance noise. The RF seeker is then taken to obtain the LOS rate which is proportional to the misalignment of the target tracking error. The latter is the error between the output of RF detection and that of positiorz sensing in the seeker. A rate-sensing feedback is also needed to improve stability and reject disturbances. The sensor used for sensing the servo angular positions are: synchro (Ac excitation), resolver (AC excitation), and potentiometer (DC excitation). A gyroscope/tachometer sensor uses gyroscopes or tachometers to measure the angular rat, of the seeker servo. A gyroscope is usually A C excited, and hence needs to be accompanied by a demodulator. In the IR detection mode, a signal detected by the IR detector is also contaminated by disturbance noise. An IR irnage processor is needed to provide a two-dlmensional image with target and background. The IR image processor consists of a head assembly, scanners, IR optics, 1R detector, IR cooling, preamplifier, amplifier, LED array, synchronization, and display. A T V camera of C C D type is usually used to acquire a two-dimensional image for T V detection. The video-tvackitig servo consists of an auto video tracking (AVT) unit and a tracking servo. AVT can generate a target-tracking error with respect to the tracking gate and thus always keep the target at the center of the tracking gate (or tracking window). AVT also provides the gate positions in the elevation and azimuth axes. The tracking servo has the feature of tracking on stabilization. That is, based on stabilization, the elevation and azimuth gate errors given by AVT are thus polarity correct. A targettracking algoritl~ttiis then used to take the LOS rate from the aforementioned multimode seeker. The target-tracking algorithm output is the input to the guidarlce algoritlztn. Finally, an accelcration (orflyensation, an autopilot, and an actuator together cause the flight vehicle to maneuver so as to intercept the desired target.

6.4 GUIDANCE ALGORITHM

A guidancc algorithm is formulated for the sake of ensuring pursuer-target intercept. As can be scen from Figurc 6-1, guidance algorithms can be classified according to whether thcy are preset or direct, both of which are investigated in this section.

6.4.1 Preset Guidance This section d~scusscsthc coticept of a prcsct guidance algorithni based on an csccllent description of guidancc techniques appearing in Vriends [1Y87]. Thc most important input for a missile guidance system relates to the targct position. A preset guidancc lncthod makcs use of prelaunch information on thc target position and motion provided to thc missilc system's fire control subsystcm. During missile flight, no targct information is scnscd. O n thc contrary, flight path corrections to the missile are dcrivcd from missile statc infortnation obtained by navigational or

Sec. 6.4

Guidance Algorithm

311

flight instruments inside the missile. With a prcsct guidancc systcm, the missile is guided along a desired cruise flight path to the target intercept position. The actual position of the n~issilcis compared with the prestored flight profile data and, if necessary, commands are generated for the missile control systcm to return to the required trajectory. Inertial guidancc is one such example. In the vertical or pitch plane, the nlissilc is controlled by an altitude-hold control loop using a simple barometric altinlctcr, and in the horizontal or yaw plane by a hcading command loop using an angular ratc gyro which monitors any change in hcading to keep the n ~ i s s ~ l c in the prcsct flight direction. More sophisticated prcsct gutdance systems for targets at medium or long range contain an attitudc and hcading refcrencc systcm (AHRS). As shown prcviously in Chapter 5 , the AHRS is a strapdown unit consisting of three ratc gyros, three low-cost accelerometers, and the associated electronics. The ratc gyros provide input signals for attitudc estimation and for autopilot rate damping (flight stabilization). Accelerometers are used for autopilot feedback and self-leveling of the strapdown system, and can be used in combination with other navigational sensors to yield accurate estimates of missile attitude and heading. An example of a reliable, accurate and cost-effective AHRS is the ring-laser gyro system or IRS. The laser gyro senses angular rate with respect to inertial space and has n o moving parts, making it insensitive to errors due to mass properties (which affect rotating mass types of gyro). The most accurate preset guidance method can be obtained with an INS. The primary sensors of the INS are three accelerometers and three rate gyros mounted orthogonally. The accelerometers sense the nongravitational acceleration of the missile while the rate gyros sense the angular rate of the missile. A gravity computer calculates the acceleration due to gravity from actual information on the missile's position. Together they provide data to determine by integration the missile velocity and position in a navigational frame of reference. In preset guidance, the frame of reference is either physically or analytically established within the missile. Accelerometers placed on a stabilized, gimballed platform are physically isolated from the pitch, yaw, and roll motion of the missile. Platform stabilization is provided by a set of gyros which can be torqued to establish the desired navigational frame. When the accelerometers are mounted directly on the missile structure, they can be isolated analytically from missile rotational motion with the help o f a set of strapdown rate gyros. The missile airframe angular rate with respect to inertial space is provided by the rate gyros and is used to calculate the attitude of the accelerometer measurement frame with respect to the navigational frame o f reference. This attitude information is used-in a coordinate transformation algorithm to compute the acceleration in the navigational frame. Although less accurate than the platform system, the strapdown system offers a cost-effective solution for the autonomous inertial guidance of guided weapons in their midcourse guidance phase. Furthermore, since the gyros and accelerometers take measurements in an airframe coordinate system, their output signals can be used for autopilot functions. Most ballistic missiles are guided in their boost phase by a platform INS because of the high accuracy demanded. Inertial navigation, such as a dead-reckoning method, is afflicted with navigational errors that increase with time of flight. TO

312

Guidance P r ~ ~ e S s i t I g Chap. 6

cope with this problem, position and/or velocity updates can be made to reinitialize the position and velocity integrators and even to estimate observable errors of the INS (Kalman filtering). Update systems or navigational aids such as radio and satellite navigation systems, star trackers, and earth reference systems are used. These last can make use of a mapping radar, an IR imaging sensor, or a T V sensor onboard the missile to obtain position fixes. A Doppler radar can provide ground-speed data for velocity updates. An earth reference system developed for long-range cruise missiles to update the INS periodically is TERCOM. Along the preestablished missile flight path, several check areas are chosen to reinitialize the INS. Above the check area, a radar altimeter with a good horizontal and vertical resolution measures elevation of the terrain, resulting in a sequence of altimeter readings along the missile track. The sequence is correlated with a prestored digitized map of the check area taken from the memory unit to determine the best estimate of the actual position and subsequently to correct the INS. Guided weapons with preset guidance are passive, at least over the unaided portion of missile flight, and hence are difficult to jam. After launch thev are autonomous: the launching" station or the fire control unit is not involved in the guidance phase. The weapon has a fire-and-forget capability. Preset guided weapons are used against stationary targets or slowly moving targets. For moving targets there is a requirement for a direct guidance method in the terminal phase of the engagement.

6.4.2 Direct Guidance Methods Direct guidance methods, which are characterizcd by the use of turning target information in the prelaunch phase and during the guidance phase, make it possible to intcrccpt moving targets. The guidance system is dominated by a guidance law (see Scction 6.5) that formulates the relation between sensed target or missile information and guidancc commands for the missile, and is implemented in the guidance computer or the correcting network. Referring once again to Figure 6-1, those algorithms that fall under the category of direct kuidance include command guidance, beam-riding guidance, and homing guidance (see Section 6.4.2). A real-life guidance system is also presented in Scction 6.4.2 to exemplify a direct-guidance application. At least one of the guidance sensors maintains LOS contact with thc targct and provides targct illformation to the guidance and control system [Vricnds, 19871. The basic principles of guidance-loop dcsign for missile systcms arc well cstablishcd, and future systcms arc unlikely to differ conccptually in any significant way from prcvious systems. That is, the fundamental limitations of dircct-guidance methods remain the same although the implclncntation of such systcms will, of course, be crucially dependent on the currcnt sratc of hardware development. LOS guidance. There are two subsets of the LOS guidance which diffcr basically in their n~cchanisms.LOS guidance is a silnplc conccpt and can bc ilnplclnented in terms of a command to line-of-sight (CLOS) (Figurc 6-23a) or bcam rider (Figure 6-23b). I n CLOS guidancc, a n uplink is uscd to transmit guidallcc

Sec. 6.4

Guidance Algorithm

313

signals from a ground controller to the r~iissilc.l'hc iitnction of the beam-rider guidance scheme is to track an E O beam directed a t the targct. The missile is first guided near the ccntcr o i t h e beam and theti flies toward the targct. The bcatn used to direct the missilc to the targct has two fi~nctions.It can be used to automatically track the targct, or it can be directed to point at the targct by sonic other means. As noted in I'astrick ct al. [1981], the bcani-rider tiiissilc requires onboard autopilot cotiipensation since the projector docs not know the missile's location once it is en route. l ' h e CLCIS scheme, however, docs track the missilc and thus cotiipcnsatcs for its position prior to transmitting the guidance signal via the wire link. A typical LOS guidancc trajcctory is illustrated in Figurc 6-23c where the missile performance depends on the gconlctry and the nlissile-target speed. As the targct speed increases, the acceleration of the missile must also increase. A rear receiver is required for these guidancc schemes. With the rnissilc remaining in the beam that tracks the target, relative positions of the target and missile are shown at various tinics in Figurc 6-23c. The individual targct and missile velocities during a real cngagctncnt determine xvhat the actual trajcctory will look like. What is dcmonstrated in this figure ~,is that LOS guidance should not imply to the reader that the rnissilc has only to fly up the bcam to intercept the target. Quite the contrary, the missile must be able to turn or develop velocity that is equal to the LOS rate and perpendicular to the LOS. as stated previously, in order to intercept a fastmoving targct. The angle between t h e tracking bcam and the missile axis demonstrates the requirement for a broad-beam rear antenna [Heaston and Smoots, 19831. LOS guidancc, in contrast to homing guidance, does not require a seeker in the front of the missile. This allows the nose to be shaped so as to minimize the aerodynamic drag [Heaston and Smoots, 19831. As shown in Figure 6-23c, LOS guidancc requires continuous maneuvering and the miss distance increases with range. The resulting miss dlstance is iLJ = R ~ ~ E+ : E;,, where E, is the targettracking error and €,, is the missile-tracking error [Hiroshige, 19861. A detailed development of the beam rider and CLOS implementation is provided by Clemow [1960]. A description of the bang-bang approach to LOS guidance commands can be found in Harmon et al. [1962]. Ivanov [I9751 studied radar-based guidance methods and typical implementations of missile-borne seekers. A CLOS guidance scheme is employed by Kain and Yost [I9761 for a defensive ship, where optimal linear filters are used to reduce the inherent beam jitter. In Thibodeau and Sharp [1969], a pulse-duration scheme is proposed for a wire-guided missile. LOS guidance works very well for missile interception as long as a stationary target is being tracked. In this case, almost perfect guidance can be achieved without the man-in-the-loop tracking error. However, improper tracking occurs in a realistic situation which leads to errors. The performance with respect to miss distance is shown to be better than 1 ft with a 90 percent confidence level, assuming that a reliable round is fired. -

-

Command guidance. Command guidance is the guidance of a vehicle by means of electronic command signals generated outside the vehicle and sent to the receiving mechanisms contained in the vehicle. The basic concept of command guidance is to study the relative position and velocity between target and missile, and

Guldance Processing

314

Chap. 6

PROJECTOR

ACCELERATION

.-

----------------- --OPTICAL LINK

EU~ER

----- -- --- - - - - --- ------ - ---.

I

1

ACCELERATION YORE C O Y L E X

-

(B)

Figure 6-23 (a) CLOS Scheme (b) Beam Rider Scheme (c) Beam Rider o r C L O S (i) Beam RiderICLOS Guidance Geometry and Equatlon (ii) Beam RideriCLOS Guidance Trajectory / / a / & Ibjfrotn [Pastrick er a / . , 19811, t Z 1981 AIAA (0(ii).fiotrl [Hrasmn and Sinoors, 19831 m i ~ hpermissionfro~~iG.4CI.4Ci

to uplink commands to the missile to adjust its flight in order to intercept the target a t a certain position. Missile. target, and track are required to be tracked precisely by a guidance (radar) station on the earth's surface or in an aircraft. T h e information sensed is used to shape guidance commands, which the missile may receive in the following ways: by a guidance wire or radio-frequency data link, via the radar or laser beam used to track the missile, or over very thin hairlike wires that uncoil from the missile as it flies. That there is n o need for a seeker onboard the missile is the chief advantage of command guidance. Its chief limitation, however, is that as intercepts occur at greater distances away from the radar there is greater degaccuracy as well. As a radation in measurement accuracy and hence in guidance result, system firepower is increasingly limited. The nlissile accuracy is limited by the noises in the systems. The effect of thesc noises for command guidance is k n o w to be strongly dependent on the range to the object tracked; the dependence on range to intercept is much wcakcr for either active or scnliactive homing guidance. Command guidance systcms can be less expensive than homing guidance systems. Ncverthcless, few command guidance systcms arc currently being built because it is bciicvcd that command-guidance miss distances arc too largc for successfu~interceptions. In order to identify those conditions where the less expensive commatld guidance systcni accuracy may be satisfactory. a systematic analysis of commandguidance miss distance would be most useful. Such an analysis can be found in Chapter 8.

Sec. 6.4

Guidance Algorithm

Guidsna Equation % = Vmf,

KgY') (%Ia ) non

5

-?$EeZn

the bcom

olzia

s) = Filter and lead network Om = Missile angle = y 0 = Target angle = y T ,,R, = Range to missile R.,. = Range to target = Rl + VT tg

R

R

I

-- Intaapt range

i) Beam RidedCLOS Guidance Geometry and Equation T ~ g aPath

'inttrccpt

$am&

/ /Tracking Beam

Equal 10 2 'limes lhrget Velocity

From [Hcaslon and Smmts, 19831 with permission from GACIAC

ii) Beam RiderICLOS Guidance Trajectory (C)

Fgiure 6-23

(Continued)

> &\'

Missile T r a c k i n g 2 \ Radar 8 Command Link I COmrnand

Link

H

\\\\ Target Tracking Radar

I I

computer

(A)

316

Fan Beam Scanning In Azimuth

Figure 6-24 (a) C;eneric Comnland Guidance (Surfacc-to-Air) (b) Track-WhilcScan Radar for Command Guidancc (Surface-to-Air) (From /Heasfor#and Smoofs, 19831 with prnnirsionfrorn GACIAC)

Sec. 6.4

GufdanceAlgorfthm

31 7

Two-Radar Command Guidance. A conlnland guidance system, as shown in Figure 6-242 [Heaston and Snloots. 19831, requires a t least two radars, one to track the targct and the other to track and send guidance comnlands to the missile. As this is taking place, the ground conlputer drives the two sets of readings (bearing, elevation, and range for each) into exact coincidence, thereupon finding the warhcad. This generic command guidance is based on separate target and missile tracking radars, with the command link included in the missile tracking radar. This was one of the first methods to result in a practical SAM, although it is now regarded as obsolete. In the previous figure, thc solid arcs emanating from the tracking radars are the transmitted radar signals, which are reflected by chc target in all directions. The dashed arcs represent the reflect~onin the direction of the target tracking radar. In order to determine the target's range and angle, the target tracking radar processes this reflected signal, sending the information to the guidance computer. (A beacon, or radar transmitter, carried by the missile is triggered by the signal received frqm the missile tracking radar.) The beacon transmits a strong signal back to the missile tracking radar, illustrated by the solid arcs emanating from the missile. This guarantees accurate tracking of the missile and ensures a data link from the missile to the ground. A ground computer receives the missile's range and angle, computes the missile trajectory necessary to intercept the target, and generates commands which are sent to the missile via the command link. This link is illustrated by the r jagged line between the missile tracking radar and the missile. Missile guidance for target intercept is achieved by monitoring the target and missile positions and refining the trajectory calculations throughout the engagement. A critical computation compares target and missile position data which are then used to send a burst command to the missile that detonates the missile's warhead. Because of its complexity, this type of radar command guidance is restricted to long-range SAM against highspeed, maneuvering targets. Low-data-rate radar command guidance is suitable for updating an inertial navigation midcourse guidance system. The advantage of command guidance is that it uses the more powerful radar and fire control computer on the ground to solve the intercept guidance problem, thereby simplifying missile electronics. However, the accuracy of command guidance falls off with increasing missile range, which leads to a requirement for either a large missile warhead or a shot-range e n g a g e m e n t d h e system is to be effective [Schleher. 19861. In the 1950s, the United States employed radar command guidance in the NIKE air defense svstem. Each site where the svstem was deoloved had a number of radars , to track the target and missile. Commands were sent to the missiles via the missile tracking radar. The PATRIOT SAM is an example of an advanced radar command guided system with a multifunction phased array (electronically scanning) radar. The radar performs several fire control and guidance functions simultaneously in a time-sharing mode and gives the system a multiple target engagement capability to make it more difficult to saturate the air defense system. The PATRIOT system, in particular, uses a radar command link. In this system, a semiactive radar target tracker resides in the missile that relays all of its signals to the ground for processing by a powerful computer. The process of tracking via the missile, referred to as TVM, is applied to prevent the accuracy problem previously mentioned. The computer

.

Guidance Processing

318

Chap. 6

generates commands and returns them to the missile to both guide and control the radar target tracker. This data link must be very secure to prevent jamming or interference from influencing the system's operation. Wire command links are Used primarily with antitank missiles that possess a relatively short range. A more modern implementation of radar command guidance is shown in Figure 6-24b, where the operation of a track-while-scan radar is depicted. T w o fanshaped beams are scanned at right angles to each other to obtain elevation 2nd azimuth angles and the ranges of both target and missile. Using this information, a computer calculates the trajectory the missile should fly in order to intercept the target, and commands are then sent to the missile via a command link to control the missile. For successful operation, the missile must remain in the volume scanned by the two tracking beams [Heaston a n v m o o t s , 19831.

Command-to-line-of-sight( c ~ sguidance. ) Command-to-line-ofsight (CLOS) or line-of-sight (LOS) guidance represents a derivative of the command guidance technique. CLOS is a realization of the three-point LOS guidance law. In this method, the missile approaches the target along-the line joining the control point and the target. At the control point, the target is tracked by a radar, an optical system, or an electro-optical system. The missile on the other hand is being tracked either by the same tracking system (differential tracking: radar or EO tracker) or by a separate missile tracker (EO or IR "hot spot"). Thus, it is sometimes called three-point guidance, where the three points represent the tracker, missile, and target. If the missile can stay on a straight line between the tracker and the target (that is, the LOS), then it will hit the target. These guidance sensors provide the angular error signals E of the missile fro111 the LOS td the target in elevation - 0,,,,and E is n~ultipliedby and in azimuth. As shoxvn in Figure 6-23c, E = an estimate of the actual missile range R,,, resulting in linear displaccn~cntfrom the LOS. The lateral acceleration commands a, arc shaped by a suitable crror compensation network G(s) to guarantee stability of thc guidance loop and a fast response to the linear displacement errors. That is,

er

In con~pensatingfor a displacement error through an accclcration command, an anticipating network is required to stabilize thc guidancc loop. Consequently, carc must bc takcn in selecting a statistical filter to scparatc thc noisc from thc LOS mcasurcmcnts. The accuracy of thc CLOS guidancc loop in tcrms of thc stcady-statc crror is improvcd by adding a fccdforward command to thc crror compcnsation command a,. The feedforward command u,,,is the demandcd latcral accclcration for thc missile to stay on the rotating LOS. This accclcratior~command is dcrivcd from mcasurcmcnts of the LOS rotation and estimates of missile range, velocity, and accclcration in the direction of flight. That is, I,,,, =

v,,,+,,, whcrc y,,,

= 2@r+ R,,, eTl V,,,

(6- 1b)

is the mancuvcr rcquircmcnt [Hiroshigc, 19861. Thus, thc guidancc accclcr~tio~~

Sec. 6.4

Guidance Algorithm Missile airframe FeedboCk loop

,

Correction/ a~lpenwtion network

t

Feedforward loop

Infrared sensor >

I

Observing

I Figure 6-25

4

Aiming

Operolor

I

Semiautomatic CLOS lor an Antitank Missile illepri~rfedtuirll per-

command u, is equal to a, plus u;, in a CLOS guidance system. Figure 6-25 shows a functional block diagram of an antitank, semiautomatic CLOS missile, depicting the feedback guidance loop and the feedforward loop. The IR sensor, which is boresighted with the optical axis of the sight, senses the angular deviation of the missile from the LOS to the target. The feedforward loop accounts only for the Coriolis acceleration. Additional feedforward acceleration terms are required when large LOS rates occur, as is the case with CLOS for short-range air defense. If a track-while-scan radar or another method is used to provide more information to the guidance computer, it becomes possible to implement guidance laws other than the three-point method. Trajectory shaping can be exploited to reduce the demanded lateral acceleration by inserting more lead angle capability or to increase warhead effectiveness by approaching the target's most vulnerable side. CLOS is best suited for use in short-range air defense and antitank systems. Inasmtuch as the guidance equipment at the control point is dedicated to only one target throughout the engagement, guidance accuracy is inversely proportional to range and firepower is sharply restricted [Vriends, 19871.

Beam-rider guidance.

Beam-rider guidance, which is a three-point method, utilizes a sensor located in the missile to generate internal signals responsible for ensuring that the missile remains within a beam that is aimed at the target. The beam may be either radar or laser in nature, and is generally maintained by a subsystem on the launcher. A narrow laser or radar beam is projected accurately towards

Guidance Processing

Chap. 6

Beam Riding Missile

I

. I

Missile I-ouncher

Illumination Radar ( T r a c k s Target

Figure 6-26 Beam-Riding Guidance (Fvoi~i[Heastorr a ~ ~Slnoofs, d 19831 with perinissiotr j o m CACIAC)

the target using radar or an EO, automatic, or manual tracking unit. The beam is modulated and the missile carries a rearward-looking beam receiver so that the missile can determine its up-down and right-left positions in the beam and guide itself to the bcam axis. The result is an interception trajcctory according to the threepoint method which is also used for CLOS [Vriends, 19871. Radar beam-riding guidance is illustrated in Figure 6-26 in which the illumination and target-tracking radars are shown as a single unit. The transmitted signal, which the missile receives and uses for guidance, is represented by the solid signify the targct reflection in thc direction of arcs. Thc dashed arcs in the figure the radar, which picks up this echo signal and processes it to ensurc that the radar beam always remains pointed in the direction o f the target. In order to guarantee that the signal is received as the missile turns, the missile receivcr (sensor) antenna needs to have a wide enough beamwidth in the rcar aspect. This situation occurs in the case of certain intercept geometries like those associated with high-speed crossing targets. Here, the missile attains a large velocity component perpendicular to the center of the bcam as it attempts to follow the beam's angular motion during targct tracking. Thc type of missile trajcctory found with a bcam-rider guidance system is drawn in Figurc 6-23c which is thc same as that with a C L O S guidance system. A disadvantagc with beam-rider guidance is that angle tracking crrors in the guidance beam result in posttlon errors that arc dircctly proportional to thc range bctwccn the targct trackcr and the targct. Conscqucntly, an above-standard tracklllg system that has slnall angle tracktng errors is csscntial for accurate guidancc a t 101% ranges [Hcaston and Smoots, 19831. Laser bcanl-ridcr guidance is uscd for antitank (SSM and ASM), atltihclicoptcr (AAM), and short-range air dcfcnsc n~issilcs(SAM) bccausc of its high clcctronic countcrmcasure rcsistancc. MMW bcanl riding is an all-wcathcr altcrnativc for laser bcam riding. Radar bcam riding is still uscd in older, medium-rangc air dcfcnsc and antiship n~issilcsystcms [Vricnds, 1987).

Sec. 6.4

Guidance Algorithm

321

Example 6-11: Talos beam-riding midcourse guidance system. This example is based on an excellent article by Gulick ct al. [19821. In a beamriding system, anytime the missile leaves the axis of a conically-scanned radar beam, acrodynan~icsurfaces will deflect so as to cause the missile to return to the beam axis. In essence, the vector that indicates the angular dircction and distance to the scan axis must be determined by the missile. The amplitude of the modulation produced by the conically-scanned beam can be used by the missile to determine this distance, while the phase of the amplitude modulation with respect to the scanfrequency rcfcrcncc signal determines thc angular dircction. The conically-scanned radar b c a ~ nshown in Figure 6-27a was employed by the beam-riding system for Talos (see Example 6-2). This, coupled with a sinusoidal variation of the pulse repetition frequency, provided the missile-borne receiver with the necessary signals for measuring missile angular distance and direction perpendicular to the nutation axis of the guidance beam, as illustrated in Figure 6-27b. A free gyro was used to roll stabilize the missile in flight so that error signals would be directed to the correct aerodynamic steering surface. Pulse groups were radiated by the guidance transmitter at a nominal rate of 900 pulses per sec, and each group consisted of three pulses that were coded by intcrp~~lsc timing to identify the guidance beam to the missile. Transmitting nutation position information was accomplished by frequency modulating the pulse group rate at the nutation frequency of 30 Hz with a deviation of k 5 0 groups per sec. An inertial system was necessary onboard a ship in order to compensate the pulse group modulation for roll and pitch. Figure 6-27c illustrates a simplified block diagram of the beam-rider system. Once detected, microwave pulses were passed through a decoder to produce one pulse for each valid code group. The decoded pulses were subsequently applied to a 30-Hz amplitude detector, a 30-Hz frequency modulation detector, and a beacon transmitter. Outputs from the frequency modulation detector became references for the steering channel phase comparators. The latter served to resolve the in-phase component of the amplitude modulation detector output to obtain steering-error signals for each wing plane. At some point in flight, the firecontrol computer programs the direction of the guidance beam so as to cause the missile to fly the desired midcourse trajectory. By automatically range tracking the missile-borne beacon pulses, it was possible- to determine missile range. This was then used by the fire-control computers to control the beam program and compute the time at which a homing enable pulse code was transmitted to the missile, allowing the homing system to acquire the target. As early as 1947, subsonic beamriding along a fixed beam was demonstrated, and in 1950 the first supersonic beamriding Talos was demonstrated.

Homing of aerospace vehicles.

As previously defined, homing involves guiding a vehicle to a specific target, as in the case of an interceptor. The vehicle's motion is con rollgd with a homing system, which operates on signals reaching it from th get. The vehicle is equipped with a sensor which detects the target in either a passive or a semiactive way, and with an autopilot which leads

d

Nutation angle = 0.85'

Desired missile course is along nutation axis

B~~~ center PRF

Viewed fromgehind radar

1

950 Dashed line is FM C

.-0

t; c

.d

m

D

3

u

"-e

3E

T

.-3.3d

:900

-91a

2

P

a

850 0

90 180 270 Beam center position (degrees)

360

Sec. 6.4

Guldance Algorithm

323

the vehicle to the targct [Pelegrin, 1987(a)]. Homing guidance systems considered in this section refer to those systcms that "zero in" on a targct while still guiding. Homing systcn~sarc categorized according to the frequency band of the electromagnetic signals, as either radio or EO systerns. Radar systcms make up nearly all systems operating at radio frequencies and operate on signals reflected o r radiated by the targct. Generally, EO systems opcratc using TV, laser, or Ill techniques. The problem of designing a homing system bccon~csless complicated for the case in which the targct itself radiates elcctron~agncticenergy. It consists o f building a receiver together w ~ t hthe appropriate computer subsystems. and then integrating them into the pursuer, so that the required control functions arc carried out with sufficient accuracy. If the targct must be illuminated in order to crcatc reflected target signals, the homing system must also include a transmitter, which may be placed on the ground. Auxiliary control systems are used to steer the pursuer and are connected to the receiver-computer subsystem [Maksimov and Gorgonov, 19881. The role o f guided missiles in warfare has grown steadily since World War -11. Section 6.4.1 discusses the development of the inertial guidance systems that were the prelude to the ability to accurately deliver long-range ballistic and other types of missiles for which the target is a k n o ~ v nset of earth coordinates. These systems were not meant to guide missiles against highly maneuvering targets, as this cannot be accomplished without the ability to sense the target location in real time and to respond to rapid changes. Using homing guidance, in which an onboard sensor provides the target data on which the guidance is based, modern air defense missiles possess such capability. Because of the manner in which the quality o f target information improves as intercept is approached, the level of accuracy that can be achieved with homing guidance cannot be obtained by any other type o f missile guidance [Fossier, 19841. As presented in Chapter 2, to track the target, the homing guidance sensor in the seeker head reacts to reflected or radiated energy in one or more of the following regions of the electromagnetic spectrum: microwave, MMW, IR, visible, and ultraviolet. The type of information which a seeker head can provide depends on the mounting of the sensor and the guidAnce energy used. Missiles guiding on radiated electromagnetic energy from the target contain radar receivers which detect this energy and use it to track the target. Thermal or heat homing devices operate on the principle that IR radiation of a heated body possesses intenFigure 6-27 (a) The guidance beam for the beam-rider missile was conically scanned at 30 hertz with a clockwise rotation, as viewed from behind the radar antenna. The radar pulse rate, nominally 900 pulses per second, was varied by ?50 pulses per second in synchronism with the conical scan. (b) Signal modulations detkcted by the beam-riding receiver were processed to determine missile position with respect to the guidance beam. Angular direction and distance from the scan axis were determined by comparing the phase and amplitude ofthe 30hertz amplitude-modulated signal with the frequency-modulated reference signals. (c) The beam-rider receiver detected and procedsed guidance beam signals to generate commands for the missile control planes and the radar beacon responses. (0 1982 by Johns Hopkins APL Technical Diyesr)

Guidance Processing

324

Chap. 6

sities that are different from those o f its surroundings. These radiations are detected by IR detectors and translated into voltage changes which, after amplification, are used for guidance. Light-seeking h o m i g systems contain photoelectric devices that detect light reflected from the target, much like a T V picture. T h e perfortnancc of homing missiles depends on the location of the transmitter o r primary source of the guidance energy: inside the missile, on the target, o r elsewhere as a part of the guidance system [Vricnds, 19871. A passivc seeker detects electromagnetic or acoustic energy radiated by the target, for example, a heat or IR-sensitive detector as used on the Sidewinder. An active seeker employs its own transmitter in order to "illuminate" the target, as in the case of AMRAAM. A semiactive seeker relies on the launching station for radar illumination o f the targct, and then homes on the reflected signal, as in the case of Sparrow. An advantage that results from having the tracker in the missile is that the positional accuracy as the missile nears the target improves. Generally, the opposite occurs with command systems, and the problem can become particularly acute in the case of long-range air defense systems where stnall angle tracking errors result in large distance errors at long ranges. In the case of using missilcs with active or passive seekers, autonomous guidance, or 110 guidance at all, the aircraft's automatic guidance ceascs the moment launch occurs. With command guidance (that is, guiding the missile by radio, for example), or with scmiactivc missile seekers, guidance of the aircraft docs not cease until the missilc reaches the targct or some close proximity thcrcto. For command guidancc in particular, the trajectory along which the aircraft flies must be such that it allows the aircraft both to sense thc target and to communicate with the missile. With semiactivc tnissilc guidance, the aircraft's trajectory must be such that the aircraft is ablc to illuminate the target. In using scrniactivc radar missiles, the aircraft's trajectory is dctcrmincd by cnsuring that the rnissilc is ablc to track thc targct rcliably [Maksimov and Gorgonov, 19881. The three rr~odcso f l ~ o r t r i tin~ ~guidancc ~ are activc, scmiactivc, and passivc. Radar homing systems in particular arc classified this way, depending on the location of the primary source of the clcctromagnctic radiation. Each type is dctcrnlined by the source of energy used for targct tracking, as illustrated in Figure 6-28. O f the three modes of homing, activc homing is characterized by an autonomous terminal modc, which increases firepower, and multidimensional multitargct tracking and resolution. Howcvcr, rangc is somewhat limited by power aperture product. and the design can bc complcx and cxpcnsivc. With scmiactivc homing, thc high-power i l l u n ~ i ~ ~ acxtcnds tor rangc, but thcrc is 1i111itcd sysrcm fircpo\vcr (that is. Icss mu]titargct tracking capability than the activc homing) and the targct illu~ninatoris exposed. Thc design is of modcratc complcsity. I'assivc h o ~ n i n g which , is thc third mode of homing, rcquircs continuous targct visibility. It prcscrvcs 1lF (laser) ~ 1 1 1 i ~ sion control, and is of si~nplcto tnodcrate design cotnplcxity.

Y

Active homing guidance.

A hornins guidancc systcm that transn~its crgy in the form of signals or waves to be reflected by a target and subscquclltl~ processed for guidancc is known as activc guidance. In activc guidance, the vehicle

Sec 6.4

Guidance Algorithm

325

M~ssileHoming On Reflection From Target

Torget I I I Launch ~ i i c r a f t Target Tracking And Illumination Radar

\ Missile Homing On Reflection From Target (B)

0

-=L------Missile Homing On Radiation From Target

Target

('3 Figure 6-28 Air-to-Air Homing Guidance (a) Active (b) Semiactive (c) Passive (From [Hearton and Smoors, 19831 with prnnbsionjo~nG A C I A C )

requires both a transmitter and a receiver. The vehicle's own transmitter illuminates the target, and guidance is based on the reflected electromagnetic energy. The term active is applied to a device or system that generates or radiates energy. Active homing of a vehicltconsists of transmitting energy waves such as radar from the vehicle to the target and receiving the reflected energy to direct the vehicle toward the target. With active systems, the guided weapon functions as a self-contained guidance system by carrying on it the target illuminator/designator. In active radar homing, the pursuer carries its own target-seeking radar with electromagnetic transmitter which illuminates the target, and receiver which senses the reflected energy. Figure 6-28a illustrates active radar homing guidance. The signal transmitted by the radar in the missile is denoted by the solid arcs originating at the missile, which also represent the beamwidth of the missile radar antenna. The arcs originating at the missile are drawn in broader fashion in Figure 6-28a than those representing the beamwidth of the aircraft radar antenna drawn in Figure 6-28~. This

Guidance Processing

326

Chap. 6

was done intentionally to highlight the fact that'the missile antenna in the confined space of the missile will generally be smaller than the antenna used on the aircraft, resulting in broader beamwidths. However, going to higher frequencies will result in narrower beamwidths. Referring again to Figure 6-28a, the dashed arcs represent the portion of reflected energy-which itself is actually reflected in all directions,_ that is reflected in the direction of the missile. T h e magnitude of the reflected signal and its apparent source is a function of the aspect from which the target is viewed, Viewing aspect constantly changes as a result o f the target's motion, and this is responsible for amplitude variations or scintillation, and angular variations or glint in the signal received by the missile. This signal is also referred to as target echo. Problems with scintillation and glint exist witl, all radar systems including the semiactive system to be discussed later. The received target echo is processed by the missile radar, which then generates guidancc commands to the missile to bring about intcrcept [Heaston and Smoots, 19831. Active guidance is usually employed only in the final terminal phase of an attack, with some other form of midcourse guidance acting to direct the missile to the target's proximity. For example, when the missile is able autonomously to search for and lock onto its targct after launch (LOAL), it is possible to cxtend the standoff range of the n~issilc.More and more interest is bcing focused on launch-andleave and fire-and-forget weapon delivery, resulting in a greater dcmand for active guidance systems. The advantage of the active guidance system, illustrated in Figure 6-28a, is that the launch aircraft can maneuver as ncccssary im~ncdiatelyaftcr missile launch is accomplished. This is especially true in the air defense or air superiority role, where the target's valuc is judged to be o f high enough value to justify the incrcased cxpcnsc and complexity. There is also intercst in utilizing active radar guidancc in air-to-surfacc applications using M M W bccausc of thc ability to go to narrow radar bcamwidths. Ho\vevcr, because large signals arc rcccivcd as a result of reflections from surrounding objects including the ground, the problem of target recognition and discrimination becomes morc difficult. Investigations are bcing conducted in the area of developing electronic procedures for target recognition and separation of target return from clutter return. An active radar seckcr that operates with a coherent pulse waveform can provide range and range-rate information as well as the LOS angular ratc. Active radar homing is being applied to air as well as surfacc intcrcept and posscsscs an all-wcathcr and fire-and-forget capability. The large spatial (rangc and anglc) and frcqucncy (rangc-ratc) resolution of M M W scckcrs has cxtcndcd thc capabilitics of surfacc intcrccpt activc radar h o ~ n i n gfrom ship targets to ground targets such as armored vehicles. Synthetic apcrturc radar-mapping techniques arc uscd to update thc INS [Vricnds, 19871. In this tcchniquc, the designated targct is tracked and illuminated by a radar or a lascr (or morc gcncrally, any convenient type of radiation). The illuminating transmitter is located on a different platform, (for examplc, on the ground (scc Figure 6-4b) or in an aircraft (scc Figure 6-28b), and the rcccivcr is it1 the pursuer. The vehicle is equipped only with a sensor

Semiactive (or semipassive) homing.

See. 6.4

Guidance Algorithm

327

(receiver) tuned to the target-reflected, illuminating signal, and is automatically piloted to reach the target. A semiactive missile is therefore dependent on the target illuminator located in the launcher to detect the targct and to provide a reference rear signal for it. As shown in Figure 6-4d, a high-power, continuous-wave shipboard radar illuminator is one example ofan energy source. In Figure 6-4b, a ground radar is illuminating the target and the reflected electromagnetic energy is used by the vehicle for guidance. A semiactive missile must be guided by its launcher throughout most of its flight until interception, thus restricting the launch flight vehicle's maneuvers and heading in the case of an air-to-air missile system. The radio-frequency (RF) energy emitted by the targct is passively homed by the missile receiver to handle electronic countermeasures. , For a semiactive homing guidance with a radar seeker as illustrated in Figure 6-28b, the solid arcs in the figure originating at the launch aircraft signify the targetilluminating radar signals. The signals, confined to a three-dimensional solid angle by the radar antenna bandwidth, are also detected by an antenna located just behind the missile to serve as a reference. The dashed arcs in the figure signify those signals that are reflected back in the general direction of the missile and launch aircraft. In reality, of course, signals will be reflected in every direction by the target. Reflected signals are processed by the missile seeker so that the missile can be guided to intercept the target. These same signals are also picked up by the radar in the launch aircraft, which uses them to ensure that the illuminator is continually tracking or pointed a t the target. As the ability to guide either some or all of the flight by making use of targetreflected energy, homing is one of the characteristics of both the SM-1 and SM-2. Providing this energy is a high-power, continuous-wave, shipboard radar illuminator, a fraction of the signal of which is received directly by the missile for use as a Doppler-frequency reference. In Figure 6-4b, it is shown how this can be done. Essentially, the illuminator antenna combines a narrow beam directed toward the target with a broad reference beam that encompasses the missile all during its flight. While the transmitter and receiver are linked, they are not collocated and, hence, this type of guidance is termed semiactive. A semiactive missile needs also to be able to home passively on electronic countermeasures signals radiated by the target. The use of such signals ranges from screening the target-reflected signal to deceiving the semiactive receiver. Should the target skin return be unidentifiable, o r if incoherent energy is present whose frequency and angle of arrival are approximately correct, then home-on-jam processing is automatically selected [Witte and McDonald, 19811.

Semiactive Laser Homing. The shorter wavelengths in the millimeter range and optical spectrum of these advanced systems mean a smaller beamwidth of illumination. In fact, by eliniinating reflections from the surrounding area, it became possible to illuminate an area smaller than the size of the target. Such capability was introduced by the laser and led to the development of air-to-surface and surface-to-surface semiactive homing systems for use against tactical targets.

Guidance Processing / /

/ /

/

/

Missile Homing On / Reflection From Target

%,'///

/

Chap. g

,--, -- -

,.

/

/ -

.---

Figure 6-29 Semiactive Laser Guidance (Air-to-Surface) (From [Heaxton arid Smoots, 19831 with permissio~l from GA. CIAC )

Owing to their large angle resolution and the ability to designate a specific target, the laser-based systems started being referred to as designators, as opposed to the term illuminator given to radar devices. They also brought about the term PGM introduced in Book 1 of the series. The only drawback to the very small beamwidth of these devices is that it makes them poorly suited for use against a maneuvering target. Basically, they serve the same function as the radar illuminator but are used in a slightly different manner. Typically, a human operator aims the laserloptical device using an imaging system or some form of optical sight, which means that these systems are used mainly against stationary or slowly moving targets at short ranges. Using an operator to aim the designator is lower in cost than slaving the same system to an automatic tracking system. For this reason, such systems have found many guided-weapon applications including bombs, guided projectiles, and guided missiles [Heaston and Smoots, 19831. Weapons that us; semiactive laser seekers can be launched close to the laser designator or from a remote site. When launched from a location close to the designator, the seeker may be locked on before launch. When launched from a remote location, it must undergo a search and acquisition phase and lock-on after launch. The use of multiple designators in a region containing several targets can be made effective by coding the laser designator signals so that a particular missile only tracks its intended target. One form of semiactive laser guidance consisting of a groundbased laser designator and a remotely launched missile containing a laser seeker is illustrated in Figure 6-29. A series of small arcs spanning the distance between the designator and target indicates a very narrow designator beam. The designator is aimed at the target by an operator using an optical sighting device. Owing to its surface roughness, the target scatters the laser energy striking it in all directions, as indicated by the series of dashed arcs originating at the target. The laser seeker positioned at the nose of the incoming missile receives and processes the reflected signal, and in turn provides guidance commands to guide the missile to impact with

Sec. 6.4

GufdanceAlgorithm

329

the target. Not yet a fanliliar acronym, scmiactivc laser homing (SALH) provides pinpoint accuracy onto tanks and similar retlcctive targcts. The designator can be aimed by a soldier or by the attacking aircraft [Gunston, 19791.

Pros and c o n s o f semiactive homing. In scmiactivc systems, a groundbased illuminator is either a CW or pulsed Doppler radar which continually tracks the targct and provides a phase reference for the seeker in the missile. Hccausc thc broad bcamwidth of the tracking radar and illuminator results in only adequate ability to distinguish targcts, semiactive systems using radar have bccn designed specifically for use against air targcts. The missile-scckcr antenna bistatically tracks the reflcctiotis frotn the targct using a narrowband filter, which accepts thc 1)opplershifted targct signals, while rejecting direct signals from the illun~inatorand reflected signals from radar clutter, which arc a t different Doppler frequencies [Schleher, 19861. The characteristic features of aircraft make them well suited for illumination with practically no nearby clutter to intcrfcre with or diminish the capability of the homing seeker. Although aircraft in formation can pose a difficulty, the terminal homing receiver is still typically able to resolve and focus on a particular target as the range decreases. Succcssf~~l ernploymcnt of a semiactive system rcquires that the targct be illuminated for the entire duration of guided-missile flight. As a result, the illuminator or designator is not available for use with another missile during this period of time, and the missile firing rate is consequently reduced to one engagement a t a time. This leads to the development of previously discussed active homing guidance systems in order to deal with the case of simultaneous multiple target engagements [Heaston and Smoots, 19831. A high kill rate is possible only when the semiactive radar homing system operates in a sampled-data mode using an electronically-scanning phased array radar to track and illuminate several targets simultaneously. For the final phase, requiring a high data rate, an autonomous guidance method can be used (for example, active radar homing as mentioned earlier). This methodology has been used in the AEGIS system for the STANDARD missile. The advantages of command guidance can be added to the semiactive missile concept through the TVM guidance technique. This technique has been used in the PATRIOT missile system. Semiactive laser homing operating in the near-IR region (1.06 Fm wavelength) is restricted to fair weather and short-range application (atmospheric attenuation) and to surface intercept (pulse repetition frequency limitation). In an air-tosurface application with automatic laser target designation and tracking, the guidance method offers a launch-and-leave capability to the aircraft which performs both designation and weapon delivery. Semiactive radar homing which has an all-weather capability, is restricted to air intercept and antiship applications because of ground clutter. Against low-altitude air targets the ground clutter is separated from the target returns using the Doppler principle. T o provide the missile with reference frequency information, the missile is fitted with a rearward-looking antenna and either continuous-wave or interrupted continuous-wave radar is used. For crossing air targets and for ship targets, the pulse waveform is preferable [Vriends, 19871.

330

Guidance Processing

Chap. g

Passive homing. In passive homing, the target must have a radiation characteristic which isolates it from the other objects in the environment so that the vehicle's sensor can be tuned to the radiation. For example, the IR spectrum emitted by a tank on the battlefield can be used. Passive homing makes use of electromagnetic energy emitted or radiated by the target to direct itself toward the target. The type o f guidance where the missile homes on electromagnetic energy, heat, light, or sound waves emitted from the target, or energy originating from natural sources such as the sun that is reflected from the target without the missile itself having- to radiate energy is usually referred to as passive guidance. Passive systems may employ TV, or IR sensors. While active systems have greater control over target infor. mation, passive systems are for the most part subject in their operation to the source of target radiation. Passive guidance systems are grouped into the previously defined category of fire-and-forget or launch-and-leave weapons because they do not necessitate further intervention from the launcher after they are locked on to the target. Furthermore, they could conceivably be developed to the point of possessing autonomous acquisition capability. Figure 6-28c illustrates the concept o f passive homing guidance, xvhcre radiation from the hot exhaust plume is picked up by an IR seeker, which then processes this signal to furnish missile guidance. Possible guidance energy is spread over the clcctromagnctic spectrum from microwave to ultraviolet. e , IR tracker that senses IR radiation from the jet With ir$ared (IR) ~ ~ r i d a t j c an engine of an aircraft provides one of the earliest examples of a passive guidance system. Such a system is found in the Sidewinder missiles, which are still being used. IR homing is attractive for any missile aimed at aircraft, though a MIG-25 in full aftcrburner is a n~illiontimes easier than a small turboprop or helicopter. Modern IR-homing missilcs can attack from any angle, even from dcad ahcad [Gunston, 19791. IR (hot-spot) guided \x7eapons contain a sccker which is able to track an IR radiating source using only onc or a few detectors. Target discrimination is performed cither by a suitable optical modulation (using a reticle in the focal plane) or by scanning the total ficld of view with a small instantaneous field of view (image scanning). Spcctral discrimination by sensing in diffcrent IR wavelength intervals o r in an additional ultraviolet interval improves the target discrimination capability. IR hot-spot guidancc has applications in short-rangc air interccpt (SAM and AAM), using as IR sourccs the aircraft cxhaust pipe, the cxhaust plumc, and for high-speed targcts thc acrodynamicallv heated leading edgcs [Vricnds, 19871. Anothcr typc of guidance is known as i r ~ t c l l i ~ r rT~lrl / b ~ q q c Thc . technology uscd in these passive systems is that of TV, \vhcrc the system functions by capturing the light rcflcctcd from thc targct. TV scckcrs havc bccn utilizcd in both command guidancc and homing systcms. In the latter, thc opcrator sclccts thc target by POsitioning a tracking gatc ovcr it on thc display. Thc tracking gatc consists of an electronic circuit that crcatcs a cursor or indicator on a display scrccn. As soon as the cursor is positioncd on a targct, thc tracking gatc auton~aticallytracks it and gcneratcs information concerning its position rclativc to somc rcfcrcncc locatioll in the sccnc, which is typically thc ccntcr of thc display. At this point, the TV seeker

Sec. 6.4

Guidance Algorithm

331

generates the appropriate signals that allow the autopilot to command the missile to hit the target. Inasmuch as a T V seeker is rendered ineffective at night, current effort is being spent to develop an analogous Ill imaging seeker [Heaston and Smoots, 19831. Much of the current research on image processing is directed toward the use of irrtelli.yrtri krrowlrd~qe-bajrdsysrrrtts ttclrrriqrres (presented in Book 4 of the series) to identify the most vulncrablc part of the target. Given the right approach to software, it has been possible to incorporate this capability on processors suitable for small and even low-cost missiles [Barnes, 19871. In radiort~~rer ilrrd i2.1~\.ICC'~q~riri~~rrce, the type of seeker device is a mdiomcter, which senses the low-level thermal radiation emitted from all objects. Such a device may prove to be very usef~11for detecting and tracking particular targets. The small dimensions of MMW equipment make application to submunitions and cannon. . launched projectiles possible. Autonomous target acquisition occurs in an active mode, whereas close to the target a passive or radiometric mode takes over to prevent large miss distance caused by the glint phenomenon [Vriends, 19871. Electro-optical (EO) prided ~veaporrsarc equipped with an imaging seeker that operates in the visual region (TV and low-light level TV), in the middle IR atmospheric transmission window (3-3 km), or in the far IR atmospheric window (8-13 km). The imaging IR or thermal-imaging seekers react on the temperaturedependent IR radiation of the target and its background. They have a day-night capability and offer a better performance in poor weather and battlefield (dust and smoke) conditions than T V seekers and sensitivity and resolution in fair weather are better than for M M W radiometric seekers. The target-tracking capability depends on the target signature and the logic used for automatic tracking, for example, contrast edge, contrast centroid, and area correlator tracking. EO weapons which are locked onto the target before launch have a fire-and-forget capability. The standoff range of weapon delivery can be improved with a video data link allowing lockon after launch and target monitoring [Vriends, 19871. With antiradiation homing (ARH), antiradiation missiles (ARM) for use against communication and radar emitters carry one or more internal or external antennas which provide directional information on target location over a large signal bandwidth [Vriends, 19871. ARM home on the microwave energy transmitted by target radars and are passive in operation. Their receivers, however, differ from the true passive systems which typically depend on thermal radiation or reflected light, and are actually more akin to the receivers used in the systems that are to be described next. Examples of ARM projectiles that have been developed andlor studied include the SHRIKE, HARM, and Antiradiation Projectile (ARP). A similar method is used when a radar-guided missile operates in a HOJ mode. -

-

-

Example 6-12: CHAPARRAL guidance.

The CHAPARRAL missile (see Figure 6-30a) is a supersonic, surface-to-air, passive IR homing missile. The missile system is composed of a guidance section, fuse, warhead, rocket motor, fixed wings, and movable canard fins. The airframe is an in-line cruciform configuration with two pairs of fixed wings aft and two pairs of movable control fins

WINOS (41 WARHEAD

SECTION

7

ROCKET MOTOR

r ---------------(A)

TARGET POSITION

I

SIGNAL

PROPORTIONAL TO SIGHTLINE

b GEOMETRY

ENABLEMCNT

TRACKING LOOP

Y

(

RESTORING MOMENTS

GUIDANCE SECTION HARDWARE

---J

L----------,,MISSILE LATERAL ACCCLCRATION )ONEMATICS

4

1

FIN ' DEF AERODYNAMICS

ECTION i

(B)

Figure 6-30 (a) CHAPARRAL MIM-72C Missile (b) CHAPARIIAL Guidance (c) ROSETTE Flying-Spot-Scan (From /I+'aire, 1983) u-irh pentzirsiotl from S C S )

I

I

Sec. 6.4

Guidance Algorithm

333

forward. The guidance ~ t t i i t ,which is mounted on the forward end of the missile, consists of a scckcr section, a solid-state electronics section, and a fin-servo section. The current guidance unit hardware uses Ill conical-scan optics to provide PNG. The basic operation of the CHAl'AI
Talos: A direct-guidance application. The Talos beam-rider midcourse guidancc system is discusscd extensively in Exan~ple6-1 1 . This section gives a detailed treatment, based on the excellent work of Gulick et al. [1982], of the Talos homing guidance system. Terminal guidance. From the time the Talos development program was first initiated, it was envisioned that there would be a terminal-guidance phase following the midcourse beam-riding phase. A homing system would have to be developed that would satisfy constraints imposed by the ramjet diffuser and that would rapidly acquire the targct without the need for accuratc missile-to-target LOS positioning information. Achieving a capability of intercepting small targets at distances up to 100 nautical miles meant that high receiver sensitivity in addition to good targct resolution was mandatory. Some factors that influenced the selection of the interferometer homing system for terminal guidance were the following: (1) ramjet inlet constraints were satisfied by the widely spaced body-fixed antennas; (2) the body-fixed antennas were less complicated than a gimbal dish antenna; (3) missile-to-target angle data were not required for rapid target acquisition; and (4) there was a desire to have the largest possible aperture and, hence, the most accurate measurement of the LOS angular rate. Figure 6-31a illustrates the basic principle of the interferometer system, where the nomenclature was first introduced in Figure 2-10. The composite antenna pattern of two widely spaced antenna elements of an interferometer is made up of a series of peaks and nulls, which are moved by a phase shifter. This results in amplitude modulation of the target signal. An output is generated by a discriminator tuned to this modulation frequency that is proportional to the angular rate of the target LOS. A P N G homing trajectory to the target is executed if the target LOS rate with respect to an inertial reference is kept at zero. It remains only to control the missile turning rate so as to maintain a zero LOS rate and guarantee an intercept. The first Talos homing system.

The first Talos homing system was based on a guidance concept consisting of body-fixed, widely spaced antennas to be used as a radar interferometer. It employed an independent interferometer channel for each wing-control plane. As shown in Figure 6-31b, the receiver used a scanning interferometer system. Target signals at the antenna were

X I = A, sin wt,

X2 =

A2

sin(ot

+ +)

= A2 sin

(1sad:ta /v>!ny?a~.7dv s u ! q d o ~suyo(Kq 2861 0) .D01 leuo!~~od -old ,vu4!s r 11, pallnsa~~ I I P)(10h\13Il ~ol13ede3-1o1s~sa~ e i(q pa~e!~ua~aa!p sem i a ~ a m o : ~ u a ~ o d all] 1llOlJ l ~ r d l ~ ra111 o - l a ~ . ~ r t r o ! ~ c ~e. ~JO~ oudu r JI(I o1 p a l d n o ~oslc sem IJeqs 1erl.L (ON 1 0 u!s) ( y j p s ~ 01 ) papuodsal~o31cg1 a(S11e ue lc IJeqs lar\[osal aql pauo!~!sod oA1aS 8u!mol[o3 -aseqd y .lole~auaZ23UJl2JilaJ aqlJo aseqd aql g!qs 01 pasn seM (mlo~leldaiqe~s)o1A2 aal3 e 01 p a l d n o ~Alle~!ueq3am J ~ A ~ ~ S ~ J - ~ y I ~ .Iaurleq> ~ U A S I O J I U O ~alZu!s e S M O ~ Sa ~ n Z ga q l w d l n o J ~ ~ ~ ~ ~ J ~ J J ~ I u ! - ~aql u !m I o I lI jI uo!~om ~ ~ s Lpoq Su!ldno~ap 103 poqlam aspald alom e sem S ~ . J ~ V L(3) S .+I 01 [euo!]~odo~dleuZ!s e sem ~oleu!lu!.13s!p aql urn13 lndlno a q l ' 7 0 so330 zlrmpsa IIE S E M N m1a1 aqL .sleuZ!s l a ~ a m o l a ~ l a aql ~ u !uo uo!lom A p o q ~ o aql aldno3ap 01 pasn srm cr~A4ale1 palnnor~~-.(poq e t u o l ~l n d l n ~-aueld [on1109 [euo%nq)~o aq] 103 pasn sem III.>IS.~S lr.>~lu.>p! I I V ..>IIPI,~ 10111111.1 .11311!5 e JOJ SI!M .1.1.>11 IIMOIIS III.>ISAS ~11.1,. J ~ I ~ I I I O J ? ) J ~ I I I ! 3111uue~s r pas11 ((0~61) ~ . > ~ ! ~ . > a ~ S t ~ !soleL r u o q "13 2 1 1 (q) ~ J ~ I ~ C I I O I ~ J J (c) W I I€-9 a~n%!a

Guidance Processing

336

Chap. 6

where o is the microwave frequency and 6 is the electrical phase difference at the antennas. A , and A2 denote the amplitudes, which could be different. The phase of X2 is advanced by an amount @(t) by the scanning phase shifter according to @(t) = o,t (P where o,is the scan frequency and (P is the initial phase shift. X, can then be expressed as

+

X3 = A2 sin

o

nd . + o, + 2 uL cos UL A

Adding XI to X3 gives X4, which can be expressed as

X4 = A1 sin wt

+ A2 sin

o

7 ~ d. + o, + 2 UL cos UL A

This last result shows that X4 is a carrier signal at w and is amplitude modulated a t nd . o, + 2 - UL cos

U'.

Although the carrier frcquency was changed by the heterodyne process in the receiver from microwave to a lower frequency, this did not alter the basic u L information since the nlodulation was unaffected. Moreover, since thc desired information was the frequency ofthe n~odulation.the nleasurcment ofuL was not directly affected by changes in the amplitude of the signal. A body-mounted rate gyroscope was used to decouple body motion to provide the LOS rate measurement u,and the gyro output frequcncy modulated an osc~llatorto produce a carrier frequency wo with a deviation proport~onalto the missile-body turning rate 0. Because a fixed value of K,xvhose term in the gyro channel is an esti~natcof cos DL, for all flight conditions turned out not to be satisfactory. Talos relied on a sclcction of two values based on thc crossing component of the target speed. This value was sct into the missile at laut~chtime. The scanning ~ h a s shiftcr e frcquency o,\vas eliminated fro111 the final term by the use of double modulation, which allowcd a frcquency discriminator tuned to w,, to providc an output that was proportional to (2ndlA) ( ~ -8bL cos uL). Ignoring biases, the discriminator output was a good approximation proportional to the dcsircd u.

Stable platform phase follow-up system (STAPFUS). The first T a l a homing system suffcrcd from t\tro critical problcms, the first of xt~hichwas that a bias resulted from anv oifsct bct\vccn the voltarc-controlled oscillator and the discriminator center frcqucncy. The second major problem was that it \tias difficult to maintain a stablc gyro gain factor equal to (2ndlA)h!. The solution to both of these problems came in the way of thc stablc platform phase follow-up system (STA1'FUS). Figurc 6-31c shows the scannit~gintcrfcromctcr using STAI'FUS. AS writh thc initial systcnl, a motor \vai used to drive the phase shiftcr and a rcfcrcncc crator. I'hasc corrections for nlissilc body motion were llladc by a singlc-dcgrccof-frccdotn frcc gyro coupled to a synchro-resolver. The gear ratio bctwccl1 the gyro and rcsolvcr providcd a constant gain factor for thc gyro coupling. The difc,

Guidance Processfng

Chap. 6

Range

(A)

Figure 6-32 (a) Pulse radar signals from a moving target can be obscured by large, stationary reflecting surfaces such as the sea or land masses. (b) Continuous wave (CW) radar signals from a moving target can be detected by filtering at the Doppler frequency. (c) The rear reference receiver used a phase-locked klystron and a groundaided acquisition (GAA) loop. The klystron frequency was coarsely corrected by the discriminator to the frequency of the illuniinator signal, followed by exact frequency control by the phase-locked loop. The voltage-controlled oscillator frequency was offset from the intermediate frequency by the j ~ a . 4loop and caused the klystron frequency to be offset precisely for the target signal entering the front receiver (see Fig. d). (d) A target signal having a Doppler frequency equal to.f~fcs.< would pass through the amplifiers and narrow-bandpass quartz crystal filters and be detected by the acquisition circuit. Upon detection, the discriminator output readjusted the frequency of the voltage-controlled oscillator to keep the target signal in the center of the receiver bandpass. The narrow-bandpass filters removed the land and sea clutter signals entering the antennas. (e) The monopulse seeker was a refinement of the original C W seeker. The target andlor jammer signals were processed through a two-channel receiver (per guidance plane). Narrowbanding occurred alniost immediately following the microwave mixers. The intermediate frequency (IF) amplifiers \r.ere hard limited on receiver noise. The Doppler tracking e oscillator and the first mixer. Predicted loop was closed through the n ~ i c r o \ ~ a vlocal Doppler inforniation provided by the ship was used to aid target search. The homeon-jamming niode eniployed the same narrow bandwidth and angle processing circuits used for tracking the target echo. (0Guidance-control logic used to ilnplement guidance switchover from midcourse to homing, to gate the steering information once homing began, and to control Doppler-search routines was based on the coherency of the angle data. The signal used for that was obtained from a phase detector that monitored the STAPFUS phase-tracking loop. (g) Multiple jammer flight tests were conductcd to demonstrate dichotomous angle-tracking capabilities inherent in the Talos interferometer guidance system. Briefly, if signals froin tiiultiple sources were present siniultaneously, the n~issilewould track one of them if a power differential of 2 decihcls or greater cxistrd within the narrow bandwidth of the receiver. In practice. the niissile ufouldnearly always intercept one ofthe targets. For thc riiultiplejanin~ertest illustrated here, the primary target was a P-4Y aircraft with a rioise jammer. After achieving a near miss on the aircraft, the missile successfully intercepted the westernmost jammer on the ground. (h) The effect of selfprotection electronic counternieasures (ECM) o t ~riiiss distance is illustrated. The curves were derived from miss distance data obtained froni laboratory ECM tests conductcd a t the Naval Ordnance Laboratory. Corona. Calif.. froni the monopulse flight tests against jamming targets, and froni historic test data of Talos flight tests against nonjamming targets. Note that the pcrforniancc against the janiming targets was superior t o that against the tionjamming targcts. (0 1982 b y Joh~tsHirl>kittrAPL Techni(a1 Digc~.sr)

-

7 To STAPFUS measurement system

Target tracking loop

Bandwidth = 1600 Hz >

Target -C

F i wnptifii with Mixer

namM

VJ*

2.5 MHz_ crystal filter

fihu

Interferometer antenna system

2 3 MHz

-

Rear antenna

F i g 10

Figure 6-32

(Continued)

64

Guidance Algorithm

-

T a m reaiver Doppler rcquirition m d mek

Reference receiver -Predicted Dopplw (GA.4)

Rear

1

!

antenna

-TO

autopilot pier search control Switchover. midcourse to homing

(F)

Figure 6-32

(Continued)

Guidance Processing

0.4

V"#K&&

Laboratov ECM tests

--

Flight tests (monopulre) with ECM. 9 tests. 6 direct hits Flight tests without ECM (Tala history)

0.2

0.1 Miss distance

(H)

Figure 6-32

(Continued)

6.4

Guidance Algorithm

343

Figure 6-33c shows a block diagram of the phase-locked klystron loop and the ground-aided acquisition loop. Just before launch, the klystron was electrically pretuned to the approximate frequency of the illuminator. Approximately 15 sec prior to intercept, the ship-based CW illuminator began radiating, and automatic frequency control pull-in occurred. Using radar data, the target Doppler frequency, fD, was computed and then transmitted to the missile by a 400-kHz plus fD frequency modulation on thc illuminator frequency. The ground-aided acquisition loop used this estimate of the Doppler to establish the initial frequency for the voltage-controlled oscillator. By doing this, the klystron frequency was not very far off from the correct value, and the search required to acquire the target was minimal. The front receiver with its narrowband quartz-crystal filters and discriminators, together with the Doppler-tracking loop, is shown in Figure 6-32d. The function of the loop was to control the voltage-controlled oscillator and, therefore, the klystron so as to keep the target signal in the center of the receiver's narrow bandpass. The SAMN-6cl missile with its CW homing receiver was well suited to a countermeasures environment. First, the narrowband characteristics of the seeker and the short (610 sec) homing time made for limited exposure to hostile environments. Second, the high-speed phase-tracking loop in STAPFUS provided a nominal angle-tracking sensitivity (volts per deg per sec) when it was tracking, independent of the SIN. Finally, the Talos phase-interferometer angle processing was intrinsically capable of resolving signals from two or more separate sources such as multiple noise jammers if there was a small power difference ( 5 2 dB) between them. There was, however, some uncertainty as to whether the CW guidance system would perform well against surface targets. In this application, the target signal and the sea clutter signals do not have a useful Doppler frequency separation. However, 'the combination of the 800-Hz and 100-Hz filter bandwidths maintained a signal-to-clutter ratio suitable for guidance, and the Talos system with CW interferometer guidance performed well against surface ships and boats.

Monopulse Homing System. The performance of the original SAM-N-6cl missile was exceptionally good against several types of jamming. Unfortunately, the missile was limited in its capabilities against certain types of deception countermeasures owing to its RF sequential lobing (scanning) angle processing, its slow automatic gain control and its time-consuming Doppler search routines. One of the foremost objectives of the rnonopulse seeker design was to enable the missile to win any o~e-on-oneencounter with an aircraft, regardless of the form of AM andlor FM noise or deception jamming the latter might employ. The result of this effort was the development by the mid-1960s of a seeker that was virtually unjammable by an attacking aircraft. One requirement for the Talos rnonopulse seeker was that it be compatible with the existing STAPFUS, which, as described previously, re(2adlA)sin uL)signal. As shown in quired a scan reference (w,) and a scan (w, Figure 6-32e, the Talos monopulse receiver was designed to provide these two signals. The receiver is somewhat similar to the scanning receiver, with the exception that a pseudo-scan was introduced at IF following narrowband filtering, which

+

344

Guidance Processing

Chap. g

involved offsetting the frequencies of the second IF amplifiers by an amount equal t o the pseudo-scan. By choosing the scan reference to be greater than the bandwidth o f the narrowband input filters, it was made certain that the pseudo-scan would be invulnerable to ECM. A deficiency of the Doppler search routines in the original C W seeker was that target reacquisition was slow or in some cases could be prevented altogether. T w o factors that played a significant role in this deficiency were the relatively slow sweep-repetition rate and the complete reliance on memory for positioning the sweep. The design of the original search patterns was based on providing a high first-look acquisition probability under minimum signal-level conditions. This imposed a limitation on the maximum search speed that opposed the ECM requirement for fast reacquisition. The reason for this last requirement is that a jammer can cause the seeker to lose acquisition repeatedly throughout the homing phase. For the monopulse seeker, a compromise was pursued in which the sweep speed was increased such that the single-look probability for the small signal case was lower than previous but, after 1 or 2 sec, was comparable as a consequence of the increased number of looks. Late in the homing phase, the probability of single-look acquisition would be essentially unity because of the relatively high target signal-to-noise ratio and, consequently, the desired fast reacquisition was achicved. Also, taking advantage of the predicted Doppler information provided by the ship during the homing phase helped solve the problem of positioning the Doppler search center. The control of critical guidance functions was based on the coherency of the angle data, which made it possible to resolve a s~gnalemanating from essentially a point source forward of the missile (targct skin ccho or a jammer) from other signals such as sea clutter, scattered chaff, reccivcr noise, and so on. An HOJ served to delay guidance switchover from midcoursc to homing in cases where acquisition of the targct ccho was not achieved immcdiatcly. This averted immediate HOJ on a stand-off jammer. Figurc 6-33f sholvs the guidance control logic.

Tests of Continuous-Wave and Monopulse Homing System. The results of cxtcnsivc laboratory tcsting of the CW and monopulse homing systcms strongly indicated that the Talos missilc with the tnonopulsc sccker could not be bcatcn by virtually any sclf-protcction noise or dcccption jammer. Twenty-fivc of the 26 valid flight tcsts of thc Unificd Missile wcrc SUCCCSS~UI. Ninc of thcsc 25 were with the ~nonopulscscckcrs aqainst -. a wide sclcction of ECM typcs and paramctcrs. Two of the successful tcsts \trcrc against multiplc jammers, one of \vhich is shown in F i ~ u r c 6-32s. Talos' pcrforn~anccagainst both jamming and nonjalnming targcts is illustrated in Figurc 0-32h, in which it can bc sccn that Talos pcrformcd bcttcr against jamlning targcts. This is not altogether surprising \vhcn onc considers that a jam111cr generally provides point-sourcc cnhanccmcnt of the targct. 111 othcr words, the janlmcr is literally providing a beacon on u~hichthc missilc can homc. Dec~elopmentof Antiradiation Hotninx ( A R H ) Missile Guidance. As disC U S S C ~in Gulick ct al. 11982], it wras not long aftcr the Victnatn conflict brokc out that thc nccd for an cficctivc long-rangc nntiradiation missilc (AIIM) to suppress

Sec. 6.4

Guidance Algorithm

345

enemy radars became evident. Because of its long-range capabilities and the ease with which it could be adoptcd for new missions, Talos was seen as both a logical and dcsirablc choicc for radar supprcssion. The Talos ARM program rcquired the development of a unique operational concept because the ship could not conceivably track the targct and implement missile guidance as it did for the cngagcment of air targets. The techniques for providing ARM with the appropriatc targct information cot~sistingof !geographic location and RF emission characteristics (ircqucncy, pulsercpctition frecluency, and so on) were as follows. First, several navigational techniqucs, including the Navy Navigation Satcllitc System, were used to establish ship coordinates. The ship then directed the missilc to thc vicinity of the targct using the beam-rider midcourse guidance. Upon approaching the targct, the missile was put into a dive and the homing system was activated. The missile approached the targct in one of two possible terminal geometries, either an approximate 45-deg dive or a near-vertical dive. Because the low pulse-repetition frequency of some target radars was not compatible with the scan frequency used by the carlicr semiactivc scckcrs, it was required foremost of the ARM homing system that it be monopulse. It was also desired to have a system with a high sensitivity so as to provide continuous guidance on the low sidelobc levels from thc targct. Finally, thc ARM guidance system had to be compatible with the existing Talos airframe. Figure 6-33a shows a simplified block diagram of the ARM seeker. The receiver used two parallel IF amplifiers (per guidance plane) with subsequent in-phase and quadrature processing. By using limiting amplifiers, a very large instantaneous dynamic range greater than 120 dB was achieved. Circuits were carefully designed so as to maintain the differential phase shift between these amplifiers at a low level, and good phase tracking was made possible with input peak power levels greater than 10 watts. For very low-frequency targets (L band), it was considered undesirable to have a guidance error associated with even 10 electrical degrees differential phase shift between the receiver channels. By adding microwave and IF transfer switches that, on a pulse-to-pulse basis, allowed radar pulses to be processed alternately through one channel and then the other. the error was all but eliminated. With these switches. the internally generated errors would average zero. Figure 6-33b shows the measured angle error arising from differential phase shift plotted against the signal power level. The two curves in the figure show the errors with and without the transfer switches operating. The acquisition and discrimination channel was placed in parallel with the angle channels, and the receiver was self-gating. A signal that met the radiation and pulse repetition frequency requirements designated at launch would provide a gate to the angle channel. The seeker could discriminate between two signals as long as these differed in their operating frequencies by at least 3 MHz. In order to minimize the effects of multipath reflections near the target, leading-edge gating was used. The outputs from the angle phase-comparator were multiplied by a 400-Hz signal to provide an amplitude-modulated signal to drive the STAPFUS resolvers. The seeker measuring the LOS rate to the target at the time of acquisition provided geometric discrimination. The ship, utilizing the midcourse beam-riding guidance, caused the missile to dive toward a point approximately four miles beyond

Guidance Processing -Guidance -Gating -Frequency

Ckp. 6

signals and switching control

B channel guidance plane

..,

~

.

I

ating ignal

Target reject from angle rate renror

I

-", 0.8 u

'0

f

0.4

I

-

-1 5

I

I

I

I

I

I

I

T

400 Hz

I

Transfer switches out ---Transfer switches in

-25

-35

-45

-55

-65

Input Power (decibel meters) (B)

Figure 6-33 Talos A l l H Missile Guidance (a) T h e Talos Antiradiation Missile (AIIM) seeker used t w o parallel IF amplifiers per guidance plane with subsequent in-phase and quadrature processing. A signal that satisfied the prelaunch frequency and pulse repctitiot~frequency requirements provided the gating signal to the angle channel. (h) Angle bias resulting from diffcrential phase shift bct\vcen the recciver channcls was virtually eliminated with the use of radio frequcncy and of IF transfer switches that operated on a pulse-to-pulse basis. 1C 1982 by Jokns Hopkit~sAPL T r ( h ~ i i ( aDixcst) l

Sec. 6.5

Guidance Law

347

the intended target. The resulting missile-to-target LOS rate was in the down direction. In the cvent of a failure to detect that downward angular rate during the target acquisition process, the signal was rejected and the seeker continued to search for another target. If the target was accepted, the missile executed a down maneuver to the target. In this way, the elevation angle at which the missilc would intercept the target was nearly 90 deg. According to Gartcn and Dean 119821, although specific details of Vietnam combat applications cannot be released, the radiation-seeking capability of the Talos ARM (RGM-8H) was demonstrated by the fact that shore batterics were silenced by missiles fired from ships placed in the Gulf of Tonkin. Because of its target discrimination capabilities, Talos was also able to effect long-range kills of MIG aircraft over land. The long-range intercept capacity put enemy air tactics in check whenever a Talos ship was nearby. The opportunity to use the surface-to-surface capability never presented itself.

6.5 GUIDANCE LAW

In order to intercept the target, a missile must constantly travel in the proper direction. The direction in which a missile travels is dictated by an algorithm built into the guidance processor known as the guidance law. Many different guidance laws have been developed over the years, and with the advent of highly maneuverable airborne targets, research on improved guidance laws is continuing [Heaston and Smoots, 19831. Guidance laws currently in use on existing and fielded missiles may be inadequate in the battlefield environments envisioned for the 1990s and beyond. Performance criteria will probably require applications of newly developing theories. which in turn will necessitate a large " comvutation capabilitv relative to classical guidance technology. Until very recently this task has been relegated to large computers, in many cases ground based. However, advances in microprocessors will allow increased use of onboard computation [Fraser, 19811. Geometry and other parameters for homing guidance are drawn in Figure 2-10. Currently, several guidance laws of the two-point (target and missile) or three-point (target, missile, and station) type are implemented in homing missiles. As shown in Figure 6-1, current guidance laws, whether of the two-point or three-point type, can be classified as LOS guidance (Section 6.5.1), LOS rate guidance (Section 6.5.2), or advanced guidance (Chapter 8). It must be pointed out that a guidance law design, to be made complete, requires guidance filter techniques such as those presented in Chapter 7. Also, sensitivity analysis and a comparison of these guidance laws will be presented in Section 6.5.3 and in Chapter 8. Figure 2-10 shows the basic geometry. In command guidance, the target and missile dynamics are separated because the radar tracks them separately. The command guidance geometry is described in Figure 6-23c for the head-on case while the semiactive homing using the standard PNG is described in Figure 2-10. The dynamics are linearized about the LOS vector at the start of the terminal portion of guidance.

Guidance Processing

348

Chap. 6

6.5.1 LOS Angle Guidance The guidance law definition used by Locke [I9551 is followed here and depicted in Figure 6-34. The LOS guidance (either CLOS or beam-rider system) in this figure is designated as a three-point guidance system. As shown in Figure 6-23a, a beamriding missile generates its own commands internally whereas a CLOS missile re, ceives its commands from a remote station. These two schemes, however, have essentially the same lateral acceleration command mechanism. From Figure 2-10,

=

- vmsin ~ l ~ < d / R ,

or

sin ul,,d

;=

- R,uIV,

(6-5 b)

Differentiating Equation (6-5b) gives Since A, = Vm(6 - ul,,d),using Equations (6-5a), (6-Sb), and (6-5c) A, becomes

A,

=

2bVm + R,ulcos ulecrd+ Vmtan

UI,,~

(6-5d)

which represents the lateral acceleration required by the missile to stay on the LOS. Combining Equations (6-ja), (6-5b), and (6-5d) yields the kinematic transfer function

where 0, replaces u to emphasize that this is the angle subtended by the missile (see Figure 6-23b) [East, 19841. The LOS guidance loop which is a two-integrator system becomes unstable under small perturbations. A stabilizing compensator is thus required [Durieux, 1984; East, 19841

A LOS guidance loop is shown in Figure 6-35a. The loop gain is maintained constant by incorporating the missile range R, in the feedback loop. This in turn ensures a sufficient loop bandwidth. The beam stiffness K, is adjusted to trade-off the tracking response and stability against sensor noises. Finally, the filter G(s)in Figure 6-3% is used to provide cnough phasc stability. The inverse kinematics in a feedforward fashion (Figure 6-35a) arc includcd to allcviatc the need of high loop bandwidth One example implc~ncntationbased on Equation (6-5d) was donc in Equation (6l b ) whcrc 0r replaces u.

6.5.2 LOS Rate Guidance The homing guidancc systcm, which contrasts with the LOS guidance, is designated as a two-point guidancc system and is implemented mostly as LOS rate guidance.

Sec. 6.5

GufdanceLaw

Figure 6-34 AGARD)

Guidance Law Types (Froin /Goodirein, 1972(b)/ wirh p e n n i s i i o ~ ~ j r o ~ n

Figure 6-35b depicts a conventional LOS rate guidance loop for both command and homing systems [Alpert, 1988; Durieux, 19841. For easy comparison, the LOS rate guidance in this figure is arranged correspondingly to the LOS guidance in Figure 6-35a. In other words, a derivative network is cast in the guidance computer. . Since there is only one pole at the origin, the loop stability is not likely to be a problem as in the case of the LOS guidance, at least until near interception when l / ( V c t g* ) 30. The trade-off between tracking performance and high frequency stability is thus alleviated. However, it is still highly desirable to keep the LOS rate guidance loop bandwidth as small as possible [East, 19841. If K, = A t k = A/ ,,t ki = 0, then the LOS guidance in Figure 6-35a is identical to the command guidance using a synthetic PNG in Figure 6-33b(i). These gains are not sufficient to keep the missile within the tracker beam when t, is large. Similar to dynamic lead guidance to be presented later, it is thus necessary to reserve the synthetic P N G (see Figure 6-35b(i)) for the intercept phase and to use LOS guidance during the initial phase [Durieux, 19841. LOS rate guidance is defined by

where y, is the commanded flight path angle, h is the navigation gain, and C is the constant or initial flight path angle yo. The guidance information is illustrated in Figure 6-36a. As shown in the figure, a body-fixed or strapdown seeker senses the look angle, UL; a gimballed, inertially stabilized seeker senses the LOS rate, u;and an airstream stabilized seeker senses the lead angle, ale,&Radar seekers can be used to sense range and range rate (closing velocity), as long as there is amitable waveform and reference signal information. Other information, such as that dealing with missile airframe motion, may be required to implement a guidance law. There are

,--------------

1I

Guidance Computer

I

I

I Range Stiffness L Estimate -------------A r--------1

em

I

Kinematics

I

L--------J

(A)

r - - - - - - - - - - - - I I

Guidance Computer Derivative Network G(s) s

1 I I

-

-

t = Y (Approximate for Figure (A)) Qr = el.f\T - Om 8 = R e c o n s r m c t e ~ v e dLQS Angle For Synthetic PNG, K, = M, = u, = A V, 8

(i) Command Guidance r---_-___1 ----

!

Guidance Computer

I

I

a = Geomebical LaSAngle For Standard PNG, Kg= &'$ = u, = A V, b Independent of R

(ii) Homing Guidance

Airframe Dvnamics -8.

or q,,,

-

-

0-7

I

Target Motion

'Feedback with Body Pitch Angle 8: Anirude Pursuit "Feedback with Flight Path Angle y: Velocily Pursuit

(C) Figure 6-35 (a) LOS Guidance (b) Conventional LOS Ratc Guidance (c) Pursuit Guidance ( ( a ) from [East, 19841 with permission from ACARD; ( b ) from [Durieux, 19841 with permissiotl from ACARD)

Sec. 6.5

Guidance Law

(iuidancx: Law

With l'erfect Response

C:ummond to 1- (hidance (&.am R i k r o r CUX)

r =O

1 Atlitude Pursuit

Guidance Equation Lateral h l e r a t i o n Command U, = Vm f c I(g G(s)lR, €1

look angle oL = 0, u = 0, A = 1. C = 0

= 0, o = y. k = 1. C = O

Vclwily Pursuit

lead angle u

Ckvialed Pursuit

lead angle olead = preset v n l u olCs4

$'"=

ft off the beam $(u-8)

$ole,,= s ( o - Y )

- ole.4)

('onslant b a r i n g Proportional Navigation tiiicbnce (PW,)

o = cro = constant. A = q C = 0

Kg ( t -~ u,)

; = 0, = A h A =

V, cos yo A

C o ~ ~ s rPNG al

A=O

A V, & 1cos uL

Extended PNG Velocity Compensated PNG

&=O

AV,~Icoso,,-V,tanoL

&=o

A V, &

(vcpffi)

constant

;o r

A V, &

- i~,,, s i n atead

Figure 6-36 (a) Geometry and Parameters for Homing Missile (b) Basic Homing Guidance Law ( ( a ) reprinted w i t h p e r m i s s i o n ~ o m(Vriends, Systems ?t Control Encyclopedia), Copyright /1987), Pergamon Press PLC)

several useful guidances which are popularly used in missile-guidance systems, as shown in Figure 6-36b. Many guidance laws currently implemented in homing missiles are of the two-point (target and missile) type, and these can be based on pursuit, proportional navigation, or both [Vriends, 19871. Pursuit guidance and PNG can be implemented in a guided-missile system using only LOS angle and angle rate information, respectively. However, an additional ground station is required for a command to LOS guidance scheme where the missile attempts to follow an EO beam directed at the target (see Figure 623c(i)). These three classical guidance schemes are shown in Figures 6-35 in. the

Guidance Processing

352

Chap. 6

form of loop flow charts, and are compared through simulation in Table 6-1. Both command to LOS and pursuit guidance laws are shown to have limited capability to engage maneuvering targets. O n the other hand, P N G guidance can be implemented fairly easily. PNG and its variants reduce the maneuver requirements and can produce good miss-distance performance [Hiroshige, 19861. It has a long history of acceptable performance and has been used successfully in several fielded-missile weapon systems. Nevertheless, it also has its limitation in engaging targets with a target-to-pursuer velocity ratio that is significantly greater than unity, or targets with higher maneuvers. The maximum normal acceleration force required by PNG is much less compared to that required for turning a pursuit or command to LOS guided missile under any combination of disturbances. If no filter is provided in the PNG, more drag and control command saturation can occur in the case where the guidance system responds to seeker noise than occurs in the case where it responds to true geometric guidance information. Both pursuit (attitude and velocity) schemes are sensitive to biases. The miss distance is proportional to bias errors. However, when there is no lag, that is, the guidance system time constant T, = 0, the sensitivity to biases theoretically goes to zero for the PNG law since differentiation of the constant bias of the LOS yields zero. The performance is limited by the control lag T, in practice in any general LOS rate guidance system [Durieux, 1984; Neslines and Lin, 19871. For a sampled data system where the last command is held for an interval of time, decreasing the miss distance does not zero out the bias error [Leistikow et al., 19671. Table 6-2 summarizes the advantages and disadvantages of the different guidance laws when they are used in combination with classical low-pass noise filters. The state-of-the-art pneumatic actuator weight increases with sideTABLE 6-1 COMPARlSOtl OP GUIDANCE LAWS POR SHORT-RANGE TACTICAL MISSILES Pursuit Ability to engage Urgnr Armracy (It CEP) Mmncuvmb'~lity A d d i d d criteria[ Yd(f,) l ~

~

~.~~ -~. ~.

State required~ On-board gyro (re0 Gimbal mshanization (seeker) On-boud clstronics ~

~

Cost (on-bwd) S m r requirements

Airfrunelpopulsion requirements

T a i u l considerations "nre and forget"

From [Pastrick

et

21..

CLOS

Beamrider

Attitude

Velocity

PNG

Optirml

<2 Low No

<2

> 30 Low No

>M

Low No

Low No

<5 Const velocity No

Accelaatins Yes

LOS No No Same

LOS No Air vane h e

LOS 2 DOF Gyro sdme

Full 2 DOF Gyro MicrocomPU~er

Ar:A~ Atl~tude No Same

Ar:A~ Att~tudc No Same

0.8

0.9

1 .O

I .4

1.6

1.8

Wire link

Optiul link

Sceker (wide FOV)

Sceker (narrow FOV)

Scekn

Seeker (sutec measured .nd estimated)

Low

Low

High

High

Low

High

No

No

Possible

Pouible

Possible

Possible

1981). 0 1981 A l A A

See. 6.5

TABLE 6-2

Guldance Law

355

COMPARISON OP CLASSICAL GUIDANCE LAWS

Guidance law 1. Command-to-Line-of-Sight

Advantages

I>isadvantagcs

N o terminal seeker required.

Very inaccurate against moving targets and with winds. Data link required. Inaccurate against moving targets and with winds. lriaccuratc against accelerating targets. Stability is sensitive to noise. Between 2 and 3.

Guidance

2. Pursuit

Noise insensitive. Easy to use with strapdown seekers. Accurate against constant velocity tugets.

3. Proportional

4. Pursuit

+

Pro Nav

5. Dynamic Lead

Between 2 and 3 in terms of accuracy. Between 2 and 3 in terms of accuracy. Easy to use with strapdown seekers.

.

Between ? and 3. Stability problems if transition to pro nav occurs when significant noise is present.

From [Gonzalez. 1979(b)]with permission from AGARD

force g requirement [Boykin and Jordan, 19711. The number of g's turning acceleration is limited by system requiremcnt weight allowances, and this leads to limitation in the accuracy of the system when employing any guidance.

Pursuit guidance.

Perhaps the first guidance law that was implemented is known as pursuit guidance, resulting in a pursuit course, or the hound-and-hare course. The pursuer aims directly at the target throughout the encounter like a dog chasing a rabbit. In pursuit guidance (see Figure 6-37a). the angle between the longitudinal axis ofthe pursuer (attitude pursuit) or the velocity vector of the pursuer (velocity pursuit) and the LOS to the target is driven to zero or some constant value (deviated pursuit) [Vriends, 19871. In both attitude pursuit and velocity pursuit, the missile is commanded to turn at a rate equal to the LOS rate. Therefore, the missile is constantly turning during an attack unless it is head on or tail on. While it permits a simple implementation and is less sensitive to noise, pursuit navigation is considered impractical as a homing guidance law against moving targets owing to the high maneuver requirement to end the attack in a tail chase. This is not strictly true in the case in which the speed of the missile is very great compared with that of the target, as in the case of a ship. In this circumstance, the small miss that results from an intense last-instant maneuver is such that the missile still intercepts the target. Figure 6-34b shows the missile velocity vector aimed at the target in three positions along the trajectory. The direction of the missile velocity vector must be sensed to steer the missile with this guidance law. Figure 6-37a illustrates target intercept based on a pursuit course. Both pure and deviated pursuit trajectories are shown for the case of constant missile and target velocity, with the missile's velocity being twice that of the target. For the deviated pursuit course, a lead angle ~ r , , d , of 5 deg was used. For this particular encounter, it can be seen that the deviated pursuit

Target I

2

3

4

5

6

7

Pure

Pursuit ( 5 O Lead Angle

Missile Velocity Equal To 2 Times Torget Velocity

Missile Velocity Equal To 2 Times Target Velocity I

I

(--~issile (B)

Figure 6-37 (lrd

(a) Pursuit Goidancc (b) Constant Hcaring (;uidancc and PNG (Frorn [Heasrorr

Srrloors, 198.11 u,itlr penrrissiort.fiorrr GACIAC)

8

Sec. 6.5

Guidance Law

355

course results in a slightly shorter missile flight and earlier intercept [Heaston and Smoots, 19831. Consider the equations of motion (2-2). For the following devclopments t w o assuniptions are made: ( I ) the target is nonmaneuvering, so that V T = constant, constant, and V,, m constant; and (2) since y~ = constant, it is convenient to let y , = 0. With regard to the first assumption, an area of current interest in modern control theory is the problem of determining thc optimal pursuit and evasion stratcgics for a target and pursuer with arbitrary acceleration capabilities. The problcm is fornlul~tedin the mathematical discipline of differential game theory. Such problems arc vcry difficult and, thus, one must assume acceleration histories to obtain analytic information (as is done here) [Powers, 19761. With assumptions (1) and (2), the equations of motion (3-3) become R = VT cos u - V,,, cos(a y) and Ru = - V T sin u + V,, sin(u - y). For pursuit guidance, y = u; thus, R = V T cos u - V,,, and Ru = - VT sin a. Note that u = 0 if u = 0 or T. Therefore u will always be varying unless a = 0, a, which implies a head-on o r tail-on path. Consequently, since y = a, the pursuer must be continuously turning if u f. 0, a,a situation which is undesirable. Now, the pursuit equations are solved dR v,,, as - = ( - cot u + - csc a) da. Assuming that 0 5 u 5 a, then, upon inteR VT gration, en R = - tn(sin a)

+

)*

1 en(tan - a) 2

+ constant

(6-7)

where p = V m lV T . When 0 5 u a is assumed, the target is essentially assumed to be "above" the pursuer. If the target is below the pursuer, then it is assumed that 0 2 u 2 -a, and cot u = -cot I u ( and csc a = -csc I u 1. Equation (6-7)

i 59)'

is then integrated, giving R sin u / tan

=

C. This relation leads to the property

of pursuit guidance which terminates in a tail chase, that is, u(tf) = 0, for all cases except the head-on case. Assuming a nonhead-on mission, has the following range of values in the neighborhood of the terminal time:

+

+={

mifp>2 Oifk<2

and j~ =

if p > 1.5 0 if p < 1.5

For the head-on pursuit, k = 0 is possible if V T > 0. For the nonhead-on pursuit, the pursuer must go faster than the target to be able to catch it in a tail chase. Thus, > 1. As the objective o f the attitude pursuit guidance is to keep the centerline of the missile pointed at the target, the velocity vector must always lag in the vehiclepointing direction while the missile flies at an angle of attack when maneuvering. T o reduce the miss distance, the missile must be equipped with a high-g capability for this attitude pursuit. Pastrick et al. [I9811 present an example of a simplified control system diagram for both attitude and velocity pursuit laws, as illustrated in Figure 6-38a. In the former, a wide-angle target sensor is required since it is typi'cally mechanized to be body fixed. In the latter, a narrower field-of-view (FOV) sensor

GUIDANCE

GUIDANCE

W L E AND OLD

ACTUAlOR M

1-4 .

-

AlRFluw EOUATIOI) OF YOTION

'.E--,-

1I

y ----I I

--7 ~

I

I

1

I

I

$ loo

E

1!w >

E m 0

s 8

3

w

9

g

5ti " 8

fj

20

0

1 TAROET V E W I T Y M ( I

0

Rf

TMOCT 0.2 ACCELEMlloU U IOU

M

(LI

(8)

Figure 6-38 (a) Block Iliagram ofOnc Axis-Attitude and Vclocity l'ursuit Guidancc (b) I'crfornlancc ofclassical Guidance Laws: Miss 1)istancc vs. Targct Vclocityi Acceleration (Control Authority Indicated in g's) (Frotr~ll'asrritk ct ol., 1YRI1, 0 1981 AIAA)

Sec. 6.5

Guidance Law

357

may be utilized since the body is dccouplcd frorn the scnsor mount in a manner described. In each case, K,\ is the forward guidancc gain, and it will differ for each as will the feedback damping gain K,,. The guidance filter in Figurc 6-38a indicates that higher guidancc gain in the velocity pursuit law requires some smoothing to inhibit noise of the target scnsor optics and its associated electronics. The performance that may be expected from pursuit guidance is indicated in Figurc 6-38b. The data were obtained for a tactical wcaoon of the class known as close support antitank. Although these are simulation results, cxpericncc with flight hard\varc over the past several years has validated the simulated perforrnancc to a high degree of confidence. Pursuit guidance applies mainly to medium-range missions that arc also characterized as launch-and-forget a t lock-on. The avionics allow for simple processing, but there is a requirement for a missile homing sensor. Inertial sensors, wind vanes, or angle of attack meters can be used for velocity vector direction establishment [Goodstein, 1972(b)].

Attitude pursuit guidance. In attitude pursuit, the longitudinal axis of the pursuer needs to be always kept along the LOS ro the target, that is, the guidance algorithm t i , as shown in Figurc 6-36b. The two mutually perpendicular projections of the angle between the longitudinal axis of the pursuer and the LOS, into the planes in which the pursuer is steered in pitch and yaw, form the look angles of a L and o,. (see Figures 6-36a and 2-10). Given that it is only required to measure UL and a,,,the primary advantage associated with this guidance method is the way in which it facilitates the task of obtaining the error parameters. The measurements of UL and a,,are generally provided by radar or IR angle scnsors. The major deficiency of this guidance method is that the aircraft's vclocity vector relative to the LOS is not anticipated. When the aircraft is equipped with a homing missile, the choice of guidance method takes into account whether the missile's antenna or EO sensor is able to move relative to the missile while it is attached to the aircraft. Fixed sensors are oriented along the missile's longitudinal axis, and the missile may locate the target within a window bounded by a small solid angle. The bisector of such an angle is approximately measured by the aircraft's longitudinal axis. Attitude pursuit must be used in this case in order for the sensor to acquire the target. When the aircraft employs ASM, the choice of guidance method ;akes into aicount the maneuverability of the missile. If the missile has restricted maneuverability, then attitude pursuit guidance is used, where the aircraft heading will be directly toward the target prior to the missile's launch and initiation of its own guidance. During this stage of flight, the aircraft's altitude is generally held constant. This technique profits from the fact that the missile is not required to undergo complex maneuvers in the horizontal plane subsequent to launch. However, because the aircraft must be in close proximity to the target, this increases its vulnerability [Maksimov and Gorgonov, 19881. With a missile that gets maneuver from body angle of attack, and with perfect response u = 0, using Equation (8) of Table 2-2, the guidance equation (shown in Figure 6-36b) becomes y, = u / ( l + T,s). That is, the guidance

Guidance Processing

358

Chap. 6

system has a time lag o f T, even assuming a perfect response to the guidance error equal to ( a - 0). The formulas for the miss distances are:

M

=

Ay V,,,T, [I - e-"Tm] for initial heading error or seeker bias equal to A y

M

=

AT,T% [[IT,

-

1

+ e-"Ta]

for target acceleration AT,

(6-9) Pursuit guidance with body-fixed seekers has poor miss-distance performance for missiles that get their lift from body angle of attack [Hiroshige, 19861. Velocity Pursuit Guidance. The use o f attitude pursuit is feasible with an ASM guidance system when encountering stationary and slowly moving targets. However, guidance errors arise even in the absence of wind as a result o f the generally curvilinear nature o f the missile's reference trajectory, and the fact that the longitudinal axis of the missile and its velocity vector are not always coincident. The best results are obtained using velocity pursuit guidance, o r pure guidance, as it is also called. If there is no skid, and if the error paramctcrs in heading and pitch are generated in a roll-stabilized system of coordinates, then the form of the pursuer trajectory control algorithms in thc horizontal plane is shown in Figure 6-36b, wherein UI,,~and u$ are the lead angles between the projections of the line of sight and the required aircraft velocity vector into the horizontal and vertical planes defined as [Maksimov and Gorgonov, 19881 u,,,~ = UL - a sin and u$ = u,, a sin 6 where a denotes the angle of attack and dcnotcs the roll angle. The essence o f this guidance process is to align the aircraft's actual velocity vcctor with the required vector. The prcccding equations show that the velocity pursuit technique requires n~casuretncntsof the angles UL, u , . a,and 4. O f thcsc four quantities, the last two arc measured rather casily. Also, the scckcrs for attitude and velocity pursuit are somewhat similar in rcspcct to size and weight.

+

+

Deviated-pursuitlfixed-leadguidance.

O n e aspect in which the velocity pursuit method suffers has to do with the fact that it does not consider the effects of wind on the pursuer, thus introducing additional guidance errors. Wind effccts can bc offset by employing PNG or the fixed-lead mcthod. The fixed-lead guidancc is also rcferrcd to as dcviated or modified guidance. Fixed-lead guidatlce is dcfincd to bc the casc in which the angle bctwccn thc LOS and the pursuer's vclocity vcctor rcnlains constant (scc Figurcs 6-36 and 2-10), that is,

-

urrur(1) - y(1) = ulrc,,,constant = Ulr,,d, [prcsct (partial) lcad angle]

(6- 10)

The pursuer flics with a preset (partial) lead anglc, anticipating the future target position and speed. The guidance algorithm is shown in Figure 6-36b. The additional lcad anglc is proportional to the LOS rate. That is, the anglc UL is changed by an amount proportional to the LOS rate in the same plane. The guidancc objective is to drive k , u ~ + k; 15 to zero, wherc k, and k; arc gains of the guidance command processor. For a nonmancuvering target, thc values selected for these quantities are

Sec. 6.5

Guidance Law

359

such that the reference trajectory is approximately a straight line. With this guidance algorithm, it can be seen that the required measurements for the fixed-lead guidance method in the horizontal plane include U L and u. That is, V T . V m ,and y r need to be known to determine the fixed-lead u,,,a. However, such a scheme might be useful if it is known beforehand where the target is conling from, and a "spray" of pursuers is sent up (for example, a crude antimissile missile system). If, as the missile remains suspended from the aircraft, the missile sensor is able to move, then it may follow the target with the help of signals from the aircraft's information subsystenl. Such capability allows the missile seeker to acquire the target over a wide range of angles relative to the aircraft's axis. If the target's motion is constant, the missile may be guided to intercept by having the aircraft fly directly toward the intercept point of the missile and target. Assuming straight-line uniform target motion, guiding the aircraft to the best intercept point involves straight-line flight and the computation of the proper lead angles u,,d< and u , (see Figure 6-36b) [Maksimov and Gorgonov, 19881.

Constant-bearing guidance.

A more efficient and less demanding course than the pursuit course can be followed by predicting the target path and directing the missile toward the predicted intercept point. This works well if the prediction is correct, and can be accomplished by controlling the missile's trajectory so as to keep the direction (LOS) to the target constant, that is, known as a constantbearing course. If an object appears to someone on a ship, for example, to be stationary while increasing in size, then that person realizes that he or she is on a collision course with the object in question. Such a condition is also pertinent in a missile-target engagement, and is known as constant-bearing navigation, which is a two-point guidance (three-point realization). It is brought about by steering the missile such that there is no rotation of the LOS. Figure 6-37b illustrates a constantbearing course for the same initial missileltarget conditions shown in Figure 6-37a. This is the optimum course that can be followed for the conditions stated: constant missile and target velocity, and nonmaneuvering targets. As shown, the LOS between the missile and the target maintains a constant direction in space, that is, the LOS from missile to target always remains parallel to itself [Heaston and Smoots, 19831. Thus, constant-bearing guidance is defined by u 0 (that is, the LOS does not rotate). By proper orientation of V,,, at guidance initiation, constant-bearing guidance will always intercept the target, even if the target is maneuverable. This is easily seen as follows. Suppose, first, that the target is flying horizontally with speed VT and the pursuer is located so that u(t) = uo and the guidance algorithm is shown in Figure 6-36b. Without assuming constant speeds, to keep u = UO,from Equation (2-2b) the components of the velocity perpendicular to the LOS must be equal. That is, VT sin(u - y ~ =) V,,, sin(u - y). Therefore, the relative targetpursuer velocity is actually being aligned with the LOS. The pursuer can catch the target in the case V,,,> V T . The only indeterminacy left is the choice for y , which can only be one of two possible choices since V,. sin(u - y ) is specified. The choices f

360

Guidance Processing

Chap. 6

are yl or a - yl since yt = sin(a - y,). This amounts to choosing the pursuer velocity vector so as to close the gap along the range vector (pursuit) as opposed to widening the gap (evasion). If the target does not maneuver, then the sit;atiO" shown in Figure 6-37b occurs. The reason is that the components of velocity perpendicular to the LOS are equal so that the two missiles are going the same Speed . . in the ~erpendiculardirection. However, at the same time, the radial distance is decreasing continuously and eventually goes to zero. This is also apparent from the equation for R (even for the maneuverable case). That is, suppose V , cos(a - y) > VT COS(O.- YT),

and the integrand is always negative (which implies Ro - I $ [ ] d t I should eventually equal zero, that is, a definite interception). Missiles that possess sufficient maneuverability permit the launch aircraft to be guided with a constant or programmed bearing angle [Maksimov and Gorgonov, 19881. The missile will have to perform postlaunch maneuvers in the case of a nonzero bearing angle, and some of its fuel resources will be depleted in an effort to get on the required trajectory. It appears that constant-bearing guidance might be the solution to all of the problems since it will intercept a maneuvering target and is not as wasteful as pursuit guidance. However, constant-bearing guidance requires the pursuer to be able to detect and correct instantaneously any changes in the LOS direction. Realistically, this would be asking too much of any guidance system. Since constant-bearing guidance has so many good qualities, it seems feasible to seek a slight modification of the scheme \x.hich will make it easier to implement while preserving its good intercept characteristics. O n e possibility is to treat u as an input error signal which the pursuer drlves to zero by adjusting y. A simple way of doing this is to require j = X u since y = constant when a = constant and y changes until jr = 0. Actually, such a guidance scheme is called PNG and, indeed, it is the most popular homing guidance scheme. The development of PNG was a major breakthrough in homing missile guidance. The fact that a constant-bearing course results in a collision course led to the development of the PNG law. PNG is one in which the missile heading rate is made proportional to the LOS rate from the missile to the target. The purpose of such a course is to counter the tendency for the LOS to rotate and, hence, to approximate a constantbearing course. The initial turning rate of a missile launched on a collision course and using PNG is zero. For a nonmaneuvering target, if the missile and target velocities remain unchanged, the missile will continue on a collision course, having achieved constant-bearing navigation. If it happens that the missile is not launched on a collision course, there will be some curvature associated with the resultillg trajectory that depends on the navigation constant. For a small navigation constant A, the missile corrections are small early in the flight, yet may become quite Pronounced as the missile nears intercept. The situation is reversed for larger values f

Proportional navigation guidance (PNG)and its variations.

Sec. 6.5

Guidance Law

361

A, where the collision course errors are corrected early on in flight, and maneuvers conscqucntly are kept at reasonable levels in the terminal phase. For values of A 5 2 (A = 1 is pursuit guidance), there is an infinite accclcration required in the region of terminal flight. A lower limit therefore corresponds to A = 2. For valucs of A close to 8, the missilc steers in response to very high-frequency noise as well as to lower-frequency signals. This situation is characterized by gross oversteering with a constant dithcr o f the controls and high drag [Trottier. 1987). A reasonable value of h based on experience is usually set at between 2 and 4. 'The trajectory of a pursuer operating with a PNG law is also plotted in Figure 6-37b. A purser turning rate of three times that of the LOS rotation rate was used to plot the trajectory. As in the Figure 6-37a, the missilc was launched in thc direction ofthe target. T h e rotation of the LOS is measured by the seeker, which causes conlmands to be generated to turn the missile in the proper direction. With PNG, the proper direction o f flight is established shortly after launch, and the missilc then flies on a constant-bearing (collision) course to intercept the target [Heaston and Smoots, 19831. If the LOS does not rotate (in inertial space), a collision will eventually occur, as shown in Figure 6-39a [Fossier, 19841. Substituting j = A u t o Equation (2-4) yields

Ru

+ [?R + =

V,,, cos(a - y)A] I?

V,,,sin(u - y)

+

YT cos(u - y ~ j~ ) -

vT sin(o - 7,)

(6- 12)

If the pursuer velocity is constant and the target is not maneuvering, the right-hand side of Equation (6-12) vanishes, and the behavior of u(t) is governed by a homogeneous differential equation. If the gain factor h(t) is set as A = - A,,R~[v,,, cos(u - y)] = A,,, where A,, > 2, and substituted into Equation (6-12). then R u - R[.A,, - 216 = 0, which can be solved analytically, giving .A,, - 2

b =

(k)

6,

and hence j = jo

(6- 13)

The solution o f this differential equation tends to zero for the pursuer-target closing, that is, d ( t ) < 0. Equation (6-13) shows that b(t), which is maximum at the beginning o f the flight, decreases linearly to zero for A, = 3 and approaches the value of zero asymptotically for A,, > 3. The collision course condition of u(t) = 0 is satisfied exactly at the final point R = 0 with a vanishing turning rate j = 0. Hence the disadvantages of pursuit guidance are avoided by thc P N G law with the gain factor A(t). As mentioned earlier, u(t) for P N G tends to the collision-course condition u(t) = 0 for increasing values of A,,. Therefore, the constant-bearing guidance is viewed as a special case of P N G in this approach, which is reached in the limiting case of A, -* m.

Example 6-13. P N G is selected as the steering law for the T G S M of Example 6-10. For the sake of simplicity, a linear model is considered for the homing head (perfect information) and all TGSM characteristics. Such simplification still makes it possible to obtain useful qualitative results. In the case of a fixed target, it

INTERCEPT

MISSILE LlNE OF SIGHT

\ UNDERSHOOT

MISSILE &TARGET L.O.S.

A

OVERSHOOT

MISSILE

L.O.S.

MISSILE TRUECTORY BASED ON w (DECELERATION PHASE)

TARGET

CONSTANT VELOCITY TARGET TRAJECTORY

ENGINE BURNOUT

A'

MORE DIRECT MISSILE PATH

NOTE: IN THE DIRECT PATH. LINE OF SIGHT RATE IS POSITIVE BEFORE BURNOUT AND NEGATIVE FOLLOWING BURNOW

Figure 6-39 Line-of-Sight Motion o f lntcrccpt ((aj.fio111[Fossirr, 19841, 8 1984 A I A A (~).FoIII[COIIZ~IPZ, IY7YIh/] ulifl1 pcrr~lissio~l.ho~s AGARD)

Sec. 6.5

Guldance Law

363

is possible to analytically solve the PNG equations to find siniplc relations bctwccn the initial altitude and look-down angle, and bctwccri the maximum acceleration and impact anglc [Trottier, 19871. From Equation (6-13). the heading rate and, consequently, acceleration for A,, ? 2 are maximum at the start of flight, diminishing to zero at its end where R = 0. Equating Equations (6-13) and (2-2b), one has for the end of the flight y, = - y,,l(A,, - 1). From the target's perspective, the final heading o f the pursuer is always at an angle y,,l(i\,, - 1) above the initial LOS. Figurc 6-40a is a diagram of pursuer altitude i t versus the down range x of the target when the terminal phase is initiated. As sho\vn in the figure for :I,,= 1 and C;, = I00 miscc, curvcs of constant impact angle arc straight lines emanating from the origin, and curves of m a x i n ~ u macceleration form circles that are tangent to the origin. Curvcs of constant impact angle are given for 20, 40, 60, and 80 dcgrees, and curvcs of constant maximum acceleration correspond to 3, 3, 10, and 15 g. Figurc 6-4Ob isolates that portion of Figure 6-40a in the pursuer's range of opcration where the thick line with the arrow at an altitude of 130 m shows the trajectory traced out by the pursuer during the search phase. Impact angles for this trajectory can be anywhere between 30 and 60 dcgrees, depending on how much time is required for acquisition. Thc acceleration requirement for this trajectory ranges from 9.8 to 13.6 g. The PNG eq~lationspresented here are valid for a stationary target and provide the pursuer's required acceleration at the initiation of the terminal guidance phasc in order to effect an intercept without acceleration saturation. This is not to say that the occurrence of accelcration saturation eliminates the possibility o f realizing a hit. Also, similar to the existence of a maximum acceleration capability required to hit a stationary target at x = 150 m from an altitude of 130 m without acceleration saturation, there also exists a minimum acceleration below which intercept is not possible a t all. The former corresponds to 13.6 g, while the latter corresponds to 6.8 g, at which acceleration the pursuer will fly in a circular path o f 130-m radius.

PNG System Design and Analysis. The preceding discussion of P N G assumed constant missile speed and a nonmaneuvering target. Upon removing these assumptions, Equation (6-12) is no longer a homogeneous differential equation. The solution will not tend to zero unless the guidance law of PNG is extended to compensate the driving functions o f Equation (6-12) as follows:

This type of guidance law is denoted by extended PNG (see Figure 6-36b). T h e first term in the brackets represents the well-known drag compensation, whereas the other terms provide target tnaneuver compensation [Goodstein, 1972(b)]. In most missiles, normal acceleration is commanded instead of turning rate. Since-A,,, = V,,cos y o +, the guidance Equation (6-6) is expressed as

u , = A,,,, = Vm cos yo A6 = lateral acceleration command

(6-15)

Guidance Processing

2m

I

1

1

I

400

ba)

800

lax,

Chap. g

DOWNRANGE (m) (A)

260

A L T I 1

u

2CX)

160

D E

(m) 60

60

100

260

Figure 6-40 (a) Curves of Constant Impact Angles and Constant Lateral Accelerations (b) Zoonl of Figure (a) in the Neighborhood of the Target (From [Trorrier, 19871 u8irh prnnissinn j o t n AGAKD)

Guidance Law

Sec. 6.5

385

The navigation ratio k;, used in Figure 6-35b(ii) is an inappropriate gain term. The guidance gain should be kept srllall for most of the flight to maintain stability and large a t the end of the flight to achieve a small miss distance. Assuming the overall navigation loop gain K, = All, ( i l = effective navigation ratio) in Figurc 6-35b(ii), the guidance equation becomes rr,

=

A liu

= ilI.., u and similarly in vertical plane

rr,,.

= AV,

6,

(6-16)

1,

The missile's lateral accclcration history is generally invariant. The invariance is not with A but with A = A I*',,, cos y,, 1 l,',, which is obtained by comparing Equations (6-15) and (6-16). This indicates that the value selected for A should be proportional to the missile-target closing rate. The rcsponsivcncss of the missile in correcting for LOS rotation is thus determined by A. That is, a higher closing rate necessitates a more responsive missile. Equation (6-16) shows that the missile velocity vector's required lead angle results when the accelerations A,,,,. and A,,,: of the missile reach rr,. and rri,,. In order to employ PNG, it is required to measure V,. as well as A,,,, and A,,,:. An automatic target tracking system is used to measure 6 and u,. A is varied to make the missile attain the degrce of rcsponsivcncss which is compatible with its response, tracker noise, targcr s~gnaln o w , and target maneuvering capability. One of the important features of PNG is the generation of a nearly straight-line flight path for the missile to a slo\vly moving target. Another very important feature of I'NG lies in its ability to cffectivelv deal with all attack altitude and aspects. That the missile flies along a straight path with constant velocity, given that the target also travels uniformly along a straight line, is most easily seen when A is very large. For i l 1, the LOS moves parallel to itself during the guidance process as shown in Figure 6-3Ya. This type of LOS motion can occur only if the projections of the missile and target velocity vectors V,,,and VTonto the normal to the LOS are equal to each other. It can be shown through using elementary geometrical constructions that these projections are only equal in the case of straight-line missile flight, with V,,,equal to a constant and with uniform, straight-line target motion. In practice, A 2 3 results in a nearly straight-line missile trajectory. It is not recommended to increase A too much, in order ro straighten the missile's trajectory, since guidance accuracy degrades as A is increased. This is a result of the manner in which the effects of internal perturbations are made more severe as A is increased. Because the missile control surfaces, used for steering in the horizontal plane, for example, can be deflected by any angle required to make Amyequal to i r , , PNG can be used to attack at all attitudes. Higher altitudes require greater deflections to produce the required value of A,, for a given value of u,. Therefore, PNG-based guidance systems are able to adapt to varying external conditions. Owing to the presence of the closing velocity V, in Equation (6-16), P N G possesses the ability to attack at all aspects. This can be illustrated by assuming that the missile is steered in the horizontal plane, and using Figure -3-11b. The system may be kept stable for any attack aspect i f the guidance signal t i i generated by the guidance algorithm Equation (6-

*

Guidance Processing

366

Chap. 6

16) is a linear function of the closing velocity V( [Maksimov and Gorgonov, 19881, Applying Equation (2-6) to Equation (6-16) yields

The expression in parentheses in Equation (6-17) represents the miss distance that would occur, lacking any target maneuver, if the missile underwent no further Yr -=-A A

tZ =Y

A ( A - 2)

tit, (A- l)(A- 2)

+

A(A-l)(A-2)

I

e

A -2

A-2 0

A=3 A=4

1.O

0

Illf (B)

Figure 6-41 (a) Normalized Relative Trajectory (b) Missile Acceleration Required to Overcome Step Function Target Maneuver (c) Missile Acceleration Required to Overcome Initial Heading (d) Miss Distance Due to Initial I'osition Error y , (c) Miss Distance Due to Target Maneuver ((b)-(r)jotn [Fosrirr, 19841, 0 1984 AIAA)

Sec 8.5

GuIdance Law

From Eq. (2-6), $=initial LOS rate = V ~ Y , ( O ) / R * (+~ Y,(O]/R(O) ) = -v, HVNO). assuming y,(O) = 0, yr(0) = -V,HE Ax2

0

0.6

1.0

t/tf (C)

(E)

(Continued)

corrective accelerations. This miss distance is termed the zero effort miss perpendicular to the LOS. Thus, PNG can be considered as a guidance law in which the commanded accelerations are inversely proportional to the time-to-go squared and directly proportional to the zero effort miss [Nesline and Zarchan, 19811. The normalized form for the miss distance due to a step maneuvering target is shown in Figure 6-41. A normalized relative trajectory y, for different A is shown

368

Guidance Processing

Chap. 6

in Figure 6-41a in which the ~ c r f e c sccker t and noise filter are assumed, and a zero lag dynamic system is also assumed, that is, T , ~ = 0 in Equation (2-35) Uerger, 19601. As shown here, a smaller position error will result from increasing A during flight. The normalized form of the acceleration required by the P N G law in order to intercept a step maneuvering target is shown in Equation (2) of Figure 6-41, while its trajectory is illustrated in Figure 6-41b. It is shown that increasing thevalues of A vl,ill decrease the missile nlaxi~numacceleration requirements for target intercept. Parasitic effects and noise considerations impose a practical upper limit on the maximum value of A. From Equation (2) of Figure 6-41b, the maximum missile acceleration required can be obtained at the intercept point as (II,),,,,, = I. AT,,l(A - 2). Thus, three times the target maneuver acceleration at A = 3 is required by the missile. More acceleration capability of the missile is generally required when other noise inputs plus system dynamics are included, as shown further on in Figure 6-43a. The normalized form of the acceleration required by the PNG law in order to intercept a targct with an initial heading error is shown in Equation (3) of Figure 6-41, while its trajectory is illustrated in Figurc 6-41c. The examples of Figures 641b and 6-41c show that, for a target maneuver and an initial heading error, thc lateral accelcratiori history is proportional to the disturbaticc. Further~norc.the shape of this curvc is dctcrnlincd strictly by A. Using a simplified two-time lag control system A,,,/e = Alf,r/(l + 7,,5/2)' it is possible to analytically dctcrrnine miss distance for thcse cases as shown in Figures 6-41d and 6-41c. Following Travcrs [1965], Figure 6-42 dcfincs thc normalizations associated with several coniponcnts of the error budget. Only one deterministic miss coniponent is listcd, the miss duc to a step of cross velocity. It is a function of normalizcd flight time rfl~,,.The remaining nor~iializedadjoints arc stochastic and depend on the signal (targct accclcration) and noise I'SD, 4. Thc dil~~cnsionlcss normalizcd adjoint cocfficicnrs in Figurc 6-32 arc indcpcndcnt of 7, and I]<. and are functions only of the ordcr of the system (for exan~plc,fifth, in Equation 3-35) and the navigation ratio A. Thcse coefficients can thcrcfore be computcd numerically if A, the ordcr of thc systcm, and the distribution o f thc time constraints are given. Thcse adjoints can also be dcvclopcd using the tncthodology of thc appendix of Neslinc and Zarchan [1984(a)]. It is generally acccptcd that a fifth-order system (sce Equation 2-33) is the lou~cst-ordcrmodcl that can be used to realistically model a missile. The stability and pcrfornlance of the PNG, as discusscd previously, lvill bc dccrcascd in rhc prcscncc of the systcm lag 7,. Advanccd guidancc law to bc prcscntcd in Chapter H compcnsatcs this systc'tn lag. AgAii, thc total RMS miss distancc is obtained by applying Equation (3-30). As sho\vn in Figurc 0-9f, illcreasing 7, beyond the o p t i ~ n ~ ~valuc t i i (7,,,,,, incrcascs thc tniss on account 01111creased target mancuvcr and rcccivcr noisc contributions. O n the other hand, dccreasing T,,below (T,,),,,,,incrcascs the tniss o\ving to increased glint noise contributioll IAlpcrt, 1988; Goodstcin, 1972(b); Ncslinc and Zarchan, 1984(a); Wcidcn~allnq 19841. For shorter flight time, thc miss distancc tends to be rcduccd by increasing A. Howcvcr, noisc can increase the niiss distancc. A = 0 would givc thc best accuracy for a mancuvcring targct in thc case of sinlulati~ig\vithout noisc. Larger

-

Guidance Law

Sec. 6.5

Normalized Miss Adjoints

M, =

5 ,(t,) I Tgt Mvr
a,, MY8

Normalized Acceleration Adjoints

(5) ( Glint

$y

=

8

a (r ) ( Active RDN

MmA=

.t~, <12 Q",'.,112 f

eR:<

TJ: RDN: a,: Yg: myD: y

4. I Active RDN

2

q,Jrf)I Semiactive RDN M m u 12

0 . ,(t

- IGlint - '=8.j/2 LQ112 8

f

):

a,,"ISemiadive RDN

9m.S~

Target Maneuver Range-Dependent Noise (RDN) Range-Independent Noise (RIN) Glint Noise Deterministic Miss Due to a Step of Cross Velocity RhB Miss RMS Acceleration

Figure 6-42 Normalized Miss and Acceleration Adjoints (From [Travers, 19851 wifk permission from AACC)

A will cause the missile motion to have low-amplitude oscillation and the miss distance to increase. These types of analyses made it clear that A should be maintained at a value between 3 and 4 to avoid saturation in the endgame. It was also concluded that, if such a value of A is maintained, a flight time of at least tenfold that of T, is adequate to lower the miss caused by any disturbance to a very small value, as long as there is no saturation of missile maneuvering capability (see Figures 6-43a through 6-43c). A particularly nice feature of homing missile guidance is that, as long as such saturation does not occur, a linearized analysis of the intercept problem is quite accurate. If there is saturation, however, the miss distance quickly grows too large [Fossier, 19841. Studies have shown that A = 4 is a good compromise for low miss distance (high A) and small time-to-go before instability occurs. Chapter 3 analyzes the situation in which saturation occurs. An optimal-control based derivation of PNG results in a time varying A, as is shown in Chapter 8. Figure 6-39b illustrates P N G and a more direct approach path in which A will vary with the closing velocity. Thus, A is picked for a tail, beam, or head-on chase, and the performance degrades

Guidance Processing

-

Chap. g

Homlng time = 10 seconds Glint spectral density = 100.8 f t 2 1 ~ ~ Glint time constant = 0.7 seconds

200 -

RMS miss distance (ft)

79 = .2 sec

100 -

-

Missile acceleration limit (g) Figure 6-43 (a) The Missile Maneuver Capability Should Exceed Three T ~ m e sthat of the Target (b) Terminal Homing Tlme to Reduce Effects of Step Target Maneover (To reduce effects of step target maneuvers, a terminal homing time equal to 10 times the total system time constant is required.) (c) Terminal Homing Time to Close Out Heading Errors (To close out heading errors a terminal homing time equal to 10 times the total system time constant is required.) (Courrcsy of K. H i r o s h k r , 1984)

for other cases. Radar missiles can directly measure V, and are therefore able to keep A constant in spite o f varying target aspect. Another scheme uses filter theory to provide an estimate of V , based on LOS rate and inertial body motion (rates). This method could be applicable to passive seekers. Through Equation (2-36). the fin deflection due to target maneuver and heading error becomes a matter of scaling Equations (2) and (3) o f Figure 6-41 by k i ' , while similarly determining RMS fin deflection due to target maneuver 2nd measurement noise is done by scaling o,,in Figure 6-42 by k ; ' . Through Equation (2-38), it is now possible to find fin rate response due to a step target maneuver by differentiating Equation (2) of Figure 6-41 and then scaling by k i t . When this is done, the result is

Sec. 6.5

Guidance Law

---

Slngle time lag Two equal lime lag

A = Navigation ratlo = Total system tkne constant = Target acceleration

T~

"I.,

Figure 6-43

(Continued)

From this result it can be seen that the maximum fin rate due to a step target maneuver occurs at the beginning of flight. As noted in Nesline and Zarchan [1984(a)], the fin rate standard deviation due to target maneuver and measurement noises is obtained by dividing u,, in Figure 6-42 by T, and kl and then replacing A, with Fi.As is true of acceleration, increasing A or decreasing T, always increases fin rate activity. Another form of modified PNG is VCPNG, wherein a compensation term of missile velocity change is introduced. For a coasting missile with no velocity change (as caused, for example, by aerodynamic deceleration), PNG will work ideally against a constant-speed nonmaneuvering target. Since in reality every missile experiences veloc'ity changes, it seems reasonable that some improvement could be gained by compensating the velocity change, particularly when it is large. By introducting the lateral component of this velocity change, the missile-guidance acceleration command is given in Figure 636b [Kuroda and Imado, 19891.

Velocity compensated PNG (VCPNG).

372

Guidance Pr~~t?sSing Chap. 6

A = Navigation ratio tg=Total system time constant HE= H e a d i error ~

--

Single time lag Two equal time bg

'd7g (C)

Figure 6-43

(Continued)

Pursuit plus PNG. O n e approach to combining elements of pursuit guidance and PNG into one guidance law is to compute guidance signals based o n both laws, provide a time-varying weighting factor for each, and sum the result. Such an application usually weights pursuit guidance heavily at long ranges where the noise problem is most severe and the accuracy rcquiremellts less severe. O f course, a knowledge of time-to-go or range is required [Gonzalez, 1979(b)].

Dynamic lead guidance.

Dynamlc lead guidance provides results sinlilar t o the weighting technique, but for different reasons. At small LOS rate frequellcies (which typically occur at long ranges), the guidance law behaves like pursuit guidance; at large LOS rates (which typically occur at short ranges), it behaves like PNG. The advantage is that no estimate of range or time-to-go is necessary. T h e behavior transitions "automatically" based on the frequency of the input signal. It also has the advantagc of better performance in atypical situations (for cxamplc, largc LOS

Sec. 6.5

Guldance Law

373

rates a t long ranges). However, stability problenls can occur if significant noise is still present when the guidance law transitions to a I'NG-type behavior [Gonzalez, 1079(b)].

6.5.3 Sensitivity and Comparison of Guidance Laws Factors which increase the miss distance are greatly manifested during the terminal part of the homing phase due to the error sources caused by the mano target and the homing sensor target acquisition capability [Cloutier, 19901. Although H guidance law may call for high turning rates, the missile may not becapable of responding and will not be able to follow the intercept trajectory. In this regard, an advantage of PNG over pursuit guidance is the fact that course corrections are made early in the missile flight. If the called-for acceleration exceeds the missile's capability, there may be enough time left to achieve the proper flight path with the missile turning as position errors that are directly proportional to the range between the target tracker and the target. Consequently, an above-standard tracking system that has small angle tracking errors is essential for accurate guidance at long ranges. Laser beam-rider guidance is used for antitank (SSM and ASM), antihelicopter (AAM) and short-range air defense missiles (SAM) because of its high electronic countermeasure resistance. MMW beam riding is an all-weather alternative for laser beam riding. In pursuit guidance, the maximum turning rates occur at the end of the flight when there is little time left. As shown in Chapter 8, with the development of highly maneuverable targets, pursuit guidance has become outmoded and PNG has become somewhat of a marginal guidance law. New guidance laws are being developed that utilize more information about the missileltarget encounter such as range, closing velocity, time to intercept, missile acceleration, and target maneuvers, and incorporate the optimization of the missile's trajectory and expansion of missile launch envelopes [Heaston and Smoots, 19831. The nominal conditions of the air target engagement are displayed in Figure 6-44a. The interceptor to the left is assumed to have a speed V, of 2,000 ftlsec, and be 5 degrees off the LOS to the target in a top view. The analysis is for the horizontal plane only. The range at the start of the simulation is 10,000 ft. The speed V r of the target at the right of Figure 6-44a is assumed to be constant at 1,000 fps, at 0 degree with the LOS, so that the nominal engagement time is about 3.5 seconds. The parameters that affect miss distance as a function of guidance law are shown in the boxes of Figure 6-44a. They are: target, varied off the LOS; target speed; magnitude of target acceleration for evasion capability, measured as a target turn capability to the target's right; sensor bias in the measurement of the LOS angle; sensor noise from all sensor causes at two different levels of target generated noise; and the average magnitude of a wind gust from one side whose instantaneous magnitude is a random function of time. The study results for air targets are displayed in Figure 6-44b. The sensitivity of the guidance laws to target heading is shown in Figure 6-44c. The PNG's use of LOS rate measurement produces larger corrective action earlier than the LOS law or the pursuit law. The latter two respond

Guidance Processing

F

TARGET

Chap. 6

I

Figure 6-44

(a) Nominal Conditions (b) Guidance Law Trends for Air Targets (c) Guidance Law Sensitivity to Target Heading (d) Guidance Law Sensitivity to Target Speed (e) Guidance Law Sensitivity to Target Acceleration (1) Guidance Law Sensitivity to Sensor Angle Bias (g) Guidance Law Sensitivity to Noise (h) Guidance Law Sensitivity to Wind Gusts (i) I'roportional Guidance Law Variations fj)Guidance Law Trends for Surface Targets (k) Pursuit Guidance Law Variation (From (Goodstein, 1972(b)] upirh pcnnission j o ~ nACARD)

too late when the missile is not going to come quite close as the last few seconds begin. When plotted to the scale of 100 feet for maximum miss distance, the sensitivity of the guidance laws to target velocity over a range between zero and double the nominal speed seems small, as shown in Figure 6-44d. Close examination shows, however, that PNG, subject to the errors in angle rate measurement, produces a larger miss distance than the other laws as target speed increases. Figure 6-44e shows

Sec. 6.5

Guldance Law

Figure 6-44

(Continued)

that, as target acceleration increases, all the guidance laws produce commands which lead to similarly increasing miss-distance magnitudes. However, if the gain for PNG is raised by one third, the miss distances fall noticeably, as shown in the lowest curve marked as having a higher proportionality constant. Similar changes to the LOS and pursuit laws do not produce similar improvements. Bias errors in the LOS measurement can come from mechanical installation, boresight procedures and changes, radome errors, electronic component changes, and mechanical changes. For the LOS and pursuit guidance, which depend on steering directly at the target, increasing bias errors in LOS measurement will cause increasing miss distances.

Guidance Processing

1

2

Figure 6-44

1

1

Chap. 6

LINE-OFEIOHT

6

(Continued)

I'NG, however, is essentially insensitive to bias errors, since they produce no angle rate error. This trend can be seen in Figure 6-44f [Goodstein, 1972(b)]. While P N G can handle a fixed- or bias-angle error, it can be prone to causing erroneous steering signals as a result of noisy angle sensing, leading to very noisy angle rate information. Figure 6-44g displays the effects of sensor noise on miss distance, showing LOS and pursuit guidance to be less affected by the angle noise. The solid curves are for sensor noise with no glint or noise from the target. As shown in Figure 6-44h, the response of the LOS guidance law to wind gusts is slow. Pursuit guidance, with its sensing of the velocity vector direction, and PNG,

UIO

/'

olnun~ (F7RMSl

UI-

---

0

5

15

10

20

NO TARGET NOISE

'20 111~;' TARGET NOISE

25

NOISE I Y R I M ~ ~ I

(G)

MIS DISTANCE IFT RMSl

WIND GUST MAGNITWE t

WIND O W MAGNITUDE I K W l S l

(H)

Figure 6-44

(Continued)

in which the LOS rate change is detected quickly when the missile is blown off course, are both less sensitive to the wind. As summarized in Goodstein [1972(b)], the LOS and pursuit laws are seen to have poor or average performance in several categories, but the avionics equipment cost and complexity is less for these laws than for PNG. The PNG performance is good in all categories except in its response to noise. Angle rate measurement noise, homing sensor noise, and target noise all cause responsiveness which generates steering signals driving the missile all over the sky. A guidance law analyst should select the lowest cost, simplest guidance law

Guidance Procasing

Chap. 6

FOR TAROIT w I m PEED ADVANTAOE

"EARLY A N D " U T L " BIRD FDR WARHEAD EFFECTIVENESS

L O0

TIME TOGO REDUCE POWER

wwm1w

REDUCE OLCWWIYN

AVMO CLOUSR BERER TAROCT SIQNATURC B E n c n FOR WARM^^ E r r E c n

(K) Figure 6-44 (Continued)

which can meet miss-distance or kill-probability requirements. Against air targets, simpler engagement situations can use LOS or pursuit guidance. PNG is employed for more difficult engagements. When the straightforward implementation cannot provide sufficient performance, modifications may be required at the expense of additional equipment and complexity. Two examples of modifications to PNG are shown in Figure 6-441. The advantages to adding a bias to the steering signal are listed as follows. The bias can bring the missile more nearly into a head-on situation, assisting against fast targets and missile loss of speed due to control application. Some targets are more vulnerable to warhead effects from the top or bottom, and a correct bias can make the missile arrive early or late-high or low-for an anticipated target and closing geometry. Another technique in use is to change the guidance constant as a function of time-to-go to intercept. This reduces control activity and, with it, power consumption and speed loss. In the same manner as sensitivity studies and applicability trends for guidance laws with air target situatiolls have been studied, studies of surface targets can be performed. As target motion is not involved, the presence of land or sea can call for warhead detonation above the

Sec. 6.6

Single-Mode, Dual-Mode, and Multlmode Guidance

3 70

target in somc cases, rather than impact, as an additional consideration. In Figure 6-44j, a tabulation of general trends of guidance law performance against surface targets is displayed. For LOS steering to the target, the trajectory tends to flatten and, for low approaches, raises the probability of clobber. Sensor angle bias has a similar effect on LOS and pursuit guidance. The overall assessment leads to pursuit and PNG have similar performance. With smaller avionics costs, pursuit guidance is frequently chosen over PNG for surface targets. As with air targets, many variations are used. Applying a bias to the steering signal when still far from the target provides a higher trajectory as shown in Figure 6-44k. This leads to less chance of clobber, a better view of the targets, and, in somc cases, much improved warhead lethality.

6.6 SINGLE-MODE, DUAL-MODE, AND MULTIMODE GUIDANCE Referring to Figure 6-1, after analyzing the guidance processor, guidance algorithms, and available guidance laws, the designer is faced with the decision of which guidance mode to employ. The majority of homing systems that operate in both active and passive modes include semiactive-passive and active-passive systems. The target signal used in a homing scenario depends on the particular situation at hand, the goal always being to provide the best system performance. System performance here may denote longest operating range, accuracy, antijamming performance, and so on [Maksimov and Gorgonov, 19881. Single-mode guidance has an insufficient launch range for most missions. Most conventional guidance systems operate only one seeker (either RF or IR) in a dense, environment degraded, and electronic countermeasured environment. Multimode guidance schemes can be regarded as the process of combining several seekers to enhance target tracking and target classification capabilities. The technology required for such alternate or supplemental guidance modes are active transmitters, multiband antenna systems, and IR domes for high-speed flight. Besides the need to physically integrate multiple sensors into a guidance package, one also has to define the multimode guidance system control logic. 6.6.1 Single-Mode Guidance Inertial guidance, although extending launch range, is not adequate for maneuvering targets. Command inertial guidance also extends the launch range and reduces terminal mode acquisition errors. However, it is not sufficiently accurate for terminal guidance, and requires a data link to the missile. Command guidance, while characterized by a simple and inexpensive design, limits system firepower.

Home-all-the-wayguidance. When a missile derives its own steering commands throughout flight based on the processing of signals received from the target, this is known as a home-all-the-way guidance policy as noted in Witte and

380

Guidance Processing

Chap. 6

McDonald [1981]. In order to realize this type of guidance policy, the target must be acquired either prior to or just after launch. This type of guidance is illustrated in Figure 6-4c, in which the various phases in the firing of an SM-1 are shown. The missile is supplied on the launcher with a postlaunch prediction regarding angle and Doppler frequency of the target. Subsequent to an unguided boost phase, either target-reflected energy from shipboard illumination or jamming energy emanating from the target is acquired by the missile seeker. As shown in Figure 6-4b, making up the seeker are a gimballed antenna and a radar receiver that measures the angle of arrival of the received signal. Closing velocity, as determined from the Doppler frequency, together with LOS angle information is supplied to the missile-guidance computer by the seeker as it tracks the target. With the help of a PNG law, the guidance computer generates steering commands that in turn maintain a collision course for the missile. Numerous missile systems make use of semiactive, home-all-the-way guidance, one reason for which is that this type of guidance provides rather good intercept range. Good intercept range is the result of high-power ship-based or ground-based illumination coupled with the use of up-and-over flight trajectories. The latter, which better maintain available kinematic energy, are achieved by launching the missile at a somewhat steep angle and allowing it to climb above the target early in flight. However, inasmuch as one of only a few illuminators must be committed to a single target for the duration of missile flight, home-all-the-way guidance suffers a drawback in its relatively low firepower. Figure 6-4d shows a second limitation of this type of guidance that centers around the numerous signals that may be found simultaneously at the missile's receiving antenna in a jamming and adverse weather environment. It is certainly much more difficult for the missile receiver to pick out the target signal immediately after launch than it would be if acquisition could be delayed until later in flight. 6.6.2 Dual-Mode Guidance

Midcourse plus terminal guidance. Midcourse guidance generally refers to an intermediate flight phase during which the missile receives enough information from an external source so as to be able to guide or steer toward the target without the need to home. T w o types of midcourse guidance commonly found in practice are command and inertial systems. With the former, there is an accepted coordinate frame in which missile steering signals are directly supplied. In the lattcr, information externally supplied to the missile is used to guide it toward a point or a succession of points in inertial space. Assuming that the ship, aircraft, or a launch site is capable of providing the neccssary support, a substantial improvement in firepower and missile intercept coverage can be realized by the use of midcourse guidance followed by a relatively short period of terminal homing. The fact that firepower is increased over home-all-the-way guidance is evident given that semiactive homing is confined to the terminal portion of flight. ~ o n s e ~ u e n t l ~ y a more efficient use can be made of thc shipboard and airbornc illuminators, making

Sec. 6.6

Slngle-Mode, Dual-Mode, and Multlmode Guldance

381

it possible to engage a larger number of targets. An adequate kinematic capability on the part of thc missile coupled with sufficiently accurate midcourse trajectory control allows a small target to be engaged a t a longer range than could occur if seeker acquisition were necessary early in flight, as happens with home-all-the-way guidance. The smaller missile-to-target propagation distance a t the start of the terminal homing phase also makes for increasing target return relative to some of the sources of interference shown in Figure 6-4d. Specifically, it is more likely that the missile will "see" the target in the presence of standoffjamming following a period of midcourse guidance than it is immediately subsequent to launch [Witte and McDonald, 19811.

Dual-mode concepts.

A particular case of multimode systems is that of dual-mode guidance systems. Guidance mode options in midcourse phase include inertial, common position grid, command inertial, command, command LOS (beam rider or CLOS), ARH, correlation matching, semiactive, and so on. Dual-mode midcourse guidance combinations include ARH together with outband home-onjam (HOJ), or HOJ in conjunction with RHOJ, or any number of other possible combinations. Guidance mode options in terminal phase include semiactive, active, IR, wideband HOJ, ARH, HOJ, and so on. Dual-mode terminal combinations include semiactive terminal supplemented by either active or IR, active terminal supplemented by IR, ARH combined with outband HOJ, HOJ combined with RHOJ, and so on. These systems have become very attractive inasmuch as they achieve a synergistic combination of sensor capabilities. It is possible, for example, to reduce the required performance of either sensor by using an inertial guidance system, which has high-frequency accuracy, with LORAN or GPS, which has lowfrequency accuracy. By coupling the long-range, low-accuracy, and all-weather capabilities of ARH with the short-range, high-accuracy, and fair-weather capabilities of T V or IIR, it is possible to increase the overall system capabilities beyond what could be obtained with either one of these sensors. Moreover, the resistance to countermeasures can be increased by combining ARH with T V or IIR, or MMW with IIR. The disadvantage of using a single-mode (inertial only) guidance system is highlighted in Figure 6-45a, which illustrates how shutdown can hamper this guidance system. Figure 6-45b illustrates the use of dual-mode guidance for a flight mission profile [Hardy, 19861.

Integrated navigation and guidance system.

Some of the benefits that can be gained from applying an inertial navigation system (INS) to modern air-to-ground weapons include making these weapons invulnerable to countermeasures, reducing their size and cost, and reducing the performance requirements of terminal guidance seekers. Conceptual advanced air-to-ground weapons generally require a low-altitude launch, followed by a midcourse flyout to the target vicinity, and autonomous acquisition of the target from an onboard seeker. In spite of much effort having been spent in the last decade on the development of autonomous seekers for both fixed and mobile targets, there is not currently a sensor mechanization that

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-.-*:.

F

= 2 6g 'r c

fn 0

:-

2 a 2

---

C G V)

In

0

a6 C

.O

.-L i z -

Sec. 6.6

SIngleMode, Dual-Mode, and Multlmode Ouldance

CROSS HAIRS BLINK

I I

I INITIATE LOCK-ON TO TARGET RF

COCKPIT DISPLAY A T INITIATION

FLY I N

\ -

-

\

BLIND

COCKPIT DISPLAY A T ARH LOCK-ON

b

lMPACT TARGET

(8)

VEHICLE

PSEUDO DELTA RANGE AIDING RECEIVER

MATICS

DESIRED TARGET COORDINATES

PSEUDO RANGE PSEUDO DELTA RANGE

r --I--

KALMAN FILTER

I I

+

IMU

I

MEASURED VEHICLE ' MOTION I

-------- - GUIDANCE PROCESS OF^^

i *

I

INERTIAL NAVIGATlON

I

VEHICLE POSITION AND VELOCITY

ESTIMATE: GUIDANCE LAW NAVIGATION

STEERING COMMANDS

1 FLIPPER ,AUTOPILOT

CORRECTIONS

t

3

CLOSED LOOP GUIDANCE

VEHICLE MOTION

I

i

MISSILE DYNAMICS

'

FLIPPER MOTION

ACTUATOR DYNAMICS

-.

(C)

Figure 6-45

(Continued)

1 COMMANDS

I I

Guidance Processing

384

Chap. 13

has gained widespread acceptance. Studies have concentrated on passive IR, active MMW, laser and SAR seekers, and autonomous target acquisition algorithms; hoivever, no generally applicable seeker hardware has entered the production stage. The proposed technologies have failed to conquer problems in the areas of robustness - of acquisition algorithms to diverse targetlbackground scenarios, weather, and countermeasures. Unlike air-to-ground seeker technology, navigation concepts for air-to-ground weapons are well developed and ready for near-term production. INS, Doppler-aiding devices, and global positioning system (GPS) hardware developed by several parties have proved reliable, and are decreasing in both their packaging volume and cost. By using a combination of these three navigation components, it is possible to achieve a high degree of scenario independence and countermeasure resistance [Kain and Cloutier, 19891. Figure 6-45c shows an example of guidancc applications using dual-mode, integrated GPSlinertial guidance system. As shown in Chapter 7, the integration of an inexpensive strapdown I M U and a single-channel P-code GPS receiver results in an accurate navigation solution with in-flight IMU calibration and alignment capability. In this integration scheme, INS velocity is used to aid GPS acquisition and tracking, - while the GPS measurements are used to estimate IMU errors. By employing a data-decoupling technique, separate GPS and INS Kalman filters arc able to operate independently on the satellite LOS pseudorange and delta-range measurements. In addition to its modularity, the system permits GPS-alone or 1MU-alone navigation by using separate hardware for the GPS receiver and INS computer. The GPS receiver, moreover, uses baroaltimeter data to extend its operation during periods of fewer than four SV coverage [Mickelson and Carrico, 19891. -

-

6.6.3 Multimode Guidance Applications

Most conventional guidance systems operate only one seeker, either RF or IR, in a dense, environtncnt-degraded, and ECM environment for target sensing andlor tracking. Multispcctral guidance uses at least two wavelengths for target sensing and/or tracking. The following are some examples of multispectral guidance: the use of two-color IR, dual-frequency RF, IR and RF, or IR and ultraviolet as in the STINGER-POST air defense missile. Multimode midcourse guidance combinations include command inertial supplcmcntcd by HOJ and RHOJ; command inertial together with sen~iactive,HOJ, and RHOJ; and other possible cotnbinations. Mu[timode tcrnlinal guidancc combinations include active conlbincd with HOJ, active combined with scmiactivc and HOJ, active cornbincd with 1R and HOJ. and other possibilicics. The use of two frequcncics or \n,avclengths rcduccs thc effect of countcrmcasurcs, cnhances target detectability, dccrcascs false alarnls, and increases accuracy. Difficulties may be cncountcrcd when implcmcnting multispectral methods because of thc complexity in sclccting and handing over from onc sensor to another as thc target is approached. Midcoursc and homing guidance systems are commonly included in a single missile. Much cffort has recently been aimcd at using multimodc systems that em-

Sec. 6.6

Single-Mode, Dual-Mode, and Multimode Guidance

385

ploy more than one guidance mode in the ternlinal homing phase. The purpose of these systcn~sis to defeat enemy countermeasures, provide higher accuracy, reduce false alarms, or enhance target dctcctability as compared to a single-mode system. This tern1 is occasionally used erroneously to describe systems that use more than one portion of the clectromagnctic spectrum for their operation. For instance, a system may employ both active millimeter waves and passive IR, which is both multimode (active and passive) and multispectral (MMW and IR). l)uc to the additional sensor and signal processing requirements, the onbonrd electronics and processing rcquircmcnts increase in complexity and cost with multimode systems.

Multimode guidance: radar and IR. In general, radar signals alone may not meet the accuracy requirements. The performance of a radar-guided missile has been shown to be considerably limited to dense electronic environments. An IR guidance can be added on to a radar-guided missile to improve the performance of a tactical radar-guided missile. This added-on IR guidance is used to overcome the effects of radome-induced instability at high altitudes, to increase immunity against jamming, and hence to reduce the miss distance. 1R and optical signals are limited by the weather and E O environments. In a guided missile using only IR, in adverse environment the SIN at the IR receiver decreases. The low SIN might cause the IR seeker to lose track of the targct. Therefore, the adverse E O environment has a dominant effect in degrading the IR seeker operation. Since a radar-guided missile is insensitive to weather or environmental changes, a radar guidance added on to an IR-guided missile can provide accurate target tracking. Thus, a multimode seeker system can operate under all weather conditions and environments, is immune t o jamming, can overcome the effects of radome-induced instability a t high altitudes, and hence achieves better intercept performance over a single-seeker system. The configuration and the function block diagram for IR and RF guidance systems are discussed in Book 1 of the series. A multimode cracking and guidance loop is the only control scheme of accurate homing on target under adverse RF and E O conditions. Figure 6-46a shows a conventional approach to the design of this tracking and guidance loop, while Figure 6-46b illustrates a more innovative approach. The noise effects are attenuated by the midcourse target tracking algorithm, thus reducing heading error at handover. A multimode guidance consisting of RF and IR seekers is considered in this figure. Either active or semiactive RF or both with a common aperture antenna design may be used. RF-IR homing systems operate with semiactive radar during the initial stage of the homing phase, and with IR during the subsequent stage. In situations with active radar, both RF-IR modes function together. Based on two tracking outputs from RF and IR seekers, a multimode target tracking algorithm including signal processing and decision logic is implemented as shown in this figure. The outputs from the target tracking algorithm are then passed to the advanced terminal guidance algorithm to provide the appropriate guidance steering commands as shown in this figure. The function block diagram of the multimode seeker system has been illustrated in this figure as the upper part (above the broken line). Figure 6-46a shows that the target is tracked in the RF and IR

Guidance Processing

386

Seeker Head Signal

-

I I + I

*

Gimbal + Target Detector Drive Unit __* Multimode Target Tracking 4 Algorithm IR (Chap. 8.12 + Advanced Receiver & Book I ) Terminal Guidance Unit 4 -4 Algorithm

-1 I

Signal

/

b

+

I 1

Digital signal ~rocessor

- - -1/

Range & Range Rate Measurement

1

Radome Couplin# (Chap. -)

Sigal

Transmit1 Digital h i v e -C Signal Unit Processor

Chap. 6

-

I I . I

'

uplink S19al

Receiver

-

8.1-8.5) (Chaps.

Motion

Accele-* ration Compensator ( . b k3)

f

-*

FCSi Airframe (Book 3)

9

LOS Rate

- - - - - - -Target Tracking Algorithm (Chap. 7 )

'

7 Advanced Midcourse Guidance Algorithm (Chaps. 8.6 & 8.7)

-

Terminal Guidance - -Midcourse Guidance

(A)

Figure 6-46 (a) Multimode Tracking and Guidance Loop Design (b) Optimized On-Sensor Fused IIFIlH Guidance Svstem

modes simultaneously for the active RF. This figure also applies to semiactive RF handover to IR search a t which point the missile homes on the target using only IR after lock-on. An integratcd RFIIR sensor asscmblv of a missile system shown in Figurc 646b has been developed by A G N C [199(;] to enhance targct dctcction and ECM capabilities. In the past, the RF seeker and 1R seeker \Irere designed indepcndcntly. The purpose of this optimized on-sensor-chip-fused RFIIR guidance system (OFRIGUS) is to integrate the front-cnd targct dctcction and ECM circu~nvct~tion subsystems by sensor signal fusiorl at read-out stage and on-sensor-chip analog signal processing (ASI'), thus improving continuous targct tracki~~g/rccog~~ition 2nd ECCM capabilitics over current state of the art. Benefits of this integrated sensor include: ( I ) fast signal processing-Integration of RF and I11 sensors in the same assembly improves signal processing speed and reduces the weight; (2) compactness-More room can be allocated for the IR focal-plane array (FPA) and for follower signal processing electronics; (3) fault tolcrancc-Dual-mode sensor is rcliablc

Guidance Processing

Chap. 6

against failures. It also has a robust ECM circu~nventioncapability by employing an onfoff slyitching mode of active/passi\~cM M W radar; and (4) a ~ ~ u r a ~ y - C o o l i ~ ~ can be extended to the read-out and ASP electronics, thus improving the accuracv, The integrated sensor is assembled primarily out of off-the-shelf components. 1"novative technologies such as uses of a staring M M W imaging FPA sensor to reduce the scanning time, EHF MMICs and superconductors to improve the passive MMw sensor performance, faster ASP chips for the on-sensor-chip signal processing, and a parallel processing algorithm for algorithm implementation are also accounted for,

Multimode guidance using inertial guidance. With RF-inertial systems, the necessary information is collected from radar and inertial units, the latter t w o working together as a combined system. Depending on the setup, the inertial unit is corrected by the radar data, or the inertial and radar systems correct one another [Maksimov and Gorgonov, 19881. A superaccurate refinement used as an add-on subsystem in one important missile is terminal radar; the missile flies most o f its trajectory on inertial guidance, but in the final seconds, as it plummets down on the target, it compares stored information-in effect it looks at detailed radar pictures of the target area, though these pictures are probably stored as digital information-with thc scene its own active radar actually sees, and corrects its trajectory until the two scenes match exactly. O f course, in any case where the target is itself an cmittcr-of radar or radio signals, visible light, heater (in theory if not yet in practice) noise-it is relatively simple to make a missile home on to it automatically. The accuracy of an incrtial guidance system is typically expressed in terms of drift rate. For most tactical systems, since thc flight time is normally rather short, low-accuracy inertial componcnts may achicvc satisfactory pcrforn~ance.If longcr flight times arc required, thc missile's position may be updated by cmploying another kind of position-fixing tcchnique. As presented in Chaptcr 5 , position updating may be achicvcd by utilizing satellite data fro111 the GIJS; from landmarks sensed by either radar, IR, or clectro-optical devices and used for area correlation; through terrain profile data correlated with stored data, as in cruise missiles using T E R C O M ; or through some othcr technique. Cruise missiles invariably have inertial guidancc but back this up with TERCOM for near-perfect final accuracy. Sensitive altimeters n~casurethe profile of the ground directly below and chcck thc result against storcd information. Each set of rcadings is uniquc to onc strip of land. INS is cmploycd in ICBM, whcrc the targct location and thc launch position arc known. These highly accuratc 2nd cxpcnsivc navigation systcms arc not only used i n lnissil~s but also on long-range aircraft, ships, and submarincs. This totally sclf-contained systcm is idcal for ICBM and SLBM. Its accuracy depends on how precisely the launch position is known and falls off with time (however, ballistic missiles arc very fast). Submarine irlcrtial navigators h a w to be updated by othcr means to avoid errors in launch position. Early intcrcontincntal cruisc nlissilcs updatcd thcir inertial guidancc by astronavigation. Star trackers kept nlcasuring the cxact azimuth and elevation of selected stars to provide a running chcck [Gunston, 1979).

Sec. 6.6

Single-Mode, Dual-Mode, and Multimode Guidance

389

Multimode guidance using command guidance.

The accuracy of command guidance decreases at long ranges because of the trnns1.1tion of the angular pointing errors in the trackcrs into position errors that arc dircctly proportional to range. Thus, as the distancc from the tracking equipment that produces the commands increases, the target and missile position errors also increase. T o increase command guidance accuracy a t intercept, the rnissilc can carry the target sensor, such as a T V camem. For example. TV svstcrns are utilized bv the WALLEYE ASM and GBU-1.5 elide bomb. In this case, the T V signals arc transmitted back to the operator who uses some kind of data link to control the missile. Radio links have been used in the past, but, as shown in Book 1 of thc series, fiber optics will be . . employed in the future with the optical fiber uncoiling from the r m m k a s Ltflies: A maximum missile range 11m1tis imposed by the fiber optics concept because of the limited amount of cable that can bc carricd. In either case, the resolution enhances as the missile approaches the target, reducing the errors in sensing the relative position of the target with respect to the missilc.

-

C

Environment sensing guidancelcorrelation matching. T h c following discussion is based on Heaston and Smoots [19831. In environment-sensing guidance, a PGM, while in flight, obtains information on its own absolute position from an external source other than the launcher (fire-control equipment) or the target. The PGM then uses this update of its position to adjust its flight path, if necessary, to hit a preselected target. Inertial guidance is generally used to control the missile between updates. In order to accon~plishenvironment sensing. the clectromagnetic spectrum is used in one Lvay or another. One approach is to have an onboard star (celestial) sensor which is used for a navigation update. This has generally been associated with exoatmospheric missiles and has been too expensive for incorporation into a tactical PGIM. However, continued research in this area promises to make such a system affordable in the future. A related technique would be to use a satellite system such as NAVSTAR-GPS for a position update. A more popular and potentially less costly meshod is to have some type of map stored within the PGM that is made during prior reconnaissance. This reference may be on film o r digitized (converted into an electronic signal) for storage in a computer and ease of comparison with real-time data obtained during the missile's tlight. Because of the large amount of information in the reference maps, a very large computer storage capacity is required for digital storage. Wlth a radar map and a radar sensor, a periodic comparison of reference scenes can be used to determine missile position during flight. This approach is called radar area correlation (RAC or RADAC) or radar area guidance (RADAG), and when applied in a passive mode at M M W frequencies is called microwave radiometry (MICRAD). RADAG is being employed for terminal guidance in the PERSHING I1 missile. Optical photoslmaps may also be employed as position references. They are compared to scenes obtained during a missile's flight tb update the inertial guidance system or to provide terminal guidance. One system using this approach is known as the digital scene matching area torrelator (DSMAC), and is intended for use in cruise

390

Guidance Processing

Chap. 6

missiles. Another variation on this theme is TERCOM which uses radar to measure the contour of the land over which the system is flying and make a comparison with stored contour data. This concept is used in the cruise missile. Generally, environment-sensing guidance is used for systems with a range greater than 300 km, where precise terminal guidance is required and an accurate midcourse update is needed to assure the desired CEP. Otherwise, the added cost and complexity cannot usually be justified. One approach to autonomous target acquisition that is possible against fixed tactical targets is image correlation or map matchitfg (referred to as matcher/correlator guidance), as mentioned earlier. As a result of prior reconnaissance, a map of the target area is made and processed to mark targets. A version of this map is then stored onboard the missile in a computer. As the missile flies along, it uses a sensor similar to the one used to generate the map to take a live image to compare against the stored image. If an exact comparison is made, the missile can fix its position. Four error sources must be considered in designing matcher/correlation systems: geoketrical distortions, systematic intensity changes, quantization errors, and possible enemy jamming. Geometrical distortions, in turn, result from four causes: synchronization, rotation, scale factor, and perspective. These difficulties have been overcome sufficiently to be used in the cruise missile as DSMAC and TERCOM and in the PERSHING I1 missile as RADAG. Success with matching/correlation and advances in technology are being used to develop imaging IR, another popular guidance concept. In this case, instead of having a stored map, the missile has a built-in algorithm for feature extraction that xvill permit target recognition and autonomous acquisition. There is intense activity in this area for application to future missile-guidance systems. Map-matching guidance (see Chapter 5 for details) refcrs to the use of radarscope film previously obtained by a rcconnaissance flight over the terrain of the route in question. This type of fill11 is used to guide an aircraft or missile by allowmg the missile through proper command generation to align itself with radar echoes received during flight from the ground below.

Example 6-14: STANDARD Missile-2 upgrade program. Even after the initial version of the SM-2 went into production, the increasing threat of highspeed, high-altitude, antiship missiles launched from aircraft led to considering ways of extcnding the SM-2's guidance system ability to counter such threats. Exploiting recent advances in digital processing techniques a t the time, the SM-2 guidance section was provided with a fast Fourier transform digital signal proccssor, which was functionally cquivalcnt to a set of contiguous analog Doppler filters, each having a small bandwidth. With such processors the goal was to enhance acquisition 2nd tracking of returns from targets in thc prcsencc of obscuring jamming noise. Improvcmcnts to the signal processing wcrc accompanied by incrcased n~issilcpropulsion to provide thc kinematic capability for successful high-altitude intercepts. Changes were then required in thc radome to offsct thc effects of increased acrodynamic heating. Rccent studies concerning the STANDARD missile involved the feasibility of multiple-guidance modes (Mitre and McDonald. 19811.

Single-Mode, DuaCMode, and Multimode Guidance

Sec. 6.6

391

Example 6-15: Multlmodelmultiband HOJ system. Following Gulick ct al. [l0871, although Talos, because of its long-range capabilities, was well suitcd as a weapon to engage standoffjamrners or to force them to remain at grcat distance, there rerrldined a guidance problem. When jammers, from long ranges. iammcd the fleet's surveillance and fire control radars to the extent that these were unable to provide good information about the location of the jammer, it became impossible to program the midco~irseguidance to put the missile in close enough proximity to the jamming aircraft for reliable semiactive homing. The primary objective for the common aperture antenna with multimodc multiband HOJ homing system was to render the missile capable of homing on jammers in S, C, and X bands. If the nlissilc could home on one of the jammers, the midcourse guidance requirements could be relaxed. All of the semiactive and on-frequency HOJ capabilities of the monopulse homing system presented in Section 6.4.2 were to be retained. Another less critical objective centered around providing the missile with the ability to home on radars operating in those bands. The last objective was to make the homing system compatible with exist~ngTalos missiles and to keep to a minimum the degree of retrofitting that would be required. Figure 6-47 shows a simpllficd block diagram ofthe homing system. The basic system was the semiactive m o n o p ~ ~ l seeker. sc For the most part, the n~odificatlonsconsisted of new, wideband antennas and nlicrowave mixers, and the addition of a wideband local oscillator. The original local oscillator Lvas used for seminctive homing. There were also added to missile body motion (0) and guidance gain scalar to ad,ust for the sensitivity of the interferometer to wavelength (frequency). The concept behind the multlband HOJ design was relatively simple. Basically, if an intermittent signal such as a blinking ECM or pulse radar excited the narrow Doppler filters of the semiactive system at a sufficiently high rate, the output of the filters would be uninterrupted CW signals that could provide excellent angle information. T o make certain that the target signal would be able to excite the Doppler filters without necessitating additional target recognition, acquisition, and automatic frequency control tracking circuits, the microwave local oscillator was swept at an appropriate rate over the RF band of interest. The result was a wideband seeker. the bandwidth of which

.

-System modificationsladditions

-

for multiband HOJ Original monopulse seeker

-local

1

band HOJ

oscillator signal

I

1

1

Figure 6-47 Talos MultimodeiMultiband HOJ System (The multiband HOJ system was basically an expansion of the semiactive monopulse homing system. The microwave portions of the original system were changed to provide the desired wideband operation. Frequency seIectlon (typically 30 or 200 megahertz bandwidth) was by digital message prior to launch Convent~onalsemiactive hornlng could be activated in fllght.) (0 1982 by>ohnr Hopkins APL Technical Digest)

392

Guidance Processing

6

was determined by the extent of the local oscillator sweep. This width was generally either 30 or 200 MHz. When the selected band contained multiple signals, guidance favored the strongest signal. A digital message was sent to the missile to tell it which RF frequency to select for the multiband HOJ operation. It was possible to initiate semiactive homing i n flight, if desired, in the same manner as for a conventional antiair engagement. Based on the assumption that the multiband HOJ feature would be employed in cases where the target range was unknown, the missile would flv by design at a cruise altitude to conserve fuel until seeker logic initiated hominp, Homing always started at this altitude. For those cases in which a radar target (on land or ship) at long range (over the horizon) and of known location was to be engaged, a midcourse trajectory employing a terminal dive was used. Although flight hardware was fabricated, the multimode/multiband HOJ system never camc to be flight tested. In 1965, however, flight tests were successfully conducted against radar targets with Talos missiles using a guidance concept identical to that of the multiband HOJ system.

6.7 DEFENSE AND OFFENSE SYSTEMS 6.7.1 Performance Parameters

Description o f a SAM intercept o f an ASM. SAM systems, composed of several radar systcms and high-velocity interceptor missiles, prevent key targets (defended site) from being destroyed by short-range, high-velocity ASM. When an ASM flies through enemy defenses employing SAM, either the ASM reaches its target or the SAM intercepts the ASM. An unopposed ASM flight typically starts with launch a t maximum range and proceeds o n a preplanned flight profile until the target is reached. For an SAM to intercept the ASM, a sequence of defensive phascs must happen prior to the ASM arriving at its target. T o establish whether an intercept attempt would be successful. various interactions between offensive and defensive elemcnts and among defensive elements themselves must be analyzed [Kostreba ct a]., 1984). - -

Defense considerations. Figure 6-48a depicts thrcc basic dcfense phases relating to an ASM trajectory along which the missile is moving towards slx SAM sites located conccntrically around a target. As shown in the figure, the three consecutive phascs arc acquisition, tracking, and SAM launch and intercept. ~cquisition is accomplished with radars that scan large volumes of airspace and detcct and determine the dircction of the approaching ASM. Having established directions, the tracking radar is activated. The latter is characterized by a narrow radiating beam and provides detailed information to be used in navigating an SAM to intercept the ASM. When thc tracking radar has determincd the required information, the SAM intcrccpt missilc is launched. Each of these phases requires what is tcrmcd a delny time, which is a parameter of the subsystem for that phase. Whcn the acquisition radar detcrmincs dircction and approximate location of the ASM, an acquisition

Set. 6.7

Defense and OIfense Systems

393

delay time is rcquircd before the track radar can pinpoint thc ASM. After the tracker pinpoints the ASM location, a tracking delay timc is required to determine information necessary for SAM launch and navigation. Subsequent to its launch, the SAM flies toward the ASM, t h e timc of flight depending on the rangc from ASM and spccd of the SAM. The defense seeks to nlinimizc the sum of delay times and to improve radar detection capabilities for earliest possible detection of the ASM [Kostrcba ct a].. 19841.

Offensive considerations.

From the offcnsivc standpoint, important factors include ASM flight time, profile, and dctcctability. The time history of altitude velocity, launch range, and angle of attack of the ASM define its trajectory. Figure 6-48b shows hypothetical altitude versus range for ballistic, boost-glide, and cruise trajectories, all of which can be used by an ASM. Figure 6-48c shows hypothetical velocity profiles for those trajectories. As the figures illustrate, differing trajectories are characterized by different average and terminal velocities that determine thcir vulnerability to the SAM. For example, the average velocity of a ballistic trajectory, in comparison to boost-glide, may be greater, thereby stressing SAM system rcaction time [Kostreba et al., 19841.

Target coverage and missile survivability. Following the treatment of Young and Cormier [1983], ASM and SSM systems are key elements of any strategic arsenal for attacking a defense system. A wide variety of operational objectives can be met by having missiles of varying propulsion systems and aerodynamic features that in turn allow numerous possible trajectories to be flown. Figure 6-48b illustrates several types of missile trajectories. The trajectories of cruise missiles are usually characterized by low-altitutde, terrain-following profiles that maximize survivability or by high constant-altitude profiles that maximize missile range. Rocket motorized missiles generally fly ballistic and semiballistic trajectories characterized by high altitude and speed so as to minimize flight time and missile vulnerability to defenses. Once they are launched, these missiles will fly either in the direction of the launch axis or off the launch axis, the latter corresponding to turning flight. T h e off-axis trajectory often uses an over-the-shoulder profile like that depicted in Figure 6-48d. Selecting candidate missile systems is based primarily on target coverage and survivability, both of which are desired to be maximized. This is particularly true in the case of selecting an ASM design and trajectory. However, these two design goals may conflict with one another (that is, maximizing target coverage may not allow survivability to be maximized), and in this case a trade-off must be sought. A missile's target coverage with respect to a given altitude and launch direction is measured in terms offootprint, which is the locus of points formed by the missile's maximum range 360 deg in azimuth around the launch point. The maximum raqge of a missile is a function of aerodynamic design, the propulsion system, and the shape of the missile's trajectory. Given that the missile will encounter terminal defenses, missile survivability is affected by its trajectory. Hence, survivability is determined to a large extent by the missile's physical characteristics and by its tra-

Guidance Processing

Chap. 6

Allitudc

Low~8llitudrCruise

\

Ground Rants from Target

Target

Ground Range lo Target

Figure 6-48 (a) Hypothctical ASM Flyby Geotnctry for Target in SAM-llefended Area (b) Hypothetical Missile Trajectories (c) Hypothetical Missile Velocity Profiles (d) Hypothetical Off-Axis Trajectory "Over the Shoulder" (e) Hypothctical Missile Footprints (1) Stand-off Launch o f ASMs (g) RCS vs. Aspect Angle (h) Stylized Defended Area of a SAM Complex (Froin /Kosrrcha ef a / . , 1984; Y o u i q aiid Cormier, 198.51 upirk pc.rmissioiljroin SCS)

I

\

9

Defense and Oflense Systems

Sec. 6.7 Hiph.Altitude Launch

,

Low.Altitudc Launch

ASM Cendid8teZ

Cronrangs

Altitude

,

.'.

,---*

Radar Crou Swtion (Square Moten)

\

'150

- SAM Envelope

0-

ASM Trajectorin

Tarnet

-

Range Aspect Angle (Dlpnerl

(G)

Oirwtion of Attack

0 S A M Site

V

-----.... ..., .

,

I Downmnngc (NM)

(H)

Figure 6-48

(Continued)

ASM Candidate 1 ASM Candidata 2 ASM Candidate 3

\

7 180

396

Guidance Processing

Chap. 6

jectory through enemy defenses. The former include the inissile's radar cross section (RCS) arid IR signature, both of which are functions of the aspect angle between the ASM and SSM and the defense system. Aspect angle in turn depends on the missile's trajectory (a,y, and + ) Additional trajectory characteristics that bear on survivability are missile speed and altitude, where survivability generally increases with speed. Low-altitude flight can increase survivability by exploiting terrain masking (using terrain features to shield the missile). As previously mentioned, it is generally not possible to design a missile or trajectory that maximizes both target coverage and survivability at the same time. Assessing missile designs is made difficult by the numerous combinations of propulsion systems, aerodynamic designs, and trajectories. Another problem that is by n o means trivial is that of calculating the data to evaluate three-dimensional flight, a problem which takes on added complexity as one attempts to shape trajectories to satisfy certain operational requirements. Given such complexity in modeling three-dimensional flight coupled with the large number of trajectories and designs that must be simulated to conduct a complete trade-off analysis, there is a tremendous need for a versatile missile trajectory simulation. ,l4irsile ratlge is likely the missile performance parameter of greatest importance, determining for the most part target coverage and defining the maximum standoff range for the launching aircraft. The maximum standoff range figures significantly in aircraft survivability. Thosc factors that affect missile rangc include missile aerodynamic and propulsion dcsign and missile trajcctorics. It is rcquired to be able to display missile rangc in all directions since niaIiy applications o f ASM involve off-axis flight. Using a trajcctory analysis program that computes down-rangc and cross-range cstirnates, it is possible to construct a ritissile foorpritlt, a diagram that shows 360-dcg co\,erage based on missile rangc from a givcn altitude and launch direction. Such missile footprints arc shown in Figurc 6-48c for t w o diffcrcnt inissilcs lau~lchcdfrom high and low altitudes, and can be used to dctcrmiric sensitivity to both propulsion dcsign and flight trajcctory. Missile rangc also establishes launching aircraft standoff constraints. In an ideal situation, thc launching aircraft would be able to attack a targct from thc outside of terminal dcfcnscs. Figure 6-48f depicts a SAM cnvclopc that shows how an aircraft might attack a targct with a long-range ASM, thus remaining outsidc thcsc defcnscs. .Idissile n t t i t ~ r d edctcrmines the effective arca prcscntcd to the SAM early xvarning or targct tracking radar. The survivability of a n ASM flight is going to bc Iargcly affectcd by tiiissilr spczcd. Thc greatcr the ~nissilc'sspccd \%.it11which i t flies. thc greater thc strcss is placcd 011 the SAM dcfcnsc systcnt [Young and Cormicr. 1'3851. Bcsidcs velocities and trajcctorics, I
Sec. 6.7

Defense and Offense Systems

397

side of the radar (90-dcg azimuth), the larger surface arca will result in a larger signature. Offcnsivcly, it is dcsircd to find an ASM t h n t will get to its target quickly (high velocity) and with short detection ranges (small IlCS, little broadside exposure if possible). Speed trajcctory, aspect angle, and liCS function of angle arc the key offensive elements, and these must be traded off in any ASM design. Owing to the dynamic nature of thcse factors for an ASM, it is not easily estimated which ASM candidatc yields to the SAM the smallest dcfcndcd arca. The complexity of the situation is made evident in Figure 6-48b. For cxaniplc, consider an arc that centers on the SAM site and cuts a t a constant distance t h r o u ~ hthe three typical trajectories. At this distancc from the SAM, each profile has a diffcrcnt aspect angle and therefore prcscnts a diffcrcnt RCS. Consequently, one candidatc may have been detected while the othcr has not. Morcovcr, cach candidate is moving at a different velocity and different trajcctory, so thc timc to target will not be the same. A change in ASM position changes the azimuth angle relative to each radar, entailing a change in RCS observed by cach radar. T o accommodate thcse changes, calculations must be performed for successive incremcnrs of time, each on thc order of a second. For each timc incrcment, the relativc positions arc determined and it then becomes neccssary to detcrniine whether the radar can detect the ASM. This action r e a ~ ~ i r cseveral s steps. The sequence for a given SAM site is: (1) the acquisition radar must detect the ASM; (7) after the acquisition delay time, the ASM is handed over to the tracking radar; (3) thc tracking radar mccrs its delay time: and (4) an SAM is launched [Kostreba et al., 19841.

6.7.2 Low-AltitudeAir Defense Systems The discussions here are based on two excellent articles [Fossier, 1984, 19881, in which it was noted that the first guidance systems designed for low-altitude air defense suffered serious technical problems. Some of these involved target detection in a cluttered environment, lvhile others had to do with applying the technology of the day to equipment that had to be capable of operating in the severe missile environment.

Homing problems. As early as 1950, active continuous-wave (CW) radar was used on ex~erimentalmissiles. While this form of active radar was not to endure much longer thereafter, offshoots such as semiactive C W radar and semiactive and active pulse-Doppler radars would soon pervade antiaircraft missile systems. O n e of the problems encountered in designing C W radar has to do with aircraft, such as the Kamikaze, that would typically attack from low-altitude flight. The fraction of radar energy directed from a missile toward an aircraft target that actually reaches the target and is reflected is very small. Moreover, the fraction of reflected energy that is intercepted by the missile antenna as a signal is smaller still. The missile, in addition to receiving signals from the target, is receiving signals that are reflected from the ground or sea, which reflects a much larger signal than an aircraft, as shown in Figure 6-49a. Without the ability to discern one from another, the small target signal is hidden by the ground (or sea) clutter. A C W radar transmits a pure

A

CLUTTER

EEOTHROUGH

FREOUENCY

'0

Vm A

10

lo.

V,

A

lo

.- 2VT+V, A A

A CLOSING SPEED

(B) AMPLITUDE

TRANSMITTER

ANTENNA

RECEIVER

(C)

AMPLITUDE

I

BIRDIES

FREOUENCY

.::'

.a z*

HORIZONTAL POLARIZATION

.. ........ . . .. . .......

7

e c - 8 .5

:s

.3

VERTICAL POLARIZATION

.1 I

0

1

2

3

4

6

6

7 6 e 1 0

GRAZING ANGLE (DEGREESI

(G)

Figure 6-49 (a) Signals Received by CW Homing Missile (b) Missile Signals: Amplitude vs. Frequency (c) Fecdthrougli I'roblcm in CW Radars (d) Noise Sidebands in Ilopplcr Radar (e) "Birdies" ill Doppler Radar (0 Low-Altitude Gcometry (g) Reflection Coefficient over Smooth Sea (A = 3 cm)(h) llcflection Coefficients over Land,' Vertical Polarization (A = 3 cm) (i) Fcedthrough Nulling Block Diagram (i) Noise Degeneration Block Dia-grarii (k) Inverse Moriopulse Ilecciver (From [Forsirr, 1984 & 1988/, 0 1984 G- 1988 AIAA)

1

REFL COEF

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GRAZING ANGLE IDEG I

0 R Y . S L l G H l L Y ROLLING WITH GRASS I TO 18 IN HIGH

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6

(H) TRANSMITTER

ANTENNA

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I

REFERENCE

REF

ANTENNA

Figure 6-49

(Continued)

Guidance Processing

400

Chap. 6

tone o f microwave energy. By reason of the well-known Doppler effect, reflections from targets in its beam will be altered in frequency. The problem as far as the missile is concerned is fundamentally one of selecting the frequency.corresponding to the moving aircraft target and spectrally rejecting the frequency that corresponds to the stationary ground, as shown in Figure 6-49b. The problem is made difficult in one respect by the fact that the Doppler shift is only about 20 Hz/ft/sec of relative velocity at a radiated frequency of 10 GHz, which is not a very large shift at all. The desired frequency resolution for separating the target signal from the interference is about lo3 Hz (50 ftlsec). Implementing such a narrowband filter was realized by beating the received signal down to video frequencies, using- a sample o f the transmitted signal as a reference for the incoming signal. By heterodyning various parts of the spectrum sequentially into a highly selective fixed-frequency 1 KHz wide filter, using a variable frequency oscillator as the mixer reference, this spectrum could be analyzed automatically. O n e of the other two remaining significant problems dealt with what is known as feedthrough, as shown in Figure 6-49c. Even as a C W radar is transmitting, it is simultaneously trying to listen for a received signal that is approximately 10-l6 of the transmitted signal. The problem is one of ensuring that none of the transmitted power is leaked into the receiver [Fossier, 19841. While pulse radars shut bff their receivers when their transmitters are on, C W radars transmit continuousl~.As a result. the receivers of C W radars must detect the small target signal in the presence of leakage from the transmitter. which can often be significantly many orders of magnitude larger. Unlike the target signal, this fecdthrough has no Doppler shift, but its large magnitude can easily saturate the reccivcr. In addition to thc feedthrough problem, the radar must cope with the problem of noise. N o micro\x,ave oscillator gcncrates a perfect sinc wave, and dcviations thcrcfore may be thought of as frequency sidebands on the dcsircd pure tone. When feedthrough or cluttcr signals arc prcscnt they will carry thesc sidebands, which may appear in thc Doppler spectruin and, hence, compcte with the target signal (Figure 6-49d) [Fossicr, 19881. The C W radar is thercforc continually attempting to minimize noisc in the transmitter and in receiver oscillators used to heterodyne (or convert) the receivcd signal down to intermediate frequencics. The other main problcn~,which is somewhat similar in nature to the problem of feedthrough, concerns thc magnitude of the cluttcr rcturn. Unless the transmittcd signal is a pure tone, it will have frcqucncy sidebands that will tl~odulatethe large clutter signal, resulting in components a t frequencics ~vithinthe rcceivcr bandwidth. Modulations at a frcquency rqual to thc diffcrencc in frcqucncy between the targct and cluttcr llopplcr frequencics will nlodulatc thc cluttcr frequcncy to crcatc a sideband a t a frcqucncy that is identical to thc dcsircd targct signal. Given thc potentially high ratio (10') of cluttcr powcr to the dcsircd targct signal powcr, the noisc o n tllc transmittcd signal must be kcpt to a tninimum [Fossier, 19841. Still another problem can bc labeled birdi1.s. Thcsc arc coherent unwanted signals (that is, having a sinusoidal waveform) that may be introduccd by harmonics of powcr supplies, by microphonies arising from rcsonant n~cchanicalclcrnolts, or as "beat" frcqucncics resulting from mixing proccsscs (Figure 0-4%). T h c radar may rllistakc thesc birdies

.

<,

for raigrrs ii'thcy srv ir, the parts ofthc Onpplcr spcawm olintcresr 2nd arc dctc.tcd, Since they arc i.i,trrru.silygmrrattd, rhr:y are abvi~ttttyo-f no value cs rnissiir guiitsncc, Fundamcn~timplcmer~tarioixprobic.m?tbq3n EQ k c -~nderst~>;sd in tki.IB#k., but the stat< oi'rfzc art in hardware w35 not cnnducivc to sarlsfactory scrlucinni. rite microwave xnutccr; of L!SS t i m e \fft'rc ptinlzrily mxgnctrom 2nd wtrc guitc r ~ ~ l y (ahbough pcxl'zctiy sacisf;rcrvrp ibr c,nvcnriani( nor\cof:tront pulse z-gdsr i?prratiottf, A h 4 rhc :\rc~isi~>g v~cuutn rubes a t;zih hlc *t that r h r wcr: Inre9 {and miirophtmic) {Fossjcr, t988\,

Atr dg.f:enw horn{- ~3;sC:em. C3i1c nfrhc adv;lnriti;cs c3f 2ii active ~ccker is is aabiiity t~ operate atzt~tr~rltuusly stscic~utl\ r;anirnitrrd 2nd recctwd signah must f i ~ w ) the , fte&krt.itgh and ~ l r s ~ c i x t d noise prnbicms \were rcdured b-9 ,>I-dtrsvf ?aaFxi~crdd-This inrfe:\sz.sdthe ma::i&um irzmsmirtzr p0:ver ict~.lthat cwld be eo%aarecI in thc activr radar fa Prvr vatrr;? by c C ~ d e rr&r~?rgnitvdr, 5 a:d, c~:untbincd, wi& ~1x6rf~u.ihlarger rfitejfna in the r?rrrafs uficrl ro iltunrinare thc ~z.rger.pn?vidc'd s<::ticitr-it :r;ickir.,g r;?nge t-& perrrric t-vet.er, d~ i f did nat allay fears &at the ef%kton g~rldance zcrcurxy of radar rrflettiurts f c ~ t ~chs t . ract?~,:rrmed rheimagr or muf+&t:k~roble#, would be severe tlno~rghtto clurc the rrtissileti; k m t . .;tsrnrtvhere br:.vxcet~zhrs tlrrger and its intrige &and thus consistet3dy collide wirh the grouad beqepr *ctachirjg chc targtt {see Figwe 6-49Q. At t h r tiwe thc prohiem was beijrg tmestChed, the design team af rhe Hawk sysrcn lcamtd that hurizantn~~y pdrrrhd microwa~t-e erwrgy can reflect af~tlostccon1.plctc2y
402

Guidance Processing

Chap. 6

problem that was predicted. The large amount of attenuation of vertically polarized energy (see Figures 6-49g and 6-49h) seemed to indicate that the problem could be solved if the grazing angle could be controlled. From the ensuing analysis, it was determined that launching at elevation angles in the range of 15-20 deg accomplished several objectives: (I) it keeps the reflection coefficient, which is simply the fraction of the incident energy reflecting in the forward scatter direction, below 0.5 in every instance, resulting in an average pointing direction of the antenna at the real target rather than the image and ensuring that the magnitude of the disturbance is well bounded; (2) it holds the frequency of the disturbance well above the autopilot bandwidth, so that the bounded disturbance established previously does not to a large degree affect the homing trajectory and therefore substantiates the simplifying assumptions used in the analysis; and (3) it enhances the aerodynamic range performance of the missile to some extent because the reduction in air density at higher altitudes more than compensates for the longer path to the target. T h e systems analysis approach that was adopted in the design of the Hawk system was a big factor in ensuring the development of a successful missile system, and brought attention to the contribution that system analysis can offer in the solution of practical problems. The hardware design conformed to the theory that governed the problem at hand, and the analysis was subsequently validated in every respect through the use of actual flight tests. This same approach is still adopted even now in the development of all low-altitude homing missiles. As mentioned earlier, if a missile launched against a low-flying aircraft is to be successful. the aircraft must first have been detected and tracked t o maintain illumination during missile flight for semiactive homing. This brought into focus once again the problems of an active Doppler radar. It was clear that success depended on the ability to separate the transmit and receiver functions (as done in semiactive missiles) as much as possible. The result of such effort was two-dish radar, which demonstrated the feasibility of detecting and tracking targets in clutter with an active ground-based C W radar. The radar design was modified in the development program to have two four-foot-diameter dishes side by side on a common pedestal, the configuration that is still used today. The transmitter selected for the tactical radar was a 200 W magnetron stabilized by an external cavity. In addition to the illuminator, a continuously rotating CW acquisition radar was designed with similar electronics but using two eight-foot-long antennas mounted above each other. This radar continuously scanned the horizon, providing the initial acquisition of targets in clutter necdcd to start the engagement process. The preceding solution of separating transmitting and rccciving functions with separate antennas ~vhich made the Hawk successful is not suited to an aircraft radar because of the limited frontal area available in high-speed military aircraft. Fundamental solutions \t7crc necdcd to all of the original Lark radar problems, especially for the problem of fecdthrough. If a single antenna is to be used, the amount of transmitter power that leaks into thc receiver must be diminished considerably. Using the newly available ferrite device, a system was developed called fecdthrough nulling (Banks, 19751 that purposciy leaks transmitter powcr into the receiver under the control of fcrritc

Sec. 6.7

Defense and Oflense Systems

403

rotators, as shown in Figure 6-49i. In principle, the ferrircs were controlled to leak transmitter powcr of the correct an~plitudeand phase to balance out thc unwanted power that inevitably leaked in. Thus, by nulling techniques, using phasc detectors to sense the remaining feedthrough, the fccdthrough could be reduced to thc extent of the gain in thc loop [Fossier, 1988). The close-out or stop frequency of thc fcedthrough nulling loop had to be kept bclow the Doppler band of interest to avoid cancelling out target signals. Transmitter noise in the Doppler frequency band in the severe vibration and acoustic environment of an aircraft noise rcmaincd a difficult problem. In one solution that was developed, noise cancellation techniques were employed, based on fecdback principles. T o achieve stabilization of the transmitter, one simply modulated the output of the klystron master oscillator to null out the inadvertent modulation (Figure 6-49j). Operating in a manner similar to that of feedthrough nulling, the resultant modulation was sensed as in-phase and quadrature imbalances in a microwave bridge, with a mechanically-tuned cavity serving as the stable reference. Although further airborne C W radar work ended not too long after this, the techniques of feedthrough nulling and noise cancellation provided a way to build a robust, highpower version of the Hawk tracking radar. As mentioned in Fossier [1988], the attractions of C W radar had been recognized bv Great Britain, and a number of critical components were being developed by their laboratories. The development of a high-power illuminator (HPI), as the radar came to be known, came not long after a 2-kW low-noise two-cavity klystron was developed by Ferranti. Although subsequently replaced in the operational design, it nonetheless allowed the HPI to be successfully developed and deployed in 1962. replacing the original low-power illuminators in Hawk systems.

Further missile developments.

The clutter rejection capability of the original Sparrow (and Hawk) design was about 40 dB and was limited by the dynamic range of the video amplifier that contained the entire Doppler spectrum, including both target signal and clutter. In several Hawk flight tescs over lava beds in 1963, the lava's high reflectivity generated clutter sufficiently high to prevent the missile seeker from detection of the target. As noted in Fossier 119881, a simple modification was undertaken that satisfied the Hawk requirements nicely. Frequency shaping was added to the missile video amplifier to attenuate frequencies at which clutter could occur (determined by the maximum velocity of missile, which was accurately known). This reduced missile sensitivity for tail attacks, where it is not needed, and optimized the clutter rejection for forward-hemisphere attacks, where it is needed. By going to this technique, an effective clutter rejection of about 60 dB was achieved, which proved sufficient to errn nit successful flighrs (although at reduced performance) under these severe conditions. In the early 1960s, a different receiver architecture was conceived that greatly increased the clutter-rejection performance of the missile receivers while also eliminating the ECM vulnerability. The resulting inverse monopulse receiver utilized the newly available crystal filters to accomplish inverse monopulse, incurring little

increase in carnplexity compared with the conventional receiver. As shown in Figure 6-49k, a three-port monopulse antenna was employed with the outputs representirlg the sum beam and the two-difference beams (pitch and yaw) that determine the txget lacations in rectangular coordinates. FoIIov~ingenough fixed-gain ampliftcation ta tstabCisk noise figure, each output went through a n identical I-kHz-wide fitet at the 1F frequency. Tlzis filter eIiminated interfering signals (such as clutte~ ar jammer modulation) that were not within 1 kHz of the desired target signal, FoUowing the filrer, the difference chantlels were intenriona2fy modufa ted onto the sum sighal (as happens in conical scanning] to allaw amplificarion and processing. Since this happens at a frequency much higher than 1 kHz, vulnerability to an~plitude-modulated jamming is eliminated because AM at the difference-channel modulation frequency is completely removed by the narrowband crystal filters before the modulation process is employed, This design led to the Improved Hawk missile prosram and. was Lam incorporated into Sparxow as well {Fossier, 19881. The 1970s uqtnessed the state of ~ h art r moviflg away firom CW radar to pulse Doppler [with rnonop~2seangle tracking). As indicated earlier, pulsed Doppler was essentially left to airborne radars such as those for the F-4, F-14, and F-15 aircraft, which only had space for a single antenna. Tbese pulse Doppler radars ernploy the Cundamental tcch~~iques dcveldped for CW radars to reject clutte'c, minimizing the problems by openltlg their receiver anly when the transrnitrer i s turned off. However, this turning on and off on the part of rhe transmitter several thousand times per second has made ~t more difficult to achieve low-noise performance. Even with the noise cancellation techniques pioneered in the HPI, transmitter noise is still a limiting factor for some appGcatibns. The application of stealth technology r o 2 i ~ a7ebjc)cs ma)' ott2r ;( ~ I C Wtcason T O US C W raiiai. Dt-cccrint; and rracking sccalth vehiclcs at 10s aititudt will rcquirc near-perfect cluttcr rejcctiot~,ivhich may bc best

achieved in ground-based radars with the supcrstability obtainable \virh two-dish CW radars and semiacrive missiles. Although the popularity of pulse Doppler radars and missiles is increasing, CW remains widely adopted. Accordirjg ro Alliatiotz M,'eek arjd Spare Terhnol~gymagazine, rhe Soviet SA-6 air defense system cmploys CW radar homing miss'ilcs. The SA-6 has been deployed in the V.S.S.R. since the 1960s and was exported to client states after the Mideast War of 1967.

6.8 FUTURE CIUIDAPfCE PROCESSING

Sanlc o l the ncwer dcsign trends conccrnirlg srrzart PGM includc high-pol~rrcd transmitters, image-based sigrral processing and ir>tclligcnt sensors, srnall inertial platforms, high cjcctrorlic packaging dcnsitv, improved cornmar~dlink$, and gvjdante-aidcd fusing. Those- arcas in which ricrc is very little change in the desig13 methodologies irtclude acrodvnarnics, srrurturcs, propulsion, radomes, and warhead. This, howcvcr, i s gcnc;a))ytrue of all flight vehicles, and not necessarily of PGM only. Signat processing rrchno\ogy will figure cvcn morc prominently in the

Ssc. 6.8

Future Guidance Processing

405

future of guidance processing as autonomous weapons will be requited to satisfy expected mission requirements. Current missile-guidance functions such as detection and tracking of point targets have already been successfully developed. Future system requirements based on accurately resolving the location and physical structure of targets will require the development of new signal processors, which will also have to decide the appropriate action to take in accordance with target identification. For example, the approach presented in Book 4 of the scries has made smart sensing and guidance possible, especially in applications that are able to take advantage of much of the research conducred in the area of image processing. The result is the use of intelligent knowledge-based systems techniques to identify to the missile the most vulnerable part of the target. Thc capability to do this resides in software that is loaded onto the missile's processors. The use of current and future missile radars and IR seekers will be enhanced by advances in autonomous target recognition algorithm development (see Figure 6-46b). Advances in patallel digital signal processing will play a significant role in the development of more optimal and robust approaches based on neural networks and multiple spatial resolution. What may be the ultimate system for missile signal processing will have multiple sensors integrated via sensor fusion techniques based on the principles of artificial intellieence. The system architecture design " will likely reflect to some extent the organization of the brain, and algorithms for target recognition based on neural networks will probably be the most natural and robust approach. In addition to radar sensor imaging, IR sensors will also be used. One approach that is applicable to image detection and segmentation, and which is based on a model of the human

-

vision system. is 1 suboptimal irnlge pmceccing architecture called the multireso-

lution spatial integration (MRSI) processing developed by Johns Hopkins Applied ~ h y s i c s ~ ~ a b o r aThis t o ~ ~can . b i developed for different antiship missile concepts (that is, completely autonomous seekers or man-aided image processing). MRSI could perform target cueing, rhus relieving the human image interpreter from having to search visually over a wide FOV. As neural-network-based recognition schemes become increasingly common, MRSI processing in combination therewith will become a missile signal processor that hHs a significant number of features of human intelligence [Boone and Steinberg, 19881.

6.8.1 Areas for Technological Advances in Signal Processing This treatment is based on the excellent report submitted to GACIAC by Heaston and Smoots [1983]. AS discussed earlier, all PGM applications will benefit from increased microprocessor capacity and the development of sophisticated target engagement techniques. In addition to these areas, other areas that have begun and will continue to be developed include detectors/sources, very high-speed integrated circuits (VHSIC), charge-coupled devices (CCD), fiber optics, surface acoustic wave (SAW) devices, integrated optical circuits (IOC), focal plane array (FPA) sensors, and microgyros. In the area of detectors/sources, PGM applications require small,

406

Guidance Processing

Chap. 6

lightweight, high-efficiency power sources. It is believed that the next decade will witness increases in power output of two to three orders of magnitude. Currently, there are also requirements for MMW sources and detectors. CCDs, which function to move extremely small aggregated units of electrical charge from one fixed position to another, will find major applications in PGM. Fiber optics is another very important area. Utilizing optical fibers that are small in size and weight, the guidance of a PGM could be a situation in which an operator receives pictures of the target area from the sensor and sends guidance signals t o the missile over the same fiber, as shown in Example 6-8. It is almost certain that fiber optics will eventually be used as data links within PGM and in their airborne and ground launcher support equipment. SAW devices find use in delay lines, frequency filters, oscillators, amplifiers, memories, and programming circuits, and may be used in the launch platform or target acquisition system of PGM. The use of I O C and optical circuit miniaturization for signal processing and image generation in PGM could bring their cost down while improving their performance. With FPA, attempts are being made to couple the FPA output to a microprocessor with a sophisticated pattern recognition algorithm to automatically recognize and identify targets.

Autonomous acquisition/image processing.

A technology that is well understood for many types of sensors has to do with automatic tracklng after acquisition (lock-on). By using LOBL it is possible to simplify the guidance system. The system in some cases may not be able to LOBL, although the conditions may make LOAL relatively easy. Such may be the case, for example, if the target is in the antenna beam and is the only target available in a clutter-free environment. Certain radar-guided missiles are designed to opcrate in this mode. With autonomous acquisition, however, the LOAL procedure is more complex, generally requiring a true arca/volume scarch, automatic target recognition, and acquisition. The postlaunch autonomous acquisition is the most difficult mode of operation for a PGM. Having been fixed into a general target area, the missile, upon entry to the terminal phase, must search for, recognize, and select a target in the absence of any operator assistance. Such a scenario is generally air to ground. Although autonomous acquisition is generally associated with image processing, it may be implemented ~e can be broken down for other types of signatures as well. ~ e n e r a l i m a processing into five classes of techniqucs that are employed therein: (1) irnagc enhanccmcllt, (2) scgmcntation, (3) featurc cxtraction, (4) featurc classification, and ( 5 ) correlation. Image enhanccmcnt involves convcrting the input irnagc data into a form more suitable for further processing. The processing operations for accomplishing this goal may includc noisc elimination, geometric correction, spatial filtering, or amplitudc spectra modification, and may be pcrformcd either in the spatial domain or in a numbcr of transform domains such as Fouricr, Haar, Hadamard, and so on. Scgmcntation is conccrncd with isolating potential featurcs or areas of an image from that which is considcrcd background. In particular, cdgc extraction is uscd to isolate or dcfine objccts. Processing associated with edge cxtraction may be uscd to enhance fcature boundaries, whilc thickcning or thinning operations may be cm-

Sec. 6.8

Future Guidance Processing

407

ployed that attempt to provide a continuous line describing the feature boundary. Although edge extraction is generally a preparatory step for feature classification. it is somctilnes also used to precede correlation steps wherein the binary feature boundaries arc correlated instead of area correlations of pixel amplitudes. In the process of feature extraction, characteristics of the segmented features are determined. These may consist of areas, perimeters, area-to-perimeter ratios, and other geometrically sensitive parameters. Texture is another type of characteristic that may be used to describe image features. Still another type of characteristic may be the entire segmented portion of the image, for situations in which template matching is employed for feature classification. In feature classification, the extracted features of segmented objects are matched to a set of known or desired features of a target object in order to identify the type or class of object appearing in a scene. In the case of a prestored target template, this consists of correlations of the segmented object with the prestored target template. However, if geometrically derived features are involved, feature classification may consist of a probability of match score of the input object with prestored target feature profiles. Correlation has to do with the mathematical correlation of a reference and input image or image segment to determine if there is a precise match or the best match for the relative positions of the two images. Either picture amplitude or phase information, threshold or binary image information, or picture edge information may be employed for correlation. Bayesian statistics allow a scene-to-scene correlation to be made to educate the seeker. In the case of midcourse or terminal guidance relying on map matching, correlation can be performed on the picture as a whole, or with multiple subareas. Correlation, when performed on segmented objects, is basically template matching for feature classification. In order for any of the image processing techniques discussed earlier to be used to their maximum potential, they must function in both ideal weather and environmental conditions and in conditions in which the target signatures are degraded.

6.8.2 Future Signal Processing for Missile Guidance This section presents an example of future signal processing for missile guidance based on state-of-the-art research being conducted at Johns Hopkins Applied Physics Laboratory [Boone and Steinberg, 19881. The typical engagement scenario in antisurface warfare, which is the focus of the treatment given here, is one in which the missile confronts several surface combatants and must pick out a high-value target from less important ones. Such a relatively small closed-set classification problem, while it may intuitively be thought of as simple, is in reality difficult. Because of signal degradations arising from both natural and man-made sources, sophisticated seeker processing is required. For radar sensors, target motion and aspect changes are unsteady. In the case of IIR sensors, signal degradation is brought about by inherently low target contrast, which is a function of target thermal signature, and by atmospheric propagation loss, which increases with range. Figure 6-50a depicting a generic, advanced missile-guidance system, highlights the most

Guidance Processfng

Chap. 6

Engagement planning system

L

Preprocessor Sensor suite

Acquis~tion extractor

Monopulse radar Synthetic apenure radar Infrared focal plane array

To guidance computer

Class~fier or image matcher

Nearest-neighbor Bayesian Neural networks Multiresolution spatial integration Fast ~ourie; transform Primitive structural features

Range bin

Range bin

(B)

Figure 6-50

(a) Signal Processing Subsystems f o r A d v a n c e d A n t i s h i p M i s s i l e Guidance

(b)

T h e V a r i a t i o n of Radar-Ilange Profiles for I l u r a t i o n s o f (i) 100 m s a n d (ii) 16 s (c) I n f o r m a t i o n Processing F l o w S h o w i n g R a n g e Profile, Feature E x t r a c t i o n , a n d N e u r a l N e t w o r k Architect u r e (C 1988 by Johns Hopkills APL Technical Digrsr)

Sec. 6.8

Future Guidance Processing Neural network Hidden layer

,1.00

7 " .-

0.75

5 0.50 E

a 0.25 0

Harmonic number

decision

IN classesl

Figure 6-50

(Continued)

important signal processing subsystems thereof. As shown in the figure, the sensor suits can be a radio-frequency imaging radar such as monopulse o r SAR, an IIR (or optical) FPA, or some combination of the two. The preprocessor and the classifier constitute other key subsystems, the former of which usually performs signal conditioning and feature extraction or segmentation. Those subsystems that may be classified as nonessential can also be part of the overall system that includes a postprocessor for decision integration and a physically separate engagement (or mission) planning system that includes a mechanism for generating reference data for classification. The architectures most likely to be found are parallel for preprocessors, sequential for classifiers, and symbolic for postprocessors. The two examples that follow are representative of advanced signal and image processing algorithms being developed, the first of which is a new technique in ship recognition now being applied to monopulse radar-derived range profiles. The focus here is on neuralnetwork-based classification algorithms. The second example is in the field of EO systems, and has to do with a multiresolution spatial integration technique for extended target detection and segmentation applicable to imaging IIR seekers.

Target recognition using neural-network-basedalgorithms. The conventional (real-beam) monopulse radar used currently by antiship missile seekers yields range profiles, or one-dimensional functions of target radar cross section versus range. While inverse SAR can be used at longer ranges for stand-off ship classification, its imagery varies to a large degree over successive looks, producing

410

Guidance Processing

Chap. 6

various perspectives such as plan and side views. SAR or monopulse imaging mav be used by future antiship missile seekers to generate ship images, which basicall\. are plan views of the radar-scatterer distribution of the illuminated targets. whichever of these sensor options may be en~ployed,it is strongly believed that future antiship missiles will have a definite need for target recognition. As a matter of survivability, the total look time required for imaging and recognition is always restricted to being on the order of the motion cycle of the targets. For reasons such as the radical changes in missile-target geometry that occur during terminal engagement, target signatures presented to a typical antiship missile will vary a great deal. Thus, classifier performance must be robust, and the training of such classifiers must include representative looks. Boone I19881 has developed a systematic methodology for classifier training that incorporates various databases (both measured and synthetic) and numerous feature extraction and classification algorithms, including traditional (statistical) and neural-network-based algorithms. When relying on radar-range profiles, there is considerable degradation of classifiers at the broadside aspect or near-broadside, since range-only profiles (having relatively low spatial resolution) will have few range cells on ship targets, particularly at broadside. One way of determining sensitivity to target aspect is to train a given classifier at a particular aspect and then test it within an angular sector spanning the training point. Better performance translates into a larger angular sector over which the probability of correct classification remains high (290%). The larger this angular sector, the less data there will be to store in the missile-guidance computer memory for real-time operation. Thus, increased classifier robustness versus aspect means less training data and smaller computer memory needs. The ability to achieve greater robustness may lie in employing neural networks. During times of signal acquisition, targets exhibit varying signatures. In the case of real-beam radars, there is usually much less than a 1-sec time interval over which signatures must be collected t o achieve an adequate SIN for classification ( 2 2 0 dB). However, over longer, muitilook intervals ( > l o sec) required, for example, for decision integration, there can be a significant variation of the range-profile structure, as shown in Figure 650b. Although image frame times for SAR may be about 1 sec, target motion may blur the detailed RCS structure of the desired targct. Moreover, look times for inverse SAR requircd to capture the best reprcsentative image of a target can easily exceed 10 sec. An effective classifier should bc able to deal with such variations in targct signature for any conceivable seeker type within the appropriate look time, without a large measure of pcrforrnancc degradation. The features, thcrcforc, selected to enable classification (that is, the input to thc classificr) should alwrays be separable in feature space to allow the output decision to bc made with a high dcgrce of confidcncc. A design goal to which considerable attention nccds to bc given is optimization of classifier performance. The clcmcnts that combine to detcrminc performance are thc spatial resolution of thc sensor, the fcaturcs cxtractcd from thc signatures, the n~athernaticalstructurc of thc classifier, and the type of dccision-integration schell1e uscd. When rclying on range-only profiles, pcrformancc vcrsus resolution is Particularly important ro targct aspect. For fcaturc extraction, onc gcncrally errlp]oYs

Sec. 6.8

411

Future Guidance Processing

Fourier harmonics of the range profile since they rcveal how rapidly the RCS of the target varies and which spectral components of the radar are dominant. With neural networks, important issues have to do with optimizing the network structure in terms of the learning algorithm (for example, back propagation) and the number of connections. Optimization with respect to the choice of a decision-integration rule is desirable when dealing with long dwells on the intended target. A higher level of confidence can be obtained by integrating sequential output decisions. As described in Boone and Steinberg [1988], the neural network al*yorithm is based on a metaphor of the human brain's capacity to process information (see [Roth and Jenkins, 19881). While the subject of neural networks encompasses many options, one of those pursued by Rumelhart et al. (19861 focuses on the multilayer perceptron, or back-propagation-based algorithm. Although such algorithms require longer times to train, the time to classify is relatively short, which can be an important tactical advantage for a missile in a typical war-at-sea engagement, where there may be little time to classify multiple targets. Figure 6-50c illustrates the implementation of a multilayer perceptron via back propagation for the recognition of ship radar signatures. After the ship signature is received, it is conditioned, digitized, and then fast Fourier transformed. The first few harmonics of the transform are chosen by reason of the sensor resolution and SIN. The number of input layers to the perceptron equals the number of harmonics. The number of nodes of the hidden layer will typically be threefold that of the input layer, enough to allow for effective separation of the classes in feature space. The number of output nodes corresponds to the number of classes, and the connections between nodes are weighted. At every node, a summation of the weighted inputs is subjected to a sigmoidal nonlinearity. In the course of training, a heuristic rule derived from the classical Widrow-Hoff technique [Widrow and Hoff, 19601 is used to adapt the weights, and the weight changes are updated, working backward from the output layer to the input layer. Afterxvards, when classification occurs in real time onboard the missile, the particular output node that coincides with the input class will be activated. ,

Comparing neural network classifiers. It is important when comparing neural-network classifiers with conventional ones to consider a particular neural-net algorithm in conjunction with its proper conventional counterpart. Lippman [I9871 submits that the single-layer perceptron is analogous to the Bayesian classifier, and the multilayer perceptron is analogous to the k-nearest-neighbor classifier. The customary basis on which classifiers are evaluated is a single-look probability of correct classification. This number can be derived from a confirsion matrix, which might look like Testing Target 1 2 3 21'

p32

I3l3

Training Target

412

Guidance Processing

Chap. 6

T h e diagonal elements of this matrix are the probabilities of correctly classifying, while the off-diagonal elements give the probabilities of misclassifying. The goal is, of course, to achieve unity diagonal terms (correct decisions) and zero off-diagonal terms (incorrect decisions) with high confidence. In actuality, one attempts to maximize the trace of the matrix, while minimizing the sum of the off-diagonal elements, where different situations may call for the use of a particular cost function to weight the off-diagonal terms. Typical radar-range profiles for a decommissioned U.S. naval combatant are shown in Figure 6-50b, and a representative set of features is shown as an input to the schematic of the neural network of Figure 6-50c. When a number of other combatants together with this one are used for training and testing, the average probability of correct classification is derived from the single-look confusion matrices. Subsequently, a comparison is made of results for Bayesian, nearestneighbor, and back-propagation-based algorithms a t various aspects, and the degradation of those single-look results versus distance from the training point is determined. While the results indicate a good performance on the part of neural networks in some cases, the fact that the algorithm, features, and decision-integration rule have not been optimized means that no firm conclusions can be drawn. The algorithms described so far have been applied to a limited closed-set classification problem and, consequently, a broader range of data sets must be used for training. A synthetic radar-signatures simulation model developed by Georgia Tech. Research Institute [Tuley et al., 19831 provides a means of generating such data sets. This tool makes it possible to accomplish extensive optimization of classical as well as neural-network algorithms. Applications of these algorithms will eventually include SAR, inverse SAR, and rnonopulse imaging radar.

Extended target detection and segmentation. Besides RF seckers, consideration is also being given to IR seekers for current and future missile-guidance roles that will include both antisurface and land-strike warfare. Some of the advantages inherent in IR technology for antiship missile application include passive operation, good resistance to jamming, and high spatial resolution. T h e last ofthese, high spatial resolution, is needed in connection with potential operational requirements for target classification. Given that there is an adequate SIN, high classification accuracy requires high spatial resolution, regardless of whether the imagery is intcrpreted by a human operator [Rosell and Wilson, 19731 or processed by a computer [Ricdel and White, 19831. The primary drawback to IR antiship missile seekers may lic in their lin~itedrange performance when conditions of degraded atmospheric visibility csist. The signal processor dcscribed as follows [Steinberg and Rivera, 19871 is designed to optimize the detection range of 1It sensors against ship targets. A single-framc signal proccssing approach is dcvcloped for maximizing II\ scnsor SIN, which is conlplernentary to earlier approaches for SIN optimizations, such as waveband optimization, advanced I11 detector dcvclopmcnts, and multiframc image proccssing. The idea of a signal processing concept bascd on a l l ~ r r n n r j r ~ i s i n r isysrcrn 111cldcl followcd a study \vhosc results indicated that ranges obtained by human observers

Sec. 6.8

Future Guidance Processing

413

of visual displays could in many instances and under very diverse conditions substantially surpass ranges obtained by a hot-spot detection algorithm. The performance of ship-detection algorithms could therefore be judged against the predicted performance of human observers. The acquisition range against ship targets and classification accuracy will be to a large extent be determined by seeker spatial resolution, or pixel size. The SIN (and, hence, detection range) is maximized by matching the pixel sizc to the target size. However, assuming adequate SIN, high classification accuracy demands that the pixel size be made much smallcr than the target sizc. Therefore, selecting the pixel size to maximize detection range means that the resolution will be inadequate for classification, while making the pixel size as small as possible to facilitate accurate classification means that the initial detection range will be very poor. Resolving the conflicting spatial resolution requirements for detection and those for classification is a matter of employing image processing methods based on a model of the human vision system used largely in the past by EO engineers to predict the performance of human operators of thermal imaging equipment and televisions [Rosell and Wilson, 19731. In this model, there is an infinite-dimensional bank of spatial filters, each of which corresponds to a possible target shape. Every conceivable shape is represented in the filter bank, plus all variants thereof that can be formed by the process of translation, rotation, and scaling. Although the original human vision system model did not lend itself directly to digital realization, MRSI was developed that approximates the human vision system in performing detections of targets seen against uncomplicated (uncluttered) backgrounds, such as are likely to be found at sea [Gasparovic, 19821. MRSI, like the human vision system model that preceded it, consists of a bank of spatial filters tuned for maximum response to objects of differing sizes and shapes. Given that the human model, because it is infinite-dimensional, is also nonrealizable, a particular challenge in designing the MRSI was to be able to achieve detection performance akin to the human system, using a low-dimensional filter bank. The noise-reducing characteristics of MRSI cannot generally be calculated analytically, and consequently a computer program was designed to numerically evaluate the effectiveness of the processor. Results of initial computer simulations measuring ship imagery as input were encouraging, even if they were not always easy to interpret. This difficulty was due to the fact that the ship images were recorded under uncertain conditions. Computer simulation was then used to determine the probability of detection ( P D ) versus SIN for three ship profiles. It was found that the SIN improvement provided by MRSI processing could be estimated as G = 0.8 qx,where A is the area of the target in pixels.

Navigation and Guidance Filtering Design The order of presentation of those elements that constitute navigation and guidance filtering design is shown in Figure 7-1. Chapter 2 has presented examples of typical target trackit~gsensors, such as IR, radar, laser sensor, TV, and tracker modeling. In this chapter, however, emphasis is placed on guidance filtering and tracking as applied to target tracking. The subjects of uncertainty in measurements due to clutter, target acquisitionldetection and recognition are presented in Book 4 of the series. For target tracking, many signal processing techniques are used to discriminate the target from its background, other targets and decoys, while filters are used to extract the geometric and kinematic variables involved in target tracking and guidance (see Sections 7.1 to 7.5).

7.1 TARGET STATE ESTIMATION As depicted in Figure 7-1, target state estimation involves stochastic filtering based on target acceleration modeling for the purpose of target maneuvcr detection and target tracking. In this context, the term lrackit{q has often been used to mean accurate estimation of target states, without consideration for the antenna pointing/control aspect of the problem [Cloutier et al., 19881. This section reviews the various filtering techniqucs.

Sec. 7.1

Target State /%timation

Target Tracking Sensor (Chap. 2.2) I Target ~ a n e u v J Modeling r Target state' Estimation Autonomous~a!get Acquisition: (Chap. 2.4) Detection and Recognition Navigation ahd Guidance Filter Design (Sec. 7.2)

Radar ?racking (Sec. 7.3) I

1

Advanced ~aviga4onSystem Design (Sec. 7.5 & Book 4) Figure 7-1

Spacecraft ~ttiiudeEstimation (Sec. 7.4)

I I

Tracking in Clutter/ ~ultitargetTracking (Book 4)

Navigation and Guidance Filtering Design

Target tracking fllter.

The tracking filter functions mainly to receive the noise radar position data and to provide smoothed target position, velocity, and acceleration estimates. These are subsequently used to generate a one-scan prediction of target position for track correlation purposes. Tracking filters can generally be classified according to memory features into the following three groups: the fixed memory filter, the expanding memory filter, and the fading memory filter. The fixed memory filter, such as the maximum-likelihood filter, typically demands the storage of all data reports which take place within the time corresponding to the memory's window. The expanding memory filter (for example, classical linear regression filter) incorporates all past data samples at the time of a new observation to form a new estimate. This characteristic is disadvantageous for estimating maneuvering targets because of the substantial fluctuations of the target state during the estimation period. Thus, fading memory filters are often favored, where the old data are used but are forgotten at an exponential rate [Schleher, 19801. Six tracking filters that are widely used in systems are the Kalman filter, the simplified Kalman filter, the modified maximum-likelihood filter, the a-P-y filter, the Wiener filt'r, and the two-point extrapolator. These filters are typically appropriate for implementation with track-while-scan and other nearly constant data rate tracking sepsors. Other types of filters include least-squares filters, polynomial filters, and adaptive filters, but these are not presented explicitly in this section. Book 4 of the series presents the topic of adaptive filters. It is assumed that the gain vectors of the first three filters (Kalman filter, simplified Kalman filter, and maximum-likelihood filter) can be calculated in real time. This enables the filters to be adapted to varying tactical environments. It also permits optimal tracking in the presence of missed data points. An implementation of the stored-gain versions of these filters can be achieved at significant computational savings. Examples of stored-gain filters are the last three filters considered (a-P-y filter, Wiener filter, and the two-point extrapolator). The first five filters just described are examples of recursive filters "with memory." Figure 7-2 illustrates

Navigation and Guidance Filtering Design

Chap. 7

Signal Processing Sensor

and

Data Association

Predicted Grget State

Figure 7-2

Block Diagram Representation of the Recursive Filter A

a general block diagram representation of the recursive filters considered. The differences in filter formulations are found in the system models and the way in which the filter gain vector is calculated. Because it requires so little memory, the filter that is easiest to implement is the two-point extrapolator.

7.1.1 Target Tracking Filter Summary The following, adapted from Cloutier et al., [1988, 19891, summarizes various types o f stochastic filters that havc been employed to estimate target motion. A comparative evaluation o f some maneuvering target t r a r k i n g a l g o r i t h m s is p r e s e n t e d in Lin and Shafroth [1983(c)]. Single-model adaptive Kalman filters can be split into t w o groups-classical and reinitializing. Classical adaptive filters (for example, Greenwell ct al. [1983]), when applicd to thc air-to-air problem, rely on a continuous modulation of the filtcr bandwidth in response to targct maneuvers (implicit maneuver dctcction). Rcinitializing-type filtcrs arc based on explicit targct maneuver detection. When target motion differs from that assumed in the model, a bias appears in the innovations process. This bias is detected b y employing statistical hypothcsis testing. During the detection process, the filter is in a nonadaptive mode. Once dctection occurs, the filter's biased-state estimates are instantaneously adjusted according to the CStimated input and the filtcr is rcinitialized. [Bullock and Sangsuk-Iam, 1984; Chan et al., 19791 assumed constant acceleration over small time intervals in developing reinitializing filters. Dowdlc ct al. [I9821 and Tang ct al. 119841 assumcd constant velocity. Lin and Shafroth [1983(b)]avoided input estin~ationby dcvcloping a rcinitializing filtcr bascd on concatcnatcd mcasurcmcnts. T h e sequential tracker cnlploys batched mcasurcmcnts and is reinitialized by a local cstimatc whcncvcr the diffcrcllce between the two cxcceds a predetermined value. Thcrc arc drawbacks to both types of single-model adaptivc Kalman filters. The classical adaptivc filtcr suffers primarily from an inhcrent lag, which is sufficicntly pronounccd as to rcsult in a significant loss o f pcrformancc. In particular, by the timc the filtcr bandwidth cxpands, the target may have transitioncd into nonmancuvcring flight. Thc primary difficulty in reinitializing filtcrs is sctting thc dctcction threshold. Thc dcsirc to havc thc filtcr

Sec. 7.1

Target State Estlmatlon

41 7

rapidly respond to maneuvers runs contrary to the goal of a low probability of false detection. As shown in Chapter 4.2.3, the multiniodcl Kalman filtcr consists of a bank of filters and is ideally suited for the cstiniatioti of systems with parametric variations. Each indrvidual filtcr from the bank is optinlally designed for a discrete parameter level. As shown in Book 4 of the series, the addptivc state estimate is obtairicd either from the conditioned probabilitylwcightcd average of the bank mcrrlbcrs or fro111 the single member which displays maximum a posteriori likelihood [Yuch arid Liri. 1984, 1985(a)l.In the case ofa semi-Markov model, switching within the bank occurs according to a Markov transition. While swift adaptation can be achieved with the multimodcl filtcr, its exponentially expanding memory requirements must be limited to irnplcmcntation purposes. This can be done in a variety of ways. Maybeck and Hentz [I9871 employed a moving bank tcchnique and recommended decision logic for moving the bank. In a nonlinear application, Verriest and Haddad [1986, 19881 used a consistency test based on the original linear regions of the nonlinear system. Gholson and Moose [I9771 made certain statistical assumptions concerning the individual filtcrs \vhich reduced the bank to a single Kalman filter structure augmented by a recursive learning term. The tcchnique employed by Blom and Bar-Shalom [I9891 is one of hypothesrs merging wherein a single Gaussian probability density function (PDF) is used to approximate the mixture of assumed Gaussian PDFs by moment matching. In the case of tracking filter design of the air-to-air dynamics1measurements structure, most target state estimation models consist only of the linear kinematic equations even though the actual target dynamics are nonlinear. As a practical matter, it makes more sense to erroneously estimate target acceleration with a linear model than with a nonlinear model. Also, in the small time interval between seeker updates, a linear acceleration model yields nearly the same target motion as a nonlinear model derived from similar assumptions. Seeker measurements, by contrast, are spherical in nature (range and angle) and are therefore nonlinear functions of the state in Cartesian coordinates. Thus, one cannot avoid a nonlinear structure here, either that of linear dynamicslnonlinear measurements in Cartesian coordinates, Figure 7-3a, or that of nonlinear dynamicsllinear measurements in spherical coordinates, Figure 7-3b, where the nonlinear dynamics are a result of the nonlinear transformation of Cartesian linear dynamics to the spherical frame. One is looking at the coordinate systems in Figure 7-3 from below the x-y plane and from the right of the y-z plane. Using" Cartesian coordinates. several researchers have transformed the nonlinear measurements into linear pseudo-measurements [Dowdle et al., 1982; Fitts, 1974; Lin and Shafroth, 1983(a);~ & etg al., 19841. Speyer and Song [I9811 compared pseudo-measurement and extended Kalman observers and found that the former was biased. They later confronted the nonlinear measurements by developing the modified gain extended Kalman filter [Song and Speyer, 19851. In this extended filter, the total variation of the nonlinear measurement function is not approximated by its first variation, but rather identically replaced by a linear structure which is a function of both the state estimate and the actual measurement. Verriest and Haddad

Navigation and Guidance Filtering Design

418

I

Chap. 7

Taraet Target ( r ! -0, - $ )

Figure 7-3 Air-to-Air DynamicslMeasurement Models /From [Cloutirr et a / . , 19891, O 1989 IEEE)

[I9881 constructed a piecewise linear function to approximate the nonlinear measurement function and, based on the linear segments, developed a semi-Markov multimodel Kalman filter. Sammons et al. 119791 and Balakrishnan and Speyer [I9861 have developed hybrid filters which take advantage of the linearity in both coordinate systems. I'ropagation is performed in the Cartesian coordinate framc while updating is performed in the spherical (or polar) coordinate frame. Just before the update, the state statistics for these filters must be transformed nonlinearly into the second coordinate frame, and the inverse transformation must be performed just after.

Filter implementation. Concerning filter imple~nentationand the corresponding coordinate system selection, spherical coordinates should be given more consideration. Examples of totally non-Cartesian filter implementations are found in Moose et al. [I9791 and Gholson and Moose [I9771 w h o selected spherical coordinates in implementing their semi-Markov models, both of which produced simpler and more accurate trackers than their Cartesian counterparts, and Aidala and Hammel [I9831 who used modified polar coordinates in a bearings-only tracker. The filtering may be performed in polar coordinates with range, azimuth. and elevation angles, and their derivatives, as the state variables. In polar coordinates the measurement system model is linear and uncoupled. However, evcn for a target moving in a straight line at constant speed, the dynamic model is nonlinear, and the linearization leads to large errors. In reality, the problem structure in Cartesian coordinates is nonlinear dynamicslnonlinear measurements, linear dynamics appear only as a result of modeling (for example, assuming constant-speed target). There is a natural tendency to avoid spherical implcmcntations, in spite of their linear

Sec. 7.1

Target State Estimation

419

~ncasuren~cnts, because thcy transform thc linearly modeled. Cartesian dynamics into nonlinear sphcrical dynamics. Cotisidcring that target dynamics arc nonlinear in any coordinate systcm, it sho~tldbe true that linearizing nonlinearly modeled spherical dynamics is no more detrimental than linearly niodcling Cartcsian dy-. namics. Such was shown to be the casc in Moose ct al. [I9791 and Cholson and Moose [19771. Finally, thcre is no compelling reason to model target acceleration in Cartcsian coordinates. Lincar modcls of inertial radical and angular accclcration could be devclopcd directly in sphcrical coordinates. In this way, thc entire filtering problem could be pcrformed linearly in that frame. This would rcquirc the nonlinear transformation of inertial strapdown outputs to sphcrical coordinatcs, and the inverse nonlinear transformation of the filter's output to Cartcsian coordinates. In dual-control applications, the nonlinear transformation of state statistics could be avoided if thc guidance law were formulated in sphcrical coordinates. This type of filtering should inspire a whole new rcsearch cffort concerning what is thc best way to model radial and angular acccleration, and what is the simplest way to incorporate target flying characteristics into such a tnodel [Clouticr et al., 19881.

Adaptive filtering. Although extended Kaltnan and various other nonlinear filters have demonstrated improvement over thc standard Kalman filter, these have exhibited only limited effcctivencss as trackers in thc air-to-air encounter. This can be blamed for the most part on inadequate target acceleration modcls. Even though much progress has been made in acceleration modeling, no single set of model statistics can accurately represent the huge set of diverse maneuvers capable of being performed by a modern tactical fighter. This fact neccssitates the use of some type of adaptive filtering for best tracking performance. Such a filter must be able to quickly respond to a rapidly changing target motion. With the proper design, the multimodel adaptive Kalman filter could be more effective than its single model counterpart, although filter complexity is increased [Cloutier et al., 19881. This performance superiority was demonstrated in Lin and Shafroth [1983(c)] in a comparison of several advanced tracking filters. In addition, Bar-Shalom et al. [I9891 employed a hypothesis-merging technique in which the mixture of assumed Gaussian PDFs is approximated by a single Gaussian PDF via moment matching. A new approach to the maneuvering target tracking problem such as adaptive autoregressive (AR) target model identification is presented in Speakman [1986]. The primary focus of this work is on the method used to update the system dynamic model and the process noise covariance. The procedure described herein assumes only that the target accel'eration is exponentially correlated. The need for heuristics in determining the target time constant and driving noise covariance is removed due to the adaptive nature of the method. A nonlinear six degree-of-freedom digital simulation of an air-to-air engagement is used to demonstrate the technique. Results indicate that much more accurate estimates of target states can be obtained using the adaptive method rather than the conventional Gauss-Markov model. Book 4 of the series presents further adaptive filter design.

420

Navigation and Guidance Filtering Design

Chap. 7

7.1.2 The Wiener Filter T h e Wiener filter is a fading-memory, constant-gain filter that is used in trackwhile-scan systems. The gain vector employed in the Wiener filter, which represents the steady-state gain vector o f the regular Kalman filter, is computed off line and stored in the computer. When steady state is reached quickly by the Kalman filter, the Wiener and the Kalman filters, exhibit similar performance as shown in previous sections. Except for the fact that no covariance elements are computed initially in the case ofthe Wiener filter, both the Wiener filter and the Kalman filter are initialized identically. Possessing constant gain, the Wiener filter does not need to solve any auxiliary equations, and requires very little computer storage. Furthermore, because its gain is derived from the Kalman filter, accounting for target maneuver statistics directly, it can be adapted to a variety of vehicles and possesses the ability to effectively track both maneuvering and nonmaneuvering vehicles [Singer and 13ehnke, 19711.

7.1.3 Kalman Filter The Kalman filter, a recursive, fading-memory filter en~ployedin track-ivhile-scan systems, is the optimal filtcr for tracking when target motion equations are known. The estimated quantities such as position, velocity, and acceleration are known as states and are embodied in the state vcctor. The statc equation is given as a difference equation for the targct dynamics in terms of state vectors. The Kalman filter makes usc of dcrcrrr~inisricrargct dynamics plus a random process that accoullts for the inexactness of the difference equations used to express the dynamics and for other error sources. An observation equation, corruptcd by measurement noisc. is employed to observe system states. A product of the Kalman filter algorithm is the optimum linear n1inimun1-error-variance unbiased statc estimate. Thc Kalman filter algorithm also cstimates thc covariance matrix of the errors included in the estimate, and therefore provides a technique for adapting the filter to changing targct dynamics. However, if the Kalman filter is not "tuned" to the appropriate target dynamics, differences in the filter output and the true cstinlatc may causc it to become unstable. This property of thc Kalman filter, prcsentcd in Chaptcr 4.5.4, is termed di~~ergcr~tr and must be considcrcd in any application of the Kalman filter algorithln [Schlehcr. 19801. T o achieve ~vhitecxcitation (~nancuvcr)noisc which is rcquircd for thc filter's optimality, the Kalman filrer e~nploysthc augmcntcd versior~of targct dynamics model. O f all thc filters studied, thc Kalman filtcr is the ]nost sophisticated, the tnost accurate, and the most cxpcnsive to implement.

7.1.4 The Simplified Kalman Filter

By simplifying thc maneuver model used in tllc Kalman filtcr, the statc vector 2nd the number of indcpendcnt components of thc covariancc matrix can be reduced. O n e way in which the nlodcl is simplified, for instance, is by incorrectly assuming

Sec. 6.7

Defense and OIfense Systems

401

for targets if they arc in the parts ofthc Doppler spectrum of interest and are dctcctcd. Since they arc internally generated, they are obviously of no value to missilc guidance. Fundamental implementation problems began to be understood in the 1940s, but the state of the art in hardware was not conducive to satisfactory solutions. The microwave sources of that timc were primarily magnetrons and were quite noisy (although pcrfcctly satisfactory for conventional noncohcrcnt pulse radar opcration). Also, thc receiving vacuum tubes available at that timc were large (and microphonic) [Fossicr, 19881. Air defense homing system. One of the advantages of an active seeker is its ability to operate autonomously subsequent to missilc launch. Taking a semiactive approach was the result of the difficulty of achieving the necessary isolation ofthe recciver from the transmitccr in an active CW. In this approach, the transmitter remains at the launch point and only the receiver is flown in the missile [Fossier, 19841. One of the early examples of a semiactive air defense system is the Sperry Sparrow I with radar beam-rider guidance, which became the Navy's first operational missilc. The Douglas Sparrow II is equipped with active pulse radar guidance. In the Raythcon Sparrow Ill, the transmitter was removed from the missilc and left in the launching aircraft. By separating the transmitter from the rcccivcr by miles rather than inches (and removing comnlon sections of waveguide through which both transmitted and received signals must flow), the feedthrough and associated noise problems were reduced by orders of magnitude. This increased the maximum transmitter power level that could be tolerated in the active radar (a few watts) by orders of magnitude, and, combined with the much larger antenna in the aircraft used to illuminate the target, provided sutlicicnt tracking range to permit homing all the way a t the required flight ranges. The Sparrow 111 was Lnally selected for operation. Even more important than its low-altitude capability, it was a homing missile that worked. It was about this time, noted Fossier [1988], that a new opportunity arose to advance the state of the art in low-altitude air defense. The U.S. Army wanted to develop the technology needed to provide a battlefield SAM to protect friendly troops from attack by low-flying aircraft. From this desire was born Project Hawk. In a symposium on low-altitude guidance in early 1953, it was shown how clutter and image (or multipath) problems could be overcome with a semiactive C W radar homing missile. According to Fossier [1981], the problem of seeing moving targets hidden by ground clutter could be uniquely solved with CW radar in the Hawk system. Even when this was proposed, however, it did not allay fears that the effect on guidance accuracy of radar reflections from the earth, termed the image or multipath problem, would be severe enough to cause the missile to home somewhere between the target and its image and thus consistently collide with the ground before reaching the target (see Figure 6-490. At the time the problem was being researched, the design team of the Hawk system learned that horizontally polarized microwave energy can reflect almost completely from a smooth earth in the forward scatter direction at the grazing angles of interest, thus increasing the likelihood of the serious image

422

Navigation and Guidance Filtering Design

Chap. 7

over position measurement uncertainty, and is defined by the three fundamental target tracking modeling parameters: track period, target maneuverability, and measurement noise

F 2 T~Q:/~/R:/~

(7-4)

where T is the time period between state updates. The optimal steady-state filter is

Y,(k + 1 )

=

Y , ( k ) + a ( k ) [ z ( k )- Y r ( k ) l

(7-5)

Under steady-state conditions, F and a ( k ) are given respectively by

F2 = 4a2(k)l[l- a ( k ) ] ,

a ( k ) = ( - F2

+

+

1 6 ~ ~ ) / 8 (7-6)

The optimal tracking performance of the first-order filter is o;,= a(k)Rk.

Sewnd-order target tracker: a-p tracker.

The mathematical model

for the second-order target tracking problem is

The optimal steady-state Kalman gain is conveniently represented in the a-P tracker problem as

The steady-state values of the components of the state estimation covariance matrix are denoted by lim PC = (P,,] lrrr

The components of the prediction covariance arc dcnotcd by lim P r =

[tni.,]

k-=

The filtcr gain given by Equation (5) of Table 4-1 thus is

The covariance update equation from Equation (6) of Table 4-1 becomes, using Equation (7-1 I ) ,

Sec 7.1

Target State Estlmatlon

423

Substituting Equations (7-7) and (7-10) into Equation (4) of Table 4-1 produces

Equating the terms of Equations (7-12) and (7-13) yields, after cancelling out some terms, am11

= 2Tm12 - T2m22

+ T'4

-Qk,

T3 2

anr12 = Tm22 - -Qk,

P m l z = T3Qk (7- 14)

From Equation (7-ll), I

= R ( 1

- a)

(7-1 5)

rn12 = R k ( P / r ) / ( l - a)

and from Equation (7-14),

Using Equations (7-15) and (7-16) in Equation (7-14), gives

Comparing Equations (7-14) and (7-15), with respect to the quantity

m12

implies

where the tracking index is obtained by using Equation (7-4). The optimal position and velocity tracking performances are obtained from Equations (7-13). (7-IS), and (7-16) as a by-product of the analysis and are given by

The resulting a-fi tracker is given by the Kalman filter equation defined in Table 4-1 as follows:

f,(k) = f,(k - 1)

+

~ { , ( k - 1)

+ a(k)[z(k) - f,(k

-

1)

-

~ { " ( k- I)] (7-20a)

Navlgatfon and Guidance Filterfng Desfgn

424

Chap. 7

Example 7-1. The following example illustrates the second-order tracking problem. Given that T = 1 sec, Q:l2 = 3 g, and R:'~ = 1,000 ft, the preceding method can be used to show that the tracking index parameter is F = 0.0966 and the steady-state Kalman gains are a = 0.3551 and P = 0.776. The detailed derivation can be found in Kalata [1984]. Third-order target tracker. As with the analysis governing the optima] second-order filter, a target modeled by position, velocity, and acceleration states yields a solution also characterized by the tracking index parameter F. The mathematic model for the third-order target tracking problem is + 1) [j~"(~ y,(k + 1 ) Y

Y

(

~

+

z(k) = [ I

' ] ; [;:::;I r] 1 +

0 0

=

W(k),

~"(k)

0 0]

The optimal steady-state Kalman gain is

p = 2 ( 2 - ( ~ ) - 4 6 = 2

a2

-

a

(7-22)

The tracking index is F L yy"/[4(1 - a ) ]while the error covariance of state estimates are at, = aRk,

a;, =

8aP

+ y(P

- 2a - 4) Rk, 8Ty"(1- a )

2 = 0..

"

Y ( ~ P- Y ) R k 4T4(1 - a )

The filter equation is then given by the equations defined in Table 4-1. The methodology formulated here can be used to improve the performance of midcourse guidance accuracy. Applying the optimal filtering theory to the target tracking problem, the tracking index is found to have a fundamental role. This is true not only in the optimal steady-state solution of the stochastic regulation tracking problcm, but also in the tracking initiation process.

7.1.6 Modified Maximum-Likelihood Filter Here, it is assumed that the predicted and measured variables are independent and Gaussian. Instead of jointly considering the target position and velocity variables, which would be equivalent to the Kalman filter, position and velocity are considered

Sec. 7.1

Target State EstImatlon

425

separately to reduce the storage requirement. This technique is called the modified maximunt-likelihoodf;Itrr.The following derivation can be found in Trunk and Wilson (19761. The joint density of the predicted position X, and measured position X, is

where K, and K,,, represent the predicted-position and measured-position covariance matrices, respectively. The maximum-likelihood estimate of the position p. represents that value of p which maximizes Equation (7-23). The partial derivative of the log of Equation (7-23) with respect to p is: K;'(Xp

-

p)

+ Kil(Xm-

(7-24)

~1.)

When the preceding partial derivative is set equal to 0 and solved for p, the maximum-likelihood estimate is obtained as

i.

= (K;'

+ KG')-'

(Ki'X,

+ Kz'X,.)

In a straightforward way, it can be shown that the covariance of

(7-23) is

A new velocity estimate can be acquired by employing the new position estimate (i and the old smoothed position. An equation with the identical form as Equation (7-25) can be employed to combine the new velocity estimate with the old velocity estimate. 7.1.7 Two-Point Extrapolator

An extremely simple filter that can be implemented with very little memory requirement is the two-point extrapolator. This filter employs the last data point to obtain vehicle range and bearing, and the last two data points are utilized to determine target range rate and bearing rate. Because this filter possesses practically no memory, previous data points do not . prejudice predictions. Thus, the maneuvering and nonkaneuvering vehicles previously covered are tracked equally well (badly) [Singer and Behnke, 19711. -

7.1.8 Comparison of Target Tracking Filters Comparison of tracking accuracy. There are two ways in which data can be entered into the tracking computer: automatic and manual. As the first classification suggests, the track data are entered into the computer by someone who views the video on a screen and specifies it for entry. In the automatic mode, no such person is involved. With manual loading, only a limited number of tracks can

426

Navigation and Guidance Filtering Design

Chap. 7

be updated by the operator in a certain time interval. This limits the effective system data rate. Also, as the operator will incorrectly tag some o f the track returns, independent additive errors in range and bearing are included in the single-look mcasurements. In an analysis conducted by Singer and Behnke [1971], the steady-stat, one-sample-ahead prediction accuracies of each of the filters described up to this point were compared on a percentage basis to that of the Kalman filter. The minimum tag time per targct for manual data entry is assumed to be 3.0 sec, and the tag position accuracy is assumed to be uniformly distributed with maximum error of 0.03 in. on a 5.0 in. display scrcen. In the case of automatic data entry, the system data rate eauals the sensor data rate and no additional measurement errors are included because of the automatic entry. The final results were obtained by determining the average of the experimentally acquired percentage degradations in each of the range, bearing, course, and speed coordinates. This occurred because the transient responses of these filters were short-lived relative to the tracking period, and because the correlation coefficient p was typically small for the data rates and vehicle types evaluated. On the average, the performance of the a-f3 filter is about 50 worse than the Kalman filter, maneuvering vehicles being responsible for the largest degradation. Because the a-P filter significantly resembles the leastsquarcs filtcr, its gain vector quickly becomes too small to correct for the large estimation errors that are produced from targct maneuvers (see Book 3 of the series). The unifornl performance of the t~vo-pointextrapolator is more than 70 percent worse than the Kallnan filtcr. Since sensor and vehicle attributes can~lotbe considered, this filter cannot be "tuned" as the others can. Both thc Wicncr and simplified Kalman filters lverc no more than 20 percent worse than the Kalman filtcr, while the modified nlasimum-likelihood filtcr \vas n o more than 10 percent \verse than thc Kalman filtcr [Singcr and Hchnke. 19711.

Comparison of computer requirements.

The conlputer time and computer storage rcquircmcnts \verc examined for the initialization and main-loop phases of cach filtcr algorithm. The results wcre normalized t o the computer requircmcnts of the Kalman filter. Thc filter implementation requirements in ascending order are as follo\vs: t~vo-pointextrapolator, Wicncr filter, a-p filter, simplified Kalman filter, Kalman filtcr. Furthermorc, the complexity of cach successive filter increases by about a factor of 2. Thesc numbers depend largely on the computer hardkvarc bcing uscd. Howcver. the relative computcr storage rcquircmcnts should not dcpc11d substantially on thc computer hardware bcing cmploycd. The rclati~~c computation times \vcrc cstablishcd for a typical computer uscd in tactical system applications. Thc times to complctc an add, a subtract, and a store wcre assumed to be equal; the timc to complctc a multiply was assumed to be five times that of an add; and thc timc to complctc a divide was assumed to be nine timcs that of an add ISingcr and Behnke, 19711. Although it has acceptable performance, thc 1110dified maximum-likelihood filtcr obtained by arbitrarily dccoupling the position 2nd velocity csti~natcsis almost as complicated as thc Kalnlan filtcr and thus \vould not be employed [Trunk and Wilson. 19761.

Sec. 7.2

Practical Navigation and Guidance Filter Design

427

In many practical systems. the tracking accuracy generated by the constantgain Wiener filter is equivalent to that of the more sophisticated Kalman filter at one third the computational cost. It isbeneficial to implement the simplified Kalman filter arid modified maximum-likelihood filter if high accuracy is demanded and the length of the transient period app~oachesthat of the tracking interval. In addition, the Kalman class of tracking filters is uniquely able to supply accurate measures of tracking error statistics, even if missed data points exist. Ncvcrthclcss, in certain situations, the implementation of highly simple filters such as the two-point cxtrapolation may bejustified by the nature ofthc systcm rccluircmcnts and the impact of tracking pcrforrnancc on thcsc rcquircmcncs.

7.2 PRACTICAL NAVIGATION AND GUIDANCE FILTER DESIGN A guidance tracking filter provides target state estimation, which can be of trcmendous aid in predicting target position and improving the accuracy of an intercept NGC systcm. Low-pass filtcrs attenuate the noise component of the sensor signal, given the frequency characteristics of target signal and noise. Optimal estimators such as Kalman filters optimally separate the target signal from the noise by using information about target (and missile) dynamics and noise covariances. 7.2.1 Guidance Tracking Filter

Function and requirements. The primary components making up guidance processing are the guidance filters and guidance law algorithms. Guidance filters, which act as estimators and track the target, provide target range, bearing, and rate. Figure 2-2 shows a guidance tracking filter whose function is to remove as many nonstationary noises as possible and to preserve at the same time signal integrity in order to achieve accurate guidance. O n e of the main functions of guidance filtering is to smooth the noise in computing the pursuer-to-target LOS rate for guidance and to separate the relative target motion. In conventional PNG, they are used to estimate LOS rate ir, whereas in advanced guidance laws they are used both to measure LOS rate u and to estimate target maneuvers AT,. An advanced guidance tracking algorithm is needed to provide estimates of the target's relative position, velocity, and acceleration to support the advanced guidance law. Guidance filtering also acts to stabilize any residual missile body-motion coupling from the seeker head space stabilization control or . from radome error type distortions at the antenna. The choice of bandwidth for such a filter will be shown to reflect a trade-off between noise transmission and performance. During flight the noise and bodymotion requirements on the guidance filtering vary. Consequently, the guidance computer is programmed in such a way that it can adapt correspondingly. The missile's flight can only be smoothed if the guidance filtering is sufficiently heavy. It must be able to counter unstable motion that is the result of large random ac-

Navigation and Guidance Filtering Design

428

Chap. 7

celerations, as it responds to the noise. Such accelerations give rise to very high angles of attack and, consequently, induced drag, all of which have a depleting effect on the kinetic energy of the missile. Still, the guidance filtering must be sufficiently light as to permit quick missile response as necessary to correct for headingerrors o r to pursue a maneuvering target. It must filter out noise to the extent required to prevent FCS problems during certain flight conditions. After clipping spikes and transients, the filtering should not remain overly heavy. Moreover, the filtering should minimize the effects of quantization and variable data rate [Yueh, 1983(a)].

Design approach.

Although there appears to be a large number o f demands on the guidance filtering, these vary during flight. In flight, tracking error noise is measured by the guidance computer, which then uses information from the FCS and inertial reference unit (IRU) to ascertain which demand should be given the highest priority. This demand is then applied to the variable filtering. During the late 1960s and early 1970s, a few missile designers did take a cursory look at applying the modern control theory developed during the late 1950s and early 1960s to tactical missiles. Basically, as shown in Figure 2-17b, such an approach would replace the classically designed low-pass filter with an optimal estimator such as the Kalman filter. In theory, this would allow one to "optimally" separate the signal from the noise by using information about the missile dynamics and noise covariances rather than filtering bascd on frequency content only. In addition, missileltarget states other than LOS rate could be estimated, even if not measured, providcd they \vere mathematically obscr\~able[Gonzalez, 1979(a)]. This, in turn, \vould allo\v one to design more advanccd guidancc laws based on optimal control theory, because such theory usually requires complete information concer~li~lg the n~issilcstates. The tracking filter is bascd on bbth the laws of motion and a stochastic acceleration n~odel,evas~vemancuvcrs, and other disturbances being modeled as pecturbations upon the constant-velocity trajectory. The resulting optimal guidancc tracking filter uses a Kalman state estimator with several comDonent state vectors in which stable and accurate estimates of the relative position, relative velocity, and target acceleration are providcd to thc flight vehicle's fire control system or the missile's guidance and control systcm. Consider the guidance filter formulation sho\vn in Fig. 7-4. The objective is to find G I ( $ )\vhosc output f r ( t ) tracks y . with ~ thc least possiblc error c ( t ) dcfincd as (1) = ( 1 ) -

( 1 )(s) = ( ( 5

-

) H ( )I

S )

+

( ) I ( )

(7-27)

Figure 7-4 sum~narizcsthe optimal solutions of C,(s) that minimize txvo diffcrcnt forms of the performance index which arc functions of the l'S1) of the position cstimatc error c.(r), +,.(LO), dcfincd as + c ( ~ )

= +<(w)

I Signal + +,(w) I Noise

-

Target Position Noise (with equivalent white noise PSD Target Maneuver $noise defined by Eq. (2-49)) Mtdcl n Spectral us Dcnsity Whik Noise Shaping with PSD +s* Filt"' H(s) (Sw Chap. 2.4.1)

Target Position Estimate

@e!!ssl

ks2+kls+ k,

G,(s) = ks + k , sz+ks+kl - 2<wns+ w; s2+ 2<wns+

-

G,(s) = k s+k

G,(s) =

4

H(s) = l/S3

* *=

(

*Minimilc

notsc , ) I 6

&e

2

p*= 0.707,

I dl = 2n

m

tun

=

a (w) dw

=

'+noise

I + 2s/F+ 2s2/F2 1 + 2s/F+ 2s2/F2+ s3/F3

F* = (+S/+"',ise)

116

E c2(t), Solutions Dcrivul in Chap. 4.6

m

**& the Gain Crossover Frequency w,, $c(w,)lSignal Figure 7-4

-

s% ks2+kls+k,

Optimal Guidance Filters

= $c(~)lNoise.

Navigation and Guidance filtering Design

430

Chap. 7

where

+.(u) 1 Signal = +.(w) due to signal = (Gf(iw) cbe(o)I Noise = +,(w) due to noise = I(Gf (jo)12

-

1) H(jw)12 4$ (7-28b)

+,,is,

Note that the total RMS miss distance u,.,(tf) is defined by

Y

]

=

uy,(tf) ( Noise =

d&

J:~+~(co)

d$ J.:

me(.)

I Signal dw. I ~ o i s d-e

Concerning the actual signal processing, the state vector is initialized at the track onset to minimize scttling time for relative position and velocity from the FCS. Modes determining gains include air-to-air, air-to-surface, surface-to-air, and surface-to-surface. Time-varying gains are determined based on weighting of measurements versus history by including: (1) continuous update based on range; (2) reinitialization based on maneuver detection (discussed in Book 4 o f the series); (3) refined gain initialization values based on track onset, targct maneuver, and coast; and (4) optimized stcady-state gain parameters.

7.2.2 Navigation and Guidance Filtering for Position Estimate It is shown in this scction that a time-varying, single-time constant noise filter as

shown in Figure 2-6 can be used in the homing loop to estimate the LOS rate signal required for PNG. The optimal noisc filter t i n ~ cconstant, being the i~lvcrsrof the = l/(k),,,, whcre (k),, is given in Figure 7-4. At a optimal frcqucncy, is (T,),, largcr range to go the rcceivcr noise +,, in the system is larger (see Figure 6-9b) and causes the optimal filter timc constant (T.,),,,,,to bc larger than that found a t a shorter rangc to go. Bccausc this noise dccrcascs as thc flight progresses, the result is that thc timc constant reduces as well. Near the cnd of the flight, howcvcr, the large value of glint noisc ncar thc intercept point would prcvcnt thc time constant from going bclow (T,),,,,,,which was calculatcd from the solution providcd in Figure 7-4 by using +,, = 0 and b2 = 0. Sincc the rcccivcr noisc is the dominant factor in the miss distance cxccpt a t a of lcss than 1 scc, thc time constant (T,),, is used to attenuate thc rcccivcr noisc which dcpcnds on thc rangc to go and is approxilnatcd as follows:

,,

(T),,

= ll(k),p,

;-

C!,

-

CHIV,

where C rcprescnts a constant dependent on seeker characteristics. Uy using a computer to both measure the noise and then adjust the filtering based on that measurement, the guidance filter gains are adapted during flight to the changing receiver noise. A signal proccssing algorithm is used to produce receiver noise having spectral propcrtics that characterize actual homing environments. In the case of trying to intercept a highly maneuvering target, it would be necessary for the optimal break frequency to increase, which is equivalent to the noise filtcr time constant decreasing. Note that the discrete-tittle s y s t r t i ~for this filter is the same as the a tracker presented in Section 7.1.3.

7.2.3 Navigation and Guidance Filtering for Position and Velocity Estimate In the case of onboard navigation systems, which receive targeting sensor position information contaminated by noise, it is required to provide these systems with vclocity as wcll as position information for guidance. Therefore, in addition to smoothing thc position signal, it is necessary to derive the components of velocity as these are not measured. Three different techniques are used to obtain this information: approximate differentiation and filtering, an aided INS, and complementary filtering. This section describes each of these systems and assesses their performance in flight based on Niessen [1973].

Differentiation of position information. With this technique (see Figure 7-5a). velocity information is derived by approximate differentiation of the sensor's position signal. However, this approach of differentiating position information alone is not capable of providing acceptable velocity information because of signal noise. There is a restriction imposed by the fact that position and velocity information used in display for N G C system applications must be practically noise free. Complementary filtering. A second technique used to provide velocity and position information for onboard navigation systems is to employ the secondorder complementary/Kalman filtering presented in Example 4-18. The resulting block diagram is the same as the one shown in Figure 4-10, but with u l = u2 = 0. The optimal gains are given in Figure 7-4. This approach is based on continuously mixing sensor position measurement X with acceleration information x obtained from onboard sensors. Inputs to the filter consist of measured or derived acceleration inputs u = x (equivalent to us in Figure 7-4), which provide high-frequency position and velocity information, and position measurements inputs z = X (equivalent to n in Figure 7-4), which provide low-frequency position and velocity information. T h e onboard flight vehicle instrumentation used to provide the acceleration information might consist of attitude reference gyros and body-mounted accelerometers. Based on the acceleration and position inputs, the filtering system first computes ground-referenced accelerations and then combines this information with the noisy

Navigation and Guidance Filtering Design

432

Chap. 7

1 x (APPROXIMATE)

(ii) TIME RESPONSE

X

sin 0

Y

sin 0

OMPLEMENTARY FILTER

1.a

z

-

zz,,

COMPLEMENTARY FILTER

Figure 7-5 (a) Transicr Function and Time Response for Approximate Iliflerentiation (b) Complcmcntary Filtering Navigation System Computations (Fro111 [.%w"r, 197.51 u,i~liprnni.rrin~r.hlmAC.4RD)

position measurement signals on a weighted-frequency basis to obtain satisfactory position and velocity information. In essence, the filtering technique is combining acceleration and vehicle position data to produce low-noise estimates of both velocity and position. Note that thc discrcte-time system for this filter is the same as the af3 tracker presented in Section 7. I .5.

Sec. 7.2

433

Practical Navigation and Guldance fllter Deslgn

The complementary-filter gains used in Niessoi [I9751 for aircraft corresponded to a natural frequency of w,, = 0.45 rad/scc with a settling timc constant of 12.5 sec, based on the timc to settle within 2 percent ofstcady state. The selection of gains was designed allowing the time constant to be long enough for suitable attenuation of noisc from the position measurement, but short enough to ensure small errors which might result from inaccuracies associatcd with the acceleration information. In regard to practical applications, the filtering system can serve to obviatc the need for more complex systcms such as inertial or Ilopplcr navigation systcnls. The noise on tlic position output is greatly reduced compared to the noise on the position input signal. The accclcrometer signal noisc is primarily due to vehicle structural vibration. As shown by the low-noise velocity estimate, this signal noisc is also essentially eliminated by the integration process within the filter. Both the acceleration and the position measurement noises are essentially eliminated by the filter. As indicated by the block diagram of the system shown in Figure 7-jb, corrections in the form of system computations are applied to the body-mounted accelerometers to account for gravitational effects and to resolve the accelerations into components along the runlvay reference coordinates. It was assumed that the pitch and roll angles were small during the final approach which permitted less stringent computational requirements. For intercept applications, applying either Equation (7-28c) or Equation (4-45) yields Total RMS miss distance = u,,(tf) =

.\/%=

INS with periodic radar position updates.

7 d2 +iL?se 4:'''

An INS using periodic radar position updates was employed for steep-angle approach work [Kelley et al., 19741. Using a former Gemini spacecraft INS as the working system, modifications were made to allow the radar position data to serve to update the navigation outputs. Sampled radar signals were digitized at a ground station at 1-sec intervals prior to being transmitted to the flighr vehicle by means of a digital telemetry link. The digital computer outputs consisting of position and velocity information were converted to analog signals and then sent to the onboard analog computers. In general, the sufficiently low noise level and high accuracy of the signal outputs made them very suitable for use with the flight director display application. Unfortunately, the gross weight (approximately 8110lb) and complexity ofthe system made it of limited applicability due to high maintenance and operational costs. The accuracy of the complementary filtering system and the INS were compared in flight with periodic radar position updates. Madigan [I9701 gives an account of the baseline performance data on the INS, while Anon [I9691 provides a description ofthis system's navigation computations, including the update logic. For long-term position and velocity information, both systems depended on the precision tracking radar, with the complementary filtering system receiving continuous position information and the aided INS receiving position updates at 1-sec intervals. The position-error drift rate of the INS was approximately 2.0 nautical m p h in the case of no updates. O n the other

434

Nauigatfonand Guidance Filtering Design

Chap. 7

hand, the position-error drift rate of the complementary filtering system was estimated to be on the order of 100 to 200 nautical mph without position feedback. This high drift rate was primarily caused by approximations which were made in resolving the accelerations as well as by computer scaling limitations.

7.2.4 Navigation and Guidance Nltering for Position. Velocity, and Acceleration Estimate The application of advanced guidance laws generally requires, in addition to the LOS rate information, an estimate of the current target maneuver level. In a general problem, one may not have knowledge of future target acceleration. However, in the problem of a missile intercepting another missile, called the target missile, which in turn is homing on a flight vehicle, the target missile undergoes a sinusoidal phenomet~on.Thus, the frequency and amplitude of the acceleration may be obtained by system analysis. The phase information may be obtained by measurements of the missile velocity. In the problem of an antiballistic missile intercepting a reentry vehicle (RV), one has knowledge of the flight path of the RV and, thus, its acceleration profile. Similarly. in a n antisatellite (ASAT) missile the target acceleration is a known quantity. Thus, the results presented in this section, which estimatc target accclcration, may be quitc useful in several scenarios [Ashcr and Matuszewski, 19741.

Uniformly distributed target maneuver. Following the third-order complemcntary/Kaltnan forn~ulationin Chapter 4.6.3 (see Exan~ple4-22), it is possible to derive a simple form of a single plane Kalman estimator by taking into account thc two ]nost important stochastic disturbances in a guidance system. random targct maneuvering and noisc. both of which arc sho\vn in Figure 2-21d. Thc optimal bandwidth for the system is sufficiently wide to follow the target motion but narrow enough to filter as much of the noisc as possible [Duricux, 1984; Ncslines, 1986; Travcrs, 1985). Approach A. With this approach, thc onboard computer can be used to derive the state and measurement equations as modeled in Equation (7-1). From Equation (4-54), thc state estilnate equations are Thc output cstimatcs arc thc LOS ratc and targct acccleratiol~cstimatcs, exprcsscd rcspectivcly as

Front Equation (4-56) or Figure 7-4, the Kalnlan filter gains are

Sec. 7.2

Practical Naulgatlon and Guldance Fllter Design

435

Figure 7-6a shows a filter which is simplified from the generalized third-order complemcntary/Kaln~anfilter in Figure 4-11. The natural frequency of the target tracking filter is defined as

Substituting Equation (7-33) into Equation (7-32) and applying Equation (4-53), the Buttcrworth form of the transfer function is as follows:

L= y,

3'

1

7

+ 2slF + 25°F' + s 3 1 ~ "

filter time constant = 21F

Figure 7-7 shows that the third-order filter has the peaking in the Bode plot at frequency F, that is, at (k),,p,which is the optimal solution of the approximate firstorder filter shown in Figure 7-4. The optimal frequency of the tracking filter is a function of the range to go since the noise magnitude is also a function of range to go. Applying either Equations (7-28c) or Equations (4-35) and (7-32) yields Total RMS miss distance = cry, (tf) =

fi = d2&~?,;?~, $,"' (7-33)

which is the minimum RMS miss distance achieved only by the optimal filter. The miss distances due to these individual sources can be computed in a manner similar to the way in which the total RMS miss distance was computed. If the relative position measurement is not readily available, then it must be reconstructed as follows. The LOS angle must first be generated by adding the integration of the seeker rate gyro to the boresight error, that is, 6 = E + ad, as was done in Durieux [I9841 and Nesline and Zarchan [1985]. Then the relative . dynamics have been totally position information is derived based on y, = ~ 6Seeker eliminated from the guidance transfer function. T o summarize, provided the range and time-to-go information are available, third-order noise filters are capable of estimating both LOS rates and target maneuvers. At the same time, implementation of PNG does not strictly require target maneuver estimates. It is in fact possible to implement PNG in the absence of both range and time-to-go data by using a simple low-pass noise filter. When a third-order noise filter is used, however, decreasing the filter's natural frequency or, equivalently, increasing the low-pass filter time constant reduces noise transmission at the expense of increased miss distance resulting from increased target maneuver noise. The point must also be made that it is of course possible to design an optimal noise filter using advanced estimation theory. The detailed derivation can be found in Chapter 4 on Kalman filtering. Finally, the discrete-time system for the preceding filter is the same as the ~ - P - Y tracker presented in Section 7.1.5.

Navigation and Guidance Filtering Design

k=ZF k, = 2F2 kl = F3

ir

-b

kl -- 9 . kl -

Chap. 7

11s

Compared to Fig.4 - l l a :

=

fir=*

A Yr

Yr = X

Amy 'u

P

P

-

u1 = u , = o

?

&Y

b

a) Approach A

Compared to Fig.4 - l l a :

*

9, = P P

Yr A

Yr

=X A

=x

by=% by= -u u 1 = u2 -- 0

b b

h?b

y

b) Approach B

+

Compared to Fig.4 - l l a : b

c) Approach C Figure 7-6 neuver

Guidance Filters for Estimating Uniformly Distributed Targrt Ma-

Sec 7.2

Practical Naulgatlon and Ciuldance FIlter Deslgn

+ 3

2

Y

3

a

'E

%

0

Z

Figure 7-7 Asymptotic Plot of Trajectory Tracking Filter

RadISec

Approach B.

This approach is somewhat similar to Approach A, except that the model in Equation (7-1) is rewritten as

Applying Equation (4-54), the state estimate equations become fr

=

jJr + k(z -

i,),

*

jJr = AT? - A,,,"+

~ I ( Z-

f,),

A,

= k,(z - f,)

(7-37) where the Kalman filter gains are the same as those in Equation (7-33) in Approach A. The output estimates are the LOS rate and relative acceleration estimates, expressed as

b=

(ir

+

t,~,)l(v, t.:),

g,

ATy -

(7-38)

The resulting guidance tracking filter is shown in Figure 7-6b.

Approach C: Angle-only measurements.

In the case the LOS angle is used as a measurement instead of y, there are three ways to proceed. Since the LOS angle is a nonlinear measurement function of the state y, the first would be to use the EKF (see Section 7.2.5). This, however, cannot accurately determine the target position and velocity unless accurate range and range-rate measurements as those in Equations (2-63) are also available. The second way to proceed for the problem is to use the statistical linearization technique (see Section 7.2.5). The third is to apply the results of Case 4 presented in Chapter 4.6.3. The measurement is assumed to be corrupted by noise so that Equation (2-5a) becomes

where v, is given in Equation (2-22) with PSD defined in Equation (2-23). Following Case 4 in Chapter 4.6.3, u = hz and h = l / ( V c & )

(7-40)

Navigation and Guidance Filtering Design

438

Chap.

where z is the relative p'osition measurement. Using Equation (4-58), the state estimate equations are given by:

(7-41) where the Kalman filter gains are also obtained from Equation (4-58) with k, k 1 1 and k l being the same as those in Equation (7-32). The resulting guidance tracking filter is shown in Figure 7-6c.

Poisson jinlcing maneuver.

The problem is formulated the same wav as earlier in this section, except the target maneuver here is modeled as white ~ 2 " s : sian noise (see Figure 2-21e). That is, the acceleration of the target is modeled as a white, Gaussian noise process. Following Approach B and using Equation (4-60), the state estimate equations become

The resulting guidance tracking filter is the third-order complementary1Kalman filter (Formulation 2) shown in Figure 4-1 1b. The Kalman filter gains are obtained from Equation (4-62) with P l l , PI2,and PIJ obtained from Equation (4-61) with 41 = q2 = 0,q3 = (br (given by Equation (2-jl)), and 7 = l/A,. The initial condition P(0) is assumed as follows:

Pzz(0) = RMS initial target velocity minus pursuer velocity Next, following Approach C and using Equation (4-63), the state estimate equation is given by

-

j, = y, + K ' ( u - 6 ) , i Y " =A T , - A,, + K;(u - 6 ) , A T ,

=

-

h y A r y + E;(o

(7-43)

-

6)

--

where the Kalman filter gains are also obtained from Equation (4-63) with K, K I , and being the same as those in Equation (7-42). Assuming some components of measurement noise, 11, in Equation (7-1 b) to be colored noise, then following the results in Chapters 4.5.5 and 4.6.3, similar results to those obtained for the two examplcs shown prc\riously {Equations (7-42) and (7-43)] can be obtained. The targct control input in the Kaln~anfiltcr cquations is assumed to posscss a zero to achicvc unbiascd estimates. A constant target acccleration can be estimated quite well. Contamination to the three state filter rcsults in the rclative ~ o s i t i o nand relative velocity cstimatcs can be brought about by abrupt changes in target acceleration. A nonzero-mean targct control input can be achieved by adding a fourth state to the three-state model, as shown in Figure 7-8. This fourth state represents the

Sec. 7.2

Practical Navigation and Guidance Pilter Design

Figure 7-8 Fourth-Order TargetPursuer Kinematics Model for Kalman Filter lmplemcntation (From (Francis and ~ M i t r l ~ u l 19831, l, 0 1983 IEEEJ

estimate of the bias due to nonzero-mean target acceleration input [Francis and Mitchell, 19831:

Practical estimator gain design. Insight into the guidance and tracking gains of advanced guidance laws can be achieved by making use of an extended frequency-response analysis. If the control gains are not determined adaptively in the modern guidance system, then implementation of the variation in these gains requires the extra range estimation or estimated t,. Following Travers [1985], the present discussion considers estimator gains for both active and semiactive homingreceivers. Variations in these estimator gains ( k , kI, and k , ) are allowable during most of the flight, but at some point they become fixed. In the early stages of intercept only thermal noise is considered, while in the later stages of intercept only glint noise is considered. The reasons for this have been discussed in Section 7 . 2 . 2 . Assuming the target has a uniformly distributed target maneuver as shown in Figure 2-21d, the target signal PSD is given by Equation (2-49). Substituting Ro = Vctf and R = Vc t, in the target noise PSD +,,i,, of Equations (2-25) or (2-48), and considering only thermal noise, the characteristic frequency F in Equation (7-33) for the semiactive homing receiver is

and for the active homing receiver is

In the last two equations, and h. =

(-) V? n$tj

+rA

ti6

Navigation and Guidance Filtering Design

4-40

Chap. 7

When only glint noise is considered, in contrast to thermal noise, the characteristic frequency F becomes F, = (+,fit;)]'/"

. The ,t at which F, = F, satisfies t,/tf =

and the t, at which F, = F, satisfies t,ltf

= (t:$vf)1'6.

It can be

~.

shown by substituting typical values into the right-hand sides of thelast two equations that a reasonable time for the estimator gains to remain constant in each case is a fixed t,ltf = 0.2. T o achieve an acceptable miss distance, a minimum flight time of ten guidance loop time constants T~ is generally required. Therefore, the required time for halting the gain variation is at t, = 27,. Table 7-1 lists the estimator gains for the active and semiactive - - seeker guidance filters of Equations (7-32) and (7-41). T h e estimator gains (K, KI, and El)defined in Equation (7-42) and the estimator gains ??;, and defined in Equation (7-43) for tracking filter that estimates a Poisson jinking maneuver, can also be approximated from Table 7-1 [Travers, 19851.

(Kt,

K;)

7.2.5 Advanced Guidance Mlter

Unlike conventional guidance schemes such as PNG, many of the new guidance laws developed rely on accurate estimates of target dynamic states. It is generally assumed that these states are relative (missile-to-target) position, relative velocity, and target acceleration. Components of these vectors in an inertial reference frame are estimated based on measurements provided by an active or passive seeker. Pursuer inertial acceleration, sensed by accelerometers, is employed as a control vector. The most commonly used algorithm for obtaining state estimates is the Kalman filter. In the event nonlinear relationships are encountered in the measurement equations, the extended Kaln~anfilter (EKF) is frequently used. In designing a Kalman filter, one generally assumes that the state transition matrix and the noise statistics are known a priori. This requires one to further assume that the dynamics of the target can be modeled in a reasonable manner and written in state-space form. T h e most commonly accepted target dynamic model in use today is the first-order GaussMarkov model [Speakman, 19861. Kalman filtering with n-state variables requires a real-time computation of an rr x n matrix Riccati equation. Furthermore, a microprocessor implcmcntation may introduce numerical truncation errors due to limitcd word Icngth. A common approach has been to reduce real-time computatioll by neglecting some of thc cross-coupling terms and prccon~putingthc filter gains. Such nlcthods arc ad hoc in naturc, and extensive simulations have been uscd to tune thc suboptimal filters. The nine-state guidance filter modeling equation is given by Equations (264a and 2-64b), which is uscd as the dcsign n~odcl.The mcasuremcnt equation, Equations (2-63a, 2-63b, and 2-63~). is assumed t o bc in thc form of a discrctc domain z ( k ) = / I ( x , k) + v(k). T w o approachcs can be uscd to cstimatc the state based on thc state and nlcasurclncnt cquations. O n c is based on the standard cxtcndcd

Set. 7.2

Practical NaulgatIon and Guldance Fllter Deslgn

TABLE 7-1 ESTIMATOR GAINS

G&

Continuous-Time Svstem & Semiactive

k kt

2h,/t, 2(h,/tg)'

k1

(haitg13

k' = kVctg

2h,Vc

Discrete Time Svstem k=a

2(h,itg)2'3 413 2(hs/tg)

k, = p i r

2

kl = ~ i ( 2 ~ ' )

(hsi'tg)

a, p, yare the gains of the a-f3-y tracker

2 1/3 2(hstg) Vc

Kalman filter (EKF) presented in Chapter 4.2.1. Since the measurement equation is nonlinear, it is necessary to form the Jacobian or matrix of partial derivatives H where H,, = ah,iax,. The second approach is the statistical linearization technique presented in Chapter 4.2.7. Both approaches are adopted in the discrete-time domain, and the discrete-time state equations are given by Equations (2-65a through 2-65g).

Extended Kalman filter (EKF) approach. The example introduced here demonstrates the nature of problems that lend themselves to EKF analysis. The nonlinear discrete measurement equations for a sensor measuring range and two LOS angles can be represented as the true values corrupted by additive noise:

[%I.

=

Rtrw = ( x t + YS + z r2 ) 1 / 2 u .,,,, = tan-'[z,/(xt y,)2 I/? ] u = tan-' ( y,lx,)

[

+

[::
I.

+ v(k)

(7-47)

where the subscript k denotes a time index. In Equation (7-47)

R,,

( k ) =

=

Ue,,,

+

+

R,

+ R, +

uef

+

ue,

f

R,i,,, (Jeyilnr

(7-48)

are the noise statistics which are dependent on the particular sensor used. In Equation (7-48) subscriptsf, glint, rn, and c denote fading noise, glint noise, thermal noise, and clutter, respectively. It is assumed that

The EKF is perhaps the most obvious approach to filtering for this system since the measurement is nonlinear in the state. The results show that, in spite of the fact that relative position and relative velocity are being subjected to large dynamic

442

Navigation and Guidance Filtering Design

Chap. 7

variations, the filter is able to track these system states well. The estimates generated by the filter of target acceleration are rather noisy, yet the filter does correctly identify the mean target acceleration after an initial transient tracking period.

Statistical linearization approach. This example again considers the three-dimensional guidance filter problem in which a better but more computationally burdensome approach is used that makes use of the statistical linearization technique. Generally, what is done in this case is to convert to pseudo-measurements. The pseudo-measurement is z(k) =

R cos u, cos u R cos a, sin a Rsinue

[

Ik

= [I3

0 O]x(k)+ v,

(7-50)

where vp is the measurement noise in inertial coordinates, and where R, u,, and u are the quantities actually measured. The Jacobian of this transformation is cos cr, cos u cos u, sin u sin u,

- R sin u, cos u -R sin u, sin a R cos u,

I

- R cos a, sin u R cos u, cos u 0

(7-51)

Thus, the first-order Taylor series expansion suggests the following approximation for the variance of the pseudo-measurement vector, Equation (7-50).

r,(x, y , z ) =

$:,[

2

u;,

u:.). ui u y i

2 ua.z u\z] = r,(R9 uC,U) = s ? ( R , uc, n)zT (7-52) uz

where ?(R, u,, u) is defined in Equation (7-49). The preceding formulation makes possible the use of the simple linear discrete Kalrnan filter with only one modification. In order to evaluate r,, the current best estimate of the position is converted back to polar coordinates using Equation (7-47) and then The uncertainty of the estimate is measured by ? ( R , s,, 6 ) and rp(2,, j , , 2 , ) . The resulting statistical linearization algorithm is close, but not identical, to the EKF for nonlinear measurements.

7.3 RADAR TRACKING Discrete Kalman filter. The t~fo-dimensionalradar tracking problem has been presented in Chapter 2.4.2 in which the radar tracking state model was defined. The radar tracking problem is conccrncd primarily with cstimating R , R, U, and u of some particular target. The radar is also conccrncd with having a future good estimate of the targct's R and u. T o this end, the radar transmits a signal every

Sec. 7.3

Radar Trackfng

443

T sec which hits the target, causing some fraction of its energy to be reflected back and received by the receiving unit of the radar. Lewis [I9861 discusses many good approaches to radar tracking filter designs, and these are summarized in this section. (Z(0). P(0)). One initializing Suppose that it is known a priori that x(0) ) ~ be taken to find E(0) and P(0). Then measurement z(0) = (R(O), ~ ( 0 )could

-

Since a well-designed filter converges to the same stochastic steady state for any -x(0) and P(O), it is generally possible, as a mxter of convenience, to select x(O) =

0 and P(0) = I. Now, one has only to enter the system and covariance matrices presented in Chapter 2.4.2 into the computer program that implements the discrete Kalman filter. The program, given the radar readings R(k) and u(k) at each time kT, will compute the best estimates of R, R, a, and 6. Equation (2-55) describes two completely decoupled systems, one for the range dynamics and one for the angle dynamics. A great simplification is therefore possible that amounts to running two parallel Kalman filters on two second-order systems instead of running just one Kalman filter on a fourth-order system. Since the number of computations required per iteration by the Kalman filter goes as the cube of the order of the system, this number can be reduced from approximately 64 to 16. Moreover, since a separate microprocessor can be assigned to each filter using parallel processing, a further reduction in computing time can be realized.

a-ptracker, and a-P-y tracker. An a-P tracker, as presented in Section 7.1, is used to estimate the position X and velocity V state of the system ~ ( t =) a ( t ) = W(T) with position X measurements z = X + v. Here U(T) is the acceleration. This corresponds precisely to the equations for the range subsystem (the first two states of Equation (2-58)) and angle subsystem (the second two states of Equation (2-58)). The Kalman filters used are therefore both a-P trackers. For a rapidly maneuvering target such that R and ii are nonzero, more robust performance can be achieved by using an a-P-y tracker. This is a Kalman filter to estimate the X, V, a state of the system i ( t ) = w ( t ) with measurements z = X + v .

Multiple-rate Kalman filter for radar tracking. Following multiplerate Kalman filter presented in Table 4-2, suppose here that the range subsystem is discretized using a period of T = Tf = 50 msec. The data sampling period T, is now the radar period of revolution, whereas this quantity was denoted simply by Tfin the other approaches discussed. Now, using the multiple-rate Kalman filter in a radar tracking problem means that range and azimuth estimates can be made available between scans. For a value of T, equal to 1 sec, updated optimal estimates can be obtained every 50 msec, although the data arrive o i l y every 1 sec. A value of T = 50 msec in the error covariance time update Equation (7-13) is used, and

Navigation and Guidance Filtering Design

444

Chap. 7

the measurement update [Equations (7-11) and (7-12)] is performed only every T , = 1 sec. In Chapter 4, techniques of continuous dynamics with discrete measurements are listed in Table 4-3, all of which can be applied to multiple-rate Kalman filter for radar tracking by propagating the estimate and error covariance between scans in continuous time and by performing the measurement updates at the next scan.

Extended Kalman filter.

Let the state be = u, from Equations (2-53) the state equations are

vTcos x2 -

XI

i l

V x 2 sin x2

= R , k 1 = R, x2 = u ,

+ wt(t)

x2

[(VTX,

-

- V T X ~ ) I X :sin ] x2 - ( V ~ l R ) x cos 2 x2

-

x2

1

+ u>r(t)

where wl (t) (0, q l ) and w2(t) (0, qz). The measurement equation for data points z(k) is given by Equation (2-58). Comparing this to the state equation in Chapter 2.4.3 in x-y coordinates, the following conclusion can be drawn. Ifthe target exhibits straight-line motion in R-u coordinates, the result is nonlinear state equations with linear measurements. If the target exhibits straight-line motion in x-y coordinates, the result is linear equations with nonlinear measurements. Hence, the EKF approach that was done in Section 7.2.5 can also be applied here.

7.4 SPACECRAFT ATTITUDE ESTIMATION The primary function of a spacecraft attitude control subsystem is the attitudc dctermination and, more generally, the state estimation (attitude of the main body. appendages, and flexible modes) [Amieux and Claudinon, 19841. Onboard gyros together with thc stabilized inertial platform provide the angular velocity of the spacecraft for this application. Spacecraft attitude is measured by sun sensors and star trackers. The attitude state, obtained from the kinematic equations, is augmented by using additional state vector components for biases in each gyro. Consequently, the gyro data are not dcalt with as observations, and the gyro noise shows up as state noise as opposed to observation noise. Rclating the spacecraft's reference axcs with the inertial axcs requires thc three Eulcrian angles pitch, roll, and yaw. Besides having physical intcrprctations, these angles arc the minimum number of paramctcrs rcquircd to rcprescnt the attitude. However. owing to the nonlinear naturc of the dynamic modcl, which contains trigonometric terms, some rotations such as occur in the gimbal-lock situation result in these anglcs bccorning undefined. As a rcsult, the Eulcr angles do not generally lend thcmselvcs well to use as primary attitude variables [Baheti, 19871. Although previous work used the nine-direction cosincs to represent thc attitude, this approach was not widely adoptcd for many reasons. For one thing, thc effcct of round-off, quantization, and truncation errors in the attitude propagation

Sec. 7.5

Advanced Navigation System Design

445

was such that the resulting attitude matrix was not orthogonal. Also, in spite of the fact that various orthogonalization schemes for the attitude matrix were developed, one was still faced with the redundancy of the nine-parameter direction-cosine representation, as well as the computational requirements. In much of the work involving spacecraft attitude estimation, use is made of a four-parameter representation of the attitude in terms of quaternions for the state-variable representation. Some of the advantages of quanternions are that they can be propagated by a simple differential equation, and that the elements of a direction-cosine matrix are quadratic forms in the quaternions, and can therefore be easily computed. Although quaternions lack direct physical interpretation, this is more than made up for in their advantages. Because no singularities are found in the quaternion representation, one does not have to deal with the gimbal-lock situation. The subject of quaternions is presented in Book 1 of the series. Some difficulty, however, is encountered in the Kalman filter with the use of quaternions as the attitude state. This arises from the fact that the four quaternion components are not independent, being instead related by a constraint that the quaternion vector has a unit norm. Lefferts et al. [I9821 examines different schemes for propagating the state vector and covariance matrix and renormalizing the quaternion vector. Applications of quaternion parametrization can be found in the NIMBUS-6 spacecraft, the high-energy astronomy observatory, and the NASA Multimission Modular Spacecraft [Baheti, 19873.

7.5 ADVANCED NAVIGATION SYSTEM DESIGN 7.5.1 Global Positioning System (GPS) Accuracy Improvement This section is based on the excellent paper presented by Kelly [1989]. Positionlocation navigation systems are generally classified as dead reckoning, position fixing, or some combination thereof. With dead reckoning, the position is kept track of beginning with an initially known position by maintaining a continuous account of the vehicle velocity. With position fixing, on the other hand, the current position can always be determined without reference to any previous position. In the simplest case, a position fix is accomplished by recognizing an observable landmark. With GPS navigation these landmarks would be satellites, the orbital positions of which are known at any time. Because of a requirement for landmarks, all position-fix navigation systems are faced with a position error mechanism known as the geometric dilution of position (GDOP). Such a mechanism occurs when the multilateration geometry of the measurement sensors generates lines of position (LOP) which are nearly collinear (rather than orthogonal). When this happens, the sensor errors brought about by receiver noise, quantization noise, propagation effects, clock bias, and so on can be unduly amplified when the sensor measurements are referenced to the navigation coordinates to determine a position fix. In contrast to an unbiased ordinary least mean-squares (LMS) estimator, a

Navigation and Guidance Filtering Design

446

Chap. 7

Ridge estimator is a biased estimator. One of the attributes of the Ridge estimator is that its mean-square error (MSE) in the presence of severe G D O P is much less than the MSE of a conventional LMS estimator such as a Kalman filter. The idea is to take advantage of the fact that the MSE is composed of t w o components, the variance plus the square of the bias. It is possible for Ridge estimators to have a smaller MSE than LMS estimators because Ridge Regression relaxes the unbiasness condition by deliberately mismatching the estimator with the assumed process model. The overall result for variance reduction is a smaller MSE than can be obtained with conventional LMS estimation. The significance of the MSE as an error criterion for a navigation system lies in the fact that it is the location of the aircraft relative to its intended flight path or to other aircraft that is important. Letting P denote the true aircraft position and its estimated position, then it can be inferred from the MSE, - P)'], that if the MSE of is small, then @ is unlikely to be far from p. When considering linear models, biased estimators lack an important feature that characterizes unbiased estimators; namely, calculating the error in the estimate does not require knowledge of the true parameter. It will be shown, however, that this is not an obstacle for the G D O P navigation problem. GPS typically uses four landmarks/satellites to establish an accurate position fix. Although GPS is characterized by a typically small noise error, the amplification of its bias by G D O P can result in substantially reduced performance accuracy. The bias as \\,ell as the variance inflation arising from G D O P can bc lowered by using a GPS position-fix algorithm based on Ridge Regression. For the GPS Kalman filter application, the Ridge Regression algorithm tunes the Kalman filter with predctcrmined Ridge parameters so the filter can achieve a minimum MSE estimatc. One must be careful here not to confuse G D O P with the problem of a mismatched model estimator since, even when these two are matched, the MSE of thc ordinary LMS can be inflatcd by GDOI'. Thc Ridge Regression tcchniquc, howcvcr, sclccts a suitable biased estimator that will reduce the MSE. The primary objective is to obtain a smaller MSE for the estimator, as opposed to minimizing the residuals. Hence, the operation is not simply one of tuning the Kalman filter process noise covariance Q in order to compensate for unmodcled errors.

~[(b

~

6

6

-

Position-fixnavigation. Traditionally, aircraft position fixing is a matter of dctcrmining the intersection of two or more LOP with rcspcct to a known rcfcrencc system. The L01' arc typically specified by & - f(3) = O where R is an observation vector that contains thc mcasurcmcnts and f)(: is a function of thc aircraft position 3. & may bc range measurcmcnts, bcaring mcasurcmcnts. rangc diffcrcnccs, or rangc sums. A particular rangc mcasurcmcnt position-fix navigation system is GI'S, for which & is the rangc mcasurcmcnts, and f(&) = I - &x I with & being the ground rcfercncc transmitter coordinates. Thc position coordinates & R of thc satellitcs, which constitute the GPS landmarks, arc downlinkcd to the uscr on a navigation mcssagc in the form of cphcmcris paramctcrs. Ensurillg a prccision fix rcquircs at lcast four rangc mcasurcmcnts in terms of signal dclay from four satellites. Three of thcsc n~casurcmcntsarc for establishing the threedimensional position, \vhilc thc fourth mcasurcmcnt is for estimating thc uscr's

Sec. 7.5

Advanced Naulgatfon System Design

447

clock-offset error. This removes the necessity for a precision clock in the GPS receiver. Jorgensen [I9841 defines an earth-centered coordinate system with respect to which the four basic range measurements are made. The equations are

where c denotes the speed of light. X, Y , Z, and T denote user position and clock bias, none of which are known. X,, Y,, and Z , are the known coordinates of the ith satellite, while R , is the pseudo-range measurement to the ith satellite. Pseudoranges are the actual range displacements plus the offset due to user time error, as shown in Figure 7-9a. In order to design a linear estimator, Equation (7-55) must uRCRAm'8 TRUCPoSrlIoN LOCATED AT INTERSECTION OF 3 CIRCLES

ILL-CONDITIONED %MATRIX EQUIVALENT TOGDOP \

at(

1

~ 3 a32 1

1

a41 0142

I

POOR GEOMETRY n l ~ QDOP n

1

E l ? (I13 1

a 2 1 a 2 2 a23

mu 1

GOOD QEOMETRY LOW QWP

I

Figure 7-9 (a) Position Location and Time Correction from Intersection ofThree Lines of Position (Two-Dimensional Navigation) (b) GDOP Concept (c) Scatter Plots: 1,000 Samples of Size 32 (Courtesy of R.J. Kelly, 1989)

Nauigation and Guidance Nltering D e i g n

Chap. 7

Bo

A+/ Bl

di

OLS ESTIMATES FOR MODEL (HI wlm NO MULTICOLLINEIRTY

OLS ESTIMATES

$; FOR MODEL (HI

mrn MULTICOWNEIRTV

~JDOPI A

(C)

Figure 7-9

(Continued)

first be linearized. This can bc done by expanding R , in a Taylor scrics about an initial point X o , YO, Zo, and time TO,which is an a priori csti~natcof the aircraft's position and clock-offset error. Such an expansion, carried out to first order, can be written

where 6X, 6Y, 6 Z , and 6T are the corrections to the a priori estimates, and dR81 = cos 0, = a,, is the direction cosine of the LOS from the user to the ith satcllitc as projected along the jth coordinate. The X , Y, and Z coordinates correspond to

ax

Sec. 7.5

Advanced Navigation System Design

j = 1, 2, and 3, respectively. Let

The dynamic process model and the measurcnicnt model are given for an unaided RF carricr receiver (that is, no velocity mcasurcmcnts). Assuming a constant-vclocity aircraft, an eight-state vector is spccificd that includes initial position correction vector 6 3 , receiver clock-offset T , velocity vector I/,and receiver clock-frequency error f . The measurement model with sampling time A T is

where 6 3 = [6Xk 6Yk 6zkIT and I/ = is given by

[.kkYkzhlT. T h e dynamic process

model

(8 x 8 matrix) assumes a constant velocity where the state transition matrix aircraft and g is the process noise and takes into account aircraft deviations from the constant-velocity assumption owing to effects such as wind turbulence, and so on. Also, the covariance of is Q, which gives the expected acceleration deviations. The estimated position of the aircraft at time k is

After every second, the direction cosines aij are again calculated using the new estimates of Xo, Yo, and ZO.From the model composed of Equations (7-58) and (7-59), a GPS Kalman filter can be constructed. Denardo and Loomis [I9881 do not employ any approximation to Equation (7-56), and thus obtain a more accurate model to develop an extended Kalman filter (EKF) for a constant-velocity aircraft. T o gain an understanding of Ridge Regression, one first considers classical regression theory, which begins with the measurement model given by

Navigation and Guidance Nltering Design

450

Chap. 7

It is assumed only that a block of data YT = [Y.I, Y2, . . . , Y,] is given. In this model x i s the measurement vector, H i s the n x p data matrix, and fi is the unknolvn p x 1 parameter vector. GDOP, which is the principal concern, occurs in the event that some of the columns of H are multicollinear. Belsley et al. [I9801 assert that the literature contains no firmly established, precise definition of collinearity. This situation is denoted by the various terms of collinearity, multicollinearity, and illconditioning. In general, p variates are collinear if the vectors that represent them are contained in a subspace whose dimension is less than p, which occurs if any on, of the vectors is a linear combination of the others. A collinearity problem can still arise even if such exact collinearity does not occur, and the latter rarely does in actuality. What is therefore needed is a more encompassing notion of ~ollinearit~ in order to deal with the problem as it affects statistical estimation. ln a less restrictive sense, then, two variates are nearly collinear if the angle between them is small. Belsley et al. [I9801 define near collinearity or near-linear dependencies o f p variates in terms of a conditioning index, which is the ratio of the largest eigenvalue of the data matrix to the other eigenvalues. Eliminating scale dependence is accomplished by normalizing the columns of the data matrix to unit length prior to computing the eigenvalues. The GDOP concept is illustrated in Figure 7-9b for the GPS application. The ordinary LMS (OLS) estimate of fi is QoLs, which is typically derived by minimizing the sum of the squares of the residuals with respect to B. That is, B is varied until SS(e) reaches a minimum, at which point B is set equal to Qors. The OLS estimate for P is given by Q0L.s

(7-62)

= (H~H)-IH=Y

For a given process defined by Equation (7-60). a data vcctor is generated such that Equation (7-62) is the OLS estimator. In order to determine an error for the estimate, the covariance oft-, C o v e ) , must be known. Suppose that E[p] = 0 and that Cove) is known, that is,

where I is the n x 11 idcntity matrix. The constant uc?lcan be estimated by -e Z g ~ l ( n- 1). Equations (7-60) and (7-62) then givc E[QOLSI

=

e

~2

Go

-

(7-64)

where QoL.7 is an unbiased cstimatc. From Equations (7-63) and (7-64), the cov(QoLS)is cov(fioLs) = E\fiOLs - E [ ~ ~ O ~ . ~ ] I E[ [~ ~O~~oSL . ~ I I ~(7-65) = ( H T ~ ) - ' ~ T ~ [ E ' ] H ( ~ T= H )u -? (' H T H ) - '

6,

The OLS cstimate can also bc reprcscntcd with scatter plots over an cnselnble of samples of sizes as shown in Figure 7-9c. Thc crrors are given by e , = P, - PI for i = 1, . . . , n. The ensemble average of thc distance 1 - 1 gives the

boLs

Sec. 7.5

Aduanced NauIgatlon System Design

451

aOLS,

precision of the estimate and is called the MsE(fioLs). T o appreciate the multicollinearity from a quantitative standpoint, the MSE for Q0L.s is calculated in terms of its eigenvalues MSE[~OL.S~ =

~ [ ( - ~b) ~ (~b o L~.-5 B)] ~ = u2TRACE[HTH1-'

(7-66)

where A , denotes the eigenvalues of H ~ H As . H.rH therefore shifts from a unit matrix to one characterized by a condition of high multicollinearity, variance inflation will increase as the smallcst eigenvalue approaches zero. With respect to the error ellipsoid, the axis associated with A, will increase as A;'/"Figure 7-9c). The form of the measurement matrix associated with the LOP matrix for the aircraft position-fix navigation problem is HTH. The situation corresponding to G D O P is in essence the multicollinearity that results from a bad measurement geometry. Put simply, the distance from @oLsto will be large if A.IIAX B A.VIIV. It can happen that even though it is at a very large distance from @, still satisfies the LMS criterion. Stated another wav. , when multicollinearitv occurs. the leastsquares sum. or goodness of fit. given by Equation (7-61) does not necessarily result in a good estimate of e . For this reason, Hoerl and Kennard [I9701 and Theobold [I9741 proposed specifying the mean-square error as the criterion on which regression coefficients should be based as opposed to the SS(g) given by Equation (761). The treatment presented here adopts the conventions ofthe navigation literature, rather than following the unit data matrix formalism of Belsle et al. [1980]. GDOP is defined in Jorgensen [I9841 as G D O P = T R A C E ( H ~ H ) - after the sensor covariance matrix W is set to the identity matrix I. With respect to GPS, Jorgensen [I9841 asserts further that: (1) GDOP is, in effect, the amplification factor of pseudorange measurement errors into user errors due to the effect of satellite geometry; (2) G D O P is not dependent on the particular coordinate system employed; (3) GDOP is the criterion for designing satellite constellations; and (4) G D O P is a means for user selection of the four best satellites from those that are visible. It is evident that a GPS signal processor which reduces G D O P is desirable from an operational standpoint.

e

eoLs,

,

+

Generalized Least Squares ( G L S ) . Assume now that the information about the model given by Equation (7-60) has been generalized to

The generalized least-squares (GLS) estimate is given by

Navigation and Guidance Filtering Design

452

Chap. 7

The covariance of bGLS is found using Equation (7-65) and replacing H by H* to get cov(fiGLS) = [ H T R - ' H I - ' . If the model given by Equations (7-60) and (766) matches the process which generated the data vector 1, then Q C L s is the BLUE of e .

Model with A Priori Information. It is assumed here that the model given by Equations (7-60) and (7-66) has experimental data about such that the average of all previous measurements equals Po. With this quantity, the model is now defined by the following two equations:

e

-

where g WS[O, Qo] defines the mean and covariance of g = @ - @o in the weaker form called wide sense (WS) which makes no distributional assumptions. Qo is a p x p positive definite matrix. Instead of being a fixed unknown parameter, @ now becomes a random variable. Equation (7-69) is rewritten in a condensed partitioned formally as a measurement matrix form using

en

For uncorrelatcd g and p, the Cov(g) is given by

fi,,

Equation (7-62) is again used as the solution along T o solve Equation (7-70) for with Equation (7-71) to obtain as the GLS solution

Matching the tcrtns in Equation (7-72) with thosc in Equation (7-70), and then using Equation (7-71). gives

Recursive estimation and the Kalman filter. In the discussion that follo~vsboth R and Ql, arc p x y and diagonal. Furthcrmore, E[zk. = E[Q, e:] = 0 for k # j and E[pl..I$] = 0. In csscncc, the cstirnate fi(k)+ takes thc place The linear rccursivc dynamic model for a p-variat~ of the a priori information time scrics Y1, , . . . , Y,, is

el,.

where thc first of thcsc cquations is thc dynamic process modcl and Q, is thc state transition rllatrix which expresses the unknown statc as a knowtl lincar transformation of thc prcvious state Bk-,plus a random error vector gk. Both H and Q, are constant y x y matrices. The starting equation for with k = 1 and Q, = I is el = + g1. 111this equation, fro is known, being given by Equation (7-69). All error vcctors arc uncorrelated. In contrast to the case of fixed but unknown

el.

e(,

el.

Sec. 7.5

Advanced ~YavIgattonSystem Deslgn

e,

453

parameters the Kalman filter estimates random parameters ek, and consequently one calculates an error covariance of @ ( k ) + fik) rather than c o v ( 6 ) . The ~ o v [ f i ( k ) +- f i k ] is denoted here by P;, where b(k) is based on all the data [&, Y , , Y2. . . . , Yk] up through k. Also, one has that &k)- = @@(k - I)+ -) - ~k where b(k)- is the predicted estimate of PI, based on observations up to k 1. Then c o v [ b ( k ) - - e k ] = P; and (fi(k)- - p) W S [ O ,PI, ] where PI, = @Pl-1 Q T + Q. Following Equation (7-70), one has as a model

-

+

-

er,

-

Although cross-sectional correlation exists among the p variates, there is no longitudinal correlation along the time axis when the estimator is matched to the model. Employing the same substitution used to obtain Equation (7-73), the estimate of Equation (7-75) is where following Equation (7-73), P; = [PC ' + H T R - ' H I - ' . For k = 1, this equation becomes PT = [Q,T1 H'R-'HI-', which is the covariance of p ( l ) + . Using the matrix inversion lemma [Sorenson, 19801, P; is transformed into the standard form, giving

+

Using Equation (7-77) and algebraically manipulating Equation (7-76), the larter can be rewritten as

From the second term in Equation (7-78) define the gain as shown in Equation (5) of Table 4-1, obtaining as in Equation (7) of Table 4-1.

Ridge Regression. Ridge Regression is a method by which the effects of multicollinearity are offset. This is accomplished by limiting the minimum values that the diagonal terms of the H T H matrix can assume, and such a technique was employed by Riley [I9551 to circumvent numerical difficulties when inverting a square matrix B, = HTH. With Ridge Regression, the diagonal components are limited to a predetermined value by adding a small positive number, K, to each diagonal component, the effect of which is to transform H'H into HTH + KI where I is the unit matrix. Essentially, one attempts to keep in check the variance inflation of ( H r H ) by finding an estimator whose variance is a function of [ H T ~ + ~ 1 l - l However, . there is a trade-off to be made in limiting variance inflation by limiting the large excursions of 1 H ~ I-'H. A bias is introduced since adding K to HrH disturbs the balance between the first-moment components of There are

-'

e.

454

Navigation and Guidance Filtering Design

Chap. 7

several methods for deriving an expression for the Ridge Regression coefficient fi,, having the form of the variance described previously. Lindley and Smith [I9721 and Vinod and Ullah [I9811 made use of a Bayesian formulation, while in Draper and Smith [I9811 is summarized a phoney data interpretation. Other techniques have been used by Marquardt [I9701 and Hoerl and Kennard [1970], the latter using the leastsquares technique with a circular restriction on the regression parameters.

Bayesian Formulation. In the treatment given here, Ridge Regression is derived in terms of a recursive filter. The derivation is motivated by developing in one step the Ridge estimator using prior information which plays a pivotal role in recursive filter applications. Following Lindley and Smith [1972], the Bayesian approach is adopted here in the context of batch processing. T o derive the Ridge estimate, one simply sets Qo = 1 / I ~and fro = 0 in Equation (7-73).

bR

+ 0 (the null vector). Thus, It is observed that as the Ridge parameter K + X , the effect of making biased is to shrink the estimates to zero. Unlike Equation (7-64), for which E[&J = POLS, this is not true for the Ridge estimator of Equation (7-80) because, for finite K, it is generally true that E[bR] f Po.

pR

GDOP bias and variance shrinkage. The material presented hcrc follows Kelly [1990]. In addition to amplifying navigation noise errors, GDOI' inflates bias components. As occurs with conventional LMS signal processing, Ridge Regression is not always able to i d e n t i ! the non-GDOP bias errors and thereby correct them. It is, however, possible to shrink the bias amplification brought about by G D O P as is now shown. To begin with, let the noise error bc composed of two components, a bias tern1 and a random term

-e

= -e~

+ &,

E[e]

=

E[go] = 0 and Cov[g] = E[go following is then true: Linear model:

Y =

LMS estimate:

bOLS=

Expectation of Bias of fiOLS:

& is the unknown =

1 bias vector

(7-81)

u21where I is the n x n identity matrix. The

HD+go+i\B

(H~H)-~H._Y = (H~H)-'H~~"

bOLS:E

n x

(7-82)

fi + (H.H)-'H.A~

+

[ ~ ~ =~ fis +] ( H T H ) - I N T O

BIAS [boLS] = ( H ~ H ) - I H ~ A J

(7-83)

Consequently, G D O P as exhibited by ( H ' H ) ~ ' can amplify the bias given by Equation (7-83) when the angle between the LOP is small. Thc MSE of is

bol..$

= BIAS @ O L S ) ~ B I A S ( ~ O L S+) TRACE [ v A R @ ~ ~ ~ ) ]

Sec. 7.5

Advanced Navlgatlon System Design

455

The first term of Equation (7-84) is a consequence of the unknown bias crror A B , and E q ~ ~ a t i o(7-84) n can be compared with Equation (7-66). The Ridge cstimate of Equation (7-83) is

PR is the Ridge matrix generalization of the Ridge scalar paramcter

K

That is, PR is a p x p diagonal matrix which permits a different Ridge parameter K , to optimally selected for each variate P,. The expectation of is E [ ~ R ]= G.HTHB G , H T M where G , H T H B = B - G,PRB. The bias of is BIAS [ Q R ] = - G , P K B + G , H ~ Q It . can be seen that there is an additional bias term, G . H T h B , which as a result of the s l ~ r i r l k q properties r of G,, that offsets the inflation of A B owing to the effects of GDOP. Using Equation (7-85), the MSE of fiR is

+

kR

kR

In Equation (7-87), the second and fourth terms are a result of the unknown bias error A B . It remains to determine the range of K , values for which

Equations (7-84) and (7-87) are used to transform Equation (7-88) into its canonical coordinates using = P T e and & I = Q T U where PT and Q T are p x p and n x n matrices, respectively. The singular value decomposition (SVD) of H is performed, yielding H X P = Q " x " Z t ' x ~ ( P T ) p where xp X is n x p diagonal, the elements of which are the eigenvalues of H denoted by Moreover, H T H = P Z T X P T where X T Z is p x p diagonal with elements h i . At this point, the transformations are performed, and Equation (7-88) is rewritten in terms of its canonical coordinates

6.

The preceding inequality becomes the result of Hoerl and Kennard [I9701 when Aai = 0. T h e preceding inequality is satisfied for each i if

Navigation and Guidance Filtering Design

456

Chap. 7

Finally, the preceding inequality can be cast into the following form:

The following four possibilities can be identified for

< 0, a, is satisfied 2. a2 > 0, a , 3. a2 < 0, al 4. a2 > 0, a, 1. a 2

~i

> 0: a ~ K i

2a2

< 0.

> 0. In this case K; > 0 does not exist such that Equation (7-90) and K; = 0, the value for the LMS estimate. < 0. In this case Equation (7-90) is satisfied for ~i > 0. < 0. Here, Equation (7-90) is satisfied by large K;. > 0. Here, Equation (7-90) is satisfied by small ~ i .

The preceding restrictions on K, must be met in order for the Ridge estimator to have a smaller MSE than the LMS estimator whenever a bias component A B , as well as a variance component u2 is inflated by multicollinearity. Equation (7-80) upon using Equation (7-90) becomes Writing Equation (7-76) for k = 1 and replacing P; - ' by P R yields the recursive Ridge estimator. Because it is not known in practice, P is replaced by its estimate bOLSin Equation (7-90). Consequently, each K , is stochastic and should be specified by confidence bounds [Oman, 19811. Matched Models versus Minimum MSE Models. When both the model and the estimator tnatch the process that generares the data, the situation is one of having an optimum unbiased estimator. With the Kaln~anfilter optimization technique, the filter is adaptively tuned until its innovations (residuals) become w h i t e . When the Kalman filter is matched to the model, there results the best linear unbiased estimator (BLUE). Still, it may happen that the MSE of the estimates is not the smallest obtainable in the class of both unbiased and biased estimators. The Ridge estimator, which is a biased estimator, is deliberately not matched in order to obtain a smaller MSE in the case of an ill-conditioned model data matrix, as occurs when G D O P is present. The following shows how the initial values of the process covariance matrix Q are selected Ridge parameters K , xvhich minimize the MSE of the estimates such that the MSE of the matched optimun~unbiased Kalman filter is larger than the MSE of the biased Kalman filter in the prescncc of strong ~nulticollinearityor GDOP.

el

Selecfing Q o .

Given what Kalman filters can do, it may at first appear that GDOP effects in the form ofmulticollinearity can be reduced by tuning theestimator process covariance Q,, in Equation (7-73) or Equation (7-85). While such a procedure identifies Qo and may match the model when G D O P does not occur, it will not optimally reduce the effects of GDOI' when this does occur because it is not

Sec. 7.5

Advanced Navigation System Design

457

possible to minimize multicollinearity effects by observing the innovations (see Equation (7-79)). Tuning Q,, is a matter of minimizing the MSE of the estimates because the effects of G D O P only appear in the parameter estimates Herein lies the essence of bias estimation, One deliberately mismatches the estimator (biased) to obtain a smaller MSE. A procedure is suggested in which a Qo is selected such that a minimum MSE estimate is achieved. In the absence of any multicollinearity, the estimator will bc matched to the process covariance Q,, when Q,, is identified by the t ~ r t i i r ! yprocess. However, should the effects of multicollinearity be greater than the cffccts of a mismatched model, the following procedure is employed. The first step is to idcntify QO and determine following which one calculates = In the next step, one uses each ci, in Equation (7-90) to obtain K , for i = 1, 2, 3, . . . , p. Finally, P R is formed using Equation (7-86), after which one sets Qrl = P K .

fi.

A,

~'fi.

Discussion.

The method presented previously for reducing the effects of G D O P in GPS navigation systems is based on incorporating Ridge Regression into the GPS receiver signal processor. In Kelly [1989], using computer simulations to compare MSE performance of an ordinary LMS signal processor with one using Ridge Regression, the theory has been shown to be capable of substantially reducing the bias and variance inflation caused by GDOP. Moreover, in addition to being applicable to GPS, the technique can also be applied to any position-fix NAVAID, for example, DMEIDME, LORAN-C, and JTIDS relative navigation.

7.5.2 Integrated GPSllNS This section is based on the notable paper by Mickelson and Carrico [I9891 of Rockwell International Corporation. Using a low-grade IMU and a single-channel GPS receiver, it is possible to obtain accurate flight control and navigation data. Such an arrangement is meant to take advantage of the long-term position accuracy of the GPS data and the short-term dynamic accuracy of the IMU. A number of integration methods can be utilized, as described in [Buechler and Foss, 1987; Knight, 1987; Nielson et al., 1986; Tazartes and Mark, 19881. All of these can generally be placed into one of two categories: those that make direct use of GPS position data to periodically update an INS Kalman filter estimate of I M U errors, o r those that utilize GPS LOS satellite measurement data to update the INS Kalman filter. While the first method is more easily integrated into existing INS systems, the GPS-filtered data can limit the performance of the INS Kalman filter. Using the second technique, the GPS and INS Kalman filters are able to operate independently on primary measurement data, so that each filter can be optimally tuned. An integrated GPSIINS system has been developed and flight tested that utilizes an LOS integration technique. By using a data coupling algorithm, the GPS receiver is able to perform all the usual GPS data processing functions such as satellite selection, ephemeris computation, and satellite data base management, perform the pseudorange and delta-range satellite measurement, and transfer the raw LOS data to the

Navigation and Guidance Filtering Design

458

Chap. 7

INS without difficult timing constraints. The use of modular hardware in conjunction with the inde~cndentGPS and INS Kalman filters allows the integrated system to provide reversionary modes to GPS- or IMU-alone navigation for added redundancy.

GPSIINS Kalman filter system design. The high dynamics tracking capability of the integrated GPSIINS demonstration system is a result of a tightly coupled design that provides INS velocity aiding to the GPS. As illustrated in Figure 7-10a, the system configuration features a GPS and strapdown IMU, which provides the basic sensor data to the navigation computer. Through a control and display unit, the user is able ro control system modes, initialize INS position and heading, and view selected system output data. A Collins SCR-100 single-channel GPS receiver tracks both the code and carrier phase to develop pseudo-range and deltarange measurements. T o enhance its dynamic tracking capability and jamming immunity, the SCR-100 accepts 10-Hz velocity-aiding data from the INS. Navigation is accomplished by proccssi~igthe GPS measurements and velocity aiding through an eight-state Kalrnan filter. The INS is able to construct the new pseudo-range and delta-range measurements through the use of a resid~raldata block output (presented later). In this \yay, the GI'S rneasurerncnts are provided to the navigation cornputcr without first b e ~ n gprocessed through the GPS Kalman filter. T h e result is that both the GI's Kalman filtcr and the GI'SIINS Kalman filters operate indepcndcntly. Thc tlvo Collins %ST-2 multiscnsors included in the strapdown I M U provrde the linear acceleration and angular rate mcasurcmcnts required to inlplcmcnt the strapdown INS function. The interface from the strapdown I M U to the navigation computer is an RS-332 data link that transmits binary data rcprcsc~~titig incrcniental vclocity and incrcmcntal angle. Also input to the computcr are the temperatures of the indi\~idualniultiscnsors. allo\ving for con~pcnsationof the primary multisensor errors. The strapdo~vnINS and the GL'SIINS Kal~nanfiltcr calculations are perfornicd by the navigation cornputcr. Scvcral ancillary functions also p r o ~ ~ i d cbyd the navigation cornputcr include a C D U interface, data recording, and velocity aiding to the GPS set. Thc INS cornputcr program performs f u ~ ~ c t i o at n sthree different iteration rates. The basic strapdolvn INS algorithms, which maintain the body to level axis transformation rnatris and transfortn the body axis vclocity increments to a locally level coordinate frame, are performed at a ratc of 50 Hz. The basic INS navigation algorithnls are performed a t an iteration ratc of 10 Hz. The Kalnian filtcr computations arc pcrforlncd a t a ratc of 1 Hz. The principal clcrnent in the design of thc integrated GI'SIINS system is the Kalnran filter. A t\venty-statc Kalnialr filtcr was sclected and i~nplcmentedalong -

-

~-

Figure 7-10 (a) Low-Cost Intcgratcd Gl'SIINS Systcrn Configuration (b) GI'SIMU Ir~tegration(c) K a l ~ n a nFilter Starc L'cctor (d) Lo\\-Cost Ir~tcgratcdGI'S'INS Sy5tcn1 Cor~figuration(c) Yurna Flight Tcst Configuration (1) INS CEP vs. Timc Aftcr l
GPS MEAS

Lq.b AIDED GPS RECEIVER

ANTENNA

OUTPUT DATA

INS AIDINO COMPUTER

.

OPS RECEIVER

m

I I

NAVIGATION COMPUTER

I GPS MEASUREMENT RESIDUALS

i

I

DECOUPLING IMU

I

PREDICTED GPS MEAS

8 STATE

UNEQF. SIGHT

LINEQFSIGHT

I

/

I

I

I

I

I

ERROR STATE

X, position error YL position error

Vertical error XL velocity error YL velwity error

Vert. vel. error XL psi YL psi Azimuth error Baro altimeter' Clock bias Clock drift x gyro drift y gyro drift z gro drift x accel. bias y accel, bias z accel. bias x accel. scl fact z accel. scl fact 1

IMU FILTER CORRECTIONS

/

UNITS

(

meters meters meters m/s m/s m/s radians radians radians meters meters m/s rad/s rad/s rad/s m/s/s m/s/s m/s/s

INITIAL COVAR (STD. DEV.) 1000. 1WO. 1WO. 10. 10. 10. 0.005 0.005 0.010 100. 100. 10. 30.e-6 30.e-6 30.e-6 0.010 0.010 0.010 0.005 0.005

Altimeter measurement not included. (C)

A 4I

I

RANDOM

WALK

I

(Unit~)~/sec 0. 0. 0. 0. 0. 0. 1.e-9 1.e-9 1.e-9 10. 0. 0.0002 0.063e-12 0.063e-12 0.063e-12 0.064e-6 0.064e-6 0.064e-6 167.e-12 167.e-12

Navigation and Guidance Filtering Design

Chap. 7

CDU

I r---I SEVS I L----A

L A

t--a I

1

, as

+---,-+',

II

STRAPDOWN IMU

ALL{

AHRS

L---J

I

DADS

L---

J

NOTES: 1. AAMP COMPUTER PERFORMS STRAPDOWN INS CALC AND OPSllNS KALMAN FILTER. 2. DASHED LINES DEPICT ADDITIONAL INTERFACES USED FOR SYSTEM TEST. 3. SEVS SIMULATION AND EVALUATION SYSTEM.

FRPA-3 CIPS ANTENNA

CDU

DlSPLAYllNlT DATA ANTENNA CASSETTE RECORDER

AHRS

(E)

Figure 7-10

(Continued)

with thc strapdown ~~lcchanization algorithm in thc navigation computcr. Simulation rcsults were confirmed by mobilc ground tcsts of the intcgratcd system. Figure 7-10b shows the rclation of thc turcnty-state Kalman filtcr to the overall system. Thc csscntial intcrfacc bctwccn thc GI's navigation function and the navigation computcr is provided by the rcsidual data block. A data-dccoupling algorithm in thc navigation colnputcr uscs this data to gcncrate the IMU measurement residual.

Sec. 7.5

Advanced Navlgatlon System Design INS CEP

TlME

CEP

PREDICTED CEP FOR TTRWECTORY X4

(F)

Figure 7-10

(Continued)

That which defines the twenty-state Kalman filter used to integrate the GPS and INS measurements includes an error state vector, state transition matrix, measurement matrix, initial covariance matrix, system noise matrix, and the measurement variance. Figure 7-10c defines the error state vector, initial covariance matrix (in the form of standard deviations), and the system noise matrix random walk values. The state transition matrix is based on classical inertial error models with the following exceptions: Coriolis coupling terms are not included in the velocity error states, and diurnal coupling terms are not included among the position error and psi error states. Although these terms were omitted to reduce computer loading, it has been shown that the resulting effect on performance for this class of inertial sensors is negligible. Finally, the coupling of the body-referenced inertial sensor errors to psi and velocity error states were correctly modeled. At the start of every 1-sec Kalman filter iteration, the estimated position,

462

Navigation and Guidance Filtering Design

Chap. 7

velocity, and misalignment (psi) errors are incorporated into the Corresponding whole-value variables. Multisensor error states (gyro bias, accelerometer bias, an; scale factor errors) have been propagated into misalignment and velocity errors prior to controlling their effects. Flight test results show that employing such a scheme to control accelerometer errors induces a velocity error in the GPS set proportional to the acceleration error. Altimeter measurement errors are taken into account b,. the inclusioi~of a baroaltimeter bias state in the twenty-state Kalman filter design'. The Kalman filter employs a special feature for incrementing the position covariance terms as a new satellite is included in the four-satellite constellation used for nay. igation. As a consequence of the inclusion of a new satellite, therc is a step chanSC in the steady-state GPS position error owing to a new satellite bias crror and nc\\. G D O P . Increasing the position error covariance terms helps prevent this new error from degrading the inertial crror states.

Data-decoupling algorithm.

An algorithm known as data decoupling Dosh and Yakos [I9861 removes the estimation effects of the GPS Kalman filtcr from the measurement residual data, thus allowing the integrated navigation filtcr to receive "raw" GPS measurement data. Using this decoupling concept, currcnt and future GPS navigation soft~varewill be able to provide the LOS data suitable for host navigation filter implemcntation. Whilc the cxtcrnal ineasurclnent preprocessing can on the one hand be minimized, the flexibility on both sides ofthe interface betwccn the GPS and the integration computer is maximized. Navigational applications have frequently requircd access to dircct LOS GI's rccei\wr data, enabling a central navigation computer to update its o\vn systcm cstimatcs independent of the GPS navigation proccss. Unfortunately, proccssing dircct ra\v data in the cxtcrnal nccessitatcs critical lntcrfacc timing in addition to thc duplication of a significant anlount of rcccivcr soft\\,arc. Morco\,cr, it docs not pcr~nitthc aiding requircd for scqucntial trackins during state-3 (undctcctcd carrier) opcration and highly dynamic nlaneuvcring. The equivalent pcrforn~anccwith a minimum of complcxity can bc obtained by using the proccsscd GPS rcsidual data. In thc past, the GI's position and velocity outputs havc bccn used by integration schemes bctwcerl Gl'S receivers and ccntralizcd navigation computers. In keeping with this approach, the currcnt user equipment rcquircrncnts call for stand-alone navigation capability and thc output of GI's position and velocity. T h e crrors in these cstimatcs are variably tiinc corrclatcd. Conscqucntly, thc fccdback time constant of thc cxtcrnal systcm filtcr using GI's position and vclocity data must bc longcr in ordcr that thcsc crrors arc rclativcly uncorrclatcd, random whitc noise. Thc implemcntation of data dccoupling rcquircs thc following algorithnl In the host vchiclc navigation computer.

(1) Statc 5 (Codc and Carricr Track) I'seudo-rang, p' r ' = I S - P'I p'=p+r-r.'+b-6'

Sec. 7.5

Aduanced Naulgatlon System Design

Delta rangc, Ag' A = r' - I S - ' At*VrI Ap' = + Ar - A r t + At(d - d') ' (2) State 3 (Code Track Only) I'scudo-rang, p' r1=IS-P'I p'=.p+r-r'+b-b'

p

a

+

(3) (from CI'S) -p, mcasuremcnt residual data -r, A- scalar range and change in range S , AS satellite position and delta position b, d clock bias and drift (4) (from - host navigation) P', V' vehicle position and velocity b', d' clock bias and drift also At is the constant value, delta-range dwell time The preceding algorithm can be used in state-3 operation, when the GPS carrier is not detected and delta range is not measured. The pseudo-range residual in GPS state 3 is not generated in the software, being rather a product of the code tracking loop. This is only significant to the host vehicle insofar as delta range cannot be used to update the navigation estimate. In those applications in which GPS clock errors are-not estimated~bythe external filter, the bias and drift adjustment terms can be ignored.

T e s t s and results.

Verification of system operation under simulated dynamic conditions was established through the use of system software tests in addition to laboratory performance tests. Figure 7-10d shows the system configuration used to conduct these tests. The simulated measurements consisting of the GPS RF signals and the incremental angle and velocity counts of an IMU procedure strapdown INS navigation results are in line with the GPS navigation results. Subsequent to performing the necessary software validation studies, numerous GPSIINS filter performance runs were conducted to determine the system response to specific IMU errors. Errors modeled by the Kalman filter as well as unmodeled errors constituted the simulated errors. It was concluded as a result of these performance tests that good navigation results in IMU error state estimates would result from any expected IMU error. A series of flight tests aboard a Rockwell Sabre 50 jet aircraft was undertaken to evaluate the performance of the system under the following conditions: (1) GPS tracking four satellites and updating the INS filter; (2) GPS tracking three satellites, incorporating bar0 altitude data, and updating the INS filter; (3) in-flight calibration and alignment o f the I M U using GPS data; and (4) unaided GPS navigation. The system configuration used in flight tests is shown in Figure 7-10e. It has been demonstrated in flight tests that an accurate navigation solution together with trans-

4&r

Navigation and Guidance Filtering Design

Chap. 7

fer alignment capabilities can be provided by the integration of a low-cost strapdown IMU and a single-channel GPS receiver. The system navigation solution that results from the optimal tuning of the GPSIINS Kalman filter is more accurate than what can be obtained with GPS alone. It is possible to obtain sufficient alignment and calibration accuracy of the IMU through numerous flight trajectories to yield the INS accuracy illustrated in Figure 7-10f. For the majority of tactical missile applications, in which terminal seekers are used, this is sufficient. The stability of INS velocity aiding is such that a single-channel GPS receiver could reliably acquire and track satellite signals on a dynamic host vehicle. Finally, GPS operation can be extended by the utilization of bar0 aiding to periods of thee-satellite coverage with a minimal degradation of accuracy.

Advanced Guidance System Design This chapter presents the analytical development of various guidance approaches used for advanced NGC systems. Where they are applicable, analyses and numerical techniques are also discussed, based on AGNC [1986(a)-(c)], as they relate to NGC system problems. The underlying theme throughout the chapter is to: (1) formulate the problem, (2) propose the model and solution, and (3) compare applicable techniques. Two dominant tasks for designing an advanced guidance system are terminal accuracy and range enhancement as depicted in Figure 8-1, which shows the task flow and structure of the advanced guidance system design. Figure 8-1 also shows the order of presentation of the material contained therein. The aerodynamic efficiency is generally not considered in terminal guidance formulations. Many practical terminal guidance laws are presented in Sections 8.1 to 8.5. Improvement of seeker model (Section 8.2), terminal guidance filter (Sections 8.2 and 8.8), and radome compensation (Sections 8.2 and 8.4) must be considered in the terminal guidance system design. An optimal controller with radome compensation (see Section 8.4) is designed to obtain terminal guidance performance enhancements in the presence of the worst-case radome error environment for high-altitude, high-speed threats. An adaptive guidance is sometimes used in real time for a tactical missile guidance system since the dynamic conditions for an air interception vary drastically during flight and the tracking errors are unknown a priori. The aerodynamic efficiency is a dominant feature in the range-enhancement guidance law design, thus leading to

Advanced Guidance System Design

Chap.

Classical Gu~danceSystem (Chap. 61

+

Advanced Gutdance System

4

+

I

Terminal Accuracy

Range Enhancement

~erminaiGuidances (Table 8-1 8i Sec. 8.1)

Seeker Model improvement tSec. 8.2)

+

I

Terminal Guidance Filter (Chap. 7.2) (Secs. 8.2 & 8.8)

o Mtnimum Fuel Consumption Other Midmvne o Combined Weighted Minimum Guidance Schemes Fuel & M~nimumTime (Sec. 8.71 I I (Sec. 6.6)

+

Radome Ermr OtherTeminal lib^^^^^ and Guidance Laws mpcnsalion (Set. 8.3) (Sec. 8.41

+

Command vs. Seml-Active Homing Guidance (Sec. 8.5) Optimal Autopilot Controller (Book3)

4 Adaptive

I

Guidance (Bopk 4)

+

Improved Time-to-Go Estimator (Scc. 8.10)

i

J

~

+

c],,~~..~, ~ , weather ~ ~

~

Effects on Guided Weapons fSec. 8.111

Midcourse Guidance System Analysts (Sec. 8.8)

4

Pulsed Mota Control (Sec. 8.9)

.

+

Multimcde Guidance Appllcai#ons (Chap. 6.8. Sec. 8.12. Book 4)

I

Implementation Issues (Scc. 8.13)

Figure 8-1

Advanced Guidance System Design T a s k

Flow and S t r u c t u r e

a very complex design problem. Perturbation approaches (Section 8.7) are used to simplify this nonlinear optimal control problem so that the resulting guidance law is nearly optimal and sufficiently simple for midcourse guidance application. Trajectory shaping (Scction 8.6) is designed such that the flight vehicle follows a desired trajectory that opti~nizcscertain performance critcria. The major contributors to terminal accuracy for practical dcsign considcrations of a guided missile include heading error at handovcr, scekcr itracking noise and bias, flight control system (FCS) bandw~dth,and radornc effects. Thcrcforc, the trajectory shaping guidance combined with a rnidcoursc guidancc filtcr (Scction 8.8) must be designed to reduce the hcading error at handovcr. The pulscd motor control technique considered in Section 8.9 cnhanccs thc rockct motor efficiency to increase the missile performance by maximizi~lgthc tcrminal vclocity. Thc timc-to-go information (Scction 8.10) is the dominant factor in dctcrn~iningthc succcss of advanced guidance applications. A nlultimode guidancc systcm (Scction 8.12) nccds to be considered for combining multiple-guidance systcms to overcome jamming, clutter, and adverse weather ef-

Chap. 8

Advanced Gufdance System Design

467

fects. Thcsc cffects arc presentcd in Section 8.11. Finally, the resulting advanced guidance systcrn rlceds to bc implcmentcd (Scction 8.13). The prcccding design tasks (Figure 8-1) arc pcrforlncd in an orderly manner with the outcon~cof each task cvaluated via analysis and simulation. A very important task in the preceding systcrn design is to perform analysis including missile system analysis (Chapters 2 and 3) and guidance stability analysis (Book 1 of the series). 1)csign algorithms presented in this book are not sufficiently advanced for designing adaptivc guidance systems. Hence, actual adaptivc guidance system design is not discussed in this book but is presented in Book 4 of the series. Further details of multimodc guidance and radomc compensation techniques arc also presented in Book 4 of the series.

Example 8-1: Advanced air-to-airmissile (AAM) NGC technology. This example provides an assessment of current AAM guidance and control technology based on Cloutier et al. [1988]. Its purpose is to review the progress that has been made over the years in AAM guidance and control technology, highlighting the more useful concepts and idenrifying those areas in need of research. Over the past fifteen years, much research has been conducted to improve AAM guidance and control performance. The areas spanned by this research include target state estimation, target acceleration modeling, target tracking, target maneuver detection, guidance law development, and bank-to-turn autopilot design. As a result, advances have been made in the areas of adaptive filtering, nonlinear filtering, parameter identification, modern control formulation, adaptive control, dual control, robust adaptive control, and differential game theory. In the problem of a modern tactical AAM that is attempting to intercept maneuverable aircraft, there is the estimation of target motion, the generation of guidance commands to optimally steer the missile toward target intercept, and the control of the coupled, nonlinear, multivariable, uncertain dynamics of the AAM. Each portion of the problem, that is, the estimation portion, the guidance portion, and the control portion, is inherently nonlinear and time-varying and all three combine to form a highly complex integrated system. A simplified block diagram of an advanced AAM system is given in Figure 8-2. Target information obtained from a seeker is processed by a modern estimation filter which assumes a target acceleration model to obtain estimates of relative missile-to-target position, velocity, and acceleration. An extended Kalman filter (EKF) is one such type of estimation filter. A guidance law based on modern control theory uses the state estimates and a t, estimate until intercept to produce a commanded acceleration. The autopilot converts this commanded input into fin commands for the actuators based on airframe aerodynamic characteristics and sensed missile-body angular rates and linear accelerations. The resulting motion produces new missile dynamics which close the three feedback loops. Air-to-air guidance consists of the midcourse, terminal, and endgame phases, as described in Chapter 6. Midcourse guidance begins at the time of launch and ends at seeker acquisition. During this phase, an onboard inertial navigation system ~ r o v i d e sestimates of missile position, velocity, and acceleration.

Advanced Guidance System Design

Chap,

ADVANCED GUIDUICE

AUmslLOl

Figure 8-2 Air-to-Air Missile Block Diagram (From [Cloulier el al., 19881 ujirb prnnissionfro~nA A C C )

Periodic estimates of target position and velocity from the launch aircraft may also be available. After the seeker acquires the target, terminal guidance is initiated. Noisy measurements of LOS angles, range. and range rate are provided by an active seeker. The last second of tcrminal guidance is referred to as the cl~dgame.It is worth treating as a separate guidance phasc since targct maneuvers arc most effective at that time. The reasons for this include the finite missile airframc time response (typically 0.25 to 0.50 sec) as wcll as the target state estimator time response (on the order of 0.50 scc for a typical EKF). The implication of this is that a well-timed target evasive maneuver has a good probability of defeating the integrated NGC system.

8.1 ADVANCED GUIDANCE LAWS The principles of controlling guided missiles are wcll known to control engineers, as discussed in Chaptcr 6. Sincc the basic principles werc extct~sivelycovered in Locke [1955], numerous control techniques have been developed to improve missile performance and to acco~nmodatcenvironmental variations. PNG schemcs are employed by most high-spccd missiles today for intercepting airborne targets during the terminal phasc. In I'NG, a fixed inertial orientation of the LOS between the missile and the targct is maintained. Under steady wind conditions and the assumption of instantaneous nlissilc response, I'NG exhibits optimal performance

Chap. 8

Advanced Guidance System Design

467

fects. These effects arc prescntcd in Section 8.11. Finally, the resulting advanced guidance system nccds to be implcmcnted (Scction 8.13). The prcccding design tasks (Figure 8-1) arc performed in an orderly manner with the outcome of each task evduatcd via analysis and simulation. A vcry important task in the preceding systcm design is to perform analysis including missile systcm analysis (Chapters 2 and 3) and guidance stability analysis (Book 1 of the series). Ilesign algorithnls presented in this book are not sufficicntly advanccd for designing adaptive guidance systems. Hcnce, actual adaptive guidance system dcsign is not discussed in this book but is prcsentcd in Book 4 of the series. F ~ ~ r t h details er of multimodc guidance and radomc compensation techniques arc also prcscnted in Book 4 of the series.

Example 8-1: Advanced air-to-airmissile (AAM) NGC technology. This example provides an assessment of current AAM guidance and control technology based on Cloutier et al. [1988]. Its purpose is to review the progress that has been made over the years in AAM guidance and control technology, highlighting the more useful concepts and identifying those areas in need of research. Over the past fifteen years, much research has been conducted to improve AAM guidance and control performance. The areas spanned by this research include target state estimation, target acceleration modeling, target tracking, target maneuver detection, guidance law development, and bank-to-turn autopilot design. As a result, advances have been made in the areas of adaptive filtering, nonlinear filtering, parameter identification, modern control formulation, adaptive control, dual control, robust adaptive control, and differential game theory. In the problem of a modern tactical AAM that is attempting to intercept maneuverable aircraft, there is the estimation of target motion, the generation of guidance commands to optimally steer the missile toward target intercept, and the control of the coupled, nonlinear, multivariable, uncertain dynamics of the AAM. Each portion of the problem, that is, the estimation portion, the guidance portion, and the control portion, is inherently nonlinear and time-varying and all three combine to form a highly complex integrated system. A simplified block diagram of an advanced AAM system is given-in Figure 8-2. ~ a r ~information e t obtained from a seeker is processed by a modern estimation filter which assumes a target acceleration model to obtain estimates of relative missile-to-target position, velocity, and acceleration. An extended Kalman filter (EKF) is one such type of estimation filter. A guidance law based on modern control theory uses the state estimates and a t, estimate until intercept to produce a commanded acceleration. The autopilot converts this commanded input into fin commands for the actuators based o n airframe aerodynamic characteristics and sensed missile-body angular rates and linear accelerations. The resulting motion produces new missile dynamics which close the three feedback loops. Air-to-air guidance consists of the midcourse, terminal, and endgame phases, as described in Chapter 6. Midcourse guidance begins at the time of launch and ends at seeker acquisition. During this phase, an onboard inertial navigation system provides estimates of missile position, velocity, and acceleration.

Sec. 8.1

Advanced Guldance Laws

4 %

with constant-velocity target. However, this scheme is not effective in the presence of turbulence, target maneuvers, and finite missile response, and often leads to unacceptable miss distances under these conditions. Earlier guidance techniques worked well for targets that are large, slow, or geographically fixed. However, these techniques are no longer effective when faced with targets that are small, fast, and highly maneuverable. Furthermore, jamming, clutter, and weather effects lead to multimodc guidance requirements, as depicted in Figure 8-1. Thus, in formulating a guidance law, these characteristics need to be considered to improve the intcrcept performance. T o intercept targets with these characteristics, a prediction term for the target acceleration needs to bc computed so that the missile can fly toward a predicted intercept position instead of attempting to continuously drive the LOS rate to zero. This scheme is termed advanced guidance law. For the preceding intercept problem, it is not a trivial matter to determine an advanced guidance law. Later in this section, several design approaches are used based on recently developed algorithms that deviate from the conventional PNG approach. They are classified as advanced guidance laws in Figure 8-3 and Table 8-1 which show all the guidance laws presented in this chapter. Classical guidance laws have been presented in Chapter 6. Sections 8.1 to 8.5 present the advanced guidance law design which includes an elegant yet practical approach based on PNG and employs modern control techniques for improvement and also makes full use of the existing guidance law and system simulation experience. Advanced guidance law design techniques are applied to interceptors. In addition, a set of unique and practical approaches is developed for application to the terminal guidance problem.

8.1.1 Optimal Guidance Law Survey Current highly maneuverable fighters pose a challenge for contemporary missiles employing classical guidance techniques to intercept these high-performance targets. Hence, advanced control theory needs to be applied to a missile guidance system Guidance Laws

f

Classical Guidance Laws (Chap. 6)

f

Modern Linear Guidance Laws (Tabe 8-1 & Secs. 8.1-8.5)

+

Modern Nonlinear Guidance Laws (Sees. 8.6 & 8.7)

L4

Pulsed Motor Control (Sec. 8.9)

Multimode Guidance (Chap. 6.8, Sec. 8.12, Book 4) Figure 8-3

Guidance Law Classification

Adaptive Guidance (Book 4)

470

Aduanced Guidance System Design

Chap. 8

TABLE 8-1 ADVANCED TERMINAL GUIDANCE SYSTEM

1st: 1st Order Filter 2nd: 2nd Order Filter 3rd: 3rd Order Filter 4th: 4th Order Filter APN: Augmented Proponional Navigation OGL: Optimal Guidance Law PNG: Proponional Navigation Guidance o: Guidance information needed A Guidance information op~ional *: Combined Seeker-Guidance Filtering in a Complementary LOS Rate Estimator t: Combined Seeker-Guidance Filtering in a Complementary LOS Angle and LOS Rate Estimator

t o improve its performance. Various authors have considered implementing advanced guidancc and control techniques in the analysis o f the terminal guidance phase of tactical nlissiles [Anderson, 1981; Gilbert. 1967; Kriegcr, 1981; Lin, 1983 (;I); Lin and Yuch, 1984(a); I'astrick ct al., 1981; Spcycr and Hull, 1980; Stallard. 1'980. 1983: York and Pastrick, 19771. Both linear and nonlinear d y r l a ~ ~ i nlodcls ic can bc uscd in the design o f the guidat~ccla\v. T l ~ former c is uscd in this section t o grncratc thc guidancc law. Optinla1 control-based techniques have bcctl uscd m u c h Inorc since the 1960s. T h i s is cnforccd by the d c v c l o p t ~ ~ eot f~ t11c t L Q G technique \vhich involves an L Q R with its dual analog, Kalnlan filtcring, and the fccdbackfort11 solution. Since then, alnlost all work in this area has been based o n the linear modcl dynamics with quadratic costs and additive Gaussian noise. l'astrick ct al. I19811 performed a survey of short-range tcrt~linalguidancc laws dc\yclopcd u p t o 1979. T h e scpararion theory for cot~tinuous-time guidance and

Sec. 8.1

Advanced Gufdance Laws

471

navigation assuming linear dynamics and measurcn~cntsis proven in Potter [1964]. The results were extcndcd to discrete-time models. Tung [1965], Templeman [1965], and McAllistcr and Schiring [I9651 uscd impulsive velocity corrections with linearized dynamic models and tcrminal constraints to apply optimal control theory to thrust-vector control in deriving several laws. Sensitivity was considered in the process. Linear missile models were considered [Rang, 1966; Rishcl, 19671 where the quadratic form was minimized to achieve a guidance law using statc feedback. Gaussian noise was incorporated in the model by Rishcl [1967]. A Kalrnan filter was also uscd to generate statc cstirnatcs, and the resulting guidance law was compared with thosc of the pursuit and 1'NG schcn~es.The stochastic naturc of the problem was expanded [Bashein and Neuman, 19681 where a linear fecdback guidance law was obtained that ex~licitlvaccounted for model uncertainties. O n the other hand, a linearized perturbation model about a nominal trajectory was used [Cunningham, 19681 to obtaln a fecdback guidance law. Dickson and Garber [I9691 presented another linear-quadratic optimal control formulation, where missile dynamics were modeled by a second-order nonhomogeneous differential equation. Solutions to the optimal controller for a missile with autopilot lag defined by two discrete-time constants is presented in Williams [1969]. Terminal guidance methods requ~ringprecomputed reference trajectories were compared in Andrus et al. [I9701 ~vhichinvolved expansion ofend constraints into Taylor series about nominal values. Linear optimal control was used [Axelband and Hardy, 1969, 1970(a), (b)] to develop an extension of PNG. However, t, was required for implementation, as with almost all linear schemes. An LQG approach was formulated [Nazaroff, 19761 where very simplified missile dynamics were assumed, but target acceleration and target acceleration rate terms were included. Stockum and Weimer [I9761 generated an LQG guidance law based on the assumption of exponentially correlated target accelerations. The feedback law obtained was analogous to time-varying PNG. A number of guidance and estimation schemes were developed [Fiske, 19771 based on LQG formulations with varying degrees of model complexity. The report gives closed-form solutions for the controllers where the effects of model parameters on the gains can be examined. The LQG formulations of guidance laws for air-to-air missiles are reviewed [Gonzalez, 1979(a)]in a similar fashion. In Wei and Pearson [1978], a planar intercept system including closed-form solutions to target velocity estimation and minimum energy control is presented. Since 1979, additional guidance laws, both deterministic and stochastic, have been proposed. Deterministic formulations were developed in Kim et al. [I9851 and Anderson [1981]. The former derived a guidance law which relied on Pronav correction to a predicted collision course. The latter compared differential game and linear quadratic optimal guidance laws, concluding that differential game formulations are less sensitive to target acceleration estimation errors than are optimal control algorithms. Closed-form solutions of three-dimensional intercept system accounting for target acceleration and the dynamics of the autopilot and airframe are derived in Lin [1983(a)]. The objective of the homing guidance problem is to minimize the terminal

472

Advanced Guidance System Design

Chap. 8

miss distance and control energy loss at the final time, and t o constrain the state values during flight. Therefore, the general optimization problem in which missdistance, states, and control effort are considered becomes that of minimizing the cost function

which is subject to the linear dynamic equation shown in Equation (3) of Table 2-1, where the state is represented by x, the control by u , the white-noise process by w , and the weighting matrices by Sf,Q, and R. The typical state equation is shown by Equation (2-60). This optimal control problem will have the following form of controller:

Simulation is done on the performance obtained from the optimal guidance law formulated as Equation (8-1). It is generally found to have better missdistance than P N G or any classical guidance schemes [Lin, 1983(a); Pastrick et al., 19811. A modified technique is presented [Speyer et al., 19821 in which the implemented guidance gains are based on linear-exponential Gaussian (LEG) theory, which is a generalization of the LQG theory. This technique combined with an adaptive state estimator an exponential cost function is to produces an adaptive guidance law. ~pecificall~, be minimized, subject to the constraint posed by Equation (3) of Table 2-1, and is o f the form p. exp[&/2] where p. is a real number and is a quadratic function defined by Equation (8-la). The form of the optimal controller using LEG is the same as that in Equation (8-lb). In fact, if = 0, LQG and LEG produce identical controllers. When I*. # 0, the gains determined from LEG have an explicit dependence on the error variance (in addition to state estimates) in contrast to the gains determined from LQG. Hence, an improvement in the miss distance (by reducing the tails in miss-distance distribution) over LQG is expected when guidance gains based on the LEG theory are implemented. However, computationally the LEG algorithm requires a tenfold increase in computer storage and a twofold increase in execution time over the LQG algorithm in guidance applications [Speyer and Hull, 19801. A number of guidance laws based on conventional LQG control principles were compared [Riggs and Vergez, 19811 which found improved performance of the LQG algorithms over PNG, especially when improved estimates of time to intercept were available. The accuracy of the estimated tR determines the performance of any realistic optimal control law in a missile application. The I, is calculated by an estimate of the range between the target and missile and the rate of change of this range, with the aid of a radar or other ranging devices. As long as the range

+

Sec. 8.1

Advanced Guldance Laws

473

and range-rate information are accurate, this process is carried out cfficicntly. However, this is not the case in many instances. More likely, the data are either covertly contaminated by noise such as in the case of radar-jamming devices, or inadvertently by the processing electronics. Thus, the estimate of the r, is adversely affected. This leads to the degeneration ofthe optimal control, which in turn jeopardizes the missile performance. A discussion of several aspects of the problem in terms of a realistic missilc application can be found in Pastrick and York [1977]. Several authors also provided computer algorithms for its solution, as well as closed-form results. Fiske [1077] also attempts to esti~natethe !,and discusses the possibility of obtaining this variable by an intensity-ranging technique. An improved t, estimator is derived in Section 8.10. Several other problems associated with the implementation of optimal guidance laws appear to be sensitive to initial conditions [York, 19781. In implementation o f optimal guidance, the system is required to be modeled accurately. It is important to select correct numerical quantities for the elements of the weighting matrices in the selected performance index, that is, Sf, Q, and R in Equation (8la). This task is shown to be difficult [Pastrick et a]., 19811. The following are some related works summarized in Clouticr et al. I19881 and Evcrs et al. [1988]. Deterministic guidance formulation [Kim et al., 19851 is derived that relics on PNG correction to a predicted collision course. For the angleonly mcasurenlerit case, application of the separation principle yields guidancc laws which show poor miss-distance performance. This is due to the fact that PNG, and its L Q C variant, produce a homing collision course in which LOS rates are zeroed by the controller. O n such a course, target position and velocity are unobservable from the angle-only measurements. Anderson [I9811 enhanced the estimation process by including the trace of the relative position covariance matrix in the guidance law performance index. Hull et al. [I9811 and Tseng et al. [I9841 reported on a similar guidance approach. In those studies, the inverse of the estimation error covariance matrix (the Fisher information matrix) was used as an observability weighting term in a minimum control effort formulation. In the design phase, the weighting term is iteratively adjusted until the desired information-enhancing effect is achieved. Later [Hull et al., 1985; Speyer et al., 19841 used the position terms of the Fisher information matrix to form a scalar observability weighting term in the performance index, yielding a less complex version of the maximum information guidance algorithm. Balakrishnan [I9871 and Hull et al. [I9871 have combined these concepts with L Q G theory to produce more refined dual-control algorithms. The former includes the partial derivative of the trace of the estimation error covariance matrix with respect to the control in the performance index. The latter allows individual terms o f the observability grammian to be weighted separately and, as a further modification [Hull et al., 19831, add a factor to enhance estimation during the earlier stages of the trajectory. T h e selection of the cost function, with the state equations viewed as constraints, theoretically determines the performance bounds of the resulting guidance law solution. Because they neglect nonlinearities associated with acceleration limits

Advanced Guidance System Design

47.4

Chap. 8

and delays in the autopilot response, many guidance formulations are suboptimal, As presented in Book 4 of the series, the specification of performance in the guidance algorithm does not necessarily translate into system performance, again because of the system nonlinearities. For longer-range air-to-air intercepts, separate performance requirements should be specified for each of the three distinct guidance phases: midcourse, transition, and terminal. These might be a minimum-time criterion (maximum F-pole) in midcourse, minimum control effort during transition, and minimum miss distance with control effort penalities during terminal. Unfortunately, merely combining these three distinct optimal solutions, each valid in an appropriate time interval, is not likely to produce a truly optimal guidance law b\. any criterion to be optimal in any overall sense. Best performance of such a system requires ad hoc tuning based on Monte Carlo analysis in a nonlinear simulation. Consequently, the specification of performance criteria in the guidance law design ~ h a s eusually relies heavily on engineering judgment and past experience. One alternative, as mentioned in Evers et al. [1988],is to formulate a cost functional for the entire intercept with the switching time between phases as an explicit control.

8.1.2 Analytical Solution of Optimal Filters and Optimal Guidance Law Analytical three-dimensional optimal guidance law. Lin [1983(a)] presents detailed results of an analytical three-dimensional optimal guidance law. LQG theory is used to design the guidance law which generates a pursuer acceleration command 14, in inertial axes. The resulting missile acceleration command in inertial axes is then transformed to the pursuer acceleration command in body axes A,,, A,.,, and A,, . Only the lateral acceleration components A,., and A,, are used to form the guidance acceleration commands in S T T missile and to form the roll rate command and normal and lateral acceleration commands in pitch and yaw channels for BTT missile. The performance index to be minimized, subject to the twelvestate guidance law equation shown in Equation (2-60). is

where R is the control \vcighting matrix in the LQG guidance scheme. If o111y 14 and the final relativc position in j arc penalized, assuming R is a diagonal matrix and Sf is a block diagonal matrix, that is, Sf = block diag{C, -0, 0,~.0}\\.here C is a 3 x 3 weighting matrix on miss distance along the x, y , and ~ d i r e c t i o n sin the LQG guidance scheme, then applying thc techniques developed in Book 3 of the series would yicld the following optirnal closed-loop guidance law 14

= (A/1:)[13,

~ ~ 1 D(tc, 3 , A),

- D(t,?,a)]?,

(,

= timc to go =

1f

-

1

(8-3

where A is the effective navigation ratio and D is the matrix function of scalars 2nd

Sec 8.1

Advanced Ouldance Laws

matrix h defined as

In Equation (8-3), 2 is the state estimate using the estimation techniques presented in Chapter 4. As the FCS lag approaches zero, that is, as ox,o,,and o, in Equation (2-62) approach infinity (this is equivalent to a perfect control loop where the

achieved missile acceleration approaches u ) , u becomes where

As RC-I approaches zero (that is, either R or C-' approaches zero), A approaches 313. If X is assumed to be equal to zero, then This guidance is known as an augmented proportional navigation (APN) with an effective navigation ratio of three. APN has been discussed in several papers where the deviations from the constant-velocity assumptions inherent in the derivation are taken into account. Arbenz [I9701 proposes a scheme that would make the closing velocity heading rate proportional to the LOS rate. In addition, it develops a closedform expression for an APN law. In Siouris [I9741 an estimate of target acceleration is added to the missile acceleration command to yield an APN law.

Example 8-2.

Assuming

the solutions, Equations (8-5) and (8-6), become u(t) = [-3tgl(3b

+ t:)][1,

tgl]x(t)

Advanced auidance System Design

Chap. 8

Combined estimator-controller: Optimal filters and optimal guidance law The APN law for the hori-

Augmented Proportional Navigation ( A P N ) . zontal plane, from Equation (8-5), is

where u, denotes/the guidance command at the y-axis. Comparing Equation (8-12) to the PNG lw Equation (6-17), the zero-effort miss expressed in the brackets of Equation (8-12) contains an additional third term due to target maneuver ATy. Optimal Guidance L a u ~ s . The generalized optimal guidance law for the horizontal plane (see Figure 2-2) from the generalized three-dimensional guidance law Equation (8-3) becomes u, = g ~ y ,+ gzir

where

14,

+

+ g4Amy

denotes the guidance command at the y-axis and the guidance gains are:

Optimal Guidance Law

Simplified Optimal Guidance Law

A

A

gz = -,

gl = y,

g3 = A D(t,, A,)

5

In

tb-'

gl , gz same as optimal guidance law;

Optimal Guidancc Law g, , gz same as optimal guidance law; with 0 ,= )

(8-13a)

r-

g3 = A

D(I,, A,.) 1;

3

g4 = 0

APN

gl, g~same as optimal guidance law;

1'N G

gi ,gz same as optimal guidance law;

where D(s,h) arc defined in Equation (8-4).

g3 = g4 = O

Sec. 8.1

Aduanced Guidance Laws

477

Optimal Guidance Law. Equations (8-13) present the general optimal linear guidance law with A (given by Equation (8-4)) as a function of t,, missile FCS bandwidth w,, and weighting matrix S f . The t, information, target maneuver bandwidth A,, and w, are required for the implementation of this optimal guidance law. The optimal guidance law, in which both the target and pursuer acceleration feedback gains have A, depends on the FCS dynamics-to further improve in the miss CC, distance over the optimal guidance law with($, = 9. Figure 8-4 shows the time variation of the optimal guidance gain A in Equation (8-4) and presents several values of thc penalty weighting RC-' and w, from Equation (2-62). In Figure 8-4a, the effect o f finite missile response is presented. A lower w, specifies a large A value of the optimal gain in order to compensate for the effective FCS and airframe lag in the pursuer acceleration. w, increases from top to bottom. As it increases, for a constant RC-', A approaches a constant gain of 3. At intercept, A diminishes to zero. However, in the case of RC-' + 0, A becomes very large as t, + 0. '\

-

~

Simplijied Optimal Guidance Law. The simplified optimal guidance law is obtained from the optimal guidance law as the target maneuvers with a constant AT,, which corresponds to A, = 0. Optimal Guidance Law with w, = m. The missile in Figure 8-4b is presumed to have an instantaneous response to an acceleration command (that is, neglecting the time lag of the missile dynamics, w, -,w). The value of RC-' decreases from left to right. As the penalty weighting RC-' decreases, A (given by Equation (8-6)) approaches a constant gain of 3. APN Law. As RC-' approaches zero, A approaches a constant gain of 3. The APN law is the special case (A, = 0 and w, = m) of the optimal guidance law. PNG Law. As noted above, A + 3 for RC-' -+ 0 and w, += m. Furthermore, if A, = 0, AT, = AT, = 0 , then the optimal guidance law is reduced to the standard PNG law with A = 3.

wy+m

3

3

Time To Go tg

Figure 8-4

0

Time To Go tg

Optimal Guidance Law Performance Tuned by RC-' and o,

0

478

Advanced Guidance System Design

Chap. 8

Combined estimator.controller based o n the separation princi. ple: optimal guidance system (OGS). In applications in which the missile is attempting to intercept a highly maneuverable target, angle-only measurements can generally be obtained during the homing phase. Classical guidance law onlv requires the LOS angle, where use is made of lower-order guidance filter to sufficiently attenuate noise effects. The implementation of an optimal guidance la\v such as Equations (8-1) and (8-3) requires an estimate of the states of the relative motion from the missile to the targct, as shown in Figure 8-5a. One way of estimating target maneuvers assuming stationary measurement noise is to use a lvcllknown third-order guidance filter described in Chapter 7.2.4 to process LOS data for the terminal guidance configuration design. This scheme contains the optima] Kalman structure of combined prediction and correction in - which - the filter is r g resented by either Equation (7-42) or (7-43). The gains (K, K,, and XI) and ( K ' , K;, Xi) which are related to the spectral densities of the process and measurement noises have been analytically derived in Chapter 7.2.4. An optimal guidance loop, termed optimal guidance system (OGS), that combines the filter Equation (7-42) with Equation (8-13) based on the separation principle is shown in Figurc 8-5b. This OGS providcs an effective approach to coping with system noise and disturbancc. The model in Figure 8-5 is for a single direction in an inertial reference frame. This same model applies for the X, Y, and Z directions; therefore, a Kalman filter is needed for each direction. Using the noisy positions and the pursuer acceleration, three Kalman filters (one for each direction) provide estimates of relative position, relative velocity, and targct acceleration in each direction. These estimates arc then used by the optimal guidance law to obtain a commanded acceleration to be applied t o the input ofthe autopilot. A block diagram of the sccker, estimation, and guidancc process is shown in Figure 8-6 [Francis and Mitchell, 19831. I t is possible and even practical to implenlcnt sophisticated estimation and guidancc functions in digital guidance systems that take advantagc of the prcsencc of onboard microprocessors. Improvement in the missile's homing performance is obtaincd using a combined guidance and estimation scheme. An advanced optimal guidancc law in conjunction with the filter based on the scparation principle is used. Equation (7-47) shows that the scckcr measurement model is gcncrally a nonlinear equation. Using thc dcscribcd intercept navigation and guidancc system estimators in Chaptcr 7.2.5, thc optimal guidancc loop can be reprcscntcd as thc optimal guidancc law (Equation 8-3) with thc EKF tcchniquc (Scction 7.3.5), and as thc optimal guidancc law (Equation 8-3) with a statistical linearization tcchniquc (Scction 7.2.;). Thc advantages of I'NG, such as cffccti\.cncss, robustness, and casc of implementation, togcthcr with its history of succcss as rclatcs to thc dcploymc~~t of missilc systcms employing I'NG, cornbinc to cause I'NG to rcmain attractive. In the scnsc that it produccs zero miss distancc for lcast intcgral square control effort with a zcro-lag guidancc systcrn whcn thcrc arc n o targct mancuvcrs, I'NG has bccll shown in studies to bc an optimal solution of thc lincar guidancc problcm. This scrvcd as thc inlpctus for attempting to dcrivc g u i d a ~ ~ claws c based on n~odcrn control thcory that would bc morc cffcctivc than YNG for nonzcro-lag systcnls ill the prcscnce of targct mancuvcrs. As shown in Figurc 8-7, inasmuch as thcy cstilnate

I

I

I I I I

I

I

Guidance Filter (Eq. 7-42)

"nh

1 1 I

Guidancc Law (Eq. 8-13)

I

Guidance Commands

I I

I I I

I+ ConvcrLs Noisy Range 4 ' LOS Anglc lu Rclalivc Posici~,ns (Eq.7-51))

I I

Anglc Measurement Noise

"c

I I

lncriial to )

-

-

I K;~lman $r 1 Filter ' a

I I

Z

?

'r

I

A,rzl

I

I I

Inertial Accelemliow

I

I "cX I

Rangc Mcasurcmcnl Noisc

I I

MY Refcrcnu: Frame 7tansli)rmation

b Z

I I

Figure 8-6 Seeker and OGS Block Diagram (From [Francis and Mitchell, 19831, 0 1983 IEEE)

I FCS T+Airlram

I I

-%'+ I I I I

&

Pn,pulsion

+"% +"nY

-&&,

Sec. 8.2

Complementaryll(a1man FIltered Proportional Naulgatlon

481

RMS miss distance (11)

Missile acceleration limlt (fUsec2)

Autopilot llme conslanl (sec)

Radome error slope (r)

Figure 8-7 The Optimal Guidance System Improves Performance Except with Radome Error Slopes (Courrrsy of K. Hiroshigr, 1984)

target accelerations and take into account lags in thc missile hardware, OGS for the most part are usually better than PNG laws and hence improve terminal guidance. OGS will always require less acceleration than PNG to intercept a maneuvering target. In order to be fully exploited, however, the application of OGS requires the concurrent optimal estimation of states, X, w,, and t,. This comes at the expense of more precise tracking requirements together with more sophisticated onboard sensors and instrumentation for the missile. The requirements on component tolerances such as radome error, band-limited glint, and other correlated unmodeled errors become more stringent with the use of Kalman filtering than with conventional PNG. Theoretically, the Kalman estimator results in optimal performance, in terms of minimal variance for RMS miss distance, when these component tolerances can be satisfied. This, however, is not always the case inasmuch as component tolerances or measurement errors are subject to a certain amount of degradation. This can be especially true for radome refraction errors. In such cases, missile performance degradation is likely to be brought on more quickly in an OGS than would happen with a conventional PNG system [Yueh, 1983(a)].

8.2 COMPLEMENTARYIKALMAPI FILTERED PROPORTlONAL NAYIaATION: BIASED PNG AND COMPLEMENTARY PNO Peak operation occurs with PNG when the target does not undergo any maneuvers. In the case of target maneuvering, it is advantageous to alter the algorithm that is used to compute the required acceleration Equation (6-16). This section and the sections to follow will investigate a complementary/Kalman filtered P N G configuration for obtaining better performance by means of adding a simple combined

Advanced Guidance System Deslgn

482

Chap. 8

estimator-controller algorithm based on the optimal guidance law derived in Section 8.1.2. Specifically, one exploits those signals already being used for flight control of the pursuer, thereby providing the derived inputs to the guidance algorithm for improving terminal guidance accuracy and reducing homing time and radome-coupling problems to thgse~characteristicof systems employing externally provided inputs. Figure 8-8 defines the complementary/Kalman filtered P N G configuration, in which the measured LOS rate u and derived inputs are passed through a cornplementary/Kalman filter to produce a smoothed LOS rate estimate & for use in the P N G law Equation (6-16) as follows:

where u, denotes the guidance command a t the y-axis computed using the PNG technique, and where is used instead of 6 . The effective navigation ratio A is computed using Equation (8-4). By contrast to most modern guidance algorithms, this complementary/Kalman filtered PNG configuration, which requires n o radical modification to the circuitry of a conventional P N G system, and in some cases, almost no externally provided inputs (such as range), significantly increases the probability of intercepting a maneuvering target or a target in a very noisy environment. The advantage of requiring almost no externally provided inputs cannot be overstressed when one considers that a pursuer will very likely face threats that will attempt to prevent it from acquiring range and range rate information. If this happens to a pursuer employing a modern guidance algorithm that relies on such information, the modern guidance algorithm may perform more poorly than PNG. Because the guidance filtering uses time-varying noise adaptive gains, which are typically functions of !, to compute an anticipated LOS rate estimate, the pursuer is constantly flying toward a predicted intercept position. This is a more effececient means of effecting target intercept than just attempting to zero out the LOS rate at all times. An optimal control law is applied to the design guidance law to enhance miss-distance performance. The optimal control law represents PNG augmented with a missile lateral acceleration term and/or a target lateral acceleration term. Also, the navigation ratio depends on time to go and control penalty function, as well as the pursuer FCS and its airframe dynamics, which take into consideration the firstorder time lag of the pursuer FCS and airframe. In addition to acceptable miss distances that are not only nearly close to the optimal guidance system (OGS) but are significantly improved over those of conventional PNG, this configuration results in system sensitivity to unmodeled errors such as radome aberration that is significantly lowcr than that of an OGS. That is,

Figure 8-8

Complementary/Kalman Filtered PNG

Sec. 8.2

Complementarylffilman Filtered Proportional Nauigatlon

483

this complenicntary/Kalman filtered I'NG system is more robust than the OGS. Consequently, compared to the OGS, increased system coniporicnt tolerances are not accompanied by decreased system pcrformance. The tremendous improvements over conventional I'NG to be gained with this system can be obtained by performing systenl modifications to conventional I'NG systems at minimal cost. This is accompanied by the fact that the number of computations required is less than the number required using the OGS. 8.2.1 Combined Seeker-Guidance Filtering

in a Complementary Filter Improved pcrformance of a seeker-guidance filter can be obtained by implementing a tracking error guidance system seeker complemented by a guidance filter. Considerable interest is found in the LOS rate guidance system in problems concerning the determination of miss distances a t lower frequencies. The seeker transfer function for this systcm which contains no guidance delay becomes uA/u = s.

Combined seeker-guidance filtering in a complementary LOS rate estimator. Following the first-order complementary/Kalman filter presented in Section 4.6.1, consider a first-ordcr gi~idanccfilter complemented by a tracking error guidance system seeker, as shown in Figure 8-Va, the result of which is that the LOS rate estimate produced is independent of the tracking-loop time constant. In this case, the measured LOS rate u is very nearly equal to the geometric u,,,,,. The transfer function for the LOS rate estimator of Figure 8-9a can be written as bicr = Ksl(s + K) which is the same model as that found in Equation (2-16) except the filter gain K n o w replaces rhe seeker bandwidth o,.Hence, the LOS rate esti-

a) Combined Seeker-Guidance Filtering Seeker Guidance Filtering

I

b) Equivalent Model

Figure 8-9 Estimator

'-@b I

Combined Seeker-Guidance Filtering in a Complementary LOS Rate

484

Advanced Guidance System Design

Chap. 8

mator with no seeker dynamic delay is easily implemented. For simplicity of discussion henceforth, Figure 8-9a is simplified to Figure 8-9b. Note that the former should be used for actual implementation. In Figure 8-9a, u = u is the derived LOS acceleration that needs to be reconstructed artificially and u b = A u is the LOS angle measurement bias error. u and abin Figure 8-9a correspond to the derived inputs in the left-hand side of Figure 8-8. The Kalman filter gain K is the cut-off frequency of the filter and it is of the form

where T, is the complementary filter t i ~ n econstant, v is the measurement variance of v ( t ) , and q is the process variance of w ( t ) . For practical implementation, Equations (8-15) and (7-29) are used to obtain K as functions of tR.

Combined seeker-guidance filtering in a complementary LOS angle and LOS rate estimator. Following the second-order complementaryiKalman filtcr prcscnted in Chapter 4.6.2. the problem is formulated (see Figure 8-10a) as x = dish angle

z=ud=x+1,

rrl

=

measured dish angle

=

KE A

u = derived LOS anglc accclcration

x2 = u = LOS angle rate cstimatc

2 = 6 = LOS angle cstimatc (8- 16) whcrc E is the measured angular tracking crror. The transfcr function from U, and u to 6 and b can bc obtaincd by applying Equation (4-41)

6 ( s ) = [u(s) + Kud(s)

ad,

+ KI]lD(s),

(8- 17) + K)&(S)- K(s + Kl)ud(s) + [s(K + KI) + KKI]u(s))ID(s) whcrc D(s) = s' + Ks + K,. Assuming u(s) u,,(s), then Equation (8-17) bccomcs 6 ( 5 ) ---- [ u ( z ) + (Ks + K,)u(s)J/D(s) = u(s), (8- 18) &(s) ( ( s + K)U(s) + rKp(s)]lD(s) = &(s)

&(I)

= {(I

-

Hcncc, thc transfcr functions arc indcpcndcnt ofthc scckcr dynamics and the transfer functions from thc measurcnlcnt LOS anglc a to 6 and u arc 6(s)/a(s) = (Ks

+

K,)lD(.c),

&(s)/a(s) = sK,lD(s)

(8- 19)

Thus, thc filtcr in Figurc 8-10a can bc simplified by a sccond-order guidance filter alone, as shown in Figurc 8-lob. For simplicity of discussion henceforth, Figllrc

Sec. 8.2

ComplementaryIKalman Flltered Proportional Naulgatlon

485

a) Combined Seeker-Guidance Filtering Guidance Filtering

b) Equivalent Model

ii

c) Approximate Model

Figure 8-10

Combined Seeker-Guidance Filtering in a Complementary LOS Angle and LOS Rate Estimator

8-10a is simplified to Figure 8-lob. Note that the former should be used for actual implementation. Applying Equation (4-40), d61dt =

b+

K(u -

b),

dbldt = 6

+ KI(u - 6)

(8-20)

In general, the best estimator poles for the unbiased filter when the measurement noise is white are the complex ones, as in the case of estimator poles sl and sz defined by Equation (4-43) when 5 < 1. However, in the case of correlated target glint and radome error, real estimator poles are most suitable, that is, 5 > 1 in Equation (443). The transfer function from u to u, is u,/u = KIAV,slD(s). The Kalman filter gains K and KI are functions of the signal-to-noise ratio (SIN). It is preferable to choose the K and KI gains from the Kalman tracking filter which depends on the signal and noise covariance calculations. For practical implementation, K and KI are functions o f t , so that the guidance filter has noise-adaptive capability. The K and KI gains are kept very small in the early stage of homing so that the heavy noise does not penetrate the filter. As the pursuer approaches intercept and the signal-tonoise ratio becomes larger, the filters are gradually opened up by increasing the gains almost exponentially as functions oft,. However, the values of K and KI are not permitted to go to infinity at intercept due to seeker-head nonlinearities, radomeloop stability, and glint noise due to maneuvering target. A simplified method using

486

Advanced Guidance System Design

Chap. 8

two first-order filters can be used to replace the second-order filter, as shown in Figure 8-10c where

Comparing Equation (8-21) with Equation (8-19). simple first-cut gains are

Since the LOS angle B is not used in the guidance law, the two cascaded simplified first-order filters presented in Figure 8-10c provide an alternative to estimating the LOS rate. Equation (8-22) can be used as a first-cut gain computation. The first filter of Figure 8-10c functions like the normal LOS rate estimator shown in Figure 8-9a, which is to generate a smooth LOS rate estimate b, while the second filter further smooths the LOS rate estimate A.

Discussion. From previous discussions, the tracking crror guidance system seeker model complemented with a guidance filter (see Figures 8-9 and 8-10) is more interesting than the tracking error guidance system seckcr model alone (see Figure 2-13b) and the LOS rate guidance system seeker model (see Figure 2-14b). The seeker models shown respectively in Figures 2-13, 2-13, 2-15, 8-9, and 810 are available to the designcr as a frequency-response tool for analyzing the stability of the guidance loop. Using these models. trade-offs must be made as they relate to relative stability in the design and evaluatiotl ofloop gains. Thc target acceleration estimate is derived mainly from LOS angular data and, in some case, also from the externally supplicd target maneuvering information supplicd by the launching platform, launching ship, and ground-defense site. Classifications of complementarylKalman filtered PNG. Bascd on Figures 8-9, t\vo possiblc structures of the complcrncntary/Kalman filtered PNG are as follows: 1. Biased PNG (Bl'NG) algorithm (presented in Scction 8 . 2 . 2 ) = ~ ~u not available, that is, 14 = 0. 2. Co~nplcmcntaryI'NG (CI'NG) algorithm (prcscntcd in Scction 8.2.3)-rr = u is availablc, and u h = 0. I'NG algorithm, 13 Conventional PNG algorithm. In the convcntio~~al = 01,= 0 in Figures 8-9 and 8-10 and thc I'NG algorithm is of thc form shown in Figure 2-6. 8.2.2 Biased PNG (BPNG) Algorithm

Technically, it is not an easy mattcr to achic1.c a sufficicntly accuratc cstimation of in Equation (8-14), as thc LOS ratc mcasurcmcnt u contains a bias crror uhbcsidcs the mcasurcmcnt noisc 71 as follows:

Sec. 8.2

Complementarylffilman Filtered Projmrtfonal Naulgatfon

487

where u,,,,. is the true LOS rate, and u h represents cithcr a constant or time-dependent bias which can be constructed from u,, = initial LOS rate (see Figure 6-41)

where 01, nz, and n are bias parameters, Aur = AT"/( VL.tV) is the change o f LOS rate due only to target accclcration, and Ah,,, = A,,,,l(Vc-t,,) is the change of LOS rate due only to pursuer acceleration. Much of the difficulty lies in the fact that thc pursuer must be provided with a n angle-sensing device with second-order astatism, prompting an often repeated suggestion that a bias be added to the PNG. When this is done. the & algorithm is of the form shown in Figure 8-9b where the LOS rate bias error u b has been removed by adding a crh correction after the integration, as shown in Figure 8-lla. Also, a, is the integration of ubof Equation (8-23b). The result of following Case 1 (position bias) in Chapter 4.6.1 is illustrated in Figure 8-llb. Basically, u and u h are operated by the equivalent filter depicted in Figure 8-11c to produce u. As can be seen in Figure 8-llc, the ubinput exactly cancels the bias term in Equation (8-23a); hence, u is purely a low-pass filter output with u,,,,. and measurement noise v as the inputs. This verifies that the formulation is correct. Thus, the idea now is to reconstruct Equation (8-2313) for ubas accurately as possible so as to achieve the relationship shown in Figure 8-llc.

-

-

Optimal BPNG algorithm. Substituting Equation (2-6) into Equation (8-13) results in an alternative optimal guidance law.

u=%

Figure 8-11 gorithm

Biased PNG (BPNG) Al-

Advanced Guidance Sustem Design

488

Chap. fj

where '-' represents the unfiltered optimal guidance law, and the guidance gains g, and g4 are given by Equation (8-13). Equation (8-24) is rewritten as

where u h must be constructed artificially. The performance of the optimal BPNG algorithm is seen to be independent of the FCS bandwidth o.This is because the second term at the right side of the second equation of Equations (8-25) attempts to cancel out the FCS dynamics.

Augmented proportional navigation (APN). From Equation (8-24) the unfiltered APN law is obtained as

Thus, the unfiltered APN law shown in Figure 8-12 defines one of the BPNG methods, that is, PNG + an extra term to account for. the maneuvering target. The bias u b = A T , . / ( ~ V , )is the additional lateral targct acceleration. Determining the bias term u h in Equation (8-26) requires that the target maneuver acceleration AT,, cithcr be known a priori and be constant or be a deterministic function of time, obtainable from the pursuer's tracking systems.

Optimal BPNG algorithm implementation. Equation (8-25) represents an optimal BPNG algorithm (bias from the targct and the pursuer acceleration) where the compensation for the pursuer guidance system bandwidth is prox~idedby the timc-varying navigation gain and an acceleration fecdback path. A block diagram of such a BPNG system loop is obtaincd by combining Equation (8-25) and Figure 8-lib, as illustrated in Figure 8-13 in which AT, is the target acceleration estimate that needs to bc obtaincd from a targct rnancuvcr estimator. One way to achieve the cstimation of AT,.is to apply the second-order complcrnentary/Kalman filter prescnted in Chaptcr 4.6.2 as follows: where b,, is gi\?cn by Equation (8-25) and thc targct maneuvcr model is as usual. The targct accelcration is the component of gra\.ity normal to the LOS in the case of an air-to-ground guidancc la\%,application, as shown in Cottrcll [1971].

Target Maneuver Iislirnalor

k y= uc Figure 8-12

Utifiltercd AI'N Guidance Law

Sec. 8.2

ComplementarylKalman Flltered Proportional Navigation

FCS Airframe 8: Propulsion

Figure 8-13

Optimal BPNG Guidance System Loop

Suboptimal BPNG algorithm with pursuer acceleration feedback. The idea of the suboptimal BPNG is to eliminate the target maneuver A T , estimator. Substituting Equation (2-9) into Equation (8-25) yields

A block diagram of such a BPNG system loop is obtained by combining Equations (8-28) and Fig. 8-llb, as illustrated in Figure 8-14 where the biased term due to u is removed by a LOS rate correction input after the integrator.

Suboptimal BPNG algorithm without pursuer acceleration feedback (referred to as suboptimal guidance (SOG)). Generally speaking, practical realization of the optimal guidance law cannot take place without information on t,,, R, V,, and pursuer acceleration. For radar-homing seekers, a Kalman filter may be used to estimate these quantities. In the case of missiles equipped with an IR seeker, however, range and rate are not accurately known, so that it becomes difficult to accurately estimate the variables necessary to utilize the optimal guidance law. It happens that a large number of short-range AAM make use of an IR-homing seeker. Based on the suboptimal BPNG algorithm with pursuer acceleration feed-

r

FCS

1 --2g, A

Airframe & Propulsion

93 g4 +

AVC

t Figure 8-14

Kg,R A"c

- Kg,R AX

g3

AC

t

v'

Suboptimal BPNG System Loop with Pursuer Acceleration Feedback

Advanced Guidance System Design

490

Chap.

back, a suboptimal BPNG algorithm without pursuer acceleration feedback is drrived in this section following Baba et al. [1988]. From Equation (2-60) A,,,, can be written as

Substituting Equation (8-29a) into Equation (8-28) and solving for u, leads to uc = {(w,

+ s)(A - 2g3 + g3t3~)/[1- 83 - 8

4 ) f~ $1) ~ 6

(8-29b)

With the assumption that s3u = 0,Equation (8-29b) gives

+

g3f-g

+ wyg3t.q)s]/[(l - 83

where g 3 = A{(Agt,q + 2)e-'~,!"

+ AytC - ?)/(A: t:).

- g4)wy

+

s])*

u, can be rewritten as

This is the resulting suboptimal BPNG algorithm without pursuer acceleration feedback, as shown in Figure 8-15 with the following gains:

As seen in Figurc 8-4, A can bc optimally approximatcd to 3. Figure 8-16a illustrates valucs of A , , T, and a as functions of !,for typical values of w, and h,. I t can be seen from the figure that these three quantities remain nearly constant in each case for values oft., greater than about 2 seconds but quickly change as tx goes below 1 second in order to cause the final miss distance to approach zero. Antiaircraft missiles, the majority of which cmploy proximity fuses; d o not strictly require a zero miss. In such instances, A l , T, and a can bc approximatcd as constant paramctcrs, thus circumvcnting the difficult task of measuring or estimating r,. The actual implcnlcntation is thcreforc grcatly facilitated. Thc guidance law given by Equation (8-30). with constant valucs of A1, T , and a , is onc of the suboptimal BPNG algorithms without pursuer accclcration fccdback which can be thought of as 1'NG with phase lcad compensation, and rcquircs only the LOS ratc 6 to be implement~d~

A1"e

Figure 8-15

Suboptimal BPNG

A~RW

rirhnl without Pursuer Accclcration Feed-

back

Sec. 8.2

ComplementarylffilmanFiltered Proportional Naulgatlon

0 ~ ' "2 " " "4 '

6

8

10

tgo(set)

+=20, % = 5

2

4

8

6

10

tgo(set)

2

4

6

8

lo

tgo(set)

Figure 8-16 Optimal Guidance Gain hi and Parameters T , a as Functions of Time to Go (A = 3) (Fror,~[Bahn et d l . , 19881, o 1988 A I A A ~

Simplified suboptimal BPNG algorithm with pursuer acceleration feedback. Substituting g3 = A12 into Equations (8-28) yields a suboptimal BPNG using a simplified optimal guidance law:

-

L(,

Avct,,. 2

= -a

AB +A,,,,,U b 2

=

B R u + - A,,2,,-

2 v,

2 v,

U;

Analogous to Figure 8-14, a functional block diagram of a simplified suboptimal BPNG algorithm is shown in Figure 8-17 in which the guidance law operates according to the measured range. The KRl(2Vc)block in Figure 8-17 is the only element which is a function of R. Substituting Equation (7-29) into KRl(2 V,) yields 0.5/C which is not explicitly a function of R . Due to the fact that the Doppler radar can measure the LOS rate and range rate information, this guidance law can be easily implemented.

Advanced Guidance S ~ s t e mDesign

K -

FCS

-

Airframe & Propulsion

S

KRI(2Vc) = 0.51C = constant

Chap. 8

B: function of tg and o

Y

Figure 8-17 Simplified Suboptimal BPNG Guidance System with Pursuer Acceleration Feedback

Simplified suboptimal BPN algorithm without pursuer acceleration feedback. Substituting Equation (8-29a) into Equation (8-32) and solving for u, leads to

With the assumption s% = 0 , Equation (8-33) gives 14, = asI?l(s + k). This is the resulting simplificd suboptimal BPNG algorithm without the pursuer acceleration feedback shown in Figure 8-18, where k = (1 - AB/2)w,, and a = hV,t,w,/2.

8.2.3 Complementary PNG (CPNG) Algorithm Thc complen~cntaryI'NG (CI'NG) algorithm is one of the comp1c~nentar)-IKalman filtcred P N G algorithnls shown in Figure 8-9 when ii is available and u b = 0. . Assuming I?,, = 0, Equation (8-23a) becomes u = I?,,,, u. T o achieve high performance in the & filtering, u = 6 = derived LOS accclcration needs to be constructcd artificially as shown in Figure 8-1Va where ii = 3 6 and & = ( 5 + K6)l(s + K) = 6 . Thus, if thcrc is no mcasurcment noise (,I = 0), and if ii can be constructcd accurately, thcn & = u compcnsatcs for thc crror in due to ii in classical I'NG. This vcrifics that rhc formulation in Figure 8-1Va is correct. The operation of the LOS ratc estimator to providc a proper unbiascd valuc o f L O S ratc estimate I? with I?,, = 0 rcquircs information on the dcrivcd LOS accclcration u.~ r a r n a t i c ilnprovemcnts in terminal homing guidancc pcrforn~anccarc rcalizcd whcn the filter bias cffcct is clirninatcd by thc usc of thc dcrivcd LOS accclcration 6 . The improved

+

0

s+k

Figure 8-18 Simplified suboptimal BI'NC; Algorithm without Pursuer Accrlcratiotl Feedback

Figure 8-19

Cornplcrnentary P N G (CPNG) Algorithm

guidance performance, through removal of the LOS rate bias effect using the CPNG law, is manifested in a number of ways, including reduced miss distance, reduced tendency for the missile to oscillate about a null value of the LOS rate, reduced missile response time, and reduced requirements for terminal homing. Figure 81% can be modified with a neiv input u/K, as shown in Figure 8-19b, which is equivalent to the BPNG algorithm in Equation (8-25.) with u = KUb.

Optimal CPNG algorithm with a complementary LOS rate estimator. Applying the combined seeker-guidance filtering in a complementary LOS rate estimator (see Figure 8-Yb) to the design of an optimal CPNG algorithm results in an optimal CPNG algorithm with a LOS rate estimator, as s h o ~ v nin Figure 8-20a. From Equation (2-8b), u can be constructed artificially as follows:

u

=

1

- [2V,U + AT." R

A

A,,,,,]

(8-34)

where AT" is the target acceleration estimate that needs to be obtained from a target maneuver estimator or uplinkrd from the launch apparatus.

Second-order optimal CPNG algorithm. One way to estimate ATyis to apply the second-order complementaryIKalman filter presented in Chapter 4.6.2 as follows:

6 = ii + K(U - b),

AT" = -hYATy+

Kl(U -

A)

(8-35)

where u is given by Equation (8-34) and the target maneuver model is as usual. Figure 8-20b shows the resulting second-order optimal CPNG algorithm. Referring to the feedback loop switch O in Figure 8-20b, the open switch enables the system to operate as a conventional PNG system, without externally provided range information in the guidance filter and without a u input to facilitate homing.

Advanced Guidance System Design

t

1.

+

1

%y

1 %.elEstimator Maneuver] ' I

a) Optimal CPNG Algorithm with a Complementary LOS Rate Estimator Oplimal Filters

A"? b) Second-Order Optimal CPNG Algorithm

r-- - - 1 Guidance La\\ Eqs. (6-13)

FCS Aimamc S: Propubion

c) Optimal CPNG Algorithm with a Complementary LOS Angle and LOS Rate Eslimalor Figure 8-20

Optimal CPNG Algorithm

Chap. fj

Sec. 8.2

ComplementarylKalman Flltered Proportional Naulgatlon

495

Optimal CPNG with a complementary LOS angle and LOS rate estimator. Another fornlulation occurs when the combined seeker-guidance filtering in a complcmcntary LOS angle and LOS rate estimator (see Figure 8-lob) is used, as shown in Figurc 8-20c.

Third-order optimal CPNG algorithm. Applying the third-order complementarylKaln~anfi!ter presented in Chapter 4.6.3 yields

The third-order optimal CPNG, whose block diagram is the same as that of the general optimal CPNG algorithm with LOS angle and angle rate estimator shown in Figure 8-20c, is an advanced terminal guidance with a Kalman filter structure. The four choices of the guidance law gains are given in Equation (8-13b). When g3 = 112, the guidance law is referred to as the tlrird-order sitnplified optirt~nlCPSG alyoritlrn~.When g, = g, = 0 are used, the configuration is reduced to a PNG with advanced filtering. A choice must be made concerning operating the system as either a second- or third-order scheme. Setting the gain KI to zero eliminates target acceleration terms. Consequently, Equation (8-36c) would not apply, and Equation (8-36b) would not have the term that is the change of LOS rate due only to target acceleration term AuTdefined in Section 8.2.2. The second-order scheme is therefore lacking as relates to prediction, since only the LOS angle u and the LOS rate u are estimated, but suffers nothing as relates to correction. This means that, although the second-order scheme does not take into account a maneuvering target, it is nonetheless suitable at high altitudes, at which targets do little, if any, maneuvering. Furthermore, when the switch O is open, the guidance computer can be reduced to a second-order LOS angle and rate filter with complex pole as in Equation (819). The gains K and KI can thus be practically chosen based on the classical secondorder filter. Finally, the third gain K1 is also chosen, empirically as a function of I,, to be adaptive to the boresight error noise level. It varies inversely proportional to T,. Following Yueh [1983(a)], the choice of varying filter gains K, K I , and KI can be somewhat suboptimal in comparison with the choice of the Kalman gains of the OGS shown in Section 8.1.2. The time-varying gains associated with Kalman filters imply time-varying constants as well. Considering its time-varying gains, one might also refer to the practical third-order system as a type of Kalman system. One difference between the two systems, however, is that the Kalman system employs recursive calculations from onboard and uplinked data to vary its gains, and hence imposes more stringent requirements on component tolerances. The practical third-order system, on the other hand, bases its gains only on the value o f t , , and hence the gains are more easily computed than the Kalman gains, and take advantage of a noise-adaptation feature. Because the data input requirements to the guidance

Advanced Guidance System Design

496

Chap.

computer are simplified for this latter guidance scheme, the gains demand less fine tuning, and the requirements concerning component tolerances become less str,,,, gent. Consequently, as intercept approaches the likelihood of degradation owing unmodeled errors such as radome errors and glint noise is decreased, while the terminal homing accuracy is nearly equal to that normally associated with truly optimal Kalman systems. Thus, the third-order complementary system functions within the limits of and is compatible with the well-understood, practical Secondorder system, and the gains are much sinlpler to compute than Kalman gains. Furthermore, the trade-off between target acceleration and r a d o ~ n ecoupling can he optimized by varying the gain parameters K, K I , and KI adaptively and pract~call~. to the noise input. The guidance filtering gains K, K J , and K1 in Equations (8-36) can also be obtained from the well-known third-order guidance filter given by Equation (743) which is used by the OGS in Section 8.1.2. Substituting Equations (2-5a) 2nd (2-7) into Equation (7-43) yields

Comparing Equations (8-36) and (8-37), K , K J , and KI can be obtained from K;, and Xi as follows:

--

K'.

- -

where ( K , K,, and K1) and K ' , K ; , and K ;can bc obtained from Table 7-1. K and KJ arc kcpt vcry small in the early stage of homing. As the missile approaches intcrccpt, their valucs increasc and then remain constant as the value of !,becomes less than double the homing-loop time constant. In Table 7-1, the estimator gains for scmiactive homing guidance systems arc functions of h,ll,, which arc defined in Equation (7-45), and which can be expanded as

where RO 2nd h . 4 arc constants. The cstilnator gains cat] be dctcrmined from LOS rate. For the tracking error guidancc system scckcr model, thc LOS rate is proportional to the borcsight anglc error and the cstitnator gains can therefore be d ~ terrnincd from thc borcsight crror.

Fourth-order optimal CPNG algorithm with decoupling feature* performance

It has bccn show11 that the PNG algorithm suffcrs a dcgradation in

whcn dcaling with high-altitude, accclcratcd, and maneuverable targets in the pres-

Sec. 8.2

ComplementarylKaiman Filtered Proportional NauIgatlon

497

ence ofenvironmental effects. A three-state vector is generally involved in the thirdorder optimal CPNG schen~efor an antiair interccptor/missile against highly maneuverablc targets. Because of the simplicity of this scheme, it has been found favorable in many applications. However, it only estimates the relative motion between the target and the missile, and cannot provide the decoupling of the target motion from the missile control in order to improve kill probability with preferred flight path angle, performance against maneuvering and possibly jamming targets, and handling of the blind mode in the endgame. A high-order scheme is proposed here that estimates the missilc and target velocity and in addition to the LOS rate 6 and the target acceleration AT,. In order to achieve decoupling of target and pursuer position and rate component, the LOS rate b in the third-order filter Equation (8-36) needs to be further decoupled into the estimated missile LOS rate component u, and the estimated target LOS rate component u T which are represented by " 1 U T = yr/R, u, = ;,/R (8-49)

i,

iT

A

The new fourth-order estimator will estimate 6, bT, b,, and A T y . The target velocity estimate y T and pursuer velocity estimate !, are in turn computed from Equation (8-40), while the LOS rate estimate must be computed from Equations (3-6) to be compatible with thc ne\v guidance filtering loop mechanizations as follows:

5 = 5 , - u,,

v<6 +R

Applying Equations (8-36a) and (8-41), the filter equation for the estimated LOS angle takes the form:

The following can be obtained by differentiating Equation (8-40) and using Equation (7-43):

Similarly, the missile component is expressed as db,,, - -v
R

R

where the missile accelerometer output is denoted by A,, and K, is theoretically set to zero before decoupling of the target and missile information. For practical implementation, Ki is much smaller than K,. The following is the band-limited target acceleration model which is exactly the same as that given in Equation (8-36c):

Advanced Guidance System Design

498

Chap. 8

Differentiating Equation (8-41) and using Equations (8-42b) and (8-42c) yields

If Ki = 0, then this equation is reduced to Equation (8-36b;) and hence verifies the decoupling of the equation of into the equations of&^ and U,,. Thus, the estimated and the target acceleration AT, can be decoupled from the target lateral velocity missile's components. This target prediction capability can be used for enhancing terminal guidance especially for a blind-mode situation in the endgame. The target flight path angle y~ and missile flight path angle y,,, can be derived from $,, and based on Equations (2-7), which, in turn, can be used in directional warhead applications to improve fusing and detonation of an aimed warhead operation to enable improvement in kill ratio. The fourth-order equations are provided by Equations (8-42) where K, K,, K,, and K, are the filter gains to be adaptive to the LOS measurement noise. In blind-mode applications, these gains should be calculated recursively as Kalman gains based on all the past accumulated LOS data that are weighted properly through filter covariance probabilistic measures for the minimum variance estimation [Yueh, 1983(a)]. In the particular case where the LOS angle u measurement is not available, it can be constructed artificially from Equation (214) as u = [ ( j T - )~,,,)lR](l I.) - 1.8 11 where the target position estimate can

iT

i7.

+

+

be approximately given by the Euler formulation as jT = rt

Fourthsorder optimal CPNG algorithm with target acceleration bias estimate. Applying the fourth-order equation in Equation (7-44) to Equations (8-36) results in the following fourth-order optimal C P N G algorithm with targct accclcration bias estimate (jib):

Suboptimal CPNG algorithm. The idea of the suboptimal CPNG is to eliminate the targct mancuvcr AT,. estimator. One way to achieve this is to utilize the suboptimal BI'NG algorithm presented in Section 8.2.2 to derive u close to the u kincrnatic equation [Equation (8-34)] so as to compensate for u h .Thus, the result is close to the suboptimal UI'NG algorithm. Thc smoothed LOS rate estimate can be obtaincd as follows by comparing Equation (8-25) with thc suboptimal BPNG algorithm of Equations (8-28)

*

K (U s+K

g=-

.3 R + Uh) = u + u* = h -.A 2x3 U + - gAV,'

83 +

+ R4

A v,

A",,.

(8-45)

]

(8-46)

hence

)

u = l[a^- v V ~(( ^-2g3)~-1 ( + R

g3

83

A,.,

This equation computes the derived LOS acceleration u input to Figure 8-19, based on A, t,. R, V,, A,, and o,. Equation (8-14) expresses the suboptixnal C P N G algorithm in which the optimal navigation law with conventional L'NG configuration can be obtained if 11 = 6 is used in Figure 8-19 and optimal A is chosen as Equation (8-4). The resulting guidance system is illustrated in Figure 8-21a in which the guidance law operates according to the measured range. The LOS angle rate 6 input,

Arframe & Propulsion

1

+

g4k3

Airframe & Propulsion

Switch

@

Open: P N F with Optimal h (B)

Decreasing FCS

GI: Approsimarz Constant G2: Function of Timc to Go and FCS Ban&%idth

Bandwidth 01, I

0

Time to Go tg

Figure 8-21 (a) Suboptimal CPNG Algorithm with Range Information (b) Simplified Suboptimal CPNG Algorithm with Range Information (c) Guidance Gain Variation for Suboptimal CPNG Guidance System without Range Information

Advanced Guidance Sgstem Design

500

Chap.

which is a measurable quantity, is shown on the left side of the figure. Referring to the feedback-loop switch O in Figure 8-21a, the open switch enables the system to operate as a conventional PNG system without externally provided range information in the guidance- filter and without the u input to aid in homing. In the case of the switch being closed, the LOS angle rate is modified by using the externally provided range information. This represents a suboptimal CPNG algorithm stru, ture.

Simplified suboptimal CPNG algorithm. Applying the simplified suboptimal BPNG algorithm Equations (8-33) instead of the suboptimal BPNG 21gorithm Equation (8-28), Equation (8-46) is simplified as follows.

Analogous to Figure 8-21a, a functional block diagram of a simplified suboptimal CPNG algorithm is shown in Figure 8-21b. This figure can also be obtained bv settingg3 = A12 in the suboptimal CPNG law shown in Figure 8-21a. From Equation (8-47),

The following is obtained by substituting Equation (7-29) into Equation (8-48): where

Gz = CB and G I = 2C

(8-50)

where range R is not explicitly contained in the gains GI and G,. A block diagram of a suboptimal CPNG algorithm without range information is obtained by applying Figure 8-10b with uiK from Equation (8-49). By using this double feedback-loop system, GI and G2 can be selected in advance. Theoretically, GI is approximated by a fixed-constant value depending on the characteristics of the guidance filter, and G2 is a function of pursuer FCS bandwidth w,, and [,. As seen from Equatio~l(850), as w, approaches m, G2 becomes

-

lim Wr

=.

GiB - G I - 3

-

2

This equation analytically presents the maximun~gain for C2. Therefore, the value o f G2 is selected so as not to be larger than GI/?. The largest possible valuc of Gz chosen at the beginning of homing diminishes to zero at !, = 0. The variation of the gain G2in Equation (8-50) versus t, is shown in Figure 8-21c. The value of the G2 gain decrcascs slightly as thc altitude increascs since the FCS bandn'idtll gcncrally decrcascs correspondingly. However, if the FCS bandwidth is designed to bc constant throughout the flight, the valuc of G2 is a function oft,, only. For a fixed C2, the cffcctive navigation ratio is proportional to G I .

Sec. 8.2

ComplementarylKalman filtered Proportional Naulgatlon

501

A conventional seeker model, as shown in Figure 2-13b is used instead of a combined sceker-guidance fitting, as shown in Figure 8-9. With the perfect dynamics of both the gimbal and thc ratc gyro, a seeker model transfer function block diagram with parasitic feedback paths from n~issile-bodyrate and body acceleration can be derived. In the absence of radome slope effects, and for a large stabilization loop gain K,, it can easily be shown that thc LOS rate estimatc and A,,,,(s)lu(s)are dependent o f the tracking-loop tinlc constant. That is, the LOS ratc estimate has a seckcr-dynamic delay. In addition, the LOS rate estimate is coupled from thc hcad dynamics since parasitic feedback loops are also coupled from the hcad dynamics evcn for a large stabilization loop gain. If the LOS ratc guidance system scckcr modcl presented in Figure 2-l4b is uscd for guidancc, then the measured LOS rate is still coupled to the tracking-loop time constant 7, and the seeker-head dynamics enter into the guidance signal. Thus, when the first-order guidance filter complementcd by a tracking error guidance system, as shown in Figure 8-9, is used in the simplified suboptimal C P N G , the results are much better than those of the preccding two classical seeker implementations. 8.2.4 Terminal Guidance System Analysis

Augmented proportional navigation (APN). The relative trajectory y, and the required missilc acceleration r r , duc to a maneuvering target can be obtained in a similar way as in Equations (1) and (2) of Figure 6-41, as follows:

As seen in Equation (8-52a). decreasing A will decrease the relative trajectory. The acceleration required by an APN law to intercept a step-maneuvering target is seen in Equation (8-52b). As time increases, the missile acceleration required to intercept a maneuvering target decreases. The maximum required acceleration using APN from Equation (8-52b) at the initial time is expressed as (u,),,, = 0.5A AT, which shows that only half as much acceleration is required by the missile with APN than missiles employing a P N G law (see Chapter 6.5.2) with A = 3 . The target acceleration is actually not known a priori. However, in the case of an A P N law it must be estimated continuously during flight. Also required for implementation of this guidance law is t,. The results of linear system analysis o f missile acceleration required for threatltarget intercept without disturbance noise are presented in Table 8-2a. Following the development of current advanced fighters, future fighter performance may be improved to achieve an evasive maneuver of approximately 7 to

Advanced Gutdance System Design

502

C b .8

TABLE 8-2 (A) MISSILE ACCELEfthTION FOR TARGET lNTERCEPT WITHOUT DISTURBANCE NOBE: LIPIEAR SYSTEM AP(ALYSIS (B\ MISSILE ACCELERATION PREDICTION FOR FUTURE TnReAT: LINEAR SYSTEM APlALYSIS

-

a) Missile Acceleration for Target Intercept Without Disturbance Noise: Linear System Analysis

tow Altitude: Target Maneuver (g) 7 7.5

High Altitude: Altitude(km) 15 18 18

Target Maneuver (g) 1.7 0.9

Accel. (g) Required with A = 3 PNG APN 23 10.5 22.5 11.3

b l . 62)Required with A = 4 APN 14 15

Accel. (g) Required with A = 3 PNG APN

Accel. (g) Required with A = 4

5.2

2.7 4.5

1.5

APN

2.6 1.4 2.3

3.4 1.8 3 .O

b) Missile Acceleration Prediction for Future Threat:

Linear System Analysis

Altitude JAW High

Target Maneuver (g) 7-9 1.5-2.2

Accel. (sl .",Rewired with n = 3 PN G APN 21-27* 10.5-13.5* 4.5-6.6* 2.3-3.3*

-1.

IE') Renuired w;% A ='4

APN 14-18* 3-4.4*

A: Navigation Ratio PNG: Proportional Navigation APN: Augmented Proportional Navigation *Other disturbances plus system dynamics wi\lrequirt even more acceleration capability of the missile

9 g. Table 8-2b prcscnts missile linear system analysis requircmcnts for futurc target intercept without disrurbancc i~oise.The preceding results show that the accclcration requjrcmrnts of APN arc much lcss than that of I'NC.

Guidance law simulation De~errriitiisticSi~t~rrlorian: The miss-disrance (lateral position error) responses to a stcp-target maneuvcr using PNG and APN arc sensitive to the FCS bandwidth. As the FCS bandwidth decreases, the pcrformancc dcgradcs. Miss-distance respollscs to a step-target rnaneuvcr using the third-order simplified optimal CPNG and thirdordcr optimal CPNG arc jnscnsitive to the FCS bandwidth. This is because the feedback loop of thc missile accclcrarion in the guidar-rcelaw is attempting to ca~lcel out the FCS dynamics when thc rcal FCS bandwidrh is known.

Sec. 8.2

ComplementarylKalman Filtered Proportional Navigation

503

I Normal Value 1

0 0

I

I

0.1

Figure 8-22

I

1 .o

1.0 A,,: Target Acceleration Bandwidth (set-l)

Guidance Law Performance vs. Target Maneuver Bandwidth

Stoclrnstic Sirnulation: The total RMS miss distances from the noise and random target maneuver are presented in Figures 8-22, 8-23, and 8-24. Four guidance laws are compared in Figure 8-22 over a range of values for the target maneuver bandwidth. The guidance filter gains are designed optimally to yield the lowest value of RMS miss distance. The major improvement in miss distance over conventional PNG is achieved through the use of the APN law which includes a target acceleration term for target maneuvers. Further improvement due to the third-order simplified optimal C P N G is achieved by accounting for missile FCS dynamics. Furthermore, the third-order optimal CPNG law provides an even lower miss distance than that

Optimal CPNG

01 0

4

I

0.1

Figure 8-23

0.5 r,: Autopilot Time Constant (Sec)

1 .o

Guidance Law Performance vs. AIP Time Constant

Advanced Guidance System Design

Chap. 8

\

PNG

.-%

Third-Order Simplified

u

-.-8 Ti

5

z

Optimal CPNG 0.

0

I

1

I

3

I

3

I

4

Missile Maneuver Capability Target Maneuver Level

Figure 8-24

Guidance Law Periormance vs. Missile Maneuver Capability

of the third-order simplified optimal CPNG at a high target acceleration bandwidth because it considers the target bandwidth. The sensitivity of the missile FCS time constant T, (inverse to the FCS bandwidth o,)on miss distance for all guidance laws is drawn in Figure 8-23. Both the third-order simplified optimal CPNG and third-order optimal C P N G law improve the missile perfornlance due to 7, over the PNG and APN laws. CPNG guidance laws are not only superior to PNG and APN in the case of large T,., but also less sensitive to T~ variation. The miss distances for all algorithms decrease as 7,.decreases. However, all of thcm become sensitive to tracking noises as T,,decreases. Thus, this stochastic simulation shows similar results as those in the preceding deterministic simulation of step-target maneuver. In the previous discussions, the designs of C P N G guidance laws are matched to the true model representation of missile dynamics, except for the neglected nonlinearities in the model. A nonlinearity due to missile acceleration limit is considered in Figure 8-24. This figure shows guidance performance as a function of the ratio of missile maneuver capability to target maneuver level. APN, third-order simplified optimal CI'NG, and third-ordcr optinla1 C P N G law offer significant inlprovemcnts over I'NG when thc ratio of the n~issilc acccleration capability to t l ~ etarget acceleration level is low. That is, they can achic\.c lower miss distances and rcquirc lower missile ma~lcuvcringcapability. The rcasoll for the slight improvement in miss distance of the third-order simplified optimal C P N G over that of the Al'N is due to the fact that the fornlcr includes the missile FCS dynamics. The miss-distance impro\~cmentby thc third-order optimal CPNG is slightly morc than that by thc third-order simplified optinla1 CI'NG in lower ratios because target dynamics are included in the third-order optimal CPNG. It can be shown that radonlc errors influcnce systcnl pcrfornlancc and thc OGS is

Sec. 8.2

ComplementarylKalman Flltered Proportional Naulgatlon

505

sensitive to the negative radomc slope. In this respect, the advanced guidance laws presented in this section are superior to the OGS in cases of severe negative radomc slope. The guidance computer of a conventional P N G systcm docs not contain prediction terms, and so 1'NG is not very effective against maneuvering targcts. The third-order optimal CPNG, conversely, provides improvcn~cntsin miss-distancc performance especially against maneuvering targets. Furthermore. it has almost all the advantages associated with convcntional I'NG with rcspcct to radome errors. C o ~ n p a r c dwith OGS, it has better radomc error tolerance and represents a trade-off in terms of ultimate miss-distance accuracy. That is, the requirctnent concerning this q u ~ n t i t yis lower than that of OGS in order to arrive at a systcm that is less sensitive to unmodcled crrors that can result in an unacceptable miss distance at the closest point of approach.

Comparison of suboptimal guidance (SOG) and PNG. Following Baba et al. (19881, simulation studies involving a linear homing system and a realistic nonlinear homing system are employed to compare the performance of SOG (see Section 8.2.2) against that of the PNG.

Performance comparison in linear homing systems. Using the linearized homing system depicted in Figure 8-23, SOG and PNG are conlparcd for performance as measured by the miss distances and the required missile lateral accelerations. Corresponding to the case in which o,.= 20 and A,. = 3 in Figure 816, the S O G parameters are given the following values: A, = 3, T = 0.26, and a = 0.215. The autopilot parameters, the time constant of the seeker dynamics, and the closing velocity are given as follows: a,, = 747336.7, a , = 62507.3, a , = 1443.72, '73 = 58.9, bo = 746214.8, 6, = 143.73, br = 63.3, T H = 0.1, and I .; = 896 (mls). In computing the rniss distnnce ( M D ) and the rnissile lnrrral ncceleratiot~d ~ r eto n launch error, it is assumed that there is no target acceleration, no noise, and that G.V(s) = 1. The results, which have been normalized, are displayed in Figures 8-25b and 823c. T h e first of these figures show that to achieve a negligible miss distance with SOG, there must be at least a 1.2-sec intercept. PNG, on the other hand, requires approximately 1.8 sec for A = 3 and 2 sec for 11 = 4. Looking at the maximum accelerations in Figure 8-23c, it can be seen that this quantity is larger for S O G than for P N G and that SOG possesses a faster response. By virtue of its smaller intercept time and faster response, SOG in this case permits a closer attack. Under the assumptions of no launch error, no noise, and C\-(s) = 1, the rniss distarlce and the missile lateral acceleration dire to n step-target acceleration are computed. The results, shown in Figures 8-25d and 8-25e, show that SOG results in negligible miss for intercept times as low as nearly one ha16 those of PNG. Moreover, the maximum miss produced with S O G is on the order of 40 percent of that with P N G of A = 4. Figure 8-25e, however, shows that the differences between the accelerations produced by SOG and P N G are small. These results indicate that SOG is well suited for use against targets with large transverse accelerations. T o compute miss distance due to white noise, the Monte Carlo simulation technique is employed with 100

Advanced Guidance System Design

Chap. 8

OUIDAfKE LAW

"1

AUTOPILOT

4sz+b,s*b, la,sJ+a,sz.a,s.+

E

10

1

fit (radlsec) (C)

Figure 8-25 (a) Block Diagram for the Linear Homing System (b) Miss Due to a Launch Error (c) Missile Acceleration Due to a Launch Error (d) Miss Due to a Target Acceleration (e) Missile Acceleration Due to a Target Acceleration (0 Miss Due to Noise (No Noise Filter) (g) Miss Due to Noise as a Function of the Cutoff Frequency of the Noise Filter (h) Miss as a Function of Cutoff Frequency of the Noise Filter (Suboptimal Guidance Law) (i) Miss as a Function of CutoffFrequency ofthe Noise Filter (Proportional Navigation) (j) Miss Due to Target's 9-g Maneuver and Noise as a Function of Spectral Density of Noise (From [Baba rr a l . , 19881, O 1988 A I A A )

loo

Sec. 8.2

CompfemenkrylKalman Filtered Proportional Naufgatlon

507

MISS DUE TO

m

a 10 R c (radkec)

4,

100

(rad2/rad/s) U)

Figure 8-25

(continued)

runs and under the assumption of zero launch error and zero target lateral acceleration. Also, a noise filter is not considered. Finally, to compensate for dynamic lags of both a seeker head and an autopilot, the P N G A was set to 4 rather than 3. The results are plotted in Figure 8-25f as a function of spectral density. As a consequence of using linear equations, the RMS miss distance is proportional to noise amplitude and thus to @N raised to the one-half power. Figure 8-25f shows that the RMS miss distance produced with SOG is about twice that with PNG. The high-frequency gain of the SOG system becomes larger than that of the P N G system, owing to the phase lead compensation represented in Equation (8-30). Consequently, the SOG system uses a second-order Butterworth filter having a transfer function presented in Figure 8-25a. Effects of the noise filter are investigated using a value of RMS miss due to noise of (PN = lo-' rad21radlsec. Figure 8-25g shows the results, which are plotted as a function of cutoff frequency a,. The RMS miss

508

Advanced Guidance System Design

Chap. 8

due to either a target 9-g acceleration or a measurement noise is plotted in Figures 8-25h and 8-251 as a function of the cutoff frequency of the noise filter. Minimizing the total miss distance (that is, miss due to noise plus miss due to target maneuver) is the criterion for selecting the optimum cutoff frequency. Figure 8-25h shows that the optimum cutoff frequency of SOG is on the order of 3 radlsec for tf = 4 set and 35 radlsec for tf = 1.2 sec. Similarly. Figure 8-25i shows that the optimum cutoff frequency of P N G is about 6 radlsec for tf = 4 sec and 40 radfsec for tf = 2 sec. From these results, a cutoff frequency of 35 radlsec is selected. Figure 8-2jj shows the miss due to 9-g target lateral acceleration and angular noise of differing < rad2/rad/sec, SOG produces smaller miss than spectral densities. For rad2/rad/sec. PNG, the opposite being true for CJN > 2 x

Application to the infrared homing short-range AAM. Following Baba et al. [1988], SOG and PNG are applied to the model of a fictitious IR homing short-range AAM. Currently existing missiles such as the AIM-9 are used to estimate aerodynamic coefficients and geometrical and inertial data, the former of which are functions of the Mach number. The simulation consists of a mathematical model to describe the equations of motion of a missilc, the aerodynamic coefficients, and nonlinear mathematical models of major missile subsystems, including the seeker, noise filter, autopilot, and propulsion systems. The overall system is the one shown in Figure 8-25a with the same autopilot parameters as in the previous linear study. Here, again, the altitude is 2,000 m , velocity is Mach 2, the time elapsed after launch is 2.8 sec, and the angular velocity of the rolleron's rotor is 2,400 radisec. The cutoff frequency is selected to be 33 radisec, and the target model is a point mass having 3 degrees of frcedom. The initial launch conditions consist of: (1) both missile and target having vclocitics of .9 Mach; (2) both nlissilc and target at altitude 5.000 m; and (3) an initial off-borcsight angle of 0 dcg. Thc simulation rcsults for the AAM intercepting a targct which is turtling at 7 g in the horizontal plane are shown in Figure 8-26a. The simulations, consisting of a Monte Carlo technique with 100 runs, result in the ~ 0 d : f o rthc case of no noise, achieving smaller misses than those with PNG. If noisc is added, however, the broadside attack and tail-chase scenarios have S O G producing largcr misses than PNG. 111thcse cases, noise contributes most significantly to thc miss. I'NG produces a much largcr miss than SOG for the headon attack. Again, although SOG is shown by these rcsults to be more sensitive to noisc than I'NG, it is also nlorc cffcctive against a highly mancuvcrable target. For a fighter attempting to cvadc a missilc, one of thc most cffcctivc maneuvers it call undergo is the high-g barrcl roll (HGB). For this rcason, simulations arc pcrformcd for the AAM against a targct ycrforming a 7-g barrcl roll. The flight pattern of a targct undergoing a HGB nlancuvcr is s h o ~ v nin Figure 8-2Ob, together with the initial position of thc missilc. Thc cngagcnlcnt gcotnetry is assumed to be hcad on. With an initial rclatiyc range of 5,000 m , thc target flies straight until thc point at which it begins the HGU. Miss rcsults vcrsus barrcl-roll rate arc sho\vn in Figure 8-26c for a targct that initintcs thc HGB whcn it is at a distancc of 3,000 m from thc AAM. Assuming 3 nl to bc

Sec. 8.2

ComplementaryIKalman filtered Proportional Navlgatlon

loo0

180 (Head-on)

0 . 13

0. 88

15. 18

509

3 . 27

3. LO

18. 19

MISSILE

TIME-TO-GO ( s e c )

-

6-

- 5E

W

U

2

L-

m 2 E

3-

2 2s'Oo'

/ LO

8 2u Q: ?i 3 0 - -

u SOG

o

8

7

-'20-

1

We

2 (rad/sec)

(C)

3

0

1

2

3

4

5

TIME-TO-GO (sec) (E)

Figure 8-26 (a) Miss and Acceleration of the Missile Intercepting a 7-g ConstantTurn Target (b) HGB Flight Pattern and Nomenclature for Coordinate Axes (c) Miss vs. Barrel-Roll Rate (d) Miss vs. Time to Go (COB = 3 radlsec) (e) Load Factor vs. Time to Go (WB = 3 radlsec) (From [Baba et a l . , 19881, 0 1988 AIAA)

6

510

Advanced Guidance System Design

Chap. 8

the effective miss for target kill, then the PNG-guided missile fails to intercept the target if the latter performs the HGB with a barrel-roll rate o f greater than 1.2 rad/ sec. ';The SOG-guided missile, on the other hand, intercepts the target for all barrelroll kates shown in the figure. Miss versus 1, when the target initiates a barrel roll with a rate of 3 radlsec is illustrated in Figure 8-26d. Gravity acts to cause the miss to change in a sinusoidal fashion. Modifying the autopilot to compensate for gravity eliminates this sinusoidal variation in miss, producing instead the broken lines shown in Figure 8-26d. In this case the miss does not depend on the maneuver initiation time. In each case the maximum load factor is shown in Figure 8-26e. Here it can be seen that the load factor of the SOG-guided missile is smaller than that of PNG.

8.3 OTHER TERMINAL GUIDANCE LAWS Some of the guidance laws that have not been covered so far have their applications in special areas such as optimization theory or, particularly, differential games for nonlinear dynamic systems. Some involve very simple and straightfonvard implementations of ad hoc controllers. H o et al. [I9651 present one of the earlier papers on the application of differential game theory to the short-range missile problem. In this paper, variational techniques were used to solve pursuit-evasion problems and an example was provided that proves that under certain conditions PNG was optimal. Pfeiffer [I9671 constructed near-optin~umguidance laws using a method of successive approximations similar to quasi-linearization. Fivc polynomial-type approximations for nominal trajectory and sensitivity matrix cor~lputationswere studied [Gemin, 19691, with particularly useful results for prcprogrammed perturbation navigation schemes. An iterative technique for the near-optimum solution of a nonlinear differential game is developed [Anderson, 19741 based on successivc linearizations of a two-point boundary value problem. This scheme was later applied [Poulter and Anderson, 19761 to an AAM guidance problern. Vastly improved S~IIIUlation results were obtained over those of a PNG law; however, these came at a cost of increased computational expense. Still later, Anderson [I9811 compared differential game and L Q optimal guidance laws. It was concluded that differential game formulations are not as sensitive to errors in targct accclcration estimation as optimal control algorithms. Most guidancc schemes making use of high-fidelity models demand excessive computational effort, drawing the attention of many invcstigators. In Liu and Han 119751 continued fractions wcrc proposcd to rcducc the modcl of a high-order nonlinear rocket. In Sridhar and Gupta [I9791 singular perturbation methods wcrc applied to an AAM in a n attcmpt to rcducc the computational cffort required. Fast mode was assumcd for thc rotational dynamics of the missile whilc slow rnodcs wcrc assumcd for the position and vclocity translation equations. Simulation results showed improvcmcnt over PNG. As pointcd out [Clouticr ct a]., 1988; Evcrs ct al., 19881, optimal guidancc of pulse motor missilc is just one arca still in need of rescarch, having bccn addressed recently by Chcng ct al. [I9871 and Katzir et al. [1988]. O n the other hand, the

Sec. 8.3

Other Terminal Guidance Laws

511

fundatnental theoretical problems associated with boost-sustain midcourse guidance appear to have been solved. Other issucs that need to be addressed include algorithm implementation and problenl formulationlsolution. Those issucs associated with algorithm inlplcnlcntation include: software design methodology selection, highorder language run-time characteristics, cross-compiler efficiencies, and hardware throughput/memory limitations [Evers et al., 19881. T h e issue of problem form~~lation/solution involves a choice between closed-form solution to an approximate optim'll formulation (LQG, LQR) or an approximate solution to an exact nonlinear optimal formulation, which entails a nonlinear two-point boundary value problem. Moreover, a numerical solution of the exact nonlinear problem is impractical for implementation reasons. With terminal and endgame guidance, there are usually problems associated with degradation in seeker measurements, resulting in a lack of target information and the need for a stochastic endgame approach. As previously mentioned, homing guidance also reduces the information available from the measurements, which suggests dual-control techniques. Additional work on guidance lawlautopilot interactions, especially with respect to the acceleration limits of the airframe and the autopilot's finite time response, should be pursued. Embedding the autopilot models in the guidance algorithm derivation is another possibly re\varding approach. Several guidance algorithms have been developed which, in some way, attempt to deal with autopilot/airframe performance constraints [Lin and Yueh, 1984(a)-(c); Yueh and Lin, 1984, 1985(a)]. A modified PNG was derived which adjusts the autopilot gains to minimize a penalty function on control effort and time to go. Other works [Aggarwal and Moore, 1984; Caughlin and Bullock, 1984; Lin, 1983(a); Lin and Yueh, 1984(a); Stallard, 19831 derived guidance laws to deal specifically with the problems of BTT control. In Caughlin and Bullock [I9881 a reachable set theory is applied to the design of guidance laws which deal explicitly xvith hard limits on airframe achievable acceleration. The endgame part of the intercept has received only limited attention in the guidance and control literature. Dowdle et al. [1982], Dowdle et al. [1983], and Lin and Shafroth [1983(a)] generalized the LQG regulator and, after appropriate model linearization via pseudomeasurements, estimated the target state with a reinitializing Kalman filter. A generalized likelihood ratio approach was applied to the innovations process as a target maneuver detector. In a more fundamental look at the endgame, Looze et al. [I9871 used CramerRao lower-bound analysis to investigate the quality of target acceleration information available from the seeker measurements. It was found that target acceleration was accurately estimated for the maneuvers considered, but that such estimates were poorly utilized by the modified PNG law. The guidance law was subsequently altered with lead compensation of the roll command to yield improved miss-distance performance. Finally, in a departure from these approaches, Forte and Shinar 119871 formulated a planar air-to-air intercept problem as a mixed-strategy, zero-sum, stochastic differential game. The cost functional of the min-max problem was singleshot-kill-probability. The optimal pursuer's strategy is in the form of a parametric guidance lawlstate estimator which demonstrates increased single-shot-kill-proba-

-

Advanced Guidance System Design

512

Chap. 8

bility against any frequency of target maneuvers when compared with any single strategy guidance algorithm. This approach has recently been expanded to a 3-D encounter [Forte and Shinar, 19881.

Example 8-3: Tenninal guidance law based on a perturbation technique. Optimization problems have always been important in the NGC field. For example, a complete trajectory can be optimized by using the calculus of variations. However, this leads to a two-point boundary value problem requiring iterative methods to arrive at the solution. Another problem in the NGC field is the perturbation technique which has been used to derive control laws for several aircraft and missile trajectory optimization problems. All the system dynamics can be optimized by these techniques by separating the system dynamics into slow and fast states. The solution obtained is then reduced in order, after which it is combined systematically to achieve a near-optimal solution to the full-order problem. Perturbation techniques are used in this example based on Aggarwal and Moore [I9841 to obtain nonlinear feedback control solutions to the terminal guidance problem of a skid-to-turn (STT) or a long-range ramjet-powered bank-to-turn (BTT) missile. T o achieve slow and fast states separation, artificially small perturbed parameters are generally contained in the system dynamics of these problems. In the perturbation technique, Taylor's expansion is used to expand a nonlinear dynamic system solution f(x, E) about the small perturbed parameters as follows:

f (x, E) = f (x, 0)

+ af + a€ €

where the perturbed parameters are represented by E and the state variables by x. O n the right-hand side of Equation (8-53). the first term is referred to as the zeroorder solution and the second term the first-order solution. The perturbation procedure is carried out in a stepwise manner as follows. The first step involves arranging the dynamics according to slow and fast states. In the second step, the zero-order solution in Equation (8-53) is obtained by setting the perturbed parameters t to zero in the state and then solving the rcduced-order problem. The initial or terminal condition of states with E = O serves as the boundary conditions for the zero-order solution in this step. The perturbed parameter is dominant in the first-ordcr solution of Equation (8-53). With the boundary conditions derivcd in the zero-order solution and the conditions that arc not satisfied in the zero-order solution, the last step of the procedure is to obtain the first-order solution using optimality conditions with fast states.

Problem formulation.

The problenl formulation for the beginning of terminal guidance is in a nonrotating coordinate frame aligned with the seeker frame A simple planar representation of the intercept geometry of terminal guidance is found in Figure 2-10 where x, is replaced by z, in this section, and the relative

Sec. 8.3

Other Termlnal Guldance Laws

513

motion has the following dynamic model:

d . - yr = 1 4 ~ "- A,,,, = AT" - A,. cos dl

+

d - i , = AT, - A,,,: = A T , - A , s i n + dr

(8-54)

(8-55)

where the missile acceleration command in the missile and target plane is represented by A,, is the bank angle, and the missile roll-rate command is denoted by $,. Zero-lag acceleration and roll-rate autopilots are assumed for the missile. It is desirable to find an optimal choice for the controls A, and 4, such that the following performance index is minimized:

+

Optimal conditions. The Hamiltonian of Equation (8-39) with respect to Equations (8-34) through (8-58) is as follows: H

=

+ A;,(-A, cos $ + AT?) + A,4, + 1 Xi,(-A, sin + + AT,) + A+$< + - (K,Az + K&:) 2

A,y,

(8-60)

Due to the independence of the Hamiltonian from y, and z,,

A,= 0,

A, = 0

(8-61)

The other three costate variables are as follows:

A.Y ,

=

- A ,,

A;,=

-A

Z 1

A+ = A,

[A;, cos

+ - A;, sin $1

(8-62)

where the boundary conditions are

Mtf) = ~"(tf).

A=(tf) =

If d is assumed to be the terminal miss distance and final bank angle, then y,(tf) = d cos

d

=

5,

dy:(tf) + zt(tf),

(8-63)

6is assumed to be the terminal

z, ( t f ) = d

-

~ ~ ( f f )

sin

5,

+ = tan-'

~"(tf)

rr(tf)

(8-64)

Advanced Guldance System Design

514

Chap. 8

The following are obtained by substituting Equations (8-64) into Equations (8-61) and (8-62): X,(t)

= d cos

-+ = constant,

A,(t) = d sin

= constant

(8-65)

The optimal controller is expressed as

+ A;,

A;, cos

sin

+ - K.A,

=

0

(8-67)

and = - AmlK*

Substituting Equations (8-66) into Equation (8-67), the optimal acceleration command is expressed as

A, =

d

- cos (& K,

(8-69)

+)tn

Using Equations (8-62), (8-66), and (8-69), A+ is obtained as

where A*(ff) = 0.As discussed previously, the optimal problem including Equations (8-54) through (8-59), (8-h4), (8-68), and (8-70), is now represented as a twopoint boundary value problem.

Optimal solution. This two-point boundary value problem can be solved by many methods. A solution for Equations (8-68) and (8-69) must be found by numerical methods because of the nonlinear dynamics. It is necessary to obtain closed-form solutions for the control commands with respect to the current values of the system states, so that practical implementation is possible. Zero-Order Solution. Since the dynamic response of a missile to the rollrate command is much faster than that of the rcst of the system states, the following assumption can be made: $Jr

= 0 for

t

> to

(8-71)

This implies that the roll-rate command here impulses at to and will bring the rolling angle $J to 6 for tirnc t > I,,. Hence Equations (8-57) and (8-70) become A:(()

=

a,

L$,(t)

and Equation (8-69) bccomcs A:' = dr,lK,

=

o

(8-72)

Sec. 8.3

Other Termfnaf Gufdance Laws

515

where the superscript o denotes the zero-order solution. The following is obtained by substituting Equation (8-73) into Equation (8-54) at 4 =

6

Equation (8-74) is first integrated from t to twice and then compared to the boundary condition in Equation (8-64) to obtain

d cos

6=

[3Kd/(3K,+

(1)

+ y r ( t ) t, + A T , t,:/2]

(8-75)

From Equations (8-54), (8-73), and (8-83), the acceleration command in the Y direction is obtained as

A,, = A:'cos$ = (Alt,:)(y,+ y,tg

+ AT,t,:/2), A = 3t,i/(t: + 3K,)

(8-76)

A similar expression can be obtained for the acceleration command in the z direction by substituting Equations (8-73) and (8-75) into Equation (8-55) as follows:

+ 4,t,, + AT,t,;/2) = V A ?+~A?: and + = tan-'

A,, = A: sin 5 = (A/t,;)(z,

(8-77)

From Equations (8-76) and (8-77), A!! (A,,IA,,). The preceding zero-order solution assumed stabilized roll. It is exactly the same as the optimal solution applied using conventional linear theory to the STT missile with zero lag FCS in Equation (8-5) with C = 1, R = K,, and A = 0. First-Order Solution. The missile is assumed nonrolling at t > to in Equations (8-72). However, it is in reality rolled near the initial time to. The transformation T = ( I - to)/K&can be made for analyzing the dynamic behavior near to. The results in the boundary layer equations can then be obtained for the missile bank angle transition at the initiation of terminal guidance. The assumption of Km + 0 leads to the roll angle being changed from = +(to) at T = 0 to $I = at T = m, while maintaining the other states in this time range. Thus

+

6

The Hamiltonian is expressed as

H I = hby,

+ A L i , + At, ( - A ? cos 4 +

AT^) (8-79)

which remains constant for an autonomous system. Hence, H I (T) = H(t,) in Equation (8-60). The following is obtained by comparing Equations (8-60) and (8-79)

Advanced Guidance Sustem Design

Chap. 8

INITIAL CONDITIONS, TERMINAL TlME 4,

-

UPDATE INTERVAL

IFtf.to~tg SET 1 = t, . t o

8

tg

, ZERO-ORDER OUTER SOLUTION

q = ~ e e s = ( ~ : ) ( y ~ + f ~ t Y ~$ 1+2% )

3

SYSTEM DYNAMICS

3

A = 3 tg / (tg

I

= tan"

+ 3 K,)

(\/A+)

I

ZERO-ORDER INNER SOLUTION

t

INTEGRATE SYSTEM DYNAMICS OVER TlME INTERVAL tg

-

I

FIRST-ORDER OUTER SOLUTION

Figure 8-27 Scqucncc of Perturbation Solution C o m p ~ ~ t a t i o tFor ~ s Siniulating MTT Missilc Trajcctorics (Irro~n/A.@ar~~,alotld .\loort, 19841 ir~irkyri*rtr~isrio~r ,fiotn AACC)

I

Sec. 8.4

Radorne Error Calibration and Compensation

and using Equations (8-71), (8-72). and (8-78):

-

A&&

1 +; - Km 6; = h;,(t,,)A!'(cos $ - cos 4) + h;,(tll)(sin $ - sin $)

Since K+ is very small, the second term in the left-hand side can bc neglected. The following is obtained by substituting Equations (8-66) at t = t,, into Equation (880): - A

= 4

- 1

- cos(6 -

+)I

(8-81)

Substituting Equations (8-68) and (8-73) into Equation (8-81) will yield the rollrate command &, equation as: Km &f = 2 ~ 1 [I' -~ cos ~(

- +)]

or

In addition, from Equations (8-69) and (8-73), it is seen that

A, = At! cos(& - I$)

(8-83)

The near-optimal guidance law for BTT missiles is due to the fact that the acceleration command A, in the target plane and the roll-rate command 4,are given by Equations (8-83) and (8-82), respectively. Figure 8-27 shows the simulation flow chart of this BTT optimal guidance law.

8.4 RADOME ERROR CALIBRATION AND COMPENSATION Missile homing performance can be seriously degraded by boresight radome errors whose slopes generally show large variations. Thus, to increase homing performance and target intercept capability, in-flight radome error calibration and compensation is essential to overcome the restrictions imposed by radome errors [Lin and Lee, 1983; Lin and Yueh, 1984(a),(b);Yueh and Lin, 1984, 1985(a), (b); Yost et al., 19801.

8.4.1 R a d o m e Error C o m p e n s a t i o n This section first describes what has been done in the past to increase radome error tolerance for guidance system and autopilot design, and the approach this section proposes to use. T o improve STT and BTT system performance, it is essential to increase the radome error tolerance. Considerable effort has been devoted to modifying guidance system and autopilot designs to achieve this goal. Chapter 2 discusses radome error and its severity on guidance and control responsiveness of a homing missile. As the speed and intercept altitude increase, the limitation imposed by the

518

Advanced Guidance System Design

Chap. 8

radome error on guidance responsiveness also increases. This is also true for broadband homing systems. Large pitch and yalv motions occur at high altitudes causing the direction of the radar waves to change through the radome. This change and the refraction produce an apparent target motion. The effect o f a given radome slope o n the miss distance is determined by the aerodynamic turning rate time T,, which is small at lo\+,altitude and does not have much effect on the radome slope. However, as T, increases with altitude, the per~nissibleradome slope range reduces which can also lead to stability problems. This radorne error-induced stability problem can cause more severe degradation for the modern guidance system than the PNG. The randomly switched radome refraction slope cannot be easily modeled and the optimal guidance system (OGS) derived sut-fers more stringent component tolerances against unmodeled errors. In fact, the allox+-ableradome slope range is one important measure used in guidance system design to specify manufacturing tolerances on the radome. When component tolerances are met, the O G S proves to be the optimal design in terms of miss-distance performance index and acceleration command history. Thus, a design that overcomes the restrictions imposed by r a d o n ~ eerror is necessary to allow increased homing missile perforn~anceand target intercept capability for a more robust guidance system optimization. In the conventional approach, the most widcly irnplc~~lcnred tactical missile systems are based on linear control theory where heavy guidance filtering is used to increase body-motion coupling stability margins. The FCS bandwidth o has a great effect on the system radome sensitivity. As o increases, system performance becomcs more sensitive to radome slope error. This leads to stricter radomc specifications. One method of stabilizing the radome-coupling effects is to introduce a lag-lead filtcr conipensation. The lag-lead filter provides the proper stability margins at low frcqucncics without causing high-frequcncy effccts sincc radomc effects are essentially a loxv-frequency problem. Some care, however, 171ustbe taken in selecting the low-frequency corner since, at low frequcncics, a large changc in the timc constant produces only a relatively small change in attenuation near the frequencies of interest. For the parasitic loop of 0, the amount of lag-lead filtering required will depend on the flight conditions. The fact that the n~issilclag-lead nctwork must be varied with flight conditions represents one disadvantage, whilc another onc is that thc timc lag induced in thc missile guidancc loop duc to thc filtering network is also critical. T o alleviate the severe radome stability problcni for a high-altitude engagement scenario, one of thc commonly uscd engineering "fixcs" is to introduce pitch-rate compensation K, with gain K, fed back to cithcr bcforc or after the guidance filtering nctwork. I t can somcwhat syni~nctrizcthe positive and ncgativc slope stability region at the cost of slowing dolvn the autopilot rcsponsc. Hence, an a]ternative nlethod of radome compcnsation is accomplished by adding a 0 feedback loop with conipensator gain K, to the guidancc coniputcr as shown in Figure 821c. This attenuates the effects of radomc slopc so that the required stability margins arc obtained for both plus and minus radorne errors. The inain disadvantagc of this compcnsation is that thc rnissilc systcm time constant is dclibcratcly rcduccd. This can be seen from tlic previous discussion involving Equation (8-41). T h e conipcll-

Sec. 8.4

Radome Error Calibration and Compensation

510

sator gain K, from Figurc 8-21c with largc stability gain can be approximated by K, = r. Since the rndome slope error is assumed validly to be constant over small regions of the look angle, thc radome compcnstor gain K, is approxi~natelyconstant. A guidancc loop that combined seeker-guidance filtering in a complementary LOS ratc cstinlator shown in Figure 8-21c is more interesting. The radomc compensation gain that is used in this guidancc loop is independent of the seeker tracking time constant. If a conventional seeker, as shown in Fig. 2-13b, is used in Fig. 8-21c instead of a combined seeker-guidance filter, thc compensator gain K, with small tracking-loop time constant and largc stability gain can be approximated by K, = r/(1 - r). Pitch-rate compensation, which biases the radomc crror toward the positive slope region, can somewhat symmetrize the positive and negative slope stability region at the cost of slowing down the autopilot response. For the hypersonic interceptor with speed beyond Mach 6, the required guidance system time constant \vill bc larger than 4 sec, which is too sluggish for a classical homing missile. Thus the simple, conventional "fix" cannot be extended to the next generation, hypersonic interceptor. Another proposed approach (although never implemented in the past) has been to estimate the radome error slope by requiring a missile body oscillation in order to correlate the LOS ratc measurement with the known dither "test" signal. The approach taken in this study is similar to the error identification concept but removes the need for dithering. Rather, the proposed optimal guidance and control technique relies on missile pitch angular velocity due to guidance commands and explicitly accounts for the actual LOS rate contained in the radome corrupted boresight data. Figure 8-28 shows an integrated NGC system with radome error compensation.

Targets

f Kinematics

-

Adaptivc Radome Error Eslimalor

(Chaps. 4, 6-81

(SC. 8.4.2)

4

4L

Intcgratcd Control System ( ~ ~ )3-51 ~ ) k ~-+ Airframe & Propulsion

4

Gain Adjustment

-

.f

Onboard Navigation System (Chap. 5)

Figure 8-28

Airframe Filter

FCS Sensors

Integrated NGC System with Radome Error Compensation

520

Advanced Guidance Sustem Design

Chap. 8

Integrated modified PNG and optimal controller with radome compensation Modified PNG with Radome Comnpensation. The radome compensation technique applied in Section 8.2.3 requires estimation of the radome slope error to correct the LOS rate measurement through missile body rate. A radolne slope error estimate has been incorporated in a modified PNG to generate a guidance Command. The guidance acceleration command becomes

where & and 0 are smoothed values of 6 and 8, respectively. A detailed derivation of Equation (8-84) is presented in Book 3 of the series. Adding radome compensator with gain K, to the guidance formulation in Equation (8-84) yields

Besides the small-trim aerodynamic correction term that is proportional to Z,/,\I, (< 0.01), there is also the pitch-rate compcnsation term to account for rhe radome error. K, is introduced in the pitch acceleration command by n~ultiplyingthe pitch rate by P - , the probability of radome error in the negative region, to be defined later. The use of P- to scale the pitch-rate compensation for enlarging the radome stability region has also dcmonstratcd reduced cxcitation in acceleration commands.

Integrated Modified PNG and Optimal Controller. The opti~nalcontroller derived in Book 3 of the scrics is related to a modified I'NG system \vith navigation ratio .A. According to the author's kno\vledgc, Equation (8-85) offers the first analytical expression for a rheorctical basis to introducc pitch-ratc compensation according to thc optimal control law provided that thc radonle error slope can be estimatcd.

8.4.2 Guidance Performance Analysis with In-Flight Radome Error Calibration Adaptive radome estimation design. An adaptive Kalma1-i filtering bank in the s\vitching environment is used to realizc the changing processes in a radome slopc. The rate of s\vitching is assumed to be cotlsiderably slo\vcr than that of thc boresight crror (observation) sampling rate. That is, by assuming the rate of positive-negative radomc slope switching to be snlallcr than thc data sampling rate, the esti~nationschcnie can be rcduccd to the design of a bank of Kalman filters, each matched to a ccrtain radomc slope configuration. Actually, follo\virlg thc adaptive filtcr formulation [Chang and Athans, 1978; Moose, 1975; Moose and Wang. 19731, thc random switching of thc unreliable plant (thc rado~ncslope) is modeled by a semi-Markov process. A scmi-Markov process is a probabilistic system that makes its state transitions according to thc transition probability matrix of a conventional

Sec. 8.4

Radome Error Callbration and Compensation

Estimated LOS Look Angles

Filter Weights

A Posteriori Hypothesis Probabilities Calculated

Figure 8-29 Adaptive Radome Error Slope Estimator (One-State, Three-Bank Kalman Filter)

Markov process. However, the amount of time spent in each state before the next transition to a different state is a random variable. It is this property of a random switching time that distinguishes the more general semi-Markov process from a Markov process. The multiple states mentioned earlier are actually chosen to typify the radome slope in the positive, zero, or negative slope regions. Thus, by proper choice of the state and modeling of the transition processes, it is desired to realize the variations of the radome slope as a randomly switching, semi-Markov process. A general adaptive filter, as shown in Figure 8-29, essentially consists of a bank of three Kalman filters, each matched to a possible plant configuration U, ( i = 1 , 2, 3). The filter outputs are weighted by a time-varying a posteriori probability to obtain the radome error slope estimate. The bank of three Kalman filters has the deterministic inputs U1 = U + = 0.02 degldeg, U2 = U" = 0.0 degldeg, and U3 = U - = -0.02 degldeg to characterize the three plant configurations in the positive, zero, and negative slope regions, respectively. That is, the filter estimates from each filter in the bank are further weighted by calculating the a posteriori hypothesis probabilities. It is the probability of a given hypothesis that the radome slope is around a certain U ,value, being true conditioned upon the past measurements. Since during actual flight the unknown plant configuration (due to polarization effect) might randomly switch at random times, the random switching of the radome slope parameters must be mddeled as a semi-Markov process. The a posteriori hypothesis testing involves the three conditional probabilities

Advanced Guidance System Design

522

Chap. 8

where Zk+ denotes the collective events of all the past measurement history up to time t k + 1, that is, Equation (8-86) actually states that for a new boresight datum z k + I that just comes in, a global hypothesis needs to be tested to see whether the datum indicates that the plant configuration, or the radome slope, is near the positive, zero, or negative slope region. It is a global instead of local hypothesis because it depends on the time history of the data, as shown in Equation (8-87). Thus, the a posteriori conditional probabilities should be calculated recursively to reflect all the past data dependence. Detailed formulations are given [Moose, 1975; Moose and Wang, 19731 and are shown below for i = 1, 2, 3, as

X

normalizing factor x

2 [_Pi,&x

Xii ( t h + ~- rk)]

j=l

in terms of the rccursive time index k. The normalizing factor is the inverse of the sum of the three a posteriori conditional probabilities. T h e transition matrix Xv from state Ui to Ui is symmetric and is based on the semi-Markov statistics. This renders a very simple form for in-flight, real-time computation and eliminates the computer storage problem that grows with increasing time in the conventional Markov process approach. The first term on the right-hand side o f Equation (888) furnishcs the adaptive learning feature of the system. It actually tests the new borcsight data r k + l to check the probability of being within each plant condition, that is, whcther it corresponds more to the positive slope or negative slope region. This conditional probability can be approximated by a Gaussian density for those cases in which thc probability of a transition occurring between any two adjacent sampling timcs is very small. As pointed out [Moose and Wang, 19731, the actual density is not Gaussian, but is in fact a weighted sum ofGaussian densities. However, it was determined expcrimcntally from computer simulation that even when the plant randomly switched as often as the duration of several systcm response ~ ~ I I I C S , the Gaussian approxinlation was quite good. The mean and variancc of the Gaussian dcnsity arc rccursivcly dctcrn~incdby the bank of Kalman filtcr estimates and the associated filtcr covariancc. The following discusscs the linear state dynamic equation in Kalman filter modeling bascd on the seeker boresight data mcasurcmcnt. Assume that or is the radome abcrration crror with explicit look-angle dcpendence. Thus, the time derivative of the mcasured LOS [see Equation (2-10)] to the first-order linearization approximation can be obtaincd as shown in Equation (2-15b). Assuming type I radio-frequency tracker, i + w,c = 6 . The measurement equation can be formalized through thc LOS ratc measurement modcl [scc Equation (2-15b)l as

Sec. 8.4

Radorne Error Calbratlon and Compensatlon

523

whcrc 2 is thc smoothcd LOS rate from thc guidance filtering (of a t least sccond ordcr) o ~ ~ t p uThe t . nlcasuremcnt partial H is defincd as

H = - 0

(8-90)

whcrc 6 is the s~noothcdvalue of 0 passing through the same guidance filtcr. In Equation (8-89) the mcasuremcnt noisc il is thc zero-mcan whitc-noise Gaussian process with the spectral density equal to the LOS measurement noisc spectral density N,that is, the boresight angular data noisc sigma value. For small look angles whcrc 1 ~ r . < l 10°, the sequential correlations of the radomc error slope can bc modelcd simply as a random walk process (the correlations arc not consistent enough to bc modeled),

where T = t k + l - tk and w is a Gaussian white-noise process with plant noise variance defined as E[w(t) . w(T)] = Q(I) S(r - T), whcrc Q(t) is given by the associated standard deviation from table look-up, if available, and may be time varying. In Equation (8-91), drldt = d(da,/au,)ldt is of higher-order dependence on 0 but is neglected in the linear filter approach. The first-order pitch rate dependent term is already included in Equation (2-13b). For actual flight the auto correlation function for the radome error slope tcnds to stay constant, rather than exponentially correlated such as that modeled by the Markov process. This is because the typical radomc error dependence on look angle is sinusoidal. Thus, the radome error slope can best be modeled as a random walk sk- I = xk + tub. The weighted plant input

.

actually represents a very smoothed version of the radome slope estimate and is used in a feedback version [that is, 5 in Equation (8-92) is used in Equation (893)] to generate a "prediction term" so that Equation (8-91) is modified to become

The prediction term accompanied by the system noise variance is derived based on UL = u - 0 at zero LOS, and hence drldt = measurement rcsidual/uL, where the measurement residual is equal to z H. This additional term has demonstrated improvements in the estimation procedure. The plant noise variance Q is generally determined on line with radome mapping table look-up. However, for simplicity it is assumed that Q is identical for the three-plant configurations and can be calculated in the first cut as the table look-up value with respect to the zero-mean radome slope. Thus, with the same boresight measurenlent noise variance R for the three filters, it can easily be shown that the three Kalman gains are identical if one starts with the same initial filter variances. Using the a posteriori probability as the weighting function to weight the estimator outputs, and assuming that the weighting

z.

Advanced Guidance System Design

524

Chap. 8

coefficients change very little from sample to sample, that is, P,, h i - I * P,,L (this assumption is not well justified if the slope is rapidly changing from sample to sample), the following can be obtained [Moose, 19751: ikil

= @

' ?k

+ ??b + Kb+l

..

'

[&&+I

f

0 '@

'

vk

+

0

-

' ub]

(8-94)

where indicates the operation of radome mapping process and becomes unity in case of n o available satisfactory mapping data, that is, without the prediction term introduced in Equation (8-93). N o w , the bank o f three filters is essentially reduced to a single filter with weighted plant input 5 given by Equation (8-92).

Closed-loopanalysis for adaptive radome estimator. The adaptive estimator results for terminal flight based on an open-loop guidance simulation are plotted in Figures 8-30a through 8-30c for a sawtooth radome error slope model. The sawtooth radome error slope model corresponds to the radome error model as a triangular wave function of uL,P that is, the radome error varies with the LOS. In Figure 8-30a, the broken curve 8/10 represents the sinusoidal body pitch-rate dependence introduced in the simulation. The dotted curve i . H is the reconstructed radome error and follows very closely the boresight error (actual data) z shown bv the solid curve, despite the large noise level chosen (at 0.5"isec 1 u level). I t has the tendency of cutting the corncrs when z goes through sharp spiking behavior. Figure 8-30b compares the actual r a d o ~ n eslope in the solid curve v with the estimated radonle slope in the broken curve i. A response lag of approximately 1 sec can be identified. The cstimatcd slopc also tends to overshoot when the actual slope changes sign abruptly. The rcason that the estimates degrade near 3 and 8 sec is a nulling c of the missile body pitch ratc, which characterizes the radonlc crror s l o ~ sensitivity and provides cscitation for thc estimator. The wcighted plant input C, as shown by the dots, follows thc general trend of the slope estimate and can be denoted as a further smoothed-out rcrsion of the estimatcd slopc. The a postcriori probabilities for each plant are shown in Figure 8-30c. In Equation (8-90), P+ + Po + P- = 1 . In actual flight only thc negative and positive slopes need calibration, and hcncc in actual implcmcntation of thc in-plane radome crror calibration scheme only the positivc slope probability P , and the negativc slopc probability P - arc used. Therefore, the rcsults herc arc normalized so that P- + P- = I . Thc zcro slope probability Po is still shown hcrc to indicatc the trend. Ucsidcs the 1-scc rcsponsc lag. the probabilitics yicld strong indications of thc polarity of the slope. For instancc, Preaches 98 perccnt a t about 1.2 scc after the slopc switchcs at 0.8 scc. It starts to dcgradc bccausc thc scnsitivity paramctcr 0 is gctting smaller and thus reduces the variance or confidcncc lcvcl (which is related to the a postcriori probability) in the changing cnvironmcnt. TO check thc adaptivc filter perfor~nanceagainst rcalistic data, hybrid simulation runs wcrc pcrformcd with all thc rcicvant angular information for the specific pla~lcrccordcd. A hllOF sin~ulationwas cmploycd with thc hcad-on cngagemcnts choscn a t 1-dcg heading error. Thcrc is a 0.5 dcglscc 1 u noise lcvcl on the LOS ratc. Thc ratc gyro bias of 0.05 dcg/scc appcarcd on the LOS ratc as ~ b , ~ The , .

Sec. 8.4

Radome Error Callbratlon and Compensation

525

radomc crror slopc was a fast-changing process with the signals switching several times. The bpdy pitch rate manifests more flailing behavior as can bc seen from thc broken line 6/10 in Figure 8-31. The reconstructed radonic crror curve i . H, as shown by the dots, still follows vcry closely the boresight crror (the solid curve Z ) . The comparison of the estimated slopc and the actual one was vcry difficult due to the anibiguity in the average and local slopc calculations during the hybrid run. Not estimating the cross-plane slopc simultancously makes the missile roll attitude contribution inseparable. After imbedding this adaptive estimator into a terminal guici'~nccforward siniulation program to perform the closed-loop analysis, encouraging results have been obtnincd. llcpcnding on the actual radon& crror and the approaching trajectory, the improvement in the miss-distance performance index ranges from 70 percent to 50 perccnt. In Yuch and Lin [1984) the estimated radome error slopc is used in guidance and autopilot commands to provide an optinial pitch rate compensation scheme for a modified proportional navigation system and an optimal controller [Yuch and Lin, 198j(a)l. Using P+ and P - , the probabilities of radome crror being in the positive and negative regions, respectively, to scale the

Figure 8-30 (a) Comparison fo Estimated and Actual Radome Errors after Matched Guidance Filtering for Sawtooth Model (b) Estimated and Weightrd Radome Error Slopes (c) A Posteriori Probability for Adaptive Radome Error Estimation

Advanced Guidance System Design

(C)

Figure 6-30

(Continued)

Chap. 8

Sec. 8.4

Radome Error Calibration and Compensation

527

pitch-rate con~pensationfor enlarging the radomc stability region, has also demonstrated rcduccd cxcitation in acceleration comn~ands.

Discussion.

A Kalman filter bank is dcsigned to enhance the dynamic response timc for the radomc error slope estimate with compensation for the seeker dynamic lag. In calculating the critical weighting coefficients (a posteriori probabilities), a measure-predict-measure techniqi~eis used when the semi-Markov statistics of a random starting proccss are used to make the intermediate predictive step. That is, the resulting estimated radomc slope paramctcr is the statistical average weighted by time-varying, a posteriori hypothesis probability, which is calculated concurrently with the recursive filter scheme by using Bayesian rule. T o reduce the computational burden, the Kalman filter bank is digitally simulated and dcsigned by tuning the noise processes, including the measurement and plant noise, to allow a one-time calculation of the Kalman filter gain. The simple one-state filter described previously can be modified to include a correlation parameter for studying the crossplane errors. The bank of Kalman filters can be increased to 3 to enlarge the dynamic range while reducing the system response time simultaneously, and to estimate the cross-plane radome error slope simultaneously for three-dimensional engagement. The adaptive radome estimator design is intended to be an add-on compensation

Figure 8-31 Data

Comparison of Estimated and Actual Radome Errors for Realistic

Advanced Guidance System Design

528

Chap. 8

network that is independent of guidance computer and autopilot design. The objective is to permit relaxation of missile bandwidth requirements by reducing error due to radome at the guidance computer output, thus enhancing missile performance.

8.4.3 Design Equations for PNG with Parasitic Feedback Following the excellent paper by Travers [1986], this section presents the design equations for PNG with parasitic feedback. In general, a good portion of guidance and control system design is taken up with iterative analysis surrounding a series o f dynamic models o f increasing complexity. T o initialize this process for PNG with parasitic feedback, algebraic equations have been developed that incorporate certain elements o f synthesis and analytic design [Newton et al., 19573. The canonic model is a seventh-order system that includes radome aberration and imperfect seeker body motion isolation. The model, the miss budget as a function of radome aberration, and the stochastic commanded acceleration and control surface rate due to range independent noise (RIN) are described by the 15 parameters that result from the design equations. Additional indicators provided to guide the design include the rate-loop gain and the lo\\.-frequency phase margin of the autopilot. Preliminary design should be very conservative fro171a stability margin standpoint, and system performance levels should be conservatively established to accommodate attrition during the design period oxving to increases in the ordcr of the dynamics.

Problem definition. Figure 8-32a. u~hichis rcdra~l-nfro171 Figure ., 2-ljb, is a block diagram of the homing process. The details of the homing controller of Figure 8-32a arc displayed in Figure 8-32b. The seeker with its target tracking and body-motion isolation subsystcn~s,as shown in Figurc 7-13b, is reprcscntcd by the multiloop section o f Figure 8-32b. Those autopilot signals of particular relevance arc 6, a,and 0 . T w o additional signals that provide a measure of the cffccts of noise on control activity arc the RMS values of u, and 8 due to RIN. The problem consists of defining all the parameters of Figurc 8-32b so that the result of a simulation will mirror thc miss distancc predicted by the stochastic inputs and the guidancc systell1 dynamics. A secondary aspect of the problem concerns computing some relevant indicators of system pcrformancc. T o this end, consider the sin~plcrdiagram of Figure 8 - 3 3 . The forward path is a fifth-ordcr system \vith the high-frcquc~lcy poles and zcros of the antenna gyro loop ncglcctcd o\\ving to the fact that they arc usually found \vcll outside thc guidance band\j,idth. The polcs of the antenna gyroloop transfer function arc likeivisc ncglcctcd in the feedback path, but the zeros (Tzs)' arc not sincc these providc thc representation of body-motion leakage. Ncglccting thc high-frequency polcs of the antcnna gyro loop thercforc allows the ordcr of thc systcm to be reduced from scvcnth to fifth. The parasitic elements in Figurc 8 - 3 2 arc a sum of the following thrcc elements: thc radolnc slope bias ( 1 . 1 ~ ) the radornc slope (r), and the leakage in the seeker body-motion isolation loop ( ' 1 . 2 ) . >'

Advanced Guidance S p t e m Design

530

Chap. 8

These elelnents are in cascade with the aerodynamic transfer function from achieved acceleration to pitch rate 0. The manner in which the parasitic elements are studied separately is as follows. First, it is assumed that v b , the value of which is not known at this point, does exist in the parasitic feedback path. For that system the effects of v 7 are investigated, following which the appropriate value of vt, is established to ensure that the best system performance is obtained when the actual radome slope is zero. Finally, one treats the effect of ( T ~ in S )the ~ presence of rr, where the signs of these coniponents are permitted to vary independently. Generally, the sign of (T>s)* in Figure 8-32c is positive. However, in the case of utilizing sight line reconstruction INesline and Zarchan, 19851 to extend the virtual bandwidth of the seeker gyro loop, the system becomes dependent on the scale factor errors of the seeker and the rate gyro. It can then happen that the sign of the parasitic feedback loop becomes negative. The zeros of the tail-controlied airframe shown in Figure 8-33c in the forward path are cancelled by the corresponding poles in the feedback path, so that they are nonexistent to the extent that feedback is concerned. However, these zeros still exist in the closed-loop response, but are neglected for the sake of simplicity in the analysis presented here.

Design equations.

In this section, the subscript 0 denotes the forward path in the absence ofany parasitic feedback, while the subscript b denotes the closedloop respo~isein the presence of r.1,. The subscript + is used for the closed-loop response in the presence of eithcr rT/2 or - I . = / - . 7 O r d e r Reduction by Truncation.

In Figure 8%32c, thc homing controller has

the following transfer functiot~:

G.~(s)=

A,,,

-) 11,

-

=

K ( 1 + 4 + 1) T ~ , , . c+. ~a,.,T:,,c2 + P.4 T.4,,s+ 1 7,. = effective autopilot time constant

-- P.4T.4,, (8-95a)

where the polcs of the autopilot transfer function arc shown norlnalized with a charactcristic tinlc constant ( T . , , ) the , cube root of the s3 tcrrn, and two-dimensionless coefficients (a, and P . 4 ) . Similar to Equation (2-42a), studies of autopilots had used this for111of the cubic because of the plot of Siljac [lY(,Y], which showed the loci of the ratio of the frequencies of the poles and the damping ratio of the quadratic in thc a, f3 planc. The guidance transfcr function \vithout parasitic effects is givcn by

A ,,, (3) u

=

A C', s C ,4 (s) (1 + ?.~)(1 +

7,,.c)

It was thought that the order of this transfcr function could be reduced from five to three to limit further the number of variables it1 characterizing the guidance dynamics, atid that the cffccrs of parasitic coupling might be plotted in the a, P plane. It happens, llowevcr, that the cubic, although sufficient for thc effects of Ya

Sec. 8.4 7'-LE

Radome Error Calfbration and Compensation

531

8-3 TRUNCATION ERRORS, 5 -+ 3

= percent error due to truncation

:\ = 3.6

From [Travcrs, 19861 with permission from AACC

and r ~ is, too low an ordcr for consideration of T2. Truncating Equation (8-9510) results in

'guidance time constant = PIjTlj= ~vherethe normalized u coefficient Po is given by T~

T,

+ T, + T~

(8-95c)

In an effort to analyze the errors arising from truncation, a comparison was performed of normalized stochastic miss adjoints prior and subsequent to truncation. Inasmuch as these adjoints are normalized to the s coefficient, which is not altered by truncation, a comparison of adjoints is equivalent to a comparison of miss distances. Some of the results ofthis study are listed in Table 8-3. The errors in two stochastic adjoints are shown for combinations of cubic coefficients and values of the ratio ( T ~+ T , ) / T ~ , which is denoted by R. The RIN miss is consistently more sensitive to truncation than the target maneuver miss. The most severe truncation errors occur for a combination of low autopilot damping ratios and low P.4 when the quadratic is the dominant term in the cubic. It was concluded that truncation errors do not generally exceed 10 percent when the damping ratio of the autopilot

Inouts Outouts Analysis:

fit

Pb

B-

Rs Kb

K+ K.

A

B+ P-

Synthesis:

Eauations

pb

R,

K+

Kb

K.

A

Note that the symbol + used in Fisure A and in subsequent expressions means that all

operations may conditionally be e~ther+ or -. The closed-loop response for the configuraration in Fisurure A is given by Kbvc s l [lt(Vc,.li,)K,(ry'-))]

4, - -

[:'~tab:)~+[~b*(~C~m)~b(~T,~)(~,~b)~b]~~t(~c~m)Kb 1 (1) For notational ease, the follo\ving paramelers are defined:

.. "c

R

q = R , K h D ; A-T,,'T, (2)

This allotvs Equation (1) to be more compactly expressed as

A" = [V,K,s] 5

where '=

=

'I

q!h

-

7

i - t CI= 7 ;

+ p,

7,

+ 11

(3) -1 3

T+s ; K r = K ~ ( I + , ~ );- 'T+ = T h ( l = q ) -1 3 ,

(ltll)

,

(

-mt h( I )

.I 3

:

B,

r

(ill, t I ~ A(l+tl)" )

The ratio of K to fl as a function of y, and 11 can be obtained from Equation (4). IK*/P*l, 1 ;11, (1-1,'V) 13

Here, 111 is defined as the value associated tvith yl and unity.

(4)

(-5)

Sec. 8.4

Radome Error Calibration and Compensation

533

is above 0.5. It is pointed out that the airframe zeros are neglected in this study. The truncation errors are recognized as being a component of the miss scale factor, S,,, which serve to bring closer together the predicted miss distance of the equations and that resulting from numerical integration.

Guidance Loop Transfer Function with Radome Slope Bias Effect and Without Radome Slope Effect. The closed-loop guidance transfer function that results from the existence of rl, in the feedback path is obtained from Equation (8-95c) as follows:

A,,, u

Y (3)

T$

-Y?

+

Al/; s I + rhA V,/ V,,, allY? + [Po + rhA(V,l V,,,)(T,I T,,)]Y,, 1 + rhA V,l V,,,

effective guidance time constant

+

PbTh

=

r y [l

(8-96a)

+ YIJ\(V,IV,/,,,)(T,IT~)] 1

+

ri,A V, I VpfZ

(8-96b)

\vhcre 7,; has the same formula as that in Equation (3-28). The relationship between the quantities subscripted with 0 and those with b is

Attitude-Loop Parameters ( A L P ) . The study of the attitude loop begins with Equation (8-96). Having represented that configuration in the forward path by the subscript b, consideration is now given to the effect of +rT12 in the feedback path. This attitude loop is shown in Figure 8-33a. The closed-loop response for this configuration is given by Equation (1) of Figure 8-33. Equation (4) of Figure 8-33 can be considered as the basic analytical equation of ALP. From the specified parameters of the open loop, K b ,A, R,, and P b , the closed-loop parameters can be derived. These are K , and P , . Those parameters that have as a subscript b can be regarded as defining the center of the parasitic zone, while those parameters that have as a subscript f can be regarded as defining the extent of the zone. From a practical standpoint, Equation (4) of Figure 8-33 becomes more useful when inverted, as shown in Figure 8-33b. Then the center of the zone can be derived from the specified limits to the extent of the zone. It should be noted that the center of the zone is not merely the mean of the zone extremes. Disregarding 2 as a variable in ALP is a consequence of the following factors. First, the normalized adjoints discussed in the next section appear to vary only slightly with a.Second, as a result Figure 8-33 (a) Cubic Attitude Loop with Radome Effects (b) Attitude Loop Parameters (Frorn [Traverr, 19861 with p e n n i r s i o t ~ j o mAACC)

534

Sec. 8.4

Radome Error Calibration and ~ ~ t n p e n s a t i o ~

of the truncation to the cubic, there was a tendency for a3 to be very close to 2 for many practical applications. Specific values for R, and A arise from the use of the synthesis form for ALP. However, it may be advantageous to depart from using these values as a matter of practical utility in design. A well-known rule of thumb is that, when the product of R, and A remains constant, their effect can be considered as remaining invariant. Letting $, which is greater than unity, be a multiplier on A and a divisor on R,, there is then a means of identifying the condition of constant product. The ratio of K to P is an indication of the incremental variability of the normalized miss adjoints. With the introduction of $, the new value for K - is smaller, as is the ncn7 value of p- . In the next section, it will be seen that the normalized adjoints vary in a counterbalancing fashion when the introduction of $ (greater than unity) results in new values for K, and p,.

Cubic Miss Adjoints (CUBAN). The cubic miss adjoints are summarized in Table 8-4. For any set of ALP, a corresponding set of normalized adjoints can be obtained from Tablc 8-4a and the renormalizations of Equation (4) of Table 84 are defined in Equation (5) of the same table. Cubic Synthesis (CUSYN). Glint is eliniinatcd from consideration for synthesis, owing to a number of reasons. First, the approximation of glint by white additive noise is one for which the miss distance nlay not cvcn be proportional to 7 [Garnell, 19801. Inasmuch as it tends to be in the direction of target length, glint miss may not have the samc impact on lethality as the othcr con~ponents.On the other hand, one presupposes that the level of glint gets vcry small as rhc target becomes very small. If glint is rctaincd in the synthcsis problcm. it \rill alxvays be a factor in determining thc optimum bandwidth of the systcm because it is thc only disturbance for which thc miss dccrcases as thc systcm becomes slo~ver.Finally, glint is removed from the synthcsis problem in order to bring to the forefront clearly the critical issues of trading the speed of responsc (improving accuracy, except perhaps for glint) with control activity and parasitic rcquirenlcnts. Retaining the glint adjoints is done so that glint as an input option can be treated in thc miss budgets after performing synthcsis. For synthesis it is required for simplicity that any dcsigll bascd on parasitic fccdback should be characterized by symmetrical pcrforrnancc at the cxtrcmcs of thc zonc. This is a consequcncc of thc fact that thc radomc slopes in practice arc gcncrally disposcd in symrnctrical fashion about zcro, so simplc cfficicncy would rcquirc that the pcrformance zonc bc sy~nn~ctrical. In this way, the cffectivc guidance timc constant that rcalizcs this symmctry can bc detcrmincd from a specification of the stochastic inputs to thc problcnl, targct n~ancuvcr,and R1N. T h c t w o componcnts of miss distance arc cquatcd as i i l , , ~ - = MGj- ~,+;./'7,:~"' = nr~,+ = Ad&+ 4 . ~ ' 2 7 ~ " 'to yield thc timc c o n s t a ~ ~ that t cffccts this equality I/'

,;/2

This cquation is suhscqucntly substitutcd into u,,,,. = I(hlTJI,b,7 x . )

'

+

Sec. 8.4

Radome Error Callbratlon and Compensation

535

a) Normalized Cubic Aaolnts

TABLE 8--4

a = ?

K = 3

p

Myg

M

K = 3.5 MTJ

Myg

K = 4.5

K = 4.0

M,r

MTJ

Myg

M,,i

MTJ

Myg

Mtri

MTJ

b) Adjoints and ALP Conditions

For the general cubic controller, the normalized miss adjoints are a function of three variables: K , a,and p identified in the transfer function for the controller, A, u

- = [KV, s]/[y3+

ay'

+

Py

+

11; y

= Ts; PT

=

T,I

(1)

These adjoints are defined for the inputs of glint, constant angle noise, and stochastic target maneuver,

which have the same formulae as in Figure 6-42. The z adjoints must be renormalized to the center of the performance zone; that is to 7;. This renormalization takes into account the ratios of P, and T f as given in the parameter q:

The renormalized adjoints, designated by a superscript asterisk, are defined as

By making iterative comparisons between ALP and typical system performance as obtained by the use of Eq. (4) in Eq. (2), it will be possible to converge on a fixed set of values in Eq. (4) for use in miss budgets. It is possible to convert the ALP zone to a performance zone using the renormalized adjoints,

and such a vrocedure lends itself to analvsis From [Travers. 19861 with permission from AACC

Advanced Guidance System Design

536

Chap.

( M ~ ~ ~ v , + ) which ~ ~ Tgives ~ ~the~ miss ~ ) with ~ ] zero ~ ~ actual ~ radome, and the result is

which gives the R M S miss in the center of the zone. Equations (8-97) determine 7; and u,,, from the inputs v&f.I2and +:I2. This gives a total of four parameters, from which any two may serve as the input parameters for synthesis. Also, from Equations (8-97) the other five equations can all be determined. The six equations involving two stochastic inputs and the RMS miss distance and 7; all constitute the synthesis equations knon-n as CUSYN that are used to initiate the design sequence. The equation for the ratio of the miss at the zone extreme to that at the center is

The importance of this ratio lies in its role in selecting the appropriate ALP, and iteration must be employed if the value resulting from the application of Equation (8-98) is not deemed satisfactory. The limited degrees of freedom associated with this simple synthesis technique may produce unexpected consequences. Experience in using this procedure has demonstrated that iteration in the C U S Y N technique may be employed to obtain reasonable values of the four parameters. A worst-case scenario, however, is one in which the synthesis procedure is so restrictive that dummy values for the stochastic inputs may have to be carried along, inserting corrections after the final step, the assembly of the miss budget. In this case the level of control activity \vould also require modification, but any modification is a simple scaling operation. Missile Siring. One of the results of C U S Y N is the identification of the stochastic target maneuver and 7;. It is required that the guidance system operate linearly. As a means of satisfying this requirement, the first step consists of identifying the step of target maneuver which is uniformly distributed over the last 7; of the intercept. Such a definition for the targct maneuver yields AT = 1 2 ( 1 ~117-/32.2(measured ) in g). For linear opcration, the maximum acceleration of the interceptor must be greater than that of the targct by a factor S , specified by the dcsigncr. This factor, termed the rnarleuvcr sfale-fanor, is gcnerally between 3.6 and 6.0, and its use defines the maximum required missile acceleration. At this point, the maximun~design angle of attack, which is a function of the missile outboard profile, must be specified by the designer. The ~nininlunlvalue of Ai, = VmZa that satisfies the requirclncnt of linear operation is given by the ratio of maximum acceleration to maximum angle of attack. From the aerodynamic definition of T= (see Table 2-2) together with an approximation, its corresponding maximum value can also be determined as T, = - 1.05/Z, and t

-

(T,),,,

5

1.05 V,,,a,, /[I845 S,q+.!'2(10 7,)t 1121

(8-99)

Sec. 8.4

Radome Error Calibration and Compensation

The ALP together with the result of Equation (8-0')) describes

537

provide thc equation that

Since RIN is also specified from CUSYN, in addition to closing velocity, it is possible to cstimate u,,,based on cstin~atesof T, and T., as fractions of 7,:that results from C U S Y N . An early cstimate of a,,,is obtained by arbitrarily setting AIKA at 4 and T, = T,, = ~,;/8.Part of the explanation for assigning these values to T; and T, is that the effect o f r ~ is , assumed to account for 318 of T,;, while the remainder is equivalent to five equal time constants. O f these, three are in the autopilot, leaving 218 to bc divided between T , and T.,. An exact expression is given for a,,,, which will be utilized a t a point further on in the design procedure, and an approximation which is used at this point in the procedure. (est) a,,.=

v ~ + ~ K,+ ~ A / z= 1.405 V,+)/" 32.2 [~T,T,,(T, + T , ) ] " ~- 7i3I2 where AIKA = 4

(8-101)

Following C U S Y N , the missile-sizing equations can be employed by the designer to judge the reasonableness of the requirements on the airframe and the radome as well as the impact of RIN on the RMS level of commanded acceleration. Some options which could be exercised a t this point include iteration through CUSYN or the consideration of pseudo-values for the stochastic inputs. Homing Controller Expansion ( H C E ) . Before expanding to the fifth order, the necessary steps must be taken to go from the center of the parasitic zone to the truncated forward path. In order to do this, however, A and r h must be determined Equation (8-96c) is substituted into (8-96b) to get

In this equation, the solution for AlKb does not depend on 7,;. Next, one approximates the cube root of A/Kb by the cube root of 1 + E, where E is much less than unity. The solution for A is then given by

Given that K b and PJ, were determined from ALP, and that 7,; is known from C U S Y N and T, from missile sizing, A is determined here from Po defined in Equation (8-95d). Also, Equation (8-96c) can then be used to determine rb

The additional lag in the expression for 7,: is a consequence of the combination of rb and T,. The required value for T , is obtained from Equations (8-96b) and (8104). (required) T, = ~ i [ ( A l K ~ ) ( l T,/T':)

+ (T,/T~)]

(8-105)

538

Advanced Guidance System Design

Chap. 8

However, the assumed dynamics to get result in an actual 7, that may not equal the required value for T, in Equation (8-105). In this event, it is a matter of scaling the values for T,, T,, and T,, and iterating the process of determining A, r b , and the required T, until the value of 7, that is obtained is that required in Equation (8105). This process may be done manually or in an automated fashion. The rule for assigning values to the time constants in the forward path may conveniently be taken as the one given under missile sizing for the estimation of u,,, although it may have to be altered if u,, is too large. Moreover, local requirements of seeker tracking may dominate in setting 7,.Since u,, is dependent on T ~where / ~ T =7 - T,, T should be made larger if possible. This completes the expansion of the order of the controller in the forward path, and determines the closed-loop response of the autopilot and T,, T,, and A. Also determined in the feedback path are r ~ T,. , and r b

Seeker Gyro-Loop Bandwidth. Because the effect of TZis greatest on the s3 coefficient of the closed-loop response, the order of the forward path should not be less than 4 so that its effects on the dynamics and hence system performance are adequately represented. In fact, a value of T2 was derived by the use of a thirdorder system for the instance in which the sign of TZ feedback was always positive. After completing the fourth-order analysis, it was discovered that the derived expression for TZcompared favorably with that derived from the third-order case. The fourth-order analysis necessitated further simplifications to avoid increases in complexity. First, only a single set of nominal normalized coefficients for the quadratic polynomial was used. Also, the case of large $ was postulated, giving large values of T,and small values of r . ~ It. is possible in this case to neglect the variation with 1.7of A and the normalizing of To. It \\.as also determined that the effect of TZon the .r2 term was sufficiently small as to bc reasonably ncglected. Then Tr influenced the 3' term and r r , the s term of the closed-loop character~sticpolynomial. It became possible to plot these adjoints in the plane of the third- and first-order coefficicnts. Additionally, these adjoints were normalized to T4 as opposed to the sum of the time constants. o r the s coefficient. As a result, the miss distances were made directly proportional to the values of the adjoints in the coefficient plane. One axis in the plane of the adjoints represented the cffcct of r T and the other, the effect of T2.It then became an easy tnattcr to examine effects of independent variation of the txvo parameters. Assuming forward path A,,,/E = YKs/[+% a443+ ~ 4 4 ' + y 4 b + I ] with 6 E 1 ; 5 , and the feedback path H,,,/,4,,, = [ t r + (&r)'](l + T,s)/( l,',,,~),thcn thc closed-loop response is

Having simplified the problc~nin thc manncr spccificd, the adjoints in the plane of the a and 6 coefficicnts wcrc normalized again rclativc to thc values at the point 0f

Sec. 8.4

Radome Error Callbratlon and Compensatfon

539

zero parasitics. Then the pernlissible excursions of Aa and A y were established empirically by analyzing a number of practical cases. Using Equation (8-106), Aa ?= ( I .:/V,,,)KT,T i / T i , then substituting T bfor T4yields

It was found that the empirically determined value of ha of 0.4 increased the miss distance on the order of 10 percent to 15 percent.

Autopilot and Airframe. The calculation of u s requires the airframe gains, and even if thcse are known a prior^, a number of restrictive aspects of the autopilot design must be given consideration. Included among those to be examined arc the rate damping loop gain and the low-frequency gain margin (LFGM) of the loop transmission. There are many instances in which the aerodynamic gains are not available a t this juncture in the problem. However, a means of deriving the gains is provided for these cases, which is based on the fact that certain factors involved in determining the airframe parameters are more easily estimated owing to the fact that they are dimensionless ratios. The ratio of Z 8 and Z , can be either remembered from experience or estimated. This is also true for the ratio of the maximum values of 6 to ct a t trim. Finally, the ratio of .21s to - V,Zs can be approximated from the length of the missile. Length e is the single physical factor involved in the ratio of mass to pitch moment of inertia for a solid cylinder. With these three inputs and V,,, and M,, the stability derivatives are obtained

The characteristic polynomial for the autopilot can be derived from Figure 8-32b and equated to the normalized cubic used in Equation (8-95a).

By equating coefficients of like order and transposing terms that do not involve the control-loop gains, three simultaneous equations are obtained, which are expressed as follows in matrix form.

+ Za

~ A O A O

(8- 109) Here, bMs denotes the rate damping loop gain, acMs denotes the synthetic stability loop gain, and a p denotes the accelerometer loop gain. Gains a, b, and c are derived from these loop gains which are designed in Book 3 of the series. The loop transmission of the autopilot is one of a conditionally stable system. At low frequencies, the phase margin due to high attenuation rates must be large enough to accommodate the phase shifts due to high-frequency lags yet to be mod-

Advanced Guidance Sustem Design

540

Chap. 8

elcd. T h e LFPM is =90 - tan-'(ac~4l\l(bMa))~.For the purpose of computing u6 due to RIN, the transfer function of the fifth-order system in Figure 8-32b is identified as

Because of the complicated form of the definite integral over frequency o f the magnitude squared of the transfer function of KA = (1 + CIV,)-' from minus to plus infinity, the tabulated formula of Newton [I9571 is used to compute this function numerically. However, an approximate literal solution of lower order maintains the dimensionality of the results and is beneficial for showing the sensitivities to some of the parameters. Such a solution is derived by defining the low- and high-frequency asymptotes of Equation (8-1 10) and then integrating under these asymptotes using the third-order noise integral formula. This gives (approx.) us = 10.433 hV,Q;"

) h4, ] ] ~ I ~ ( T , T , ) ~ ' ~ ( T A O (8-111) )~~*]

In Equation (8-1 1I), each of the time constants in the denominator is some fraction of the guidance time constant, so that a5 can be shown to be inversely proportional to T;(2;~6) . Given that the high-frequency asymptote cannot continue to infinity with a sGpe of - 1. a multiplicative correction factor is constructed by integrating from a fixed frequency to infinity under the high-frequency asymptote and then subtracting this result from the result of applying Equation (8-1 1I). This correction factor, denoted by CF, is then In the preceding equation for CF, the fixed frequency is arbitrarily given a value that is five timcs the rate-loop gain, and a typical value for C F i s 0.9. Thc preliminary design procedure should concludc with the ability to roughly size up the control surface actuator. It is not possible to estimate the hinge moment, but the product of wi and .tls can provide a relativc reference relating to the size of the actuator.

Applications.

At this point, many of the preceding cquations are used ro obtain the numerical valucs for the parameters required of simulation. The tables containing numerical valucs that follow also havc refcrcnccs to the pertinent equations, and the order in which thc cquations arc bcing uscd can be identificd from thcir scqucntial listing in thc tables. The pararnctcrs in the tablcs are uscd in a sirnulation that makcs usc of thc adjoint method to gcncratc nliss distanccs. Thcn, a comparison is made bctwecn these miss distanccs resulting from simulation and those produccd by using the design cquations. Prior to invoking the design equations, the ALP constants and C U B A N adjoints arc uscd to obtain the constants for the CUSYN cquations. For thc examples that follow, a listing of the ALP constarlts and C U B A N adjoints is givcn in Table 8-5a. For the ALI' constants, the zorlal extremes wcrc uscd to dcfinc the operating points of the centcr of thc zone. Using a value of 3.86 for the ratio a/Qa corresponds to a valuc of unity for $. The miss

Sec. 8.4

Radome Error Callbration and Compematlon

TABLE 8-5

(Prom [Travers, 19861 wlth permission from AACC) a) Prwrarnrned Constants for Examples

541

ALP (Fig. 8-33b) 5.00 Kb = 3.43 P- 1.75 P b = 3.52 3.0 A = 13.6 4.0 11. = 1/12 q = 117 CUBAN (Table 8-42 and Eq. (4) of Table 8-4) M$+ = 1.21 M,,th = 1.87 Mti- = 6.59

p*

= = K' = K- =

= 1.24 Myyb = 1.70 M = 3.43 M .

M = 1.71 M ~ j b = 0.77 M*,.. = 0.48

b) CUSYN Equations a n d Design Point for Examples Shortened notation: J

5

6;'; L

E

vC+/-JZ; M umb; T

I

'rg

CUSYN Equations (Table 8-4 and Eqs. (8-97)) Inputs J.T: L,T: M,T: M. L: M, J: L. J:

L J L J L T

= JT'i3.854 = 3.854L/T2 = M/(3.505T1") = 581.8L51M'

;M ;M ;J ;T

= 0.9096JT5" = 3.505LT1"

1.0994MlT5" (MIL)'/12.29 = ~~"J~'~13.572 ; T = 1.039(M/~)~" = 1.963(LiJ)11' ; M = 4.911(Li"lJ"4) CUSYN Design Point (AT is defined in Missile Sizing section) Inputs: M = 12.5 ft. T = 0.65s. V, = 5000 ftls Outputs: L = 4.42 ftlsiHzl" J = 40.3 ftls3/Hz112 = 3.2 g, + : / I = RSD of RIN = 885 x 1 0 - " a d / ~ z ~ ~ ~ AT MISSILE SIZING Maneuver Inputs: scale Sg = 4, a,,, = I5 deg (cases I & Ic), V, = 3500 ftls factor Missile e = 17 ft, a,,, = 30 deg (cases I1 & IIc) leneth " Outputs: see lines 2. 3, and 4 in Table 8-5c =

=

c) Parameters for Examples (and Comparisons with Simulations)

Line Number

Example Parameter

Un~ts

I -

1 2 3 4 5 6

amax

Tmmax rTmln (est)~,, aA

PA

d% sec

62

-

la --

15 2.34 0.0625 11.86 2.5 2.3

15 2.34 0.0625 11.86 1.8 1.8

IIa

11 -

-

30 4.68 0.0313 11.86 2.5 2.3

Ref.lEq. #

-

30 4.68 0.0313 11.86 1.8 1.8

(8-99) (8-100) (8-101) (8-95d)

Advanced Guidance Sljstem Design

542

Chap. 8

(Continued)

TABLE 8-5c

Example

Line Number

Parameter

Units

1

la

11

Ila

Ref./Eq. #

sec sec

-

KA Vrn)

22

deg

LFPM w, =

rad/sec2

u6bMa U",

radlsec3 g DESIGN RMS MISS

rT/2

Sm

+

1.3

31.1

0

1.3 1.3

16.3 31.7

-

*The adjoints M,r+ arc divergent.

-'

(1

+ C!

Low frequency phase margin (8- 107) 15(8-I 12)

V ~ I T ~sec- '

mi

5

RMS MISS FROM SIMULATION

Sec. 8.4

Radome Error Callbratlon and Compensation

543

adjoints listed undcr CUBAN in Tablc 8-5a arc from Tablc 8-4a. Substituting these values for the adjoints in Equation (8-100), the ratio of maximum to minimum miss is expected to be 1.9. Table 8-5b lists the six CUSYN equations based on the adjoints of Table 8-5a. Now, in starting the examples, it is noted that any two of the four system parameters can be used as inputs. A step of target acceleration of 3.2 g with a random uniform distribution over the 6.5-sic interval before intercept is listed in Tablc 8-4b. From Equation (2-20d) T, = 0.1 sec, and from Tablc 8-5b r)' (an RMS value of RIN) = 2 millirad. The remaining portion of Tablc 8-5b deals with thc inputs for sizing the missile. Two values of a,,,.,,= 15 and 31)deg are used to highlight design sensitivities to this factor. The results of using the sizing equations are given in Table 8-5c, in which 35 numbered lines of data are arranged in columns that correspond to the examples and subexamples. The first line of data is for a,,, specified for the two designs. Lines 2-4 are the data that result from using Equations (8-99) through (8-101) to obtain (T,),,,, r ~ , , , , ,and (est) ow,,respectively. The decision is made at this point to design to the indicated values for (T,),,,, which is equivalent to placing the easiest requirements on the missile. That is, the missile is required to simply meet the specified maximum acceleration with no excess. The value of 11.9 g for a,,in line 4, for a missile whose capability is by definition only 12.8 g (4 x 3.2), clearly reflects a level of commanded acceleration due to noise that is not consistent with the requirement that the missile operate linearly. As a result, the two examples are further subdivided into additional cases, and, for each of the two missiles, different dynamics are now entered for the forward path. In lines 5 and 6, a* and PA, which are the cubic operating points of the autopilot, are equal to 2.5 and 2.3, respectively. The quadratic roots of the autopilot have a damping ratio 5.4 = 0.67 whereas for the other values 5.4 = 0.4. In lines 7 , and T, are determined in an iterative process which assures that the and 8, T ~ T,, equality of the actual .i, is equal to that required in Equation (8-105). They follow the rule of 118 closely, while 7, and T, are deliberately made larger in column Ia to reduce the value of a,, in line 26. The preceding remarks also hold for the other two columns respectively. Thus, the subdivision into four columns is motivated by the need to do something about (est) D,,. As a result, A and r b are determined. In columns I and 11, (est) a,, agrees well with the actual a,,.The only way to increase T, and T, would be to make the autopilot faster. However, increasing the speed of the autopilot by scaling TA"means that faster servos would be required and the value of a* would also increase. In columns Ia and IIa, these effects have been circumvented temporarily by reducing the quadratic damping ratio when the autopilot was made faster. In spite of this, the lower damping ratio in the autopilot will only make it harder to increase the order of the autopilot with the dynamics of the actuators, instruments, and structure, and so on. The final solution may involve reducing the amount of noise or increasing the time constant of the system. In lines 11 and 12, (Zs/Z,),, and (6/a),, are inputs used to estimate the stability - V,Zs, and I*. in lines 13-17. The differences in derivatives, M a , Ma, - V

544

Advanced Guidance Sustem Design

Chap. 8

values across the columns are connected with the different outboard profiles of missiles designed for two different a,,,,,. As a result (L varies by a factor of three. Lines 18-23 are attributes of the autopilot, and line 23 in particular lists the gain crossover frequency of the seeker loop gyro, which is related to T2. Lines 24-26 are related to the computed levels of control activity, while lines 27-29 list the RMs miss distance for positive, zero, and negative radome slopes as designed and as simulated for the four cases. By examining the values across the columns in line 29, it can be seen that the largest errors occur with negative radome slopes and with higher a.Also, the simulation results in lines 30-32 show that decreasing T2 fourfold t o approximate the effects of perfect body-motion isolation causes a reduction in miss distance, which is greatest for negative radome slope. A divergent increase in miss distance can be seen in line 30, columns Ia and IIa. This divergence is brought about by the onset of high-frequency instability at positive radome slope, a wellknown phenomenon that is discussed in Nesline and Zarchan [1984(b)]. The last group of simulations concerns approximating the effect of neglecting the airframe .. zeros. To this end, Z s was made zero and the miss distance consequently decreases as compared to the design values with no miss scale factor, as shown in lines 3335. Here, the largest percent reduction occurs with positive radome slope.

8.5 COMMAND VERSUS SEMIACTIVE HOMING GUIDANCE SYSTEM DESIGN AND ANALYSIS Comparison of command versus homing guidance is presented in this sccrion. Further details appear in [Alpcrt, 1988; Durieux, 1984; East, 1984; Neslines, 19861.

8.5.1 Modeling As notcd in Chapter 6, semiactive homing missile systems and command guided missile systems both continue to be in operation today. The analysis in this section assumes perfectly known ranges and closing velocity. Figure 8-34a is the standard linearized system block diagram of the missile intercept problem that uses synthetic PNG in the command guidance mode, which rcquires range estimate R to obtain the LOS angle. As shown in Figure 6-35b, synthetic PNG differs from standard PNG only in that the former rcquires LOS from the missile to thc targct to be mathematically cor~structcd bascd on fire-control radar mcasurcmcnts. Conscqucntly, i t contains the radar's mcasuremcnt errors of the missile and target. It is not rcquired to know thc rangc with homing systcms because thc geometrical LOS is obtained strictly from the physics of the situation. Therefore, homing systcrns do not have to perform any rangc measurements as is necessary in the casc of command guidance. Boresight error, which is proportional to the LOS ratc, is measurcd by the homing seeker. As command guidance docs not employ a scckcr, however, the LOS ratc must bc reconstructed from thc derivative nctwork (see Figurc 6-35b(i)). Figure 8-34a contains the firc-control radar noises prcsent in a

Sec. 8.5

545

Command versus Semiactive Homing Guidance System

MISSILE TARGET TRACKING NOISES TRACKING NOISES RANGE DEPENDENT NOISES

RANGE

b-pa&E NOISES

1 " , ' P O

-

AVC8 h T

I1

+

11s

IMPULSE ACCELERATION

Figure 8-34 (a) Command Guidance System Diagram with Noises (b) Adjoint Diagram of Command Guidance System with Range-Dependent Noise (c) Equivalent Adjoint Diagram of Command Guidance System with Range-Dependent Noise (Frotrr [Alpert, 19881, O 1988 AIAA)

command guidance system. Again, the system is assumed to be fifth order. All the noises are input at the point just before the division by the range estimate. The rate f, at which the fire-control radar tracks the missile and the target is assumed to be high enough so that the track loop can be considered continuous, and not sampled data. Before entering the guidance loop, all fire-control radar noises, which are in angular units, are multiplied by the range from the radar to the object being tracked (that is, R T for the target and R , for the missile). The adjoint system diagram (see Figure 8-34b) from the linearized system diagram, often called the forward system

Advanced Guidance System Design

546

Chap. 8

diagram (see Figure 8-34a). A multiplying factor of R'% is used for the track of the target (or R: for the track of the missile) between the point at which the signal leaves the adjoint of the homing loop as ~ C L Tand the place where it enters the integrator. System analysis of the dependence of range to intercept of Command guidance system performance is discussed in Section 8.5.2 where c l o ~ e d - f o r ~ steady-state adjoint solutions for statistical miss distance are derived for Command guidance systems that use PNG. Command.versus semiactive guidance system performance at specific ranges to intercept is discussed in Section 8.5.3.

Semiactive homing guidance dynamics. The guidance filter of Figure 7-6b is shown in discrete domain in Figure 8-35a. As in Chapter 7, y, is the input signal to the Kalman filter, which is followed by the guidance law and FCS to generate missile acceleration A,,. Figures 8-35b shows the complete homing loop dynamics for the semiactive system employing optimal guidance system (OGS). LOS angle u is sampled at a 100-Hz rate and then is converted to y, by multiplying by R. Since l? is assumed to be known perfectly, the seeker model becomes a measurement of y, corrupted by noise.

Command guidance dynamics. Figure 8-3% shows the complete command guidance loop dynamics employing OGS. The ground antenna and receiver

I

I

I I

I I I UALMAN F I L T E R

--

I------

(A)

Figure 8-35 (a) Discrete Kalman Filter and Optimal Guidance Law (b) Semiactive Homing Loop (Forward Time) (c) Command Guided System (Forward Time) (From /Nerlincr, 19861 with permirsio~ifnrm AACC)

Sec. 8.5

Command versus Semiactive Homlng Guldance System GLINT CORRELATION

547

-

ITS = SAMPLING INTERVAL. TN CORRELATION TIME CONSTANT)

GLINT NOlSE I (1 GL11121 I

9-

RANGE INDEPENDENT NOlSE

Of MISSILE FLIGHT CONTROL SYSTEM

DISCRETE ANGLE NOISE

LSGAGL+\T

GEOMETRY

SEEKER

Yr klml

100 HI SAMPLER

krml

J-% FILTER AND GUIDANCE LAW

- 4, \

-

100 HZ SAMPLER

ISEE FIG 2Al

ATMOSPHERIC NOlSE ROLLOFF FILTERS

THERMAL YOlSE

4"

-J I

RANGE INDEPENDENT NOISE

ATMOSPHERIC NOlSE I 0 ATM 1 4 I

Or DISCRETE ANGLE NOlSE

MISSILE FLIGHT CONTROL SYSTEM 1

E\GAGEUL\T GEOMEIRY

SAMPLER

GROUND RADAR

KALMAN FILTER AND GUIDANCE LAW

Ym

SAMPLER

(C)

Figure 8-35

(Continued)

I T ~ S + ~ , ~

-

Phq

-

-

100 HZ SAMPLER

TRACKER

-

ENGAGEMENT GEOMETRY

Advanced Guidance System Design

548

Chap. 8

measure O,,, and O T . Unlike homing, the radar must also measure various range infor~nationR T , R,,,, and R in order to convert Or and O,,, to y coordinates by y T = RTeT, >I,,, = ~ , , , 0 , , ,and , y,. = f T - f,,, to obtain the LOS angle u = y , . l ~(these measurements are referred to as R T , R,,,, and R ) . In the command guidance system, separate noise terms corrupt the measurements o f 0 T and O,,,. It is assumed that this missile has a transponder and therefore the measurement of O,, is perfect co~npared to that of 0 T .

Noise inputs. In addition to glint noise ~ , ~ i i , , ,semiactive , systems must deal with range-independent noise (RIN) af and range-dependent (or thermal) receiver noise v,.,, sources. The major noise sources (on the measurement of Or) in the command guidance system are atmospheric fluctuation of the signal from ground radar to the target u , ~ , RIN af on the ground radar, and range-dependent noise or,,on the ground radar. Thus, there are six major noise sources to be considered, three for the homing systeln and three for the command guidance system, each with different variance and frequency spectral content. The measurements (and hence the noise on them) are multiplied by R in the semiactive honling systeln and R T in the command guidance system to obtain y,, for input to the guidance loop. In the case o f the semiactive homing system. R, and hence the noise on y,, go to zero at the end of flight. However, the noise on 11, in the command guidance system, because o f its dependency on R T , does not fall off to zero as the missile approaches the target. The larger the distance alvay from the radar a t which intercepts occur, the larger the value of R7- and, hence. the more noise that enters the system. 111 this last respect, thcre is an advantage associated with the semiactive homing system. It is shown that the range dependence of the noisc sets basic limitations on the effcctivencss of a command guidance system. Glint Noise a,li.,. The 1'SIl +,, of the target glint )I, is givcn by Equation (2-20b). )I,, is divided by R in the senliactive homing case or R T in the command guided case to create u,,,,,,,.Uccause glint noise is quite small in the command guidance since the receiver never gcts close to the target (that is, R T is always large). this noise can be neglected in command guided systems. Range-Independent Noise ( R I N ) a,. The semiactive RIN I'SD is givcn by Equation (?-?(Id). In c o n ~ c r t i n ga to 11 the It1N noisc is multiplied by R and 11cncc approaches zero a t the end of the flight. Uccausc the ground rcccivcr is usually c'r bcttcr quality than the missilc-borne rcccivcr, the I'S1) of v, i n the command guidance is smaller than that of thc semiactive systcm. Thc I'S11 of or in con111la11d guidance is 6, = ~ , , ~ . , . l\rlicrc f, rloy = (B,,.lBs,c)', f , is the data rate in Hz, B,,, is thc bcamwidtll of radar aperture, and BsH is the heam-splitting ratio for large signalto-noise ratio (SIN). Range-Dependent Noise a,.. and Ward, 19841 as r,, =

The variance of a,,can bc computed \Barton

(B,.)'I[~K;,(SIN)],

K,,, = figure of merit

Sec. 8.5

Command versus Semfactiue Homing Guidance System

549

Since SIN is a function of range, thermal noise is likewise range dependent. SIN can be calculated from the semiactive radar range equation as follows:

wherc K is the SIN a t the reference ranges RTI)and Ro. Rn, is the target-to-illuminator reference range, and Ro is the target-to-missile reference range. Substituting Equation (8-114) into Equation (8-113) yields the same form of variance as in Equation (2-300 as r,,,.q,., = r ,,,, [ ~ r ~ l ( R m ~ =r ,r,,,,,(~lR,,)' )]~ wherc r,,,,,,, is the variance of thermal noises at R T = Rn, and R = R,,. I11 this section R m = 124,000 ft, Ro = 16,000 ft, r,,,,,,, = 10'' rad'. Then the semiactive receiver noise PSD $r,,S;I is given by Equation (2-20e). The command guided S/N is

where K, is the SIN at the reference target to illuminator range RREF. The reference range is determined from radar system parameters. Substituting Equation (8-1 15a) for the SIN into Equations (8-113) and (2-20e) gives the command guided rangedependent noise variance and PSD (variance) r,,,, = r,,, ,,*,,, (RTIRREF)~.

(8- 115b)

where

and 2 T,,, = llf,, and o r is the radar cross section in m2. Before being converted to a positional noise, the command guidance angular thermal noise, which is already a function of the square of the range from the radar to the target, is multiplied once ~ ~missile ). carries a beacon, more by the same range $,nc, = @ m ~ R E F ( ~ $ / RIf $the where $ R D ~ Jand ~ ~ RREF ~ are defined for the then $mC = beacon. Another factor that causes deviations of the apAtmospheric Noise UAN. parent target position as seen by the radar tracker is atmospheric perturbation of the radar signal. Atmospheric noise statistics are taken from experimental tracking tests from which empirical formulae are derived [Barton and Ward, 19841. This effect, while very small in the semiactive system, may be quite significant in the command guidance system. The former is true because of the short missile-to-target where @ A N , ~ F = distance. One way of computing $AN is $AN = $AN,&* ~ T A . V I A Nf~,~/~~ " R* ~ , is the path length in m to tracked object in lower 5 k m of atmosphere, w is the antenna aperture in m, (standard deviation of U A N ) rykner:=

550

Chap. 8

Advanced Guidance System Design

.44 x lo-' rad, T A N is the atmospheric noise correlation time (0.6 sec), and f, is the correlated noise coefficient (0.4, see Alpert [I9881 on correlated noise effects).

Total Angular Noise PSD 4,. The total angular noise PSD 4, for the semiactive homing guidance is given in Equation (2-23), while that for the Command guidance is given by

8.5.2 Miss-Distance Analysis for Command Guidance

The treatment described here, based on the excellent work of Alpert [1988], derives new analytical equations based on adjoint theory for statistical miss distance caused by target maneuver, range-dependent, servo, glint, and atmospheric noises for command guidance. An optimal total guidance system time constant T ,is also derived which yields the minimum statistical miss distance, taking into account realistic constraints on the minimum achievable T,. The steady-state adjoint solution for miss distance is developed here only for the RIN. Looking at Figure 8-34a, using the binomial theorem,

Using Equation (8-117), Figure 8-34c can be developed from Figure 8-34b. The output of the integrator with 1,: as a factor in the integrand reaches a steady-state value a t :,,, that is, K,,T,;;-', where K,, are the normalized adjoint coefficients given by Kt =

,

( )

t+=o

I

*

)

d *

t* = t, is the adjoint time

(8-118)

Consequently, the steady-state adjoint mean-square (MS) miss distance due to range dependence is:

This same procedure call bc used to derive the miss distances for atrnosphcric I I O ~ S C and R1N. When this is done, the steady-statc adjoint solution for MS miss distance caused by target maneuver is o ~ , , ~ . ~ ,=, , b,K,\I , . ~ 7,: whcrc K.u = 7 ,'$' 7.

I*==('

[J;

C.(J;

, ~ , ; ~ ~ ( tdl* *)

+

)

1 dl* dl*

I

dl*

Table 8-6a contains the values of normalized adjoint coefficients K,, through and K M for A = 3.0, 3.5, and 4.0. Note that KO = M:, KA, = M$,, and K2 = M:,-whcrc M,, il4i-,. and M,,-arc dcfincd in Figurc 6-42. Thc stcady-statc adjoint

Sec. 8.5

Command versus Semlactlue HomIng Guldance System

551

equations for the mean-square command guidance miss distance due to each noise source arc given in Table 8-6b together with those for target maneuver. Table 86b also lists the corresponding equations for the scmiactivc homing guidance case. It 1s assumed from here on that only RIN and atmospheric noise need be considered. This could be the case, for example, if the missile carries a beacon. The variance of the total miss dist'lnce is the sum of the variances of the miss distances duc to each noise sotlrcc and due to target mancuvcr. That is,

A good approximation to Equation (8-120) that keeps the largest terms of each polynomial is given by

xx ++

where can be identified as the sum of spectral noise densities a t the intercept range. corresponds to +,, in Equation (2-23) for homing guidance. Equation (8-121) shows that the variance of the miss due to the noises is approximately inversely proportional to T , ~and , the variance of the miss due to target maneuver is proportional to the fifth power of T,?. The optimal time constant for minimum RMS miss distance can be found by setting the partial derivative of the expression with respect to T~ equal to zero. This gives Optimal time constant:

Minimum RMS miss distance:

(T,~),,,

+ (K"C K.LI+~ ) [ ( ox 4 1.25 'I6

= 0.2

U . U I ~ ~ ~= ,~\.

.

I

)]1'2

(8-123)

For this optimal condition, maneuver miss distance accounts for 41 percent of the total RMS miss distances. Note that (7JOPT in Equation (8-122) and UMISS.~IJ.V in Equation (8-123) have the same formulae as in Equation (7-34) for T and (7-35) for a,, (9, respectively, for homing guidance.

Aduanced Guldance System Design

552 TABLE 8-6

Chap. 8

(From [Alpert 19881,O 1988 AIAA)

-

a) Normalized steady-state m i s s d i s t a n c e adjoint C) Parameter values u s e d in t h e examples

coemcients I

1 I

KI K2 K3

4.4 9.2 25.9 89.7 365.0 1707.0 3.77

K4

Ks K6 K.w

6.7 15.6 47.6 176.0 742.0 3750.0 2.38

10.0 25.7 84.8 336.0 1541.0 7979.0 1.98

b) Steady-state adjoint m i s s distance e q u a t i o n s Command guidatire .for farget fvatka

2

~ M I S S Y I . Y= ( b r K . ~ ~ ~ ;

Homing guidai~teb

BW

= 0.035 rad (2 deg) S/.YKEF = 100 (20 dB)

Beamwidth Reference S/.V a t reference range RREF RREF = 50,000 m Reference range u .r = 100, 10, 1, 0.1 m2 Radar-crosssection f s = 40 Hz Data rate BSR = 80 Beam split ratio for large S/,V U', = 10m Wing span of target TGLT = 0.08 s Glint noise correlation time UI = I m Antenna aperture 7.4= 0.6 s Atmospheric noise correlation time" Navigation ratio A = 3 v . ~ = 1500 m/s Missile \relocity VT = 450 mls Target velocity (positive for incoming target) 11 T = 19.6 mls' (2 g) Target maneuver level T,tr = 2.5 s Target maneuver timeb ( T ~ ) . \ I I . ~ = 0.5 Minimum achievable system time constant' U,,C~:LIM = 98 nils2 (10 g) Maximum allowable missile acceleration causcd by noised #

' For missile track MI' replaces I/.,.. Rb = R I ~ ~ ~ R MtheT product , , ~ ~ of the illuminator-to-target range and missile-to-target range at which thcrc is S/&cc~ on a 1 m 2 target.

' Assunied target altitude is helow 5 km, hence R' = It. For a Poisson-distributed step targrt nlancu\'er with average time between maneuvers TI., use T.rf = 0.25 T p ; here T p = 10 s. ' Used only in numerical Examples 2 and 3. * Used only in numerical Example 3.

Sec. 8.5 TABLE 8-6

Command versus Semiactive Homing Guidance System

(Continued) d) Normalized steady-state acceleration ad. joint coefIlclents K;(nr) for a flfth-ordersystem with equally distributed time constants

Asymptotic system performance. At this point (following discussions in Alpert [1988]), a set of equations is developed to describe asymptotic miss-distance performance, each of which represents U.~rlssLrr, which could be achieved if only one class of guidance noise were present. Sources of guidance noise influence mindefined by Equation (8-121). Since, howimum RMS miss distance through ever, the relative magnitudes of the terms in 2 4 depend explicitly on intercept range, at any particular intercept range one noise term will dominate. The u . t ~ l s s . , , , , that would be expected if only this dominant noise terni were considered along with the maneuver term is what defines asymptotic miss distance. At the shortest intercept range, glint dominates and miss distance is independent of intercept range. Using the notation (CLT) to indicate that the glint noise dominates,

x+

As intercept range is increased, the RIN miss distance eventually increases to be equal to that of glint noise miss distance, at a range of

Beyond this range the asymptotic miss is 1.25[K~+,(2Ko+f)5 11 / 1 2 ~r 5 / 6

~ M I S S M ~ X ~=R ~ ~ ,

(8-124c)

In this region, where the RIN dominates, the asymptotic miss distance is proportional to the five-sixths power of intercept range. In the next region, where the intercept range is larger than Rlz =

(8-124d)

+fl+~~nap

the atmospheric noise dominates and the asymptotic miss distance is ~ M I S S M I N ( A ~\)= >

5 1/12

514

~ . ~ ~ [ K M + S ( ~ K O + A] N RREI F )

(8-124e)

Advanced Guidance System Design

554

Chap. 8

The miss distance is proportional to the fire-fourths power of intercept range. T h e RIN dominates at intercept ranges larger than 113

R13 = ( 2 s T + A . V ~ E ~ - R ~ E F ~ ~ R D N H U F )

(8-124~

The asymptotic miss distance is

The miss distance is proportional to intercept range to the five-halves power. T o draw asymptotic miss distance on log-log paper, it is only necessary to assign values to the various system parameters, evaluate the critical range points of Equations (8124b, d, and Q, and compute the miss distances of Equations (8-124a, c, e, and g). Between regions, only the slope of the miss distance curve with respect to intercept range varies. These miss distance curves (Figures 8-36a, b, d, e) are asymptotes of the minimum RMS miss distance that contain maneuver and the largest noise contributor (that is, glint, range-independent, atmospheric, or RIN noise).

Numerical results presented here illustrate the methodology presented earlier. Table 8-6c contains the numerical values of system variables where 4, = r~$iT.,,is the target acceleration PSD and r 7 and ~ T,,, are defined in Table 86c. Figure 8-36a is the plot of the asymptotic miss distances for the example, while the cxact solutions are presented in Figure 8-36b as curves with dots over a copy o f Figure 8-36a. The cxact solution is calculated from Equation (8-120). It can be seen that the asymptotes and the exact solution agree well ovcr most of the curves except near breakpoints in the asymptotes. Figure 8-36c contains thc time constants, ( 7 . c ) ~[SCC ~ ' ~Equation (8-122)], corresponding to thc csact solutions of Figure 836b. Also displayed are the asymptotes of ( T ~ ) ~ ) I >calculated T from the mancuver spectral dcnsity and thc donlinatlt noise spectral density.

Example 8-4.

System time constant and missile acceleration limits.

The rcsults ofthe prcvious section arc in rcality subjcct to limitations on the minimum achievable total T,? and niaxit~lumachicvablc tnissilc accelcration. Thc dcvelopmcnt so far implicitly assumed that therc was no bound on the achievable T , nor on the achicvable nlissilc accelcration. 111 real physical systems, missile autopilots and airframcs hecomc difficult, if not impossible, to dcsign as thc autopilot-airframe response-time constant requircd bccoincs very short. This irr~poscsa lolvcr bound on the achievable (T,~),,,,,,. (7,C)c)1J-1. defined prc\,iously is only \.slid as long as it escecds (T,),,,,,,at cach

Figure 8-36 (a) A s y n t ~ t o r i cMiss 1)istartccs o f l>ornit~antNoise atid Matieuvcr k)r Examplc 1 (b) I)o~~litianrNoisc and Mancuvcr Miss Ilista~iccAsymptotcs and Exact S o l u t i o ~for ~ Examplc 1 (c) Optimal Timc Constatits and Asymptotcs for E x : ~ n ~ p 1l r (d) Noisc-Mar~cuvcrAsymptotcs \\.it11 Minimum Tirne Constant Constralllt and Exact S o l u t ~ o n ,Example 2 (c) Noise-Mancuvcr Asymptotcs with and witliout Acrclrrati(~nLimits nnd Exact Solution, Exaniplr 3 (1:rd111 /Alprrr, 191181, 0 19x8 .41.4.4J

Sec. 8.5

555

Command versus Semiactive Homlng Guidance System

\ '~lss.,.,,,,,

2 1

I,,

3

5

,

, , ,

10

20

30 40 50

INTERCEPT RANGE (km) INTERCEPT RANGE Ikm) (A)

--

(B)

25 20

I-

2' a

15

z

0 10 U

EI-

08

,0 6

2

O5

I-

n 04 0 03

< D

0

C

02 3

5

10

20

30

4 0 50

100

INTERCEPT RANGE Ikm) (D)

INTERCEPT RANGE Ikm) (C)

-

1

3

"WIS~MINI~IN,

6

10

20

304060

INTERCEPT RANOE Ikm)

(E)

100

100

Advanced Guidance System Design

556

Chap. 8

operation point (that is, intercept range). Equation (8-121) shows that, for 7 , greater than (7,)0pT, the miss distance is determined mainly by the maneuver miss. consequently, the asymptotic miss for the minimum achievable 7, is approximately u ~ ~ ~ ~ ~ < ~ = ~ ( (T + ,q~ ~. , ,, ) w~ ' ,~ ~( 7~~ )where 2 : ~ the notation MIN[(T,),,,,] indicates that the minimum miss distance is set by the minimum time constant constraint.

Example 8-5.

The asymptotic miss corresponding to (7,),,,;,, = 0.3 is plotted in Figure 8-36d, in which it appears as the horizontal line at the 4.2-m level, The only difference between this example and Example 8-4 is that the time constant limitation is now included. Exact solutions [Equation (8-120)] are represented as curves with dots, and it can be seen that there is good agreement bet\veen the asymptotes and the exact solutions. Comparison with Figure 8-36b indicates that the minimum 7, limitation increases miss distance only at short ranges. Another realistic limitation that must be taken into account, as discussed in Alpert [1988], is the maximum achievable missile accleration. If the guidance system were to achieve more acceleration than the system design limit, the resulting miss distance would probably be very large due to component failure. If internal limits imposed on the acceleration command are exceeded by noise-induced acceleration commands, then no acccleration capability is available to counter target maneuver. This would also producc large miss distance. Good design practice, therefore, avoids acceleration saturation as shown previously. Here, it is shown that acceleration saturation due to noises can be avoidcd by i~nposinganother minimum timc constant constraint that is depcndent on system parameters and intcrccpt range. The adjoint technique can be used to develop cquations for the RMS accelerations induccd by noises. Using the same adjoint systcm as for the miss distance. the impulse is instcad applied at thc point corrcspo~~ding to commanded accclcration in the basic fortvard system diagram. Thc adjoint rcsults then bccornc accclcrations instcad of miss distances. Follo\ving the previous dcvclopn~cnt,which deals with the steady-state adjoint SOlutions, equatiolls can be dcvclopcd for IIMS accelerations that are similar to the equations of Table 8-6b. A ncw set of normalized steady-state adjoint coefficients K;, arc found by applying the impulse at A, whcre A is the r, bcforc intercept at which the acceleration occurs. Thesc coefficicnts increase with dccrcasing (,, or as thc point of intcrccpt is approached. Thc rcsulting RMS comtnandcd accclcration is writtcn as a polynomial cxprcssion in tcrnls of missile and target vclocity and missilc T , ~ .Once again, keeping only the dominant terms, thc variance of the cornmandcd accelcratiorl duc to noise alone can bc approximated as uXcc:(t,v) = 4)K;,(/,,.)7,<'. It is ncccssary to hold thc comrna~~dcd accclcration bclow a specified value only until about onc 7, bcforc intcrccpt. Thc rcason is that any commands issucd aftcr that timc would have littlc cffcct. Table 8-6d c o ~ ~ t a i numerical ns values o f K;, for 1 , equal to 7 , to 3 7 , and for thc navigation ratios A = 3, 3.5, and 4, for a fifth-ordcr systctn with cqually distributcd timc constants. Thc variancc of the noisc-induccd accclcration com~nandis invcrscly proportional to the fifth power 0f T~ The minimum T , ~ ,thcrcforc, that causcs thc rnissilc to avoid accclcration saturation at the spccificd r , is givcn by T A C C L I . ~ , = ( ( ~ ~ ) [ K ; ~ ( ( , ) I U $ C C Lwhere IM])'~~

(x

Sec. 8.5

Command versus Semlactiue Homing Guidance System

557

u..,cc~f.,~ = specified limit 011 acceleration saturation caused by noises. It should be ren~enibcrcdthat this timc constant constraint is imposed only if the computed time constant is larger than (T,~),,,,,,and ( T ~ ) O I > Tto, achieve minimum miss distance. As happens with longer ranges, the timc constant must son~etirnesbe constrained by accclcration limits. In this case, miss distancc will be determined mainly by the maneuver RMS miss distancc, which is uL1,ss,,,l.,z~..,,:, :,,,,,, = ( ~ b ) b 3 ~ . + , ~ X ( t y ) l u ~ ( : ~ : ~where . ~ . , ~the notation ..ICCLl.LI indicates that thc minimum miss is set by the acceleration limit. Conscclucntly, by requiring the missile to remain below the acceleration limit. the noise level determines the maneuver miss which dominates the total miss distance.

Example 8-6. The curve with dots in Figure 8-36e shows the exact adjoint solution for the 10 dBsm target of Example 8-5. For this curve, T ~ c c L l n is l used as T whenever T.A.(:CLI.LI > ( T , ~ ) Oand ~ T (T,),,,,,, for the case where UACCLIM = 98 m l sec' at !,= T,,, the acceleration command due to noise is limited at one time constant before intercept. The plotted solid line asymptotes are the same as in Figure 8-36d, and the dashed lines are the asymptotes of maneuver RMS miss distance, ( T ~ ~ ,,, r., .~ , 2 iS , , ~ r .,,.,,,r, presented in Example 8-3. It can be seen that the exact solution is close to the maximum of the dashed and solid line asymptotes. 8.5.3 Analysis of Optimal Command Guidance Versus Optimal Semiactive Homing Guidance Using adjoint theory, this section presents an analysis of optimal command guidance versus optimal semiactive missile guidance based on an excellent paper by Neslines [1986]. The Kalman filter and optimal control law gains are both generated in forward time and read back in reverse time for the adjoint run. The command acceleration M, is an input to an analog third-order FCS which generates the achieved missile acceleration A l W yUniformly . distributed target maneuver shown in Figure 2-21d is assumed in this section. The target acceleration PSD +,is given by Equation (2-39).

Noise inputs. The noise inputs for the analysis in this section have been discussed previously except the following. All noises are considered to be on the discrete measurement o f u or 6T, as indicated in Figures 8-35b and 8-35c, the discrete measurements are made at .O1 sec intervals.

+,

G l i n t Noise. Glint noise PSD is obtained by applying Equation (2-20b) assuming a medium-size airplane target with YCLT = 25 ft2 and TCLT = O.j/Hz. Range-lndependent Noise ( R I N ) . sampled at a 100-Hz rate. Range-Dependent Noise.

ft.

+f

is assumed to be 2.5 x lo-'

Y,,,~,,,. = 4 X

rad2 and RREF= 1.28

rad2, X

10'

Advanced Guidance System Design

Chap. g,

Atmospheric Noise. Using empirical data presented in Barton and Ward [19841, the variance of atmospheric noise for a target at 20-deg angle above the earth is

where L , is the length o f radar beam in lower troposphere (15,000 m), and Wl is the radar aperture in meters (3.05 m). Therefore from Equation (8-125), r,!& = . o l mr. The atmospheric noise PSD $A.v rolls off sharply with frequency. This rolloff may be approximated by two first-order low-pass filters with frequency breaks a t f,.,which can be calculated using where f,,is the cut-off frequency of two first-order filters and V Ais the transverse velocity of radar beam through atmosphere (measured in mls). The break frequency f, calculated from Equation (8-126) for a roughly stationary radar beam and a normal atmospheric drift of about 3 mlsec is 0.46 Hz.

Kalman filter analysis. The square root spectral density &A''(defined in Equation (2-23) for homing guidancc and in Equation (8-116) for command guidancc) is plotted in Figure 8-37a for the short-range intercept case and in Figure 837b for the long-range intercept using different scales. For reasonably stationary signal and noise statistics, the transfer function from the input y, to the output y, o f thc Kalman filter is given by Equation (7-34) whose F is plotted in Figures 837c and 8-37d for the short- and long-range intercept, respectively. As already mentioncd, the homing +:I2 dcpends on missile-target range, and the variation in this quantity during the flight, for the exarnplc here, is shown in Figure 8-37a. Hcrc, 4;;' is grcatesr at the beginning of flight because of the thermal noise and dccrcases monotonically throughout the flight's duration. Thcre is an increase in angular glint noise at the end of the flight but is small compared to the thermal noisc. Because +L'%s low for short-range intcrcept in both systcms; cach has rather fast Kalman filter. Still, thc filter in the command guidance is faster. Owing to the largc amount of thcrmal noise, F of the homing is small at the beginning o f flight but then slo\vly increases as a result of the decrease in the noisc as the missilc draws nearcr to thc targct. F of the homing is, in fact, a function of thc SIN. While the state cstimatc is poor a t the beginning of flight \vhcrc F is small, it bccomcs quite good as thc missilc approaches thc targct. O n e rcason for which thc Kalman filter ~ O S S C S S C Sthc bcst pcrformancc is that, a t thc bcginning of thc flight, its large time constant for dealing with thermal noisc docs not influcncc thc miss due to target nlancuver at thc end of t11c flight. For long-range intcrccpt, 4;' in the scrniactive homing guidancc is at first larger than that in the command guidance but later bccomcs smaller. This is a result of thc forrncr's dcpcndcncc on R, which as stated bcforc, goes to zcro. F of thc homing in thc semiactive systcm increases to approxinlatcly thc same valuc as in the short-range casc, whilc F in thc command guidance remains low throughout thc flight bccausc thc I'SD o f thc rncasurcnlcllt

Sec. 8.5

Command versus Semiactive Homfng Guidance System

20

I

18

!

559

I

1

i

12

~

8 4 0

1

COMMAND

0

1

2 3 TlME (SEC)

4

E TlME (SECI

(A)

(B)

. n

2-

32

! 2.8

!

I

2.4

!

z Y a

0

-

-1

1.6 1.2

I

0.8

COMMANO

K

u 0.4 k

2

2o

'

I

I

I

0

TlME (SECI

HOMING

!

3

O O

2

4

6

8

TIME (SEC)

(C)

Figure 8-37 (a) Command Noise Is Less than Homing Noise at Short Target-toIlluminator Ranges (b) Homing Noise Becomes Less than Command Noise at Long Target-to-Illuminator Ranges (c) Commmand Characteristic Frequency Is Higher than Homing at Short Target-to-Illuminator Ranges (d) Homing Characteristic Frequency Becomes Higher than Command at Long Target-to-llluminator Ranges (From [Nerliner, 19861 with permission from AACC)

noise is a function of R r , which remains large. If the filter is slower than, say the guidance law and FCS, then its speed of response will limit the overall system speed of response. Consequently, the miss distance and required accelerations against maneuvering targets will also increase. If the filter is much faster than the other components of the system, system response will be for the most part determined by the slower components. In this case, filter dynamics will have little impact on performance.

Performance analysis.

A reasonable measure of miss distance and acceleration requirement due to a maneuvering target can be obtained by assuming a l-g target acceleration, which can occur at any time throughout the flight with

10

560

Advanced Guidance System Design

Chap. 8

unifor~nprobability. Because the autocorrelation function of such a model is the same as that of integrated white noise, adjoint theory can therefore be used to calculate RMS statistics due to this target maneuver model [Fitzgerald and Zarchan, 19781. The RSS of the RMS miss distance and acceleration requirements for all sources are used as measurements of performance. This results of the short-range intercept scenario, in which intercept occurs 15,000 ft from the illuminator, are summarized in Table 8-7a. The largest contributor to miss distance and acceleration requirement for the semiactive system is glint noise. Because of the target-dependent nature of glint noise, it is not likely that the contribution to total miss distance and acceleration requirement from this quantity could be improved dramatically by improving the system components or the Kalman filter and guidance law. Random target maneuver is second to glint noise in its contribution to miss distance and high acceleration requirement. Thermal noise, which is dependent on ( R r ) * ( R ) ,represents the smallest contribution. However, the effect of RIN is also very small. Total RSS miss distance and acceleration requirement are roughly 6 ft and 10 g, respectively. The total RSS miss distance and acceleration requirement for the command guidance in the short-range case compare closely to that of the semiactive system. In this casc, total RSS miss distance is between 3 and 6 ft, and the total RSS acceleration requirement is between 5 and 7 g, depending on the atmospheric noise components. The table shows that these components have the largest impact on total RSS miss distance and acceleration requirement. Since they are determined by the intcrcept scenario, they do not fall under the control of the missile systcm designer. Although RIN and random target rnaneuvcr also contribute to miss distance and accclcration rcquircment, their values are rather small. The results of thc long-range intercept arc shown in Tablc 8-7b. This interccpt occurs 130.000 ft from thc illuminator. The overall pcrformancc of the semiactive system at the long-range intcrccpt docs not diffcr radically from thc short-range intercept. Thc contributions from glint noise, RIN, and random target maneuver to performance basically follow the same pattern as in the short-range intercept case. The target mancuvcr portion is very nearly the same because the filter and other system components arc not much different from those found in thc short-range intcrcept casc. Howcvcr, thcrmal noisc beconlcs a bigger contributor to both miss distancc and accclcration rcquircnicnt in the long-range intcrccpt casc. The increase in each is a result of thcir dcpcndencc of S T , thc target-to-illu~ninator range. If the scckcr is of poorcr quality, thermal noisc may bc an inlportant contributor to total miss distancc at long-rangc intcrccpt rangcs. For t11c command guided system, thcrnial noisc is thc most significant contributor to high miss distance and largc g's sincc from Equation (8-1 15) command guidancc thcrnlal noisc miss and accclcratjon are dcpcndent on R$- which is still largc ncar intcrccpt. Long launch rangcs suffer from corlsidcrablc pcrformancc dcgradatiou. If thc radar bcani to thc targct is stationary, thcn miss distancc and accclcration rcquircmcnt duc to atmospheric noise can also bc quitc largc. As with scn~iactivcsystcms, thc missilc systcm dcsigncr docs not havc control ovcr this componcnt. Unlike the short-rangc interccpt case, RIN is thc lcast important contributor. I
TABLE 8-7(A) MISS DISTANCE AND ACCELERATION ADJOINT RESULTS: SHORT. RANGE INTERCEPT (8) MISS DISTANCE AND ACCELERATION ALLJOINT RESULTS: LONG-RANGE INTERCEfT Prom [Nesiines,19861 wlth permission from AACC

"om% Source

~

o

c

Comnd

.

e

nias(ft)-

l (9'8) .

I

k c e l (g'a)

I

I

Homing Thermal Noise

I

Glint Noise

.

Homing Range Independent Noise Command Thermal Noise Atmospheric Noise Comand Range Independent Noise

!

Random Target Maneuver

2.3

6.1

XIS

2.4

1.3

9.8

6.0*(3.1)'*

7.0*(5.1)'*

Low Frequency Atmospheric Noise Model

" High

Frequency Atmospheric Noise Hodel

b ) MISS DISTANCE AND ACCELERATMN

ADJOINT RESULTS

I

I

\

Earning miss (ft) Accel. (g's)

Source

j

Homing Thermal Noise

0.9

2.9

Glint Noise

5.6

7.8

Homing Range Independent Noise

0.5

1.3

Command Thermal Noise Atmospheric Noise Command Range Independent Noise Random Target Maneuver

RMS

**

I Camnand Miss (ft) Accel (9's)

I

I

I

0.9

5.8

/

2.4

8.8

L

I

- LONG RANGE INTERCEPT

Low Frequency Atmospheric Noise Model High Frequency Atmospheric Noise Hodel

14.2

45.4*(32.8)**

5.5

22.0*(19.6\**

562

Advanced Guidance S ~ s t e mDesign

Chap. 8

to miss distance and high g's as a result of a slower Kalman filter whose gains have been reduced by the large noise in the system. Overall, the performance of the command guidance has degraded in comparison with the semiactive system results for this long-range engagement.

8.6 ANALYTICAL SOLUTION OF OPTIMAL TRAJECTORY SHAPING FOR COMBINED MIDCOURSE AND TERMINAL GUIDANCE This section presents a combined midcourse and terminal guidance law design for missiles to achieve range enhancement with excellent intercept performance. Analytical solutions of a closed-loop, nonlinear optimal guidance law for three-dimensional flight for both the midcourse and terminal phases are derived. This combined guidance law can quickly modify the missile trajectory during midcourse guidance when the target direction changes. Zero heading error is achieved a t handover from the midcourse to the terminal phase. The guidance algorithm is in a feedback form, either in inertial coordinates or in seeker coordinates. It is sufficientlv simple for onboard ilnplementation and has been applled successfully for on-line operation. In the past, both L Q guidance laws [Glasson and Mealy, 19831 and various nonlinear guidance laws [Cheng and Gupta, 1986; Lin, 1983; Lin and Tsai, 1987; Menon and Briggs, 19871 have been proposed for the midcourse phase. Glasson and Mealy [I9831 constructed an approximately optimal midcourse guidance law in which the kinetic energy loss is ~ninimizedb5* time scheduling the guidance navigation ratio. Cheng and Gupta [I9861 and Menon and Briggs [I9871 used singular perturbation theory to develop implementable closed-loop guidance laws, the former based on minimum time. the latter on both flight time and terminal specific energy. As these examples indicate, research 011 midcourse guidance has focused on deterministic optimal control formulations. Details of these results that can also be applied to an air-breathing engine are presented later in this section, while other midcourse guidance results are presented in Section 8.7.

8.6.1 Optimal T r a e c t o r y Shaping Guidance A combined missile r~.ridcourscand terminal guidance law design that maximizes the final vclociry can lcad to range erthanccment of lnissiles as presented in Chapter

6. Thc midcoursc guidancc law normally represents a form of proportional navigation with the proper trajectory-shaping charlges to ensure a n~inimizationof the energy loss. Many guidance studies of long- and medium-rangc missiles indicate that optimal trajectory shaping ensures an extended range i n v o l ~ i n gmore be~lcficial cndgamc conditions. For this optimal trajectory-shaping guidance law design, the cornbincd guidance law for the midcoursc and tcrlninal phase is developed in this section using optimal control tcchr~iqucs.Ho~ve\.er,dircct application of the optil1la1 control theory will result in a two-point boundary valuc problem. The problem

Sec. 8.6

Analytical Soiutlon of Optimal 7Yq)ectory Shaping

563

can be further complicated with the lift, thrust and drag, and control constraints forced by structural and angle of attack limits. This increases the computation time so that it is not feasible to implement the resulting solution of the missile performance on a digital computer. In view ofthis complexity in the problem setup, either indircct methods [Lin, 1987(a)]or direct methods based on nonlinear programming are used to solve the sensitivity and convergence problems. However, both methods require very fast onboard microprocessor technology for real-time, on-line operation. In this section, a simpler, more easily implementable feedback guidancc law for aerodynamic control is sought that would surpass the performance of direct and indirect methods. This closed-loop, nonlinear optimal guidance law is applied to both the midcourse and terminal guidancc phases. Its solutions arc derived to satisfy the need for a real-titnc guidance law that can be easily implemented on line within the computational capabilities of current microprocessor-based onboard systems. It has, in fact, been applied successfully with on-line operation and has led to an overall improvement in missile performance. 8.6.2 Problem Formulation

The present position and velocity of the missile are expressed as ;(to) = (xo, yo, 2,))and v(t,,) = (.kg, i o , 2") where x is the longitudinal position, y the lateral position, I the altitude; the dot represents derivative with respect to time, and the subscript 0 represents the present time. The flight guidance problem of the missile is to find an acceleration command vector ii,(t) = [a,,(t), ay,(t), a,,(t)] for to 5 t 5 tf such that at if, 't(tf) and v(tf)would have reached a prespecified terminal condition with some cost function optimized. The subscript c denotes command and the subscript f denotes the final time. First, simple guidance gains for the explicit guidance command equations in inertial coordinates are derived. This technique is introduced in Cherry [1964]. Then optimal control is applied to find the analytical optimal guidance gains. The steering equations with simple guidance gains are explicit functions of the current and the desired boundary conditions (position and velocity). Hence, this guidance law is known as explicit guidance. Details of explicit guidance are provided in Cherry [1964]. Here, the derivation is briefly summarized. The equations of motion with respect to a planet-centered, inertial Cartesian coordinate system are

Integrating x(t) from the present time to to general time t will give t

) - (to) =

Jto ?(T) dT

564

Advanced Guidance System Design

Chap. 8

By integrating Equation (8-129) from to to tf,

where time-to-go t, = tf - to. From Equation (8-129),

T h e unknown function 2(t) can be expressed in a generalized Fourier series. However, Equations (8-130a and 8-130b) and (8-137) can determine only two of these undetermined coefficients. In other words, there are only t w o coefficients C1 and C2 of x(t) which satisfy Equations (8-130a and 8-130b) such that

where Pl(t) and P2(t) are linearly independent, prespecified functions of time. For implementation simplicity,

are chosen. Substituting Equations (8-131a and 8-131b) into Equations (8-130a 2nd 8-130b) [Cherry, 19641,

can be obtained where K1 = -2 and K2 = 6. Similarly,

2 ( f ) = (K~/t,)[i(tf) - i(to)] + (Kzlt;)[z(tf)

-

z(to) - 2(to)t,l

(8-132)

can be obtained. The function chosen for PI([) and P2(t) in Equation (8-131b) leads to particularly simple equations. This choice of functions is not necessarily the optimum choice, although it is probably the simplest and leads to quite a useful steering law. The gains K1 = - 2 and K 2 = 6 satisfy only the boundary conditions [Equations (8-130a and 8-130b)l. In order to satisfy the optimal trajectory of atmospheric flight, it is desirable to find the optimal time-varying K l and K2 that optimize a performance index and satisfy the constraints imposed on the solution so that a general acceleration vector can be expressed explicitly in a more gcneral vector form as

This guidance law is also an explicit function of the current and the desired boundary conditions, but with optimal time-varying gains to be derived later. From Equation (8-127) the missile acccleration command in inertial coordinates 2, is equal to 2 - 2 where ;i is from Equation (8-133). Figure 8-38 defines the nomenclature of a three-dimensional intercept geometry. The variables 6(R) and u ( R ) are functiolls of the slant range R (thc dotted line connecting and The variable S is the prcdicted velocity angle error of thc prcscnt and final vectors, and u is the heading

6).

Sec. 8.6

Analytical Solutlon of Optimal nqiectory Shaping

565

error angle. The dotted triangle defines the plane containing and R. T o reduce the effect of an uncontrolled axial acceleration command, let the first term of the guldance algorithm [Equation (8-133)] at the normal direction be (Kilt,) V sin 8. The second term at the normal direction is reduced as - (K21t,) ( V sin u/cos u ) . Since R == I, V cos u, the normal acceleration is written as

KI R

a = - V%in 6 cos u -

K2 -

sin u

R

in seeker coordinates. The instantaneous curvature defined as

K

(8- 134)

of the missile trajectory is

where 9 is the flight angle about some inertial reference and dsldt = V. From Equations (8-134) and (8-135), the optimal curvature K is thus K

=

KI . sln ti cos u R

K2 .

- - sin u R

Example 8-7: Low-altitude target engagement with grazing angle control. This example is based on a detailed presentation in Wang [1988]. The problem of target detection and tracking is always present in the case of engaging low-altitude targets. For such a scenario occurring over the ocean, it is desired that the grazing angle be kept at Brewster's angle near intercept to increase the signalto-image ratio, thus minimizing the water-reflected signal. An analytic solution of a combined optimal control and estimation for the missile shown in Example 8-2

Figure 8-38

Intercept Geometry

566

Advanced Guidance System Design

Chap. 8

can achieve a grazing angle with excellent intercept performance. T h e guidance algorithm is in a feedback form and in inertial coordinates. A quadratic performance index (see Example 8-2) is employed together with a Brewster's angle, the boundar,. constraint of which explicitly defines the state to be minimized. The performance of the guidance law is extremely dependent on the approach grazing angle. What is shown is that a significant performance improvement can be obtained with the optimal grazing angle when a good t, estimation such as that presented in Section 8.10 is employed in the guidance algorithm. Furthermore, the algorithm described in this section yields th; best and least complex shaping angle. 1n order to satisfy the optimal grazing angle, it is desired to find the optimal guidance gains KI and K2 that optimize the performance index subject to the constraint of Equation (8133). T h e grazing angle can be defined as IL = tan-' [ Z T + z,,lxY]. Referring to Figure 2-10, y T and y,,, (in horizontal plane) correspond to z~ and z,,, (in vertical plane) used in this example. The best controller to maintain the grazing angle used a curvature method. For a given curve c, the arc length s may serve as a parameter ) ~ m d t Then . 1 / is~ defined in parametric representations. It is given by ~ ( t = as radius of curvature. T h e optimal guidance law is u = - a t , . - (a2/4)zT. The first term of Equation (8-133) is the velocity error of the predicted terminal speed relative to the current velocity and optimizes the grazing angle. The second term considers relative position error and minimizes the time. Hence, the grazing angle is controlled to Brewstcr's angle. - -

8.6.3 Analytic Optimal Guidance Law The missile trajectory is generally defined in inertial coordinates with the acceleration [Equation (8-133)J as the control normal to the missile velocity; however, it is more convenient to use the guidancc algorithm in sccker coordinates [Equation (8-134)], that is, a function of thc feedback variables 6 and a,instead ofthe guidance algorithnl in inertial coordinates [Equation (8-133)], to derive the optimal guidance gains KI and K? with a performance indcx

and boundary conditions S(tf) = o ( l f )= 0. Maximizing the terminal speed in the pcrformancc index Equation (8-137) for a mancuver from launch to the intercept point is cqui\,alcnt to minimizing thc total loss in thc kinetic energy of the entire flight path and hence promising cxtcnded range with nlorc favorable cndgame conditions, which is thc goal of this optimal trajectory shaping. Equation (8-133) is a well-dcfincd guidance equation and is used as explicit guidancc law. The gains K1 = - 2 and Kz = 6 satisfy only the boundary conditions [Equations (8-130a and 8-130b)l but not thc optimality condition. Since it is difficult to dircctly dcrivc optimal gains K, and K2 in thrcc-dimensional flight as stated previously, simplified and practical solutions are obtaincd from a rcduccd-order problem. Thc analytic optimal guidancc gains K c and K2 are first dcrivcd for the vcrtical plane guidance Equations (8-132a and 8-132c). A similar derivation is thcn done for thc horizontal plane guidancc Equations (8-132a and 8-132c). Then thc gains dcrived from the

Sec. 8.6

Analytical Soiutlon 0~Optlmal7Ydectory Shaping

567

vertical and horizontal plane guidance equations are applied to a three-dimensional flight using Equations (8-132a and 8-132c) as guidance accelerations in inertial coordinates of X, Y, and Z directions, respectively. These equations are mutually independent and are three components of Equation (8-133).

Vertical plane guidance. equal to 9 and

In the vertical plane, the flight path angle y is (8- 138a)

6(R) = yf - 7 u(R) = y

+0

(8-138b)

where 0 is the inertial LOS angle. The equations for R and 0 are, respectively, dRldt = - V cos o

(8- 139a)

d0ldt = ( V sin u)lR

(8-1 39b)

and

The state equations which govern the states y and o and the control from Equations (8-133), (8-138b), and (8-139a and 8-139b) as dy1dR = - K sec u

K

are derived (8-140a)

and duldR = - K sec a

-

(tan u)lR

(8-140b)

To obtain the optimum guidance law a,([) optimal control theory needs to be applied to find an optimal control law K(R)subject to the state Equations (8-140a and 8140b) and the boundary conditions u(tf) = 0, y(tf) = yf (specified and updated in flight), and R(tf) = 0 such that the missile terminal speed is maximized. Maximizing the missile terminal speed is equivalent to maximizing

G is first found as a simplified function of the state variables u and y, the control variable K, and the missile characteristics. Then optimal control theory is applied to find the optimal K such that G is maximized. The derivation of G is described as follows. T o facilitate identification of nomenclature used in the derivation from Equations (8-142) through (8-147), the following are defined: CD, = zero lift drag coefficient

CL, = lift coefficient curve slope [dCLlda] D = drag = Do

+ DL

D L = drag due to lift = IlCL,a2qS = qL,a2 Do = zero lift drag = C D , ~ S

Advanced Guldance System Design

L = lift L,

=

=

Chap. 8

CL,aqS = Laa

lift curve slope [dLldu] = CL,S

m = mass

q = dynamic pressure = pV2/2 S

=

area

T = thrust W = weight

=

mg

a = angle of attack q = aerodynamic efficiency factor

p = atmospheric density The equations of motion, neglecting mass change due to fuel cons~lmption,are

dVldt

=

( T cos a - D)lm - g sin y

(8- 143)

Vdyldt

=

(Tsin a

+ L)/m - g cos y

(8-144)

T o obtain an analytic solution of optimal control K , the derivation of G is simplified by assuming cos u -- 1 - a 2 / 2and using Equation (8-143) so that

dV = T ,g sin .y dt

in

-

Do

[I m

+

( I ~ L ,+ T / 2 ) a 2 / D o ]

(8-145)

and assuming sin a = a using Equations (8-135) and (8-144) to obtain

a

=

tnV2(K + g cos yIV2)(T

+ La)-'

(8- 146)

Equation (8-146) is then substituted into Equation (8-145) and the result divided by Equation (8-139a) so that the range R is an independent variable. Thus,

[I +

(qL,

+

Substituting this equation into Equation (8-141) yields

DO G=-1,mV wherc

I

T / 2 ) m V 4 (+~ g cos - ~ / V ~ ) ~ / [-kDL,)*]] ~ ( T sec u

1

T/m

-pin

v

y

scc u dR

Sec. 8.6

Analytical Solution of Optimal 7?dectory Shapfng

Neglecting the effect of gravity, Equation (8-147) can be rewritten as

" T scc u dR

L

""

G, = - I ttl V

mv

where 1 =

J:

+

(I

$)

sec u dR

and

,

=

DoL,(TIL, + 1)' m 2 V4(2q + TIL,)

(8- 150)

Optimal solution for the power-off stage of solid rocket engine. From Equations (8-149) and (8-130), it is evident that trajectory shaping is influenced by variations in aerodynamic and propulsion parameters. For the power-off stage of a solid rocket engine, the second integral term at the right-hand side of Equation (8-149) does not exist so that maximizing I is equivalent to maximizing C , that is, maximizing the missile terminal speed. Hence, I becomes the new performance index. The Hamiltonian to the variation problem is

= [l

+

K'/(~F')]sec u - A, tan ulR - (A,

+ h , ) ~sec u

(8-151)

where A , fulfills the adjoint equation dA,ldt = - aH*/dx, where H* is the maximized Hamiltonian and x, the state. Since the y coordinate can be ignored, there is the integral

A, = Co The optimal variable

K

(8- 152)

is found to be K =

I t is seen that K is proportional to

A, for the unconstrained

K

A,

F2(h,

+ C")

+ CO.From Equations (8-151)

to (8-153),

is

An equation for K is formed by taking the derivative of Equation (8-153). Then substituting Equation (8-154) d~ d~ = dK

d~ =

[($

[(g

- FZ) sin u -

F.)

.in

+ +

+.I R

R

+

sec2 o,

(8-155) (I

+ sin2 u + ...)

are obtained. By neglecting higher-order terms of sin u and (8-155),

K'

sin u of Equation

Advanced Guidance System Design

570

Chap. 8

From Equation (8-140b),

d(R sin dR

5)

-RK

=

and substituting Equation (8-157) into Equation (8-156),

2 d(R sin u) R dR

d2 dR2

- ~ ' R s i n u+

- ( R sin a) - -

C

=

0

is obtained. It is found that

C R sinu = F~

+ C l e F R ( F R-

1)

+ C 2 e - F R ( F R + 1)

(8-159)

satisfies Equation (8-158). At the boundary condition t = t f ,

Rf = 0,

crf = 0

(8- 160)

Using these boundary conditions in Equation (8-159),

C = (C1 - C2)F2

(8-161)

is obtained. Then Equation (8-161) is substituted into Equation (8-159) to obtain

R sin u = C l [ e F R ( F R- 1)

+ 11 + C 2 [ e - F R ( F R+ 1) - I ]

(8-162)

By differentiating Equation (8-162) with respect to R and using Equation (8-157) for comparison, K =

+

(8- 163)

- C I F 2 e F R C2F2e-FR

is obtained. Equation (8-140a) is integrated with K from Equation (8-163). In the integration, Equations (8-138a and 8-138b) are used in the process and ( y f + y)l 2 ;= y and cos[(yf - y)/2] ;= 1 are assumed to obtain Sqf cos a dy = sin 6 cos 0. Thus, sin S cos u FR = C1(l - e ) F

+ C2(1 - e - F R )

From Equations (8-162) and (8-164), coefficients C 1 and C 2 can be obtained as follows: ( 1 - e - F R ) Rsin u

-

+ [I

- C F R ( F R+ I ) ]

+ 41 l ) R sin u + [eFR(FR- 1 ) +

sin 6 cos o F

E-~~(F + R2)

11

sin S cos u F

I/

I/

[ e F R ( F R- 7)

[ e F R ( F R- 2)

Analytical Solutlon of Optimal Trqjectory Shaping

Sec 8.6 Substituting K =

C1

5 71

and Cz into Equation (8-163),

[ F 2 ( 2 - eFR - e - F R ) ]sin ~ cr e F R ( F R - 2)

[2F2R - F(r"' - e T F R ( ~+R 2)

f

-

+

e - F R ) ]sin 6 cos a 4

Thus by equating Equations (8-136) and (8-165), the closed-loop feedback guidance gains in Equations (8-133) and (8-134) are analytically determined as

In the optimal guidance law, either in inertial coordinates [Equation (8-133)) or in target tracker coordinates [Equation (8-134)], the first term, known as velocity to go, considers the velocity error of the predicted terminal speed relative to the current velocity. The second term considers the position error of the predicted terminal position relative to the current position. The first term minimizes the energy loss, that is, trajectory shaping, and the second term minimizes the time. Hence, the combined weighted minimum energy loss and minimum time optimal intercept control trajectories are generated for a specified region of the predicted intercept point. If the final flight path angIe is not specified, then hYf = 0 and hence Co = 0 and C = 0. From Equation (8-161). C, = C2 is obtained. Applying Equations (8-162) and (8-163) yields

Optimal solution for air-breathing engine or the power-on stage of a solid rocket engine. In the previous discussion, it was assumed that the second terms of Equation (8-149) can be neglected. However, this assumption is not suitable for air-breathing engines. This integral propulsion effect is retained in the following derivatives. Rewrite Equation (8-149) as

-

The optimal solutions have been derived previously for no thrust effects of ( D O T ) 2 0, that is, with trajectory coefficients 2 0. At thrust stage, T > Do so < 0; therefore, a new trajectory-shaping coefficient F: needs to be defined as

F

F: =

-F

= L,(T

- Do)(TIL, +

l ) ' l [ m z V 4 ( 2 q + TIL,)]

(8- 169)

Advanced Guidance System Design

5 72

Chap. 8

Equation (8-169) is substituted into Equation (8-168) to obtain

c2=

"L' - $) (I

sec a d~

The Hamiltonian to maximizing Equation (8-170) with respect to Equations (8, 140a and 8-140b) and (8-169) is

2)

HI = a (1 -

sec a - A,,

tan a R

(A,

+ A Y )sec ~ a,

a, =

Do T mV

The Hamiltonian is independent of the variable y, hence A, = Co

The optimal variable

K

is found to be

+

Co. From Equations (8-171) to (8It can be seen that K is proportional to A, 173), X, for the unconstrained K is given by dh, - - a, dR

sec' a F:

[(: ) -+

F:

sin a

co

+

C=-F

2 2

(8- 174)

a1

Taking the derivative of Equation (8-173) and substituting Equation (8-174) yield

" dR

[($ + F : ) s i n U + " R" 1 secZ [(: + F;) sin + R 1' (1 + sin2 a,

"dR=

=

a

+

(8- 175) a

+

By neglecting high-order terms of sin a and K2sin a of Equation (8-175).

Substituting Equation (8-157) into Equation (8-176) yields d'

2 d(R sin a) dR

(R sin a) - dR2 R

+

+

C=0

(8-177)

-+ C I(cos(F2R) + FzR sin(FrR)] + Cz\sin(F2R)- F F$

~ cos(F2R)I R

FgRsina

It is found that

R sin a =

C

Analytical Solution of Optimal ndectory Shaping

Sec. 8.6

573

satisfies Equation (8-177). Using the boundary conditions [Equation (8-160)] in Equation (8-178) gives C I = CIF:

(8- 179)

Then Equation (8-179) is substituted into Equation (8-178) to obtain R s i n u = Cl[cos(FzR) + F2Rsin(F2R) - 11 + C2[sin(F2R)- (F2R)cos(F2R)] (8- 180) By differentiating Equation (8-180) with respect to R and using Equation (8-157) for comparison,

is obtained. Equation (8-140a) is integrated with respect to R and using K from Equation (8-181). In the integration, Equations (8-138a and 8-138b) are used in the process and (yf + y)/2 .= y and cos[(yf - $121 = 1 are assumed to obtain J:f cos u dy = sin 6 cos a. Thus, sin 6 cos u

=

- C1F2 sin(F2R) +

C2Fz[cos(F2R) - 11

F2

From Equations (8-180) and (8-182), coefficients C1 and C2 are obtained as follows: C1 = Cz =

F~[cos(FzR)- 1]R sin a - [sin(F*R) - F2R cos(F2R)] sin 6 cos u/F2 F2[2 - 2 cos(F2R) - FzR sin(F2R)] F2

+

sin(F2R)R sin u [cos(F2R) + F2R sin(F2R) - 11 sin 6 cos ulF2 F2[2 - 2 cos(F2R) - F2R sin(F2R)I

Substituting C1 and C2 into Equation (8-181), K

=

F:[cos(FzR)

-

+

1]R sin u F2[sin(FzR) - FzR] sin 6 cos a 2 - 2 cos(F2R) - F2R sin(F2R)

(8- 183)

Thus, by equating Equations (8-136) and (8-183), the closed-loop feedback guidance gains in Equations (8-133) and (8-134) are analytically determined as

(8- 184) If the final flight path angle is not specified, then A,, = 0 and hence Co = 0 in Equation (8-172) and C = 0 in Equation (8-174). From Equation (8-179), C1 = 0. Therefore, applying Equations (8-180) and (8-181) yields K = -

K2 . sin a, R

F$R' sin(F2R) K2 = . sin(F2R) - F2R cos(F2R)'

K1 = 0

(8-185)

Advanced Guidance @stern Design

5 74

Chap. 8

General solutions of optimal trqjectory-shaping guidance law, The effects of gravity are neglected previously in deriving the guidance laws. However, the gravity effects are retained in the following derivation. The trajectoryshaping coefficient is redefined as F: =

L,(Do- T+mg sin y) (TIL, m2 V4(2q+ TIL,)

A2 ) ~ + ~ ,A = g cos YIV'

+I

(8-186)

and Equation (8-186) is substituted into Equations (8-147) and (8-148), respectively, to obtain

The Hamiltonian to maximizing Equation (8-187) with respect to Equations (8140a and 8-140b) and (8-186) is H2=a2

(

I+-

+ e) s c 2R F$ K~

-

A.

tan u - (A, + A,)K R

sec u

(8-188)

The costate A, is expressed as

Integrating dA,ldR and substituting Equation (8-140a) yield L, g sin y

F:

K

(TIL,

+ I)' + A2/2

where a. is a constant. Substituting Equation (8-182) into Equation (8-188), optimal K is found to be K

= (F:la2)~,

+ a.

(8- 190)

From Equations (8-188), (8-189), and (8-190), A, for the unconstrained dk, = dR

(13,

sec2 u/F:)

[ ~ ' / 2- F:

+ f(y)] sin a +

""3 R

K

is (8-191)

An equation for K is formed by taking the derivative of Equation (8-190). Substituting Equation (8-191),

Sec 8.6

Analytical Solution of Optimal 7Ydectory Shaping

5 75

is obtained. By neglecting high-order terms of sin u and K2sin u of Equation (8192), dK - - [ F ; - f(y)I

dR

sin u

+

K

-

~

I

J

R

If the shaping coefficient is approximated by Fz P

FI

La(Du - Ti (TIL. - f ( y ) = m'V4(Zn, TIL.)

+

+ 1)'

(8- 194)

which is the same as that derived in Equation (8-168). then Equation (8-193) becomes

In the case of no-thrust effect (that is, F; 2 0 ) in Equations (8-186) and (8-194), all is an arbitrary constant when the final dive angle yf is specified. The solution is exactly the same as in Equations (8-165) and (8-166). When the final dive angle yf is not specified, a. is a specific value as defined in Equation (8-189) where X,, = 0.

The solution is of a different form from Equations (8-165) and (8-166). For F4 < dK 0 as in Equation (8-194), Equation (8-195) is expressed as R - - F:R sin u - K dR + all = 0 where F: = -F:. When the final dive angle yf is specified, a. is an arbitrary constant. The solution is exactly the same as that given by Equations (8183) and (8-184). When the final dive angle y f is not specified, a. is a specific value as defined in Equation (8-189) where A,, = 0. The solution is of a different form from Equations (8-183) and (8-184). In other words, the gravity does not affect the closed-loop guidance and gains when the final dive angle y f is specified; it affects the closed-loop guidance and gains when the final dive angle yf is not specified. This is due to the fact that once the final dive angle yf is decided, the missile trajectory is dictated. The traiectorv will be less sensitive to the gravitational effects. However. u if the final dive angle y f is not specified, then the trajectory can be optimized by taking into account gravitational effects. Hence, there are more trajectories to satisy the boundary conditions, and the gravitational effects are one of the factors that determine the final optimal trajectory. >

Horizontal plane guidance. As in the case of the vertical plane guidance equation, the analytic optimal guidance gains K1 and K2 for the horizontal plane guidance equation are also derived. All the equations and optimal gain formulations in vertical flight are applied to the horizontal flight except that all gravity terms do

Advanced Guidance System Design

576

Chap. 8

not exist. In addition, a becomes P which is the sideslip angle, y becomes JI which is the heading angle, and 0 is now the inertial LOS angle in the horizontal plane. Since the gravity term does not exist in the second term of G in Equation (8-147), G I in Equation (8-149) and the optional solutions are now more correct. For + f specified, K1 and K 2 are given by Equations (8-166) for the power-off stage of a solid rocket engine, or they are given in Equations (8-184) for an air-breathing engine. For + f not specified, K2 is the same as that found in Equation (8-167) for the power-off stage of a solid rocket engine, or it is that found in Equation (8-185) for an air-breathing engine. For both applications, K1 = 0. That is, in this case of unspecified + f the final incident angle I J f is chosen as the current incident angle (that is, I J f = +) so that the lateral velocity error is not considered in the horizontal guidance equations. Hence, the main concern is to minimize the lateral position error, that is, the final time. The preceding optimal guidance gains of the two planes are obtained by solving the guidance equations while satisfying the preceding constraints.

Mathematically probing the analytical solution of the optimal trajectory-shaping guidance. The combined midcourse and terminal guidance law design for missiles presented earlier achieves range enhancement with excellent intercept performance. Also derived was the closed-form solution of a closed-loop nonlinear optimum guidance law for three-dimensional flight for both the midcourse and terminal phases by neglecting certain nonlinear terms. Beginning with the optimal guidance formulation derived by Lin and Tsai 119871 for the particular case of solid rockets in Equation (8-ljj), [Rao, 19891 mathematically massages this in order to arrive at an analytical solution that retains terms neglected earlier. It should be mentioned that the solution presented earlier was obtained by taking the full solution to be presented now, and then making simplifications based on real-time considerations, a discussion of which is presented in Section 8.6.5. This section shows how it is possible to take a result derived on the basis ofengineering experience and expertise, that is, Equation (8-ljj), and to probe it mathematically in order to arrive at particular solutions. The differential equation to be solved in analytical form is Equation (8-155) dK =

[($

*

[($ - F')

dR

R

dR

=

+

[(:

- F') sin u

-

+

+

R

1'

+ sin2 u)

,I

I

+

(K

+ C)

P) R sin u

+

(K

R sin u

I

+ C)

sin2 u

Sec. 8.6

577

Analytical Solutlon of Optimal Trqjectory Shaping

Letting h = R sin u and using the fact that duldR = - K sec u = tan u l R yield dh du - = R cos u - + sin u = - K R . Equation (8-196) becomes dR dR

where E is the average value o i u , and f , cases:

$1,

and ,q" are defined for the following

Case 1 .

Optimal solution for a solid rocket engine based on Equation (8-

Case 2.

Optimal solution for an air-breathing engine based on Equation (8-

133)

173) f2 - (Ff

8 1 = c(l

+ F 2l sin. 2 -u ) --

+ sin's),

-Ff(1

g2 = 1

+ sin" E ) ,

+ sin2 E

Case 3. General solutions of optimal trajectory-shaping guidance law based on Equation (8-192): f2

= - [F:

+ F:

sin2 E

+ f(y)],

g l = -ao(l

+

sin' E ) ,

g2 = 1

+ sin2 E

In the treatment of Rao [1989],only Case 1 is considered. Introducing a change of independent variable, F I R = T,gives

Even if the right-hand side of Equation (8-198) is neglected, a greater degree of mathematical rigor is retained by guarding some of the terms not considered earlier in terms of their average value, thus including their effect on the behavior of the parameter A. The equation to be solved then becomes

Introducing a change of dependent variable defined by P = ( g l l f 2 ) - A, Equation (8-199) becomes

Equation (8-200) can be solved analytically in terms of the modified Bessel function of order p. T o begin with, the dependent variable P is transformed according to

Advanced Guidance S ~ s t e mDesign

578

P = U T"+ 112 and n = ( 1 dependent variable, giving

Letting p. = (2n

+

+ g 2 ) / 2 , the

Chap. 8

result of which is to make U the neIv

1)/2, one obtains a standard Bessel equation of the form

The solution to Equation (8-202) is of the form U = A I , ( T ) + B K , ( T ) where I , and K , are modified Bessel functions of order p, and A and B are integration constants, Referring back to the transformation defining P in terms of U and T, one has where p. = 1 + (8712). In terms of the original variables R sin u and R, Equatior, (8-203) becomes

R sin a = ( g ~ l f 2 -) ( f R ) " [ A I , ( f R )

+

BK,(fR)]

where p. = 1 + (S2/2),and A and B can be solved for using the conditions supplied earlier. Using the solution to Equation (8-196) given by Equation (8-204), the gains K1 and Kz presented earlier for each of the three cases will be slightly modified. While it is possible that such modified gains may show m o r e of the nonlinear effect in trajectory-shaping guidance, no convincing arguments or simulation results have ever been published to vcrify whcther this is in fact the case. As often happens in cngineering, however, extending the mathematical rigor may have a certain elegance of its own, but this does not always contribute significantly t o the solution of the problem at hand.

8.6.4 Real-Time lmplementation and Performance Real-time implcmcntation is difficult in view of the two-point boundary value probIcm as wcll as the aerodynamic nonlinearitics associated with the lift, thrust, and drag, and the control constraints forced by structural and a limits. Howcvcr, analytic solutions that havc bccn derived arc able to satisfy all optimality conditions under a wide range of statc and control constraints. Thc desircd guidance acceleration commands arc cxprcsscd explicitly in inertial coordinates by Equation (8-133) in terms of the missilc position and velocity at thc prcscnt time and the intercept point (I'll'). Thcrc arc two mcthods of implcmcnting this guidancc law in real time. First, if the inertial refcrcnce unit (IIIU) is available on board, thcn it provides the missile its position and velocity. The target-tracking radar uplink provides the target position and velocity. The computing unit computes t , and PIP by using the present missile acceleration, the missile-to-target relative velocity, and the

Sec. 8.6

Analytical Solution of Optlmal 7kdectory Shaping

579

missile-to-target range. Equation (8-133) is then uscd to compute the guidance acceleratiorl commands in inertial coordinates. The IRU quaternions arc then used to transform the111 into guidance commands in body coordinates and input to the autopilot. The second method of in~plcmentationis applicable when there is no IRU on board. Then Equation (8-134) is uscd to compute the guidance acceleration command in seeker coordinates. The control gains K , and K? are analytically derived and are nonlinear functions of the trajectory-shaping coefficient F and the range R. As noted previously, F depends on the missile aerodynamics and propulsion characteristics. The variables K , and k'. are systematically updated in flight trajectory to reduce errors in the missile state; errors in the terminal state due to measurement error, target noise, and target maneuver; and autopilot and control-loop dynamic response time lags. As pointed out previously, the heading error at handover is the main error affecting the miss distance. By radar specification, the noise distribution of the uplink data is known, enabling covariance analysis to be used to design the optimal guidance filter to attenuate noise effects (including measurement error of missile and target statcs). Appropriate midcourse guidance filter design can therefore reduce the heading error at handover. Therefore, the analytical solution solves the boundary value problenls associated with missiles intercepting targets in three-dimensional engagements. The resulting guidance law is computed in a feedback manner. The performance index and the optimal guidance law used, either in inertial coordinates or in seeker coordinates, contain equivalently a weighted combination of flight time and energy loss, and the intercept is regarded as a terminal constraint so that the optimal control will cause the missile to maneuver efficiently in terms of minimum energy loss and minimum time. A special case included is minimum time, as in the case of optimal guidance Equation (8-167), for the power-off stage of a solid rocket engine. For an air-breathing engine or the power-on stage of a solid rocket engine, the optimal guidance is given by Equation (8-185). Owing to the analytic form of the resulting feedback control law, the guidance algorithm is suitable for implementation in an airborne digital computer to demonstrate the feasibility of on-line optimal control operation. This explicit guidance algorithm was derived with different gains for solid rocket engine and air-breathing engine applications. Figure 8-39 shows an optimal-shaping guidance law structure for various applications. In general, the guidance law given by Equation (8-134) with gains from Equations (8-166) is more suitable at the power-off stage for medium-to-long-range flight. At the power-off stage for short-to-medium-range flight, it is suitable to use this same guidance law with gains from Equation (8167). For power-on stage, T - Do must be positive. Equation (8-134) with gains from Equations (8-184) is more suitable for medium-to-long-range flight, while Equation (8-134) with gains from Equation (8-185) is more suitable for short-tomedium-range flight. The vertical launch and short range to go will be discussed later.

Implementation and simulation of three-dimensional flight. In a three-dimensional engagement, the missile is guided during the midcourse portion

Advanced Guidance System Design

Chap. 8

Target Position, Velocity Data

EkI Tracking Filter

and Short Range to Go

I*Time-to-GoEstimator Integral Tables

Guidance Law

+

AIP Command

*I Time-to-Go Estimator

K, = -2, K2 = 6

I Figure 8-39

Optimal Shaping Guidance

Law Structure

of the flight by the optimal guidance law in inertial coordinates. For practicality, implementing this guidance law for three-dimensional flight in inertial coordinates is accomplished as follows.

Midcourse phase. In midcourse, three components of accelerations o f the explicit guidance law (8-133) are first computed. There are t w o approaches. In the first approach, gains K1 and K2 are calculated from Equations (8-166) for the poweroff stage, or those from Equations (8-184) for the power-on stage. The gains arc used to compute Equations (8-132a and 8-132) for the three-dimensional accelerations. In the second approach, K1 = 0 and K 2 is calculated from Equation (8167) for the power-off stage. or from Equation (8-185) for the power-on stage, to compute the minimum time guidance acceleration command. Hence, from Equation (8-128a), a, is equal to a-.C where a has the X, Y, and Z accelcration components; a, is then transformed into guidance acceleration commands in body coordinates. For S T T missiles, Y- and Z-direction acceleration commands in body coordinates arc used as commands to the autopilot. Hence, the optimal guidance gains selected emphasize trajectory shaping and minimize cncgy loss and time for the first approach. However, the second approach only minimizes the time.

Sec. 8.6

Analytical Solution of Optimal Trajectory Shaping

581

Switching phase and terminal phase.

When target maneuver is detected as the missile is approaching homing a t the end of the midcourse phase, trajectory shaping becomcs less iniportant while minimizing the position error for good accuracy in guidance becomes more important. Hence, the guidance gains are switched to either the gains given by Equation (8-167) for the power-off stage or by Equation (8-185) for the power-on stage. Thus, the guidance equation in the switching and terminal phase now minimizes the lateral position error, that is, time. Thcrcforc the I , of the missile trajectory in homing is minimized for target interception. This modification of optimal guidance gains near handover can achieve superior crossing-shot performance by turning the missile onto a favorable collision course early in the engagement and reducing heading error to zero at handover. It can quickly modify the missile trajectory during midcourse guidance when the target direction changes. Thus the same guidance law can be used for both the midcourse and terminal phases. This modified optimal guidance law is easy to implement and works especially well for all-weather long-range missiles because it is especially simple and useful and has been substantiated by real-life applications. Many realistic simulations using the proposed method have shown that very high performance is achieved in midcourse and tcrminal guidance. For SAM missile applications the launch angle and the PIP range and altitude are used as scanning parameters to compute a family of three-dimensional intercept optimal trajectories that maximize the final speed at intercept. The PIP dive angle yf determines the highest altitude required by the missile. For real-time application, the launch angle and the PIP dive angle are a table look-up of the PIP range and altitude in a proportional relationship.

Comparison between optimal trqjectory and nonoptimal trqjectory. With the optimal gains, it is possible to obtain smoother guidance acceleration command time histories than with the nonoptimal gains K1 = - 2 and K 2 = 6. For SAM applications, the normal acceleration command history using the nonoptimal gains results in a higher-altitude trajectory than that using the optimal gains. The dynamic range and magnitude of the cl and P time response using the optimal gains are much smaller than those using the nonoptimal gians. The a and p time response using the optimal gains are also smoother. Hence, the optimal gains are suitable for air-breathing missile applications where the dynamic range of the a and (3 needs to be constrained. Since the optimal trajectory has smaller a and P than the nonoptimal trajectory, it has much smaller heading error at handover. Therefore, it is easier for the missile autopilot to control and follow the optimal trajectory with minimum control surface activity. This results in improved miss distance with shorter flight time. T h e optimal trajectory presents a significant improvement in the final speed over the nonoptimal trajectory since the drag owing to the a is less than that occurring in the nonoptimal trajectory. In some SAM applications, the final speed in vertical flight has about a 15 percent improvement for a 50-km flight range. When implemented in a three-dimensional engagement, the optimal trajectory has significant

582

Advanced Guidance System Design

Chap. 8

improvement (40 percent) in the final speed over the nonoptimal trajectory. Although a SAM model is used in the guidance law design, analysis, and simulation, the guidance law can also be applied to SSM, AAM, and ASM.

Vertical launch of tactical missiles.

Figure 8-40 shows an example of a vertical launch. The realistic problem has to do with computing optimal guidance law for vertical-launch tactical missiles intercepting at short range to go with large initial heading error a. The problem formulated in Equation (8-134) with t,Q estimator is not suitable for this critical case. Due to tactical reasons, vertical launch may be preferred in a missile system. Vertical launch of a tactical missile was performed at least as early as 1967 nonsson and Malmberg, 19821. It is interesting to study this shaping guidance law. T w o of the optimal criteria that may be formulated regarding this shaping guidance law Equation (8-134) are available for vertical launch of missile intercepting at short range and low-altitude target. The first is to minimize the control deflection historv, together with the control deflection limi., cation, while the second is to minimize the time to transfer the missile position and velocity from initial status to a final status. The guidance Equations (8-132a, 8132b. and 8-132cl are available for aerodynamic control or thrust vector control for this application. The guidance computations, needed for solving explicitly the more general boundary value problem, involve a determination of the required (, which is much more different from what been described so far. The development of a steering equation to guide a missile to a desired set of terminal conditions is based on the solution of the following variational problem. It is required to find the acceleration a(/) ~vhich\vill minimize the functional J;'A"T) d~ subject to Equations (8-128b). The optimal solution of this problem is A ( / ) = bl + 621. which, after substituting into Equations (8-130a and 8-130b) yields the same optimal solution Equation (8-133) 1 ~ 1 t hoptimal gains K , = - 2 and K 2 = h. AS discussed previously, this optimal guidance law is suitable for short !,(or short range-to-go) applicatiot~ssince atmospheric effects are neglected at deriving this optimal guidance law. In a vertical-launch missile intercepting a t short-range and low-altitude target guidance law application, the acceleration command a, is a function not only of .. two-point boundary values but also of the required ,, estinlator. In general, the problem under consideration requires the missile to perform a high a maneuvcr. This, togcthcr with the control dcflcction limitation, implies highly nonlinear systc111 dynamics. For practical considerations, the r, is detcrmincd under the control dc-

7 Target

Figure 8-40

Vcrtical Launch

Sec. 8.7

Other Advanced Midcourse Guidance Schemes

583

flection limitation for avoiding control saturation. The shorter the i,, is determined, the more mnticuver is required. Therefore, the niinimutn intercept zone is decided from the accepted niini~nunii,, to intercept. Detailed structure of thc optin~al-shaping guidance law with vertical-launch capability has been shown previously in Figure 8-39.

8.6.5 D i s c u s s i o n s Cornplctcly closed-form solutions of optimal trajectory-shaping guidance have been presented that maximize the final speed at intercept. The proposcd gi~id;i~ice law has been applied successfully to three-dimensional engagements. Since the guidance law formulation for the midcourse and terminal phase is the same, and can easily be implemented, it is very suitable for onboard application. In general, it is more appropriate for the missile to be guided during the midcourse portion of the flight by the optimal guidance law [Equation (8-133)] in inertial coordinates. In terminal phase, the same guidance law, can be used. The major contribution of this guidance law is that it is very simple and useful, especially when applied in inertial coordinates. Application of this optimal trajectory-shaping guidance law results in maximization of the filial speed at intercept, improvement in miss distance, smoother acceleration command time history, zero heading error at handover, shorter flight time, smaller and smoother angle of attack and sideslip angle.

8.7 OTHER ADVANCED MIDCOURSE GUIDANCE SCHEMES As shown in Section 8.6, the optimal guidance problem is a two-point boundary value problem whose solution is generally obtained in an open-loop form. However, it is usually required that the optimal solution be of closed-loop form in the case of onboard real-time guidance. Several closed-loop guidance schemes have already been explored, as shown in Section 8.6, as a result of recent interest in onboard real-time generation of midcourse guidance commands for medium-range AAM. In addition to those presented in the previous section, Kelley [I9621 and Breakwell et al. [I9631 have studied the problem of guidance in the neighborhood of an optimal nominal trajectory, and have developed procedures for synthesizing linear feedback for optimal guidance. Higher-order feedback approximations were introduced by Kelley [1964(a)], while Weston et al. [I9831 and Visser et al. [I985 &- 19873 made use of still another approach adopted by Kelley and Well [I9831 in which intercept onboard calculations are performed that make use of Euler solutions for a pointmass vehicle model but incorporate elements of singular-perturbation theory. Midcourse guidance law design for an AAM that employs the singular-perturbation theory is presented in Section 8.7.1. In Section 8.7.2, a method of performing onboard real-time calculations for an optimal guidance approximation is described in which elements of extremal fields and neighboring extremal theory are used together with precalculated Euler solutions to establish the closed-loop guidance algorithm. -

~

Advanced Guidance Sustem Design

584

Chap. 8

This last approach is applied to midcourse guidance of an AAM equipped with boost-sustain propulsion and modeled as a point mass in the vertical plane.

8.7.1 Singular Perturbation for Advanced Midcourse Guidance In this section an advanced midcourse guidance law based on minimum flight tilne criteria is developed. The optimality conditions of the control problem result in a two-point boundary value problem which is too complex for onboard real-time implementation with near-term microcomputer technology. Nevertheless, singularperturbation technique approxin~ationsare applied to solve this complicated problem in this section. The resulting guidance law is near optimal and sufficiently simple for implementation. In reality, singular-perturbation techniques are used to simplify the optimal control problem so that the resulting near-optimal control problem does not require the solution of any t\vo-point boundary value problem. The midcourse guidance law based on this methodogy is proposed by Cheng and Gupta [1986].

Problem formulation. Assuming small angle of attack and sideslip angle at midcourse guidance, the state equation in inertial coordinates can simply be formulated as:

X.

= V cos 0 cos

4,

= I; cos 0 sin +,

x(to) = so I

)

=

(8-205a)

)1(~

(8-205b)

1; = 1.' sin 0,

11(10) =

h(,

(8-205c)

4

$(to) =

40

(8-205d)

8(10) = 0,)

(8-205e)

= L sin $/(ril l,'cos 0),

0 = (L cos 4 - r r y cos 0)1(rnl/),

where the state variables are the position coordinates x, y, and it; pitch angle 8; yaw V212g. The angle \Ir; and the specific energy E which is represented as E = h variable 6 denotes the oricntation angle and V denotes the total missile speed. The lift vector in the pitch and yaw directions normal to the missile-body axis. defincd by ma~nitudcL and oricntation 6.is the input control to be optimized. The thrust and mass rrr arc predefined fi~nctionsof time. Moreover. the thrust cornpollent normal to the body axis is assunled small and hence is ncglcctcd. For simplicity, the performar~ceindex is sclccted here as minimizing the tinlc of flight, that is3 J = I;! dr. The optinla1 control problcm is to solvc for the control input that minimizes), subjcct to the constraints of Equations (8-305) and the desired tcrnlillal condition. The Hamiltonian is

+

Here,

z7 =

(x, y , 11, 4, 0. E ) and A T = ( A , , A,. A,,, A+. A,,, A&). ~ h costatc c

Sec. 8.7

Other Aduanced Midcourse Guidance Schemes

585

variable A satisfies A = - (dHIaZ)T and A(tf) = Ar where the optimality conditions arc dHIdL = 0 and dHldd, = 0.

Near-optimal solution using singular-perturbation technique. Equations (8-205) can be expressed as the system using dimensional analysis:

R d(.rlR) --= V

lit

R n ( y l R ) = cos 0 sin JI I ' lit h,,,,., d(hlh,,,,,) -

v

= sin 0

L sin 4 L,,,,, cos 0

mV d+ -L,,,, dl

-

mV d0 L,,, dt

L -cos L",.,,

-.

dt

7'-D

2

cos 0 cos JI

4+

m,y cos 0 Lm,,

where h,,,, is the operational altitude limit, L,, the lift limit, and R a characteristic range. The typical value of h,,,/ V is less than those of Rlv and vlg and the value of h,,.,,l V is larger than that of rnVIL,,,,. The missile dynamics in Equations (8205) can be categorized into three groups. The first group is the slow dynamics of x, y, and E. The second group is the medium or altitude dynamics based on h, and the third group is the fast dynamics based on pitch angle 0 and yaw angle +.

Slow dynamic approach. Since h, +, and 0 are classified as faster variables, their corresponding equations may be considered to be in equilibrium with the slow dynamic approach. The equilibrium conditions of h, Jr, and 0 imply that 4, = 0, = 0 and L, = mg from Equations (8-205c) to (8-20%). Here, the subscripts denotes that the variables are in the slow dynamic approach. The Hamiltonian of Equation (8-206) becomes

H,

=

1

+ A,,

V, cos Jr,

+ A,,V,

sin

+, + A,,[V,(T

-

D,)lmg]

(8-207)

Note that since Hs is independent of X and Y, A,, and A,, are constants. Thus, the = A,,/A,,. Integrating Equations optimal value of is constant and given by tan (8-205a) and (8-203b), the following is obtained:

+,

+

J19

=

tan-'[(yf - yo)l(xf - XO)]

(8-208)

Equation (8-208) suggests that the yaw angle in the horizontal plane is determined by joining the current missile position to the predicted final missile position. T h e adjoint variables A,, and A,, are constants and can be defined as A,,=

1 --cos+,lV,(tf), 2

A,*=

1 . - - ~ i n + ~ I V ~ ( t ~ ) (8-209) 2

The adjoint variable AE, can be determined by the following given two factors: (1) Equation (8-207); and (2) the conditions for free final time and unconstrained = 0. T o solvc AE, without final specific energy, that is, AE,(tf) = 0 and H,(tf) reference to the backward differential equation, the values of T and it1 are approx-

586

Advanced Guidance Sgstem Design

Chap. 8

i~natedby their average valuesT and i i , respectively. This removes the dependency on time of H, and thus

H, = 1

+ A,,

V, cos

+, + A,,

V, sin

+, + AE,[v,(T - D,)l(-iiis)l = 0

(8-210)

Substituting Equation (8-209) into Equation (8-210), the following is obtained:

The optimal altitude is given by dH,lalt = 0 which becomes

after eliminating the Lagrange variables through Equations (8-209) and (8-211). Equation (8-212) determines the optimal altitude h implicitly through the term aD,/ah.

Medium dynamic approach.

The definition of medium dynamics is based on the altitude dynamics which are faster than the dynamics of x, y. and E, but slower than the pitch angle and yaw angle dynamics. T o solve for altitude dynamics, A,., A,.,, and AE, in Equations (8-209) and (8-211), the approach used in slow dynamics is applied. The values of and 0 are assumed in their equilibrium conditions since the dynamic responses of $ and 0 are much faster than those of the rest of the system states. The corresponding Hamiltonian can be rewritten as:

+

+ A,,

H,,, = 1

+

V,,, cos O,,, cos &f+ A,., l),,, cos O,,, sin 4,

hi,., l',, sin O,,,

+ A,

t1,,,[(T- D,,,)lrtq]

where the subscript m denotes variables in the medium dynamic characteristics. The optimality condition dH,,ld0 = O is reduced to tan O,,, = A,,,/(A,, cos

T and

+, + A,,

sin $,)

(8-211'

are again approximated by their average values to give the addition, condition H,,,(I) = 0 to avoid solving the backward differential cquation for Ah,,. This condttion, with the Lagrange variables elinlinated by Equations (8-209), (821 l ) , and (8-213). gives the following result: 111

The value of sec 0," is given by Equation (8-214) where O,,, is positive for h, < hl and negative for h, < h,,. At this point, given the current pitch angle 0 and yaw angle J, and the desired pitch anglc 0 , and yaw angle +,, the orientations of the current velocity vector V, and the desired velocity vector Vdare specified respectively by the following. V, = [cos 0 cos $, cos 0 sin +, sin 01, Vd = [COSern cos +,,,, cos 0, sin J,,, sin Om]

(8-21 5 )

Sec. 8.7

Other Aduanced Midcourse Guidance Schemes

585

variable A satisfies A = - (dHIaZ)T and A(tf) = Ar where the optimality conditions arc dHIdL = 0 and dHldd, = 0.

Near-optimal solution using singular-perturbation technique. Equations (8-205) can be expressed as the system using dimensional analysis:

R d(.rlR) --= V

lit

R n ( y l R ) = cos 0 sin JI I ' lit h,,,,., d(hlh,,,,,) -

v

= sin 0

L sin 4 L,,,,, cos 0

mV d+ -L,,,, dl

-

mV d0 L,,, dt

L -cos L",.,,

-.

dt

7'-D

2

cos 0 cos JI

4+

m,y cos 0 Lm,,

where h,,,, is the operational altitude limit, L,, the lift limit, and R a characteristic range. The typical value of h,,,/ V is less than those of Rlv and vlg and the value of h,,.,,l V is larger than that of rnVIL,,,,. The missile dynamics in Equations (8205) can be categorized into three groups. The first group is the slow dynamics of x, y, and E. The second group is the medium or altitude dynamics based on h, and the third group is the fast dynamics based on pitch angle 0 and yaw angle +.

Slow dynamic approach. Since h, +, and 0 are classified as faster variables, their corresponding equations may be considered to be in equilibrium with the slow dynamic approach. The equilibrium conditions of h, Jr, and 0 imply that 4, = 0, = 0 and L, = mg from Equations (8-205c) to (8-20%). Here, the subscripts denotes that the variables are in the slow dynamic approach. The Hamiltonian of Equation (8-206) becomes

H,

=

1

+ A,,

V, cos Jr,

+ A,,V,

sin

+, + A,,[V,(T

-

D,)lmg]

(8-207)

Note that since Hs is independent of X and Y, A,, and A,, are constants. Thus, the = A,,/A,,. Integrating Equations optimal value of is constant and given by tan (8-205a) and (8-203b), the following is obtained:

+,

+

J19

=

tan-'[(yf - yo)l(xf - XO)]

(8-208)

Equation (8-208) suggests that the yaw angle in the horizontal plane is determined by joining the current missile position to the predicted final missile position. T h e adjoint variables A,, and A,, are constants and can be defined as A,,=

1 --cos+,lV,(tf), 2

A,*=

1 . - - ~ i n + ~ I V ~ ( t ~ ) (8-209) 2

The adjoint variable AE, can be determined by the following given two factors: (1) Equation (8-207); and (2) the conditions for free final time and unconstrained = 0. T o solvc AE, without final specific energy, that is, AE,(tf) = 0 and H,(tf) reference to the backward differential equation, the values of T and it1 are approx-

586

Advanced Guidance Sgstem Design

Chap. 8

i~natedby their average valuesT and i i , respectively. This removes the dependency on time of H, and thus

H, = 1

+ A,,

V, cos

+, + A,,

V, sin

+, + AE,[v,(T - D,)l(-iiis)l = 0

(8-210)

Substituting Equation (8-209) into Equation (8-210), the following is obtained:

The optimal altitude is given by dH,lalt = 0 which becomes

after eliminating the Lagrange variables through Equations (8-209) and (8-211). Equation (8-212) determines the optimal altitude h implicitly through the term aD,/ah.

Medium dynamic approach.

The definition of medium dynamics is based on the altitude dynamics which are faster than the dynamics of x, y. and E, but slower than the pitch angle and yaw angle dynamics. T o solve for altitude dynamics, A,., A,.,, and AE, in Equations (8-209) and (8-211), the approach used in slow dynamics is applied. The values of and 0 are assumed in their equilibrium conditions since the dynamic responses of $ and 0 are much faster than those of the rest of the system states. The corresponding Hamiltonian can be rewritten as:

+

+ A,,

H,,, = 1

+

V,,, cos O,,, cos &f+ A,., l),,, cos O,,, sin 4,

hi,., l',, sin O,,,

+ A,

t1,,,[(T- D,,,)lrtq]

where the subscript m denotes variables in the medium dynamic characteristics. The optimality condition dH,,ld0 = O is reduced to tan O,,, = A,,,/(A,, cos

T and

+, + A,,

sin $,)

(8-211'

are again approximated by their average values to give the addition, condition H,,,(I) = 0 to avoid solving the backward differential cquation for Ah,,. This condttion, with the Lagrange variables elinlinated by Equations (8-209), (821 l ) , and (8-213). gives the following result: 111

The value of sec 0," is given by Equation (8-214) where O,,, is positive for h, < hl and negative for h, < h,,. At this point, given the current pitch angle 0 and yaw angle J, and the desired pitch anglc 0 , and yaw angle +,, the orientations of the current velocity vector V, and the desired velocity vector Vdare specified respectively by the following. V, = [cos 0 cos $, cos 0 sin +, sin 01, Vd = [COSern cos +,,,, cos 0, sin J,,, sin Om]

(8-21 5 )

Sec. 8.7

587

Other Advanced Midcourse Guidance Schemes

Also, the lift orientation

4 can be obtained by changing the angle to turn A + as A + = cos-'(V<

V,,)

(8-316)

Fast dynamic approach. The fast dynamic characteristics arc defined by the dynamics of the pitch angle 0 and yaw angle +. The Hamiltonian, with the Langrangc multipliers derived from the slow and medium dynamics, can be written as:

H.,

=

+,, + A,, V,,, cos 0 , sin +., + A, ,,,, V,,, sin 0, + hi:. V,,,(T - D,)I(rt;y) + A*L sin + / ( t t r V,,, cos 0,)

1

+

+

A,. V,,, cos 0, cos

he(L cos

-

trl'q

cos 0,)/(m V,,,)

In order to determine Ae and A*. in terms of L and +, the optimality conditions E)H,/E)L and dH,ld+ = 0 can be used. Again, approximations of T and m by their average values give an adjunct condition, that is, H,,(t) = 0. The Lagrange variables and drag terms, shown as D = q s ( C ~ ,+, CD, ( a ( + cD,2a2) and L = qSCL,,cw, can bc eliminated from this condition to dcfine the optimal life L. Furthermore, by ass~imingthe angles ($, - +,) and (0, - O,,,) to be small and by ignoring negligible terms, simplification is feasible and results in the following

Midcourse guidance algorithm.

Based on the results derived earlier, a midcourse guidance algorithm can be formulated involving the following cornputations. In the slow dynamic approach, yaw angle in Equation (8-208) and hence optimal altitude h in Equation (8-212) are computed. In the medium dynamic approach, pitch angle 0,, in Equation (8-214), lift orientation 4 and angle to turn A + in Equations (8-215) and (8-216) are computed. In the fast dynamic approach, lift magnitude L in Equation (8-217) is computed.

+,

8.7.2 Midcourse Guidance for Boost-Sustain

Propulsion Midcourse guidance is examined in this section for an AAM featuring boost-sustain propulsion.

Computational scheme.

Results for open-loop best-range guidance of a medium-range AAM are presented. The computational approach follows that discussed in Katzir et al. [1988]. In this optimal vertical-plane flight problem, the nonmaneuvering target is in a head-on engagement at the same altitude as the flight vehicle. T h e problem is illustrated in Figure 8-41, where [Dl represents the target with the launcher range at launch; [R] represents the missile range; and [S] represents

590

Advanced Guidance System Design

Chap. 8

In principle, this condition is used to solve for an extrcrnal control n*. For the dragpolar [Lin, 19801, the result is

O n e should check that the extremal control satisfies the constraints 'and that the control actually mi~zimire.cthe Hamiltonian function. The latter condition is satisfied if hE is negative. Some common boundary conditions on the costates ( t r a n s ~ e r s a l i t ~ conditions) which vary according to the problem being examined, are h,(tf) = - 1, A,(tf) = 0, E ( t f ) = E f , or h ~ ( t f )= 0. The E - hE boundary condition is complicated because it invd~vesan inequality. In practice, one solves with the specified Galue of E(tf) and then checks the corresponding costate h ~ ( t f ) A . negative value of this quantity corresponds to an extremal solution. If this is not the case, one must refigure with AE(tf) = 0 (natural) boundary condition. This will arise if the specified bound Ef is less than the "natural" value, that is, E(tf) with At(tf) = 0. A preliminary study of time-range-energy optimal trajectories shows that there is a fair ~ f t e an r initial high-g pull-up the trajcctories are amount of lofting in these almost ballistic. Extensions to three-dimensional flight have vroduced similar be" havior. Future efforts will be directed at obtaining reduced-order models that are suitablc for closcd-loop synthesis and still retain the significant characteristics of the point-mass model.

Near-optimal trajectories. The approach to onboard real-time calculations for midcoursc near-optimal guidance prcsentcd here follows the work of Katzir ct al. [1989]. I t is based on cxtrcmal fields and neighboring cxtrcmals theory ideas uscd in conjunction with precalculated Eulcr solutions and fccdback gains to producc a closed-loop guidance scheme. The optimal rcfercnce trajccrorics ("nominal") and the closed-loop gains indexing arc done in tcrms of either current or stretched time. It is assumed that, in certain time intervals, the predicted intercept point is first computed and then uscd as the new terminal condition for the missile. Thc fccdback gains are computed using methods from the previous sections. The optimal trajcctory, having been generated prcviously, is now considcrcd as a nominal rcfcrcncc trajcctory hcrc. Ncar-optimal trajcctorics, which arise as a result of the follo\ving dcsign of a pcrturbation guidancc law, arc based on this nominal rcfcrcncc trajcctory. Assuming the cxistcncc of a nominal reference trajectory togcthcr with small pcrturbations in thc initial statcs Gx(r,,) and final conditions 6x(tf) for a fixcd final tirnc, [Kcllcy, 19621 showcd that the perturbed and the adjoint systcms constitute a linear, tirnc-varying systcm. Moreover, the optimal problcm of minimizing a performance index J is rcduccd to that of minimizing the second-ordcr approximation to J. Thc ensuing problem is a lincar, time-varying two-point boundary valuc problem that has a quadratic performance index, thc solution to which is wcll dcvclopcd. The optimal control law represented by the solution to this problem is of a contir~uouslincar fccdback typc which can be written, as shown in Lin [1983(a)], as u(r) = urC,(r) + Su(t) whcrc 11 is thc near-optimal

Advanced Guidance Sgstem Design

Chap. 8

Figure 8-41

Intercept Geometry (From [Katzir e l a l . , 19881 with pennission j o r n

RA NCE

AACC)

the target with the launcher range at intercept. This scenario can be divided into two problems: 1. Maximum range a t launch ID]: Assuming no target maneuvers, r11c11 D ( t f ) = R ( t f ) + V r t f , where R is the maximum range achievable by the missilc in time t f . Maximizing the distance D leads to D l ( / / ) = R 1 ( t f )+ 1 . ~ = 0 or R ' (t,) = - V r . 2. Maximum separation at intercept IS]: Assuming the launcher does not maneuver, then S ( t f ) = R ( t f ) - V L t f .

Maximizing the d~stanceS requires that t f bc sclcctcd so that R 1 ( t J )= I/;. In this case. thc best t, occurs where the slope of the R - curve is positive. I t should be clear from this elementary analysis that the key issue is the range-time curve for the missile. Thc problcm of maximizing the missile's range in a givcn time is also considered. The fundamental objectives are to construct the maximum range-time curve for the missilc and to understand the nature of the trajectories that yicld maximum range. T o formulate an cnergy state model, consider thc general, nonlinear system .+ = f(x, u, I) in a timc intcrval \to, tr] with the initial conditions x(ttr) = xo, a target sct x ( t f )E Of, and a control constraint sct I ( ( ! ) E a. For the missilc example here, x = IX, / I , y, Ej and I ( = 11,and point-mass cquations for thc vertical-plane motions arc simplified from Equation (8-203) as x = V cos y, = I/ sin y, j = g(n - cos y ) / V , and = (7' - D ) V / W . The functional dcpcndence of T and L). the aerodynamic and propulsive modeling functions, is dcnotcd by T = T ( t , { I ) and D = D(h, E, 1 1 ) . The propulsion is of boost-sustain type, thus the explicit dcpcndencc on tirnc. The control variable in the modcl is the load factor [ n ] and is lirnitcd by two constraints: Structural limit: Aerodynamic limit:

1 rl(t)l 5 n,,,,, 1 n(r)l c~,.,(M)qSl W

Sec. 8.7

Other Advanced Mldcourse Guidance Schemes

589

At this point, the optirnal control problem can be formulated. Stated simply, it is to minimize a performance index with a control vector function u(f) E 0 and with the produced trajectory .u(r) that satisfies the initial conditions xo and the target set x(tf) E Of. The problem essentially consists of finding the load factor n(*) such that the final range X(tf) is maximized. The following boundary conditions are applied to this examplc: Initial Point

Final Point y(tf) = free h(tr) = It,, (specified) E(tf) 2 Ef (specified inequality)

X(0) = 0 h(0) = /lo (specified) E(0) = E,) (specified)

Note that since the final altitude is specified, in this example the inequality on the final energy implies a lower bound on final velocity. The basic problem (PI) is to maximize the final range with the common boundary conditions (given previously), and y(0) = 0 (specified) and tf = (specified). This leads to a two-parameter family of curves, parameterized by Ef and I / . A one-parameter family of problems, denoted P2, is obtained by leaving i f unspecified. Finally, a modified problem [P3] is investigated wherein the initial path angle is open, that is, y(0) is open and tf is specificd. A variation of this problem is to maximize the final energy for a specified range, keeping all the other requirements the same. Since the nominal trajectory is assumed to satisfy the Euler-Lagrange equations, Pontryagin's minimum principle can be invoked to obtain the optimal solution. Following Katzir et al. [1988], the variational Hamiltonian for this example is defined as: H(A,

X , tc,

t) =

X,V(h, E) cos y

+

h~,V(h,E) sin y

+

[X,gI(V(h, E))l[n-cos yl

The costate variables are governed by the adjoint equations, namely:

Ah =

- [AIcos

y

+ Ahsiny - ( ~ , ~ i V ' ) ( -n cosy) + AE(T- D)lW]dVldh

i E= - [A, cos y + X h sin y -

+ (A,

V2)(n - COSY)

+ XE(T - D)/ W]d VldE

Vl q(aDlaE)

The minimum principle requires that an optimal control, which in this case is the optimal load factor n*, must minimize the variational Hamiltonian. Mathematically, this is expressed as min H(h, x, u, t ) = min [X,glV)t~ - (AEV/LV)D(h, E, n)] "€0

l!l/S?lmdx

590

Advanced Guidance System Design

Chap. 8

In principle, this condition is used to solve for an extrcrnal control n*. For the dragpolar [Lin, 19801, the result is

O n e should check that the extremal control satisfies the constraints 'and that the control actually mi~zimire.cthe Hamiltonian function. The latter condition is satisfied if hE is negative. Some common boundary conditions on the costates ( t r a n s ~ e r s a l i t ~ conditions) which vary according to the problem being examined, are h,(tf) = - 1, A,(tf) = 0, E ( t f ) = E f , or h ~ ( t f )= 0. The E - hE boundary condition is complicated because it invd~vesan inequality. In practice, one solves with the specified Galue of E(tf) and then checks the corresponding costate h ~ ( t f ) A . negative value of this quantity corresponds to an extremal solution. If this is not the case, one must refigure with AE(tf) = 0 (natural) boundary condition. This will arise if the specified bound Ef is less than the "natural" value, that is, E(tf) with At(tf) = 0. A preliminary study of time-range-energy optimal trajectories shows that there is a fair ~ f t e an r initial high-g pull-up the trajcctories are amount of lofting in these almost ballistic. Extensions to three-dimensional flight have vroduced similar be" havior. Future efforts will be directed at obtaining reduced-order models that are suitablc for closcd-loop synthesis and still retain the significant characteristics of the point-mass model.

Near-optimal trajectories. The approach to onboard real-time calculations for midcoursc near-optimal guidance prcsentcd here follows the work of Katzir ct al. [1989]. I t is based on cxtrcmal fields and neighboring cxtrcmals theory ideas uscd in conjunction with precalculated Eulcr solutions and fccdback gains to producc a closed-loop guidance scheme. The optimal rcfercnce trajccrorics ("nominal") and the closed-loop gains indexing arc done in tcrms of either current or stretched time. It is assumed that, in certain time intervals, the predicted intercept point is first computed and then uscd as the new terminal condition for the missile. Thc fccdback gains are computed using methods from the previous sections. The optimal trajcctory, having been generated prcviously, is now considcrcd as a nominal rcfcrcncc trajcctory hcrc. Ncar-optimal trajcctorics, which arise as a result of the follo\ving dcsign of a pcrturbation guidancc law, arc based on this nominal rcfcrcncc trajcctory. Assuming the cxistcncc of a nominal reference trajectory togcthcr with small pcrturbations in thc initial statcs Gx(r,,) and final conditions 6x(tf) for a fixcd final tirnc, [Kcllcy, 19621 showcd that the perturbed and the adjoint systcms constitute a linear, tirnc-varying systcm. Moreover, the optimal problcm of minimizing a performance index J is rcduccd to that of minimizing the second-ordcr approximation to J. Thc ensuing problem is a lincar, time-varying two-point boundary valuc problem that has a quadratic performance index, thc solution to which is wcll dcvclopcd. The optimal control law represented by the solution to this problem is of a contir~uouslincar fccdback typc which can be written, as shown in Lin [1983(a)], as u(r) = urC,(r) + Su(t) whcrc 11 is thc near-optimal

control vcctor function, ir,
+

8rr(t) = - N,(t)GX(t)

+

:V,,(t)8/1(t)+ Ny(t)6y(t)

(8-218)

+ N1;(1)8E(r) + S,,f(t)S/~(t,-)+ Nl,(t)8E(tf) If the optimal guidance problem consists of specifying the terminal range and altitude, it is more expedient to consider the equivalent optimal problem that seeks to determine a control function n(*) that maximizes the final energy E(tr) for a specified final range. For this case, the feedback part of the guidance law is the same as Equation (8-218) except that, in the last term on the right-hand side, N~/(t)6E(t/) is replaced by Nxf(t)Sx(tf). It has been observed, using perturbation schemes like these, that the gains are very large near the final time. This may cause instability to occur as the final time is approached. In fact, such sensitivitv associated with neighboring-optimal guidance schemes close to the final time was pointed out in Section 8.6, and has been documented in the literature (see for example, [Lin, 1980, 1982(a)]).In reality, following Section 8.6, it is assumed that a terminal guidance law such as PNG or a more advanced guidance scheme such as those presented in Sections 8.1 to 8.5 would be employed during the endgame. I t should be noted that the guidance law developed in this section only applies to problems with fixed final time. Some work has also been performed in the area of introducing perturbations on the terminal time, resulting in a method in which a time-stretched nominal model is used. Such a method at this time appears to hold promise. Comparisons have been made using digital simulation runs between trajectories generated using the guidance scheme developed in this section with those generated from optimal-control solutions. Unfortunately, a substantial numerical effort is required for synthesis of the former's feedback law. Additional research effort in this area would therefore be required to make the feedback guidance law suitable for practice applications.

8.8 GUIDANCE FILTER DESIGN TRADE-OFF In this section, the guidance filter design trade-off is studied. One of the dominant parameters affecting the missile system performance, that is, miss distance, is the heading error at handover from the midcourse guidance phase to the homing guidance phase. The heading error characteristics at the end of midcourse guidance phase is affected by some error models. This can be designed together with the midcourse and terminal guidance filter. Elements which affect the heading error at handover include missile speed V, target range from ground-tracking radar R , time to go (t,) a t handover, autopilot time constant 7, = l l w , , and guidance filter lag time 7,.

Aduanced Guidance System h f g n

592

Chap. 8

8.8.1 Optimal Guidance Filter Design Analysis The pursuer acceleration state is augmented to the state estimate equation [Equation (7-42)] to yield

(8-219) Equation (8-219) in discrete domain is

?(k

+ 1)

=

@(k)?(k) + \Y(k)u,(k) + N ( k ) z ( k )

(8-220)

with T being the equal time interval ( t k + , - tk) for the discrete-time integration and z ( k ) being defined in Equation (7-2). The optimal guidance command u, in Equation (8-13) in discrete form is

u<(k)= g l ( k ) f r ( k ) + gr(k)$dk) + g3(k)A-rY(k)+ g4(k)Am,(k) = E(k)i(k) sl(k)

= AlT:,

(8-221)
where T k = tr - k T is the time to go from tk to intercept. Applying Equation (8221) to Equation (8-220) yields

] satisfies the propaThe covariance matrix is defined as Pk = ~ ( ? ( k ) ? ( k ) ~which gation equation in Equation (3-29) P

+N

= I

-

Qk = E [ z ( k ) ~ ( k = ) ~y?(k) ]

+

Rk

(8-223)

Pk at handover can be gcncratcd by Equation (8-223) by recursion. At handover k = H, the variation of acceleration is u:, = EP"ET

(8-224)

Moreover, following Equation (8-134), the acceleration can be expressed approximately as

&(k)

- ( A l f l l R ) sin Sy,

A defined in Equation (8-4)

(8-225)

Sec. 8.8

Guidance Filter Design Trade-off

593

where 6y is the heading error and V is the norninal missile speed generated from the closest fit to the trajectory. Taking the variation of Equation (8-225) yields mi,.,

= R a,,'/(/\ b")

Substituting Equation (8-224) into Equation (8-226), mi,., = R EPHETI(AC")

From Equation (8-223), the covariance PH at handovcr is computed recursively. As the data interval T is taken to be a constant, the standard deviation of heading error a t handovcr is a function of 1,. autopilot time constant T,.,, and filter lag. Thus, when 7.4 and ?' are assumed fixed, the filter design is off with the desired lR at handover.

8.8.2 Missile Midcourse Guidance System Analysis

A simplified seven-state covariance analysis program has been developed [Yueh, 1983(b)]that is very useful for investigating handover conditions due to initial error, sensor error, and targeting noise. The following presentations are based on this analysis program and the general filter formulation discussed in Section 8.8.1. It pcrforms lincar error propagation analysis about nominal midcourse guidance trajectories and calculates thc heading error standard deviation. The program also calculates target tracking components of the guidance error and the IRU initialization errors including misalignment and drift. The chief benefit of such a program is that it eliminates the need for repetitive simulation of the midcourse phase when performing Monte Carlo-based simulation studies. Rather, these can begin near the handover to terminal phase. thus producing results faster and less expensively. This simple error analysis program has been demonstrated to achieve comparable accuracy to the much used full-size covariance program with reduced computation and storage. In the simplified covariance approach used here, it is possible by opening the guidance feedback loop to propagate the initial IRU error. Given the prescribed nominal Mach number and velocity angle history, the covariance analysis technique proves to be quite capable of generating missile state uncertainties near terminal handover. These in turn provide correlated initial condition statistics which may then be used to set up the terminal flight simulation. This combination of a covariance approach for the long midcourse phase and a Monte Carlo technique for the much shorter terminal homing phase is termed a hybrid analysis. The approach is to generalize the error covariance analysis from missile launch to terminal handover. When determining a nonlinear time-varying systems terminal handover error sensitivity to initial conditions and tracking noise, only the perturbed states around the nominal trajectory are analyzed. I t may be required to segment the approximate trajectory using different guidance gains to ensure close fits in missile position, velocity angles, and Mach numbers over the entire midcourse phase. This ensures the validity of the small angle linearization approximation used for the perturbed states along each segment. Following the methodology presented in Chapters 2 and 3 (see Figure 2-3), it is then desired to determine the statistical

Advanced Gutdance System Design

594

Chap. 8

properties, mean and second-moment of the system state vector x ( t ) . The highly nonlinear nature of the generic missile midcourse guidance equations, involving complicated coordinate transformations from the tnissile frame or wind axis frame to the inertial frame or vice versa, make the problem nontrivial. Fortunately, the missile trajectory in the Cartesian coordinate-frame is usually very smooth-since excessive maneuvering during the early stage of flight is always undesirable. Linearization is achieved with the aid of a simple explicit guidance scheme using adjustable gains to approximate the missile position and velocity states for certain portions of the flight. In general, different tuneable gains are required to fit the boost-phase and glide or sustained-phase trajectory sections. Depending on the intercept geometry, certain diving trajectory portions also need to be segmented in order to achieve the best parabolic curve fir. From this high-level curve-fitting approach, it is possible to construct a simplified linear error covariance analysis program using explicit guidance techniques for error propagation through the various sections of the nominal missile flight trajectory. Actually, as few as seven perturbed state elements in a Cartesian coordinate frame such as x = [6X, 6Z, 6y, a,, 6Y, 64, a,jTare required to achieve meaningful results where 6 X is the perturbed missile down-range position component, 6Z is the perturbed missile altitude position component, 6y is the perturbed missile velocity angle component in the X-Z plane, a, is the perturbed missile altitude acceleration component, 6Y is the missile off-range position component, 6 4 is the perturbed missile velocity angle component in the X-Y plane, and a,, is the perturbed missile off-range acceleration component. Other simplifications, in addition to the small angle linearization approximation for the perturbed states along each segment of the flight, include the assumption of a fixed targct configuration, and the accruing of the errors associated with thc targct state to the n~issilestatc unccrtainries. Consequently, the need to estimate the target state by filtering the illu~ninatortracking errors is renloved. By holding the target parameters fixed, both the range and angular tracking errors will contribute to the missile state covariancc through the proper transfornlation. Further assumptions include symmetry between missile and targct statc for the semiactive homing missile (see Figure 2-10), a first-order autopilot, and no angle of attack involved. As a result, there is no acceleration limiter for the midcourse phase. The dynamic modeling in discrete domain is of the form shown in Equation ( 5 ) of Table 2-1 where the transition matrix Q kis given by

I

10 0 1

- T V sin y cos 6 f I Ti'

C05

y

1

0 0

0

c$

fl

0 0 00

1 0 0 0 - TV sin y sin

tan ylcos

6

TA~!(I' cos y) I -f 0 7.4f tan y tan 4 I ' cos y

0 0 0 0 I

- TI' cos y sin 4

f~ tan 6

0 0 0 TV cos y cos 4

iJ

0

1

0 0

fI 7* f l (

I,,

COS

y

COS

i

6)

1

Guidance Fllter Dgign 'ItadeOff

See. 8.8

y = nominal velocity angle

component in vertical plane

+ = nominal velocity angle component in horizontal plane and u(k) (the perturbed guidance command vector at time form (see Section 8.6) as:

tk)

is given in cxplicit

where k; = kl + k2. Here, k, and k 2 denote commonly used explicit guidance gains. The sensitivity matrix 9 can be written as:

qI

6x Su

= -

I I

- h tan ylcos 6 h f l i ( V cos ~ ~ Y)

- h tan 0

+

0

f

0 0 h tan y tan +/(TA V cos y) f , / ( V ~ cos ~ y cos +) 0 f

T 2 -f h =1 2

1

Substituting Equation (8-228) into Equation (5) of Table 2-1 yields x(k

+ 1) = G(k)x(k) + w(k)

with

G = @

+9E

(8-231)

Thus, the error covariance matrix equation can be readily written as

where the second term on the right-hand side comes from the ground-tracking error on the target. The missile state is affected by each of these uncertainties due to the symmetry generated by simultaneously tracking the missile and target with the battle-group radar tracking system. The target tracking data measurement noise matrix Rk = diag(u$T, m i z , ugL) is usually given in diagonal form neglecting the correlations in the range and angle measurement channels. U R is~ the standard deviation for the target range tracking noise, while U A Z and UEL are the RMS values for the azimuthal and elevation angle measurement noise level, respectively. The matrix K characterizes the contribution of the target tracking errors to the perturbed

Advanced Guidance System Design

596

Chap. 8

missile state through the explicit guidance command. K can thus be represented as

With 9 given by Equation (8-230) as the 7 x 2 coefficient matrix relating the guidance command vector u(k) to the missile state vector x. Symmetry in the explicit guidance between the missile and target state allows ( x r ) E i to be expressed in a similar manner to Equation (8-229):

H is the transformation matrix between the target state X T given in Cartesian coordinates and the range and angular data which are in spherical coordinates. The corresponding 7 x 3 matrix is given by

-

C O S ~ T C O S ~ T-

H =

-

R T c o s ~ ~- Rs T~s i ~ n p ~T c~o s a ~ 0 sin P T R T cos P T 0 0 0 0 0 0 c o s p ~ s i n a ~R T ~ ~ ~ p T c o s -aRrT s i n p T s i n a T 0 0 0 0 0 0

RT defined in Figs. 2-10 and 6-23c a~ = azimuthal target LOS angle

PT =

elevation target LOS angle

For missile handover from midcourse to terminal, heading error (Sy and 64) is one of the dominant parameters that significantly characterizes each individual flight. Its effect on miss distance is a function of target range, missile speed, handover time, filter lag, and autopilot response. Variations on the guidance command can be used to compute the heading error standard deviation from launch to terminal handoff.

6AY, = (k;,/Tk)(- V cos y cos @ 64 &A,, = (k;,/Tk)(- V cos y Sy)

+

V sin y sin

4 6y),

(8-234)

Thus, thc standard deviation of the heading error is related to the guidance command uncertainty through

N=

[

T k

Tk/(ki, I/ cos y) tan y tan + l ( k i , V cos y)

I

0 Tkl(k4, V cos y cos 4)

(8-235)

Sec. 8.8

Guldance Fllter DesIgn Trade-off

597

so that thc heading error variances and cross-plane correlations can be readily derived as

Error analysis.

Propagation of the initial error dispersions through the entire course of flight can be achieved through the use of the recursive error covariance matrix equation given in Equation (8-732). For instance, the platform angular misalignment error due to IRU errors at launch can be computed from this simple seven-state covariance analysis without violating any assumed constraints. Figurc 8-42a shows a frequently used way of describing the IRU errors for a flat nonrotating earth. In this figure, Vx, Vy, Vz are accelerometer biases; 4,. are misalignment angles; E,, c,, E, are gyro drift rates; AX, A Y, AZ are position errors; A V x , A Vy, A Vz are velocity errors; and Ax(t), Ay(t), Ar(t) are nominal acceleration profiles. The quantities g+, and -g+,, which are the result of IRU misalignment angles, represent incorrect gravity compensation. The simplified block diagram of the IRU errors for nonrotating flat earth in Figure 8-42a contains nine integrators, or nine states, although the accelerometer bias V, gyro drift E, and gravity compensation terms can actually be ignorcd since their effect on the terminal handover heading error disperson is negligible. Conversely, the effect of the mis+,on the handover heading error digression for the nominal alignment angles +, trajectories being considered is very pronounced. To be consistent with the simple two-control axis approach, the IRU down-range velocity and position error should also be neglected. This is reasonable since their omission does not significantly affect the magnitude of the acceleration command issued by the guidance computer. With this in mind, a diagram of thls simplified IRU error propagation scheme is shown in Figure 8-42b. Further examination shows that the terms in parentheses in the figure can also be dropped since the nominal axial acceleration Ax is an order of as 6y and JI, as magnitude larger than the lateral components. By identifying - 6+, a similar recursive IRU error covariance matrix equation may be constructed as:

+,, +,

+,,

+,

Comparing Equation (8-237) with Equation (8-232), it can be seen that the guidance loop is actually opened in the IRU error propagation case since such errors are nonexisting so far as the guidance and control system is concerned. As shown in Yueh [1983(b)], it would be a simple matter to include the accelerometer bias uncertainty for the off-range and altitude control axes into the corresponding elements of the initial error covariance matrix. The gyro drift components for the two lateral control axes can also be included by replacing the (3, 3) and (6, 6) elements of the transition matrix @ k with (1 + €,A) and (1 + €,A), respectively.

Advanced Guidance System Design

Chap. 8

Figure 8-42 (a) IRU Error Diagram (b) Simplified IRU Error Diagratt~(c) Normalized Standard Deviation of Heading Error (deg) a t Terminal Handover Due to Target Track~ng Noise (d) Standard Deviation of Heading Error at Terminal Handover Due to 1 deg IRU Misalignment (From [l'ueh, 1983(b)] ulirh permissionfrom SCS)

Results.

Thc algorithms resulting from this curve-fitting techniquc are tested with thc hclp of the gcneral explicit guidance equations as described in Section 8.6. Threc cascs arc studicd in which nominal missilc trajcctory from launch to terminal handovcr is approximated. Thcsc thrcc cascs evaluate thc quality o f thc curve f i t obtained when thc trajcctory is scgmentcd into onc, two, and thrcc sections, respectively. In the first casc, one set of gains is uscd to fit the entire trajcctory, and the results are amazingly good in terms of missilc position components. I t is mainly near the transition from boost to glidc phase that any crrors arc found, and thcsc range up to 6.2 pcrccnt of the maximum position discrepancy. Howcvcr. for thc rest o f the glide phasc thc error is always less than 1 pcrccnt. T h e fittcd vclocity angle history curves also prove to be very good approximations. T h c biggcst crrors occur near midflight at about 3 deg for both azimuthal and clcvation angle com-

--

Time To Go

Case 1

Case 2

Case 3

Figure 8-42

Time To Go

(Continued)

Case 1

Case 2

Case 3

600

Advanced Guidance S ~ s t e mDesign

Chap. 8

ponents. Due to the large launch angle chosen, the error in the elevation \relocitv angle can be as much as 8 deg shortly after takeoff, but the transient is quickl;? diminished near the power-off period. For the nominal Mach number history curvh, even though the fit totally misses the peak at the end of boost phase, the largest error is still only about 10.8 percent. The parabolic curve-fitting scheme simply rounds off the sharp kink without further segmenting the trajectory. For the second case, a division is made between the power-on and power-off points of the trajectory and the guidance gains are adjusted until a best fit and a smooth velocity transition are obtained for both sections. Except for shortly after the boost phase, the approximations of all the position and velocity components yield values correct to within 1 percent. For the transition period, a 2 percent maximum error in position fit is achieved, while the transients for Mach number may overshoot by around 5 percent. For this reason, a further segmentation of the transition period is done for the third case, and this proves to be very effective in reducing errors in the curve fit over the entire trajectory of 2 percent o r less. The gain parameters thus chosen are used to propagate covariance errors through the corresponding segments of the flight trajectory. In Figure 8-42c, the heading error standard deviations (in deg) are compared for all three cases near terminal handover as functions of time to go. The results are normalized according to us, = (uEL/ @(Table) and = [ u A Z / ( Vcos y)]u(Table) where the range tracking errors are assumed to be negligibly small. As shown in Yueh [1983(b)], the results for the second and third cases are very close, showing that this technique converges as more segments are used. For the initial IRU platform misalignment angular error dispersion, Equation (8-237) is used for error propagation alone without incorporating the target tracking noise source. The total covariance results should be simply the matrix summation of Sc and P h from Equations (8-232) and (8-237), respectively. Figure 8-42d presents the heading error standard deviation results which are obtained from Equation (8-236), by normalizing the two tilts to one.degree for each plane. Again, there is close agreement between all three cases. This implies that the effective propagation of initial condition errors is somewhat insensitive to discrepancies arising from the nominal trajectory approximation.

8.9 PULSED ROCKET CONTROL

In the discussions prescntcd in Book I of the scries, it is acknowledged that airbreathing engincs are ideal from a propulsion standpoint since thcy have the ability to continually vary thrust. This ability, however, comcs a t thc expensc of increased complexity over that of the simplcr rocket motor. Thc ideal missile propulsion system would therefore be able to combine the advantages of less complexity with a rocket motor with those of continually varying thrust capabilities associated with an air-brcathing engine. This would allow significant range cnhanccmcnt whilc at the same timc eliminating residual fuel for air-fed burning occurring past the combustion chamber during altcration of flight conditions. Bccausc thcy are in fact

Sec. 8.9

Pulsed Rocket Control

601

sin~plcr,rockets have been modified in an attempt to incorporate or at least a p proximate the capabilities of the ideal air-breathing engines. This can be seen in the pulsed rocket motor. It is appropriate here to consider first the primary disadvantage of regular rockct tnotors in regard to missile engagement with maneuverable air targets. Basically, tactical missile rockets coast toward the target, and after the first several seconds they have burned up nearly all of their propellant, When the early warning radar alerts the pilot of a missile launch, the pilot will employ a maneuver which causcs the missile to turn suddenly, thereby bleeding itself of irreplaceable energy. A pulsed rockct motor. on the other hand, not only employs double or triple burning, but also consumes propellant in a highly efficient manner. This gives the pulsed rockct motor its primary advantage over the conventional simpler rocket motor because it alleviates the fuel-depletion problem. This also gives a missile using a pulsed rocket motor the ability to significantly increase its performance in order to maximize the terminal velocity and achieve intercept at extended range. In lofted trajectories using the pulsed rocket, a second burn takes place at higher altitude and lower speed. Consequently, the drag xvhich relates to fiiellenergy depletion and back prcssure which relates to thrust are both lowered. Many early missiles were equipped with motors resembling the fixed timc delay pulsed motor. In fact, a pulsed motor with a fixed time delay has been applied in many recent missiles. Unfortunately, the delay is chosen for an arbitrary amount of time. In spice of this, an approximate time delay of the sustainer burn allows an improvement in the final velocity to be achieved. In the case of the pulsed rocket motor, the sustainer burn is delayed relative to the booster burn. In this section, optimal guidance laws determine the moment of the second motor firing and weigh the performance advantages against the added weight of propellant for the second pulse. Control expeditiously utilizes the pulse as the tactical scenario dictates. This versatility is a consequence of incorporating packed electronics and multiplexed data buses into the missile. In order to create an energy boost in the endgame, a possible three-pulse motor aided by thrust vectoring, auxiliary thrusters, or powerful aerodynamic flight controls can make a quick final trajectory correction of the missile to close on a maneuvering target. An alteration of the laser proximity fuse could initiate a course command. Therefore, the threepulse motor substantially reduces the miss distance and may allow direct hit of the target to heighten the hit-to-kill objective of missiles. A simplified model of the two-pulse motor used in the development of optimal control logic is put forth in this section. Both pulses have constant thrust values but the burn time durations for the two pulses are different. Additionally, the rate a t which propellant is consumed is defined as a function of motor thrust resulting in a linear decrease in the overall missile mass which a pulse is burning. Then thrust profile in the mathematical formulation of pulsed motor can be characterized by the following five parameters. The first two are associated with the booster burn and are the thrust level and burn time duration, respectively. The other three parameters are associated with the sustainer burn and are the thrust level T, burn time duration

Advanced Guidance System Design

602

Chap. 8

tsD, and the time t, at which the sustainer burn starts, respectively. The burn time

duration is constant. The sustainer burn therefore includes t w o parameters. However, not all of the five parameters just mentioned are free parameters as can be seen when physically imposed constraints are added to the model. In the case of airlaunched missiles there is a requirement that the missile separate from the launcher aircraft as quickly and safely as possible at launch time. his implies that the thrust level and burn-time duration of a booster are assumed to have prespecified values that do in fact ensure launch safety and satisfy this requirement. Additionally, because the total impulse from the rocket motor is a fixed parameter, this implies that the sustainer-thrust level and burn-time duration must also be fixed parameters. As a consequence, only the time at which the sustainer burn starts t , is left as a free parameter. The pulsed rocket problem is therefore reduced to the problem of determining an optimal value for t, that maximizes the terminal velocity [Glasson, 1984; Glasson and Mealy, 19831. This section presents optimal pulsed rocket control with the aim of selecting an optimal t,. Consider the dynamic system described by the following nonlinear differential equation

and a performance index of form

Adjoint the dynamic system of Equation (8-238) to Equation (8-239) with multiplier A and then define a Hamiltonian H = G + Arf to obtain J = +(x)R=o +

In

dR. The variation in J due to variations in the control variables dR u and t , for the intercept problem from current range R to intercept point R = 0 can be derived as R

( H - A'*)

The optimal control solution is determined by the following optimality conditions

aHlarr = 0

(8-241)

T o simplify the optimal control problem, the optimization problem described in Equations (8-238) and (8-239) is transformed into two separated but coupled subproblems, respectively. One is the problen~with Equation (8-241) for control variable u and the othcr is the problem for t , with Equation (8-242). An optimal guidance solution, when I, is known, is given in Section 8.6. The solution of the I, control

Pulsed Rocket Control

Sec 8.9

603

logic for maximizing velocity a t intercept is the combined optimal solution of Equation (8-242) and the intercept guidance problem described in Section 8.6. Under the assumption that the other control variables arc scheduled optimally, the optimal ignition time 1, can be judged at which

is evaluated. If a delay in pulse ignition (St, > 0) would cause a decrease in J in which case 6J < 0 in Equation (8-240). I is then a negative value and the pulse should be fired immediately. This translates into the following logic: Evaluate I with a nominal ignition time at current time t, and fire the pulse immediately if the integral turns out to be negative, that is, less than or equal to 0. Equation (8-171) is substituted into I as defined by Equation (8-243), where H i s substituted by HI, to obtain sec a dR Substituting Equations (8-139a), (8-171), and (8-135) into Equation (8-244) yields 'f

a2(Do - T ) a ~ ; dt, 2rn v4 at,

~

a = missile acceleration

(8-245)

The function F2 depends on t, explicitly through the thrust T and mass m. Hence, the integral of Equation (8-245) can be decomposed into two specific terms by using the chain rule as follows

aT

-1

a F ~ ~ a m t am at,

-+ at,

(8-246)

Figure 8-43a describes the partial derivatives of the thrust and mass with respect to the variation of the starting time which can be expressed as

{

ts

ST St, t,

t,

+ Sts 5 t < t. + tsD + tso 5 t < t, + t s +~ St,

+ St, 5 t < t, + I S D + St,

604

Advanced Guidance System Design

Chap. 8

Figure 8-43 (a) Perturbation in Time of Thrust Pulse (b) Opti~nalGuidance Pulsed Rocket Control

For a small nonzero tit,, Equation (8-246) evaluated a t current time is

where tit denotes a negative constant mass-flow rate. Fire the pulse immediately, if and only if Equation (8-247) is negative. If Equation (8-247) is positive, command ignition delay and the sustainer have not yet burned. The evaluation o f basic pulse motor ignition control in Equation (8-247) is therefore checked in the next time step to determine the sustaincr burn time. Observing Equation (8-247), an unknown variable must be predicted a t current time. An approximation procedure for evaluating Equation (8-247) during the midcourse phasc needs to be considered. By extending the optimization algorithm presented in Section 8.6, the algorithm for pulsed rocket control is obtained. Both the aerodynamic controls and sustainer burning time I, are computed by Equations (8-241) and (8-242). It is also necessary to satisfy the optimality condition and intercept constraints in Section 8.6 in addltion to choosing a14 and at, to increase J. Comparing a conventional rocket motor to thc pulsed rocket motor using simulation shows an average increase o f about 40 percent

Sec. 8.9

-

Pulsed Rodret Control

Target Position, Velociry Data

L

Pulsed Motor

Tracking Filter

and Short

Integral Table (K,, K, in Sec. 8.6) Guidance Law

K, = -2, K, = 6

I

I

+

Guidance Command (B) Figure 8-43 (Continued) in terminal velocity due to optimizing the sustainer burn time in air-launch application. However, this improvement in velocity entails a longer missile flight time, so that consequently the range at which the target is intercepted is decreased. Since the computer time and storage requirements of the algorithm are severe, the numerical pulsed rocket control optimization of Equation (8-247) is impractical for onboard implementation. Thus, the implementation of the pulsed rocket control, basically an extension of approximate optimal guidance law with the exception of a few modifications, is considered [Glasson, 19841. The pulsed rocket control structure, shown in Figure 8-43b, was developed. A modified version of the optimal guidance law consists of the optimal guidance blocks within the dotted lines in Figure 8-43b. The remaining blocks are the firing logic, and the optimal firing time is determined from a precomputed schedule. In the case of multipulse rocket motors,

606

Advanced Guidance System Design

Chap. 8

the pulse thrust level and burn-time duration are specified in the motor design, thus fixing the total impulse of each pulse. Only the pulse initiation times represent inflight controllable variations [Calise and Smith, 19821. The multipulse rocket motor optimization problem can be handled by applying to each pulse the preceding arguments.

8.10 IMPROVED TIME-TO-GO ESTlMATOR Conventional guidance law such as PNG has proven to be inadequate to counteract the threats of modern advanced fighters. A review of tactical missiles in Section 8.1 concludes that optimal control law would be the best candidate. However, implementation of most linear optimal control laws requires t,. Knowledge oft, is an important parameter when implementing optimal guidance laws. The standard technique of dividing range by range rate information is very poor when both the missile and target are accelerating. A better method for estimating tg must then be developed to improve guidance law performance. Techniques for improving the accuracy of the predicted tg estimation are discussed here. Although the evasive maneuver of a target can be predicted as discussed previously, the estimation of target maneuver is contaminated by noise, thus making the prediction of the t,, more difficult. Therefore Riggs 119781 proposed a much simpler algorithm where acceleration information was used to improve the I, estimation. The tR estimates using prelaunch acceleration information can also be used. 8.10.1 Estimation of Time to G o

A simple estimate of time to go is t2 = -Rid. Another estimate is t, = - l?. PI'/( l2 whcrc and i/ represent the relative position vector and velocity vector, respectively. If the velocity vector points directly at the target and the relative acceleration does not have any changes, then the two estimates are in agreement. The estimate of the I, can be improved using acccleration information. if an estimate of the average relative acceleration along the LOS between the missile and the target is represented by A, r, can be better estimatcd by using an improved rangc and range rate approximation

8.10.2 Time-to-Go Estimate Algorithm Riggs's algorithm is bascd on the assumption that the missile is pointing a t thc target throughout most of thc terminal guidance phase of thc flight, and that the target is not accclcrating [Riggs, 1978; York, 19791. Undcr this assumption, it is possible to estimate the closing avcragc acccleration along thc LOS using only the missilc ccnterline componcnt of the accclcration. T w o paranlcters arc used by the acceleration cstimatc bcforc launch: A,,,,, and A,,,,,,.The former is an cstimate of component of

Sec. 8.10

Improved T I r n e t a G o EstIma tor

607

acceleration along the x-axis beforc burnout, and the latter an estimatc ofacceleration after burnout. The componcnt of acceleration along the x-axis then has an average of where t,, dcnotcs the initial time ofterminal guidance, tHo the time ofengine burnout, and tf the time of intercept. Riggs's algorithm uses input values of A,,,,,, A,,,,,,and an estimatc of I / . Thcn it uscs Equation (8-249) to calculate an average acceleration A . Next, I, is obtained by solving Equation (8-148). Thcn the currcnt time is added to t,, to cstimatc t f . Finally, Equation (8-249) is uscd again to calculate an average acceleration A. The correct root to solvc Equation (8-248) is Equation (8-249) has an improved version for A as follows [York, 19791 1 '4 = {A,,,,, SIC[min(tf, ~ B O ) - 101 + A,,lttlSIG[tf - max(to, tea)]) (tf - to)

The following discussion on improved time-to-go estimation is based on an excellent article by York [1979].

Missile nonalignment. Under the condition that both the missile and target are not highly maneuverable in a terminal guidance scenario, it is sufficiently accurate to assume that the missile is aligned along the LOS. However, in the case of highly maneuverable missile and target, this assumption is doubtful. This assumption has further affected the accuracy of the guidance law used. From Figure 2-10, cos u L = cos 0, cos where 0, and I), represent the Euler angles relating the seeker coordinate system to the missile coordinate system in pitch and yaw components, respectively, of UL. An estimate of the closing relative acceleration along the LOS is required for Riggs's algorithm. With the assumption of no target acceleration, the projection of the missile X-axis acceleration component onto the LOS is (8-252) A cos UL

+,

Consequently, Equations (8-250) and (8-252) are used equivalently in the corrected version of Riggs's algorithm where there is no assumption of alignment of the missile centerline and LOS in which the parameter A in Equation (8-230) is replaced by Equation (8-252). Hence

- --

t, = 2Rl(V,

The estimate is expressed as tf

+ V V ;+

Equation (8-251)

A

4AR cos oL) Equation (8-253)

t,

tf

(8-253)

Aduanced Guldance System Design

608

Chap. 8

Updating of acceleration estimate.

Rather than using an average for the entire flight, a more effective approach is to calculate an estimate of average Xaxis missile acceleration for the remainder of the flight. This method can improve the Riggs's algorithm, which uses the simplest possible missile X-axis acceleration profile with two functions-one for thrust-on and the other thrust-off. With this approach, the average centerline acceleration is described by using four parameters ama ax)^, ( A m , , , ) r ,(AmOx)D, and (A,,,,2)D which represent the maximum and minimum accelerations during thrust-on and thrust-off. Suppose that the function of x-axis component of acceleration is f, during the thrust portion of flight and is f 2 during the thrust-off flight. The average missile acceleration along the x-axis is then calculated as

when t f z

tBO. Equation

(8-254) can be represented as

Note that when missile acceleration is constant during thrust-on and constant during thrust-off, Equation (8-255) gives Riggs's algorithm which is described in Equation (8-251). Equation (8-255) can be used to represent the average acceleration a t arbitrary time rather than the beginning of the guidance phase. Let t" be arbitrary at current time t. Then the average acceleration from current time to intercept is

where

and

= (Ami,,)~

< tso

where the current acceleration is represented by a(t)at thrust-on or by a1 ( I ) at thrustoff. Given a missile, a plot of missile centerline acceleration as a function of time

Sec. 8.11

Jammtng and Clutter Eflects on Gulded Interceptor

609

is generally needed for various target scenarios. Before power-on, a ramp is a desirable approxinlation. However, during power-off phase, another type of curve fit is more appropriate as follows.

Parameters in Equation (8-257) can be calculated on board or stored as data points for a table look-LIPmethod using linear interpolation. Therefore, the required centerline acceleration parameters a , (A,,,.,,).r, '11, and (,9,,,,,.,)D in Equation (8-236) become available.

8.11 JAMMING AND CLUTTER EFFECTS ON GUIDED

INTERCEPTOR The effects on a radar-homing inrcrceptor due tojamming and clutter environments and the effects of advcrse weather on an IR-homing missile are presented here. Further details of these are presented in Book 4 of the series, while the background material is presented in Book 1 of the series.

8.11.1 D e c e p t i o n Repeater Jammer The idea of a repeater jammer typically used in electronic countermeasure (ECM) techniques is essentially to receive, then retransmit an enemy's radar signal. Before retransmitting back the signal, it first undergoes amplification, a possible time delay, amplitude modulation, and a Doppler shift in frequency, all according to the particular jamming model being used. The objective in using a repeater jammer is to mask the true signal by transmitting a modified version of this signal to the enemy's tracking radar causing it to in turn break lock. Monopulse tracking radars are jammed by the introduction of artificial glint onto the jamming return. One of the most important considerations is the jam-to-signal ratio (JSR), which must be on the order of at least 7 to 10 dB if the jammer is to be able to capture the radar's tracking circuits. This represents a repeater gain and power output specification [Hoisington, 19811. In order to calculate the JSR at the semiactive radar receiver, an expression for the repeated jammer signal is first developed. At the self-protection repeater jammer, the transmitted radar signal spectrum received, PJR, as shown in Skolnik [1980], is P ~ R = PAG~X'GJR/[(~T)~R~RLJ] where GjR denotes the receiver antenna gain o f t h e repeater jammer, G, denotes radar transmitter gain, PA denotes average radar transmitter power, R ~ denotes R jammer-to-radar distance, A denotes \vavelength, and L denotes loss term. Multiplication of the preceding equation by the repeater's electronic gain G , gives the output of the repeater PI

Advanced Guidance System Design

610

Chap. 8

The repeated jammer signal spectrum at the missile's receiver is therefore

P,,,j

=

P j C m , h 2 G j ~ I [ ( 42 R,VJLR] n )2

(8-259)

Substitution of Equation (8-258) into Equation (8-259) yields PmJ = P A C ~ G , ~ X ~ G ~ R G ~ ~ ~ , / [ ( ~where T ) ~C RJ-r~ is Rthe R$ antenna ~LRL gain ~ ]of the repeater's transmitter, C,,, is the receiver gain of the missile, and R,,j is the missileto-jammer distance. The signal spectrum returned to the missile's receiver P R is generally given by

where RCS denotes the radar cross section of the target. JSR can now be computed from P,,]/PR = JSR = G J R ~ - r T G c ~ 2 / (RCS 4 ' r rL R L ~where ) it can be seen that JSR depends on the repeater's gains. Defining S , = JSR RCS, where this product represents the magnified RCS desired from the standpoint of the repeater, the electronic gain of the repeater can be expressed as The jammer's power can be found by substituting Equation (8-261) into Equation (8-258) resulting in

The required repeater gain and power output arc specified by Equations (8-261) and (8-262). The JSR for a given EW system is proportional to the square of the distance bentyeen the jammer and radar. The relative strength of the rargct echo to the jamming echo ultimately determines whether target detection occurs. Again, the purpose of the repeater's jamming is to provide false target information to the enemy's tracking radar to essentially mask or conceal the real target signal. For a semiactive homing receiver, the JSR may actually approach a constant value since R ~ does R not diminish as much as R,,, during the homing phase. In the case of active homing, RjR and Rng are equal so that the JSR diminishes rapidly in proportion to the distance between target and missile. This means that the semiactive receiver is more easily deceived than the active receiver in an EW environment.

8.11.2 Clutter Clutter is the name given to any unwanted radar echo. These echoes can contaminate the radar receiver signal, making target detection more difficult. In the case of aircraft detection, clutter in the radar can come from reflections from land, sea, rain, or even birds. Turbulence and other atmospheric effects can also produce clutter. As discussed in Chapter 6.7, when the target is flying a t a suficiently low altitude that the radar main beam intercepts the ground, the large illuminated ground results in a reflected power (called clutter) that is typically several orders of magnitude larger

Sec. 8.11

Jamming and Clutter E f f e c t s on Guided Interceptor

611

than the power reflected from the aircraft, and aircraft detection is thus denied. Multiple-pulse returns are added equally for clutter and target signal, and so n o advantage is gained by integration. The problem of clutter led to the pursuit of a different type of radar, one that utilized the Doppler frequency of a moving target as its basic signal, which then could readily (in principle) be separated from stationary non-Doppler shifted ground clutter. Ground clutter effects on the semiactive homing radar-guided AAM have previously been studied [Miwa and Imado, 19861. A guided missile is significantly affected by the main lobe clutter. This effect can be avoided by employing a guidance-shaping concept described in Section 8.6. Figure 8-44 shows a target scenario involving a semiactive AAM. The figure also gives the relative positions o f the missile and radar. A transmitter is located on the aircraft radar and energy is being radiated primarily to the target. Howeoer, because of the side lobes, energy is also being radiated in the other directions. A receiver antenna is located on the missile, whose speed is V,,, to direct it to the target, but this antenna also receives clutter through its side lobes. All relevant terms are defined in the figure. Skolnik [I9701 gives the clutter power spectrum dP due to a small radiated area on the earth dS, as

where A denotes the duty cycle of the radar and a, denotes the bistatic backscattering coefficient of dS. When the transmitted waveform is a rectangular pulse train of constant width T and pulse-repetition period T, the duty cycle is simply TIT. T h e typical duty cycle of a pulse radar used for aircraft detection is on the order of 0.001, while for a CW radar transmitting continuously, the duty cycle is unity [Skolnik, 19801. The product of the back-scattering coefficient a, and the

dS

earth

Figure 8-44 Geometry and Designation of Variables (a) Side View (b) Bird's-Eye View (From [Miwa and Imado, 19861, 8 1986 AIAA)

Advanced Guidance System Design

612

Chap. 8

resolvable ground area u, dS gives the RCS o f a patch of ground. The antenna patterns of the transmitter and receiver hare azimuthal symmetry with respect to the antenna axis, so that G , and G,,,,become functions o f E A and E,,,, respectively [Miwa and Imado, 19861. The back-scattering coefficient at dS with respect to the scattering coefficient from A to M has the direction of p which is a vector sum of unit vectors a and rrt%n Figure 8-44b. The curvature of the earth is assumed to have a negligible effect. The small area dS is chosen in the following manner. With the aircraft at one focus and the missile at the other focus, an ellipsoid of revolution is formed such that, anywhere on the surface, the range of the clutter path (R = R.4 + RM) is constant. The intersection of the ellipsoid with the earth's surface defines an elipse. Another elipse whose center is slightly shifted is formed by the increment of R. The dS is selected dividing area between these two ellipses. If the target has a velocity relative to the receiver, there will be a frequency shift in the received signal away from the transmitted frequency f o by the amount +- f,, which is defined as the time rate change of path d divided by the wavelength fd = dlh where the combined path dis the path from the transmitter to the target and then to the receiver. The frequency shift is specified by fd if thc distance between the target and the receiver is decreasing and vice versa [Skolnik, 19801. The Doppler-frequency shift f , due to clutter is cornposcd of t w o terms, one representing a Doppler-shifted frequency f o due to the reflection at dS, and another representing the Doppler-shifted frequency of the direct transinission to the receiver. The expression for f,is

A clutter frcqucncy spectrunl and this quantity is due to the receiver n~cchanisn~. is obtaincd by s u ~ n ~ n i nthc g dP over the entire area. The frcqucncy may be a mainlobe rcturn or sidc-Iobc rcturn. Using Equations (8-263) and (8-264) in conjunction with simulation tcchniques, it is possible to draw a plot ofspectrum versus frequency of clutter at any intcrccpt time point. The targct signal spectrum at the receiver P,,,, is computcd from

which is the rcsult as found in Equation (8-XI), ~vhilcthc corresponding Dopplerfrcqucncy shift f , is

ff = {ftl+ (

+

[',\r

cos COS

v,4.1' + *'C

*.,,T)IA)

cos

*

T.4

- { f , ~+ ( V n

+ k'.). COS vT.21

COS @.4.\1

(8-266)

- VIP,COS @.\1.4)lhI

The t w o tcrnls in Equation (8-266) reprcsent thc 1)oppler-shifted frcquency f o due to rcflcction by thc targct and thc 13oppler-shifrcd frcqucncy f , l of thc dircct transmission to thc receiver, rcspcctivcly. In a straightforward tnanncr, the Doppler spectrum vcrsus frcquency plot for target signal is easily obtaincd. It is possible to simulatc cluttcr and target-signal frequencies and spectra by using mcthods that

Sec. 8.12

Hultlmode Guidance System Design

613

combine the bistatic radar equations, that is, Equations (8-263) through (8-266). with simulation techniques. It is equally possible to obtain the solution to any type ofclutter and weapon system using sim~ldrtcchniqucs. In the case of the radar missile and targct being in line with each other, the value off, in Equation (8-264) becomes + 2V,,,/u, while the value o f f , in Equation (8-266) becomes 2 V,,,/A + Vr(cos Y cos 4'' T I I ) I X .

8.11.3 IR S e e k e r in Weather Condition An IR sensor measures the LOS angle which is used in guidance. A noise sensed by an IR seeker can bias the measured LOS anglc from the true LOS angle. For a given temperature difference AT between the target and the background, the power per arca of targct emitted P , is defined by P, = ~ K T "Tp. where K denotes Boltzman's constant, T target temperature, and p fraction in the frequency band given black body radiation. If the detector surface of IR seeker is of the area A, the power on thc detector P, is obtained by P, = Pt.Ae-SRA,l(nR') where 6 denotes attenuation factor. .4, observable target area, and R range between detector and target. The noise equivalent power P,,, which is the function of specific detectivity D and noise bandwidth w,,, is defined in Tso and Lobbia [I9861 as P,, = ( 4 w ~ ) ' l ' l D The . measured LOS angle u detected from the IR seeker is the sum of the true LOS angle u ,,,,,. and the LOS angle error a,.

in which power spectrum

+,,,

of the LOS angle error u , due to noise is given by

and Of denotes instantaneous field of view. The IR homing performance can be evaluated based on Equations (8-267) and (8-268).

8.12 MULTlMODE GUIDANCE SYSTEM DESIGN In this section, a multimode guidance system is introduced. Further details of multimode guidance system design are t resented in Book 4 of the series. T o compensate for the limitations of individual sensors such as IR or RF seekers and to improve the overall tracking ~erformance,a conceptually complete, implementably simple, and highly efficient approach called the adaptive multimode tracking and guidance ( A M T G ) system is employed. This system consists o f t w o or more heterogeneous sensors, for example, RF and IR in this case, and this configuration enables the system to work with the sensors in different modes. Each sensor, according to its design capability, is best adapted to a particular operation condition. The combination of several types of sensors can render the system to perform highly efficiently under almost all types of mission conditions. Moreover, the combined

Advanced Guidance System Design

614

Chap. 8

design can reduce equipment weight and speed up signal processing. In addition, dependability is significantly increased with respect to the case of single sensor or single type of sensor in the sense of high system reliability and wide operation condition. The functional diagram of an AMTG structure is shown in Figure 8-45a where the center-of-reflection measurements (position, azimuth, elevation, and occasionally Doppler rates) and target attitude measurements (pitch, yaw, and roll angles, shape) are fused [Yang and Lin, 19911. This structure is very relevant in a highly maneuvering environment [Yang et al., 19911. The real dimension information of the target provided by an imaging sensor can be exploited by this scheme and it is technologically feasible due to fast growing imaging sensors and computation techniques. Advanced tracking and guidance systems will be essentially constructed based on this adaptive multimode concept, that is, to integrate imaging sensors not only for target matchinglselection in the final impact but also for target attitude estimation. Another advantage of an AMTG system is the fact that an imaging sensor estimates target maneuvers more efficiently. Target attitude variables, deduced from

------- 1 I

/

/ / 'liacking. I'oinling. and + ('onln>l

'large1

-2

.-S

\

,-------

\ \

-,

I Sig~~al ['roccssing Model l-0 L - - - - - - - l

:Ki;nya:l4

a

v 4

LL c!%.

-

A

v Ki~rnlalic 1:ilIcr

I

and Clullcr (A)

Figure 8-45 Adaptwe Multinlode Tracking atid Guidancc (AMTG) Systrln (a) Sensor Level Fusion (b) Track Level Fusion (c) Partially IJroccsscdData Fusion (di Multimodc Guidancc System Using Optimal CI'NG Algorithm with LOS Angle and LOS Hatc Estimator

----------I I lmagc Altitude I'm~~sor

A

/

----

1 /

I- .rarst

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Figure 8-45

(Continued)

616

Aduanced Guidance System Design

Chap. 8

RFlIR MUL.TIMODE SENSOR INTEfiRATIONIFUSION (IR, K[.' Active/SI.miactive/Parsive)

I W: Fnvironrnent Condition SC: SensorCapabilities TC: Tgrget Cbamcteristin

Figure 8-45

(Continued)

target image, provide fast and reliable measurements for maneuvering detection and estimation. The delay in maneuvering detection using position and angle measurements only can thus be significantly reduced by using imaging sensors [Sworder and Hutchins, 1989; Yang et al., 19911. 8.12.1 WflR Multimode Sensor IntegrationlFusion The structural advantage of the AMTG system depicted in Figures 8-45a through 8-45c is further reinforced by implementing powerful supporting algorithms. Adaptation and fusion algorithms are two key components. The adaptation part is to assess for which type of sensors the current operation environment condition is best suitable. This adaptation can be achieved by an on-line identification of environment conditions and by consideration of sensor capabilities and target characteristics. This results in an optimal weightinglbalance of sensors available on board. The fusion counterpart is designed to combinelintegrate these heterogeneous sensor data in an optimal way. The optimal fusion algorithm is designed to account for each sensor's measurements, measurement uncertainties, and relative confidence of the sensor generated by the adaptation algorithm. The adaptation/fusion concept has a desirable flexibility when applied to a multimode configuration. Application of adaptationlfusion to raw sensor data results in sensor-level fusion (Figure 8-45a) which demonstrates under certain circumstances better performance. O n the other hand, its application to filtered data o f sensors, that is, thc estimates obtained by an individual sensor as it operates alone, provides a track-level fusion (Figure 8-4jb). Between these two extremities, fusion at the partially processed data level (Figure 8-45c) is possible with the AMTG system depending on the situation.

-

Sec. 8.12

Nultimode Guidance System Design

OIrIIMA1 CI'N(; Al (iOUl'l1IMWITII ACOMPI IZMI:NrARY I (kSAN(il.1, AND IUI'I;I Sl1MAN)U

Figure 8-45

(Continued)

In Figure 8-45d, a simplified AMTG system is illustrated where the RF and IR sensors are fused at the partially processed data level. This system is designed to validate both RF and IR tracking data and to integrate the IR LOS rate information into a combined sensor selection and weighting scheme. For ideal operating conditions without jamming, clutter, and severe weather environment, one or a composite set of measurements is used for guidance filter input. A sequential process of measured LOS angles (aRF) from active RF and rate (uIR)from IR seekers is passed through a signal processing testing algorithm for SIN threshold and validation region tests. The appropriate signal is then selected to be used in operation. The sensor integrationlfusion technique is to generate a composite LOS angle a, at the seeker configuration selection by weighting the respective selected signals, URF and a , (which ~ is the integration of uIR).The weighting value is chosen to quantify the relative faith in a particular measurement. This weighting function or heterogeneous sensor data fusion is obtained by optimization with respect to environmental conditions, sensor capacities, and target characteristics. In a jamming and clutter environment, the measured radar signal from a jamnler or clutter is considered as a target, thus degrading target tracking, as shown in Section 8.11. A set of precomputed thresholds needs to be stored on board. By monitoring the computed SIN of the RF and IR seeker, the effects due to the presence of adverse RF and E O environment can be examined. Computed levels of SIN below the threshold will cause the logic to offload the RF or IR measurements from further processing. The validation procedure of chi-square test can be used to detect the presence of RF jamming and clutter. This procedure employs a validation region. A measurement within this region is of interest with high confidence probability. Measurements falling outside the validation region indicate the onset of jamming

Sec. 8.13

Advanced Guidance law implementation issues

619

ance described in Figure 8-45d. The results show that larger differences between sidelobe and mainlobc magnitudes result in bcttcr performance since the effects of sidelobe jamming become less severe for an RF scckcr. Without the janlniing, clutter, and adverse wcathcr effects, the performance of multimodc guidance has very good results as expected. The performance of a seeker using R F only in j a n ~ n ~ i n g or a clutter environment demands very low sidclobc antenna, which can be costly. Likewise, the pcrforniancc of a seeker using IR only in adverse environment can be severely dcgradcd. O n the contrary, the miss-distance performance with multimode guidance in these severe conditions is sholvn to be improved. Thc performance is acceptable if severe conditions do not exist all the time during homing. However. if the sevcrc conditions persist throughout homing, an HOJ mode is used to intercept the jani~ncr.

8.13 ADVANCED GUIDANCE LAW IMPLEMENTATION ISSUES This section presents issues associated with the design and implementation of an optimal guidance algorithm in Ada for AAM. ~ e t a i l c ddiscussions can bc found in Evers et al. [1988], on which the iollowing discussions arc based. The most critical implemcntation issue involves limited n~icroprocessorthroughput, which is associated with the run-time overhead of the Ada language and the efficiency of Ada cross-compilers. Other issues center around the design and coding of the guidance software and the assessment of rhe throughput and memory requirements of the code in a simulated microprocessor environment. Conclusions from a current study on these implementation issues are also discussed. Research on the beyond-visualrange air-to-air guidance problem [Cloutier et al., 19881 has prompted much lvork in this area. The development of numerous guidance law formulations based on optimal control theory have shown performance improvements over P N G in computer simulations. However, the problems associated with implementing the algorithms on missile-quality microcomputers have not been studied closely enough. Also, problems associated with the required use of Ada (MIL-STD-1815) and the MIL-STD-1750A 16-bit instruction-set architecture for embedded computer applications by the Air Force have not been considered. The goal of this study [Evers et al., 19881 was to investigate issues like these as a first step to providing basic research results for further development. In this study, the following algorithms were implemented: singular perturbation (SP) for midcourse guidance [Cheng and Gupta, 19861, and P N G using filtered seeker measurements for terminal guidance. A Fairchild F9450 microprocessor was selected for the processor-in-the-loop simulation testing, as was done in Lin [1982(h)].

Implementation problems. In addition to the previously mentioned design issues, there remains the goal of having the guidance algorithm run in real time (for example, 40-Hz command update in terminal) on a microprocessor. One obstacle has to do with the limited word length, memory, and throughput of the

620

Advanced Guidance System Design

Chap. 8

current microproccssor~that are suited to use with n~issiles.T h e design work itself, of course, is done on large computers that are not similarly hindered. The MILSTD-1750A instruction-set architecture now required by the Air Force for embedded applications, has a word length of 16 bits and a memory address space of 64K words. An extended Kalman filter implemented in hardware that satisfies this criterion may show divergence caused by truncation effects in the covariance matrix: computations. This is particularly true when the different state covariances differ by several orders of magnitude, as may be the case with ~llissileguidance filters in LQG effort associated with the usual altcr. formulations. The extra computatio~~al natives, that is, using the Joseph form o f the covariance update equation or a squareroot filter formulation, increases throughput requirements o f the microproccssor [Maybeck, 19791. Evcrs et al. (19881 noted that the major implementation issue for complex algorithms may be associated with the design methodology and high-order programming language selected for the embedded software. For large software systems, the use o f a high-order language and structured coding techniques improves reliability and maintainability. For this reason, the Ada high-order language \vas developed for use throughout the life cycle of D o D embedded systems. The idca is to use Ada in the design, development, implcmcntation, and maintcnancc stages of the soft\varc lifc cycle. Combincd with modern softwarc dcsign 171cthods. Ada lends itself to producing a modular, structured, reliable end product. For example. guidance, estimation, and autopilot functions are dcsigned at the top lcvcl as separate semi-indcpendcnt ~nodulcs.This design is ~vrittenin Ada which, when successfully compiled, assures that thc rnodules will function in unison as dcsircd. This is followed by detailed coding of cach algorithm. Follo\ving co~npilationand debugging, the software is rcady for processor-in-thc-loop testing. \vhich involves cross-COIIIpilation of the Ada code into objcct code for thc microprocessor 011 xvhich it will cxccutc. 111 this way, the algorithm software performancc is being cvaluatcd in t11c embedded cnvironmcnt for \vhich it was desiencd. u Rcal-timc pcrformancc is strongly affected by object codc cfficiency. Unfortunately, Ada carrics an ovcrhcad penalty into thc objcct codc which affects both mcmory and throughput performance. O f course, any high-ordcr language is similarly pcnalizcd to somc cxtcnt whcn compared ~ i t optirnizcd h handwritten asscmbly codc. However, sincc Ada is a rclativcly ncw language, cross-cotnpilcr maturity will remain a critical issuc i n thc design of Ada cmbcddcd soft\varc. Ilcsults to datc indicatc that inlplc~ncntingthc SI' guidancc algorithm on a 10-bit microproccssor in Ada will not adversely affect guidancc pcrformancc. It should be pointcd o u t , howcvcr, that the F9450 has 23-bit hardwarc floating-point capabilitics. It may happen that diffcrcnt rcsults arc gcncratcd by implementing thc algorithm on a microproccssor with reduced-lcngth floating point, or in a fixed-point arithrnctic environmcnt. Althoueh " mcnlorv constraints of thc MIL-STD-17jOA instrucr~onsct should havc no impact on study results, thesc should bc considcrcd in any s y s t c ~ l ~ design. Scvcral of thc advanced guidancc algorithms dcvclopcd may rcquirc large memory.

Sec. 8.13

Advanced Guidance Law Implementation Issues

621

Throughput limitations of the microprocessor may have a significant impact on the pcrforniance of a guidance algorithm based on optimal control theory. In fact, it is not certain whcthcr such lirnitatiolls might eve11 prevent implementation altogether. This is cogpled with the run-tirne overhead associated with Ada and with the efficiency of Ada cross-compilers. Although the S1' algorithm is projected to utilize only 58 percent (with overhead allowance) of the F9450 throughput, guidance is obviously not the only function required of the nlicroprocessor in an actual tnissilc implcnlcntation. Evers ct al. [I9881 stated that in the original design, a squareroot state estimation algorithm was used which demonstrated good performance on a large computer. The same algorithm, however, contained throughput requirements which exceeded the capacity of the F9450 and had to be replaced. The interdependency of the terminal guidancclstate estimator algorithms suggests maintaining an awareness of throughput requirements from the start in any design effort. The results obtained up to this point indicate that optimal guidance algorithms which have demonstrated improved performance over PNG are suitable for implementation in current n~icroprocessors,as has been established in the processor-in-theloop studies. Such studies also determine the actual computational burden placed on the microprocessor by the SP algorithm. The F9430 has been replaced by a microprocessor that yields better than 30 percent improvement in throughput capabilities. Thus, continuing advances in microprocessors make the implementation oicomplcx guidance algorithms in embedded systems an even more realistic proposition.

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Index AAM. See Air-to-air missile (AAM) Accelerating capability, 254 Acceleration comvcnsation. 310 Acceleration estimate, updating of, 608-9 Accelerometers, 14, 176-77, 193 Active homing guidance, 324-26 Adaptive radome estimation design, 520-24 Adjoint method, 106-1 1 applications, 108-1 1 adjoint model of linear hotning loop (illlts.), 110 deterministic inputs, 111 performance sensitivity due to target maneuver (illus.), 110 philosophy, 106-8 (ill~rs.),107 RMS miss distance error budget automatically generated by (illll~.),110 steps to construct adjoint systcm, 108

stochastic inputs, 109. 11 1 Advanced air-to-air missile (A.4M) navigation, guidancc, and control technology, 467-68 Advanced guidance and control s!.stem design, 50 Advanced guidance filter, 440-42 extended Kalman filter (EKF) approach, 441-42 statistical linearization approach, 442 Advanced guidance law in~plemcntation issues, 613-15 in~plemcntationproblems, 614-15 Advanced gutdance laws, 408-81 analytical solution of optimal filters and optimal guidance law. 474-81 definition, 469 optimal ,guidance law survey, 469-74 Advanccd guidancc system design, 465 analytical solution of optimal trajectory shaping for combined nlidcourse and terminal guidancc. 562

command versus semiactive homing, design and analysis, 544-62 complcmcntary1Kalman filtered proportional navigation: biased1 complementary PNG. 481-510 guidance filter design trade-off, 591 -600 improved time-to-go estimator, 606-9 jamming and clutter effects on guided intcrceptor. 609-13 multimode guidance system design, 613-19 ocher advanced midcourse guidance schemes, 583-91 other terminal guidance laws. 510-17 pulsed rocket control, 600-608 radome error calibration and compensation, 517-44 Advanced midcourse guidance schemes, 583-91 midcourse guidance for boost-sustain propulsion, 587-91 near-optimal trajectories, 590-91 singular perturbation, 584-87 Advanced missile guidance system against very high-speed target (example) 276, 278-79 Advanced multivariable control system design, 49-50 Advanced navigation, guidance, and control: concepts, 4-7 systems, 7-8 systems design, 10, 12 Advanced navigation system design, 445-64 global positioning system accuracy improvement, 445-57 integrated GPSIIN navigation system, 457-64 Advanced terminal guidance system (table), 470 AEGIS combat system, 266, 299, 329 AHRS. See Attitude and heading reference system (AHRS) Aided inertial navigation mechanization, 213-15 advantage of augmented positionvelocity inertial, 215 cost and performance improvement, 213 (illus.), 214, 215

effect of augmented inertial mechanization, 213 error improvement, 214, 215 (illus.) Aided inertial navigation systems, 179 inertial navigation systems and external navigation devices (illus.), 180-81 Air defense homing system, 401-3 Air-to-air dynamicslmeasurement models (illus.), 418 Air-to-air missile (AAM) multiple mode guidance, (illus.) 273, 281, 283 Air-to-surface missiles (ASM), 283-84 Air-to-surface roles and missions, 284-87 close air support, 284 deep tactical strike. 285 defense suppression, 285-87 tactical interdiction. 284 Air-to-surface weapons, 283-84 ALP. See Attitude-loop parameters (ALP) Alpha-beta-gamma filter, 421-24 first-order target tracker, 421-22 second-order target tracker: a-0 tracker, 422-24 third-order target tracker, 424 Alpha tracker, alpha-beta tracker, and alpha-beta-gamma tracker, 443 Altimeter aiding, 197 (illus.), 200 Altitude rate estimator (example), 161 Altitude sensors, 32-33 American Control Conference, 3 Analysis of optimal command guidance versus optimal semiactive homing guidance, 557-62 Kalman filter analysis, 558-59 noise inputs, 557-58 performance analysis, 559-62 Analytic optimal guidance law, 566-78 general solutions of optimal trajectory-shaping guidance law, 574-75 horizontal plane guidance, 575-76 mathematically probing the analytical solution of the ootimal trajectory-shaping guidance, 576-78 . . . optimal solution for air-breathing engine or the power-on stage of a solid rocket engine, 571-73 optimal solution for the power-off

Analytic optin~alguidance law (cotit.) stage of solid rocket engine, 569-71 vertical plane guidance, 567-69 Analytical solution of optimal trajectory shaping for combined midcourse and terminal guidance, 562-83 analytic optimal guidance law, 566-78 discussions. 583 optimal trajectory shaping guidance, 562-63 problem forn~ulatioil,563-65 real-time implenlentation and performance, 578-83 Analytical three-diniensional optimal guidance law, 474-75 ANISPY-1A radar, 266, 299 Antiradiation homing (ARH), 29, 331 missile guidance, 344-47 Antiradiation projectile (ARP), 331 Antisatellite (ASAT) missile. 434 Antiship niissiles. 296 Antisubmarine lvarfare (ASW), 294-96 Antitank missiles, 296-97 APN. Srr Augmented Proportional Navigation (AI'N) Apollo program, 2, 3 Kalman filter, 128, 151 A priori information, absence of. 153 Area defense. 287-88, 292 cxaniple, 202-93 ARH. Sre Antiradiation homing (AIIH) ARP. Src Antiradiation projectile (AIII') ASAT. S F PA~ltisatellitc(ASAT) ASW. Scr Antisubmarine warfare (ASW) Asymptotic system performance. 553-54 ( l l l l l ~ . ) ,555 ATLAS missile, 178 Atmospheric noise, 558 U A ~ Mj49-5(J + Attitudc and heading rcfcrcncc systctii (AHIIS), 311 Attitudc-loop paralnctcrs (All'). 531. 533-34 Attitudc pursuit (direct guidancc), 357-58 Augmented inertial n~cchanization, effect of. 213-15 advantages of augnicntcd positionvelocity inertial, 215

cost and performance improvement, 213-15 error improvement, 213 Augmented proportional navigation (APN), 475, 476, 488, 501-2 law, 477 Autopilot, 16 Autopilot and airframe, 539-40 application, 540-44 Autopilot response characteristics necessary to achieve homing, 65-66

Ballistic missile threat. 293 Batch process in^, 454 Bayesian formulation. 454 Bayesian rule. 137 Bcam-rider guidancc, 319-20 Talos beam-riding midcourse pidance system (cxaniple), 321 Best unbiased linear estiinator (BULE). 456 Bias: m o d c l i n ~ .156-58 no, 11 = 0. 161-02 position. 159-00 rate, h = I , I01 Biascd I'NG (BI'NG) algorithm, 480-87 aug~nc~lted proportioiial navigation (ANI'), 488 implementation. 488 optimal, 487-88 simplified suboptimal BI'NG algorithm: \vithout pursucr accclcration fccdback. 492 \\.it11 pursucr nccclcration feedback. 49 1

suboptimal BI'NG algorithm: without pursucr accclcration frcdback [suboptimal guidaticc (SOG)]. 489-91 with pursuer acceleration fccdback. 489 Birdies, 400 BI'NG. Set Biased I'NG (BIJNG)

BULE. Sct. Best unbiased linear estimator (I3ULE)

C.41lET. Scv Covariance analysis describing fi~nctiontechnique (CADET) Monte Carlo and statistical linearization with adjoint method qiialitative con~parison,120-22 cost comparison for linear and nonlinear systems (illrrs.), 123 nonlinear stochastic systems analyzing and evaluating tool, 116 CCI3. Src, Charge coupled devices (CCW C D U . See Control and display unit (CUU) CEP. Seij Circul.ir error probability (CEP) Center-of-gravity (CG) latcral acceleration, estimation (example), 160 Center-of-gravity (CG) normal acceleration, estimation (example), 160 CG. See Center-of-gravity (CG) CHAPARRAL guidance (example), 331, 332 Charge coupled devices (CCD), 262 Circular error probability (CEP), 261 CIWS. See Close-in-weapon system Classical versus modern terminal guidance and control (illus.), 50 CLOS. See Command-to-line-of-sight (CLOS) Close-in-weapon (CIWS) threat. 294 Closed-loop analysis for adaptive radome estimator, 524-27 discussion, 527-28 Close-loop stability and performance, 77 Clutter, 610-13 effects on guided intercepter, 609-13 Colored-measurement noise, 157 Colored noise, 156 Colored-process noise, 137 Combined seeker-guidance filtering: classifications of complementary/ Kalman filtered PNG, 486

tilter. 483-86 in con~plerr~cntary in complementary LOS anglc and LOS ratc estimator, 486-88 in coniplctiicntary LOS rate cstinlator, 483-84 conventiotial LING algorithm. 486 Command guidance, 313-16 multimodc guidance using, 389 Command gi~idancedynamics, 546-48 Comniand guided system (forward time) (ill~rs.),546 Command midcourse guidance. 290 Command to line-of-sight (CLOS) guidance 299, 312-16, 318-19, 348 Command versus semiactive homing guidance design and analysis, 544-62 analysis of optimal command guidance versus optimal scniiactive homing guidance, 357-62 miss-distance analysis for command guidance, 550-57 modeling, 51-1-50 Comparison of statistical digital simulation methods (example), 53 Comparison of suboptimal guidance (SOG) and PNG, 505-10 CompIementarylKalman filter approach to estimator design, 1 3 - 7 3 first-order, 158-63 second-order, 163-70 third-order, 170-75 CompIementarylKalman filtered proportional navigation: biased/ complementary PNG, 481-83 biased PNG (BPNG) algorithm, 486-92 combined seeker-guidance filtering in complementary filter, 484-86 complementary PNG (CPNG) algorithm, 492-501 terminal guidance system analysis, 501-10 Complementary PNG (CPNG) algorithm, 492-501 fourth-order optimal CPNG algorithm: with decoupling feature, 496-98 with target acceleration bias estimate, 498

Complementary P N G (CPNG) (cont.) optimal, with complementary: LOS angle and LOS rate estimator, 495 LOS rate estimator, 493 second-order optimal CPNG algorithm, 495 simplified suboptimal C P N G algorithm, 500-501 suboptimal C P N G algorithm, 498-500 third-order optimal C P N G algorithm, 495-96 Computer frame (psi-angle) approach, 189 Computer requirements, Kalman filter, 149-53 software, 152-53 square-root filter, 151-52 suboptimal filter, 152 Computer unit (CU), 199 Computing, incrtial navigation systems function, 179 Constant-bearing guidance, 359-60 Continuous dynamics, discrete measurements: extended Kalman filter (tahlc), 135 statistically linearized filter (table), 136 Continuous Kalman filter, 129. 131-32 first-order altitude rate Kalman filter (example). 132-33 Continuous-\vavc interferomctcr homing system, 337-43 test of, 344 Control and display unit (CDU), 179 Conventional PNG algorithm, 486 Correlated acceleration process, 70-71 Covariance analysis, 102-6 RMS trajectory profile (ill~rs.),105 Covariance analysis describing function technique (CADET). 78 CI'NG. Scc Complcmcntary I'NG (CI'N C;) Crossover frequency, 85-86 C U . Scr Computer unit (CU) CUBAN. Scr Cubic miss adjoints (CUBAN) Cubic miss adjoints (CUBAN), 534 Cubic synthcsis (CUSYN), 534-36 CUSYN. Scr Cubic synthesis (CUSYN)

Data-decoupling algorithm, 462-63 tests and results, 463-64 Data handling system (DHS), 231 (illus.), 235 Data processing, 188 generalized navigation computer (illus.), 188 Defense and offense systems, 392-404 low-altitude air defense systems, 397-404 performance parameters, 392-97 De Gaston-Safonov algorithm, 95 Design algorithms for advanced navigation, guidance, and control systems, 9-10 Designators (laser-based systems), 328-29 Design equations for PNG u i t h parasitic feedback, 528-44 applications, 540-44 design equation, 530-40 problcm definition, 528-30 Design rcquiremcnts, 84-86 Deterministic inputs, adjoints for, 111 Deviated-pursuitlfised-lead guidance, 358-59 DHS. Sec Data handling system (DHS) Discretc Kalman filter, 129 n~ultiple-rate(tahlr), 131 and optimal guidance law (illrts.), 546 step-by-step procedure (table), 130 Discrete-time system, 431 Divergence Kalman filter, 420 DME, 33 Doppler aiding, 195 (ill~ts.),199 (illlrs.) Dopplcr radar, 179, 225 advantages and limitations, 225 Doyle bounds, 95 Do):lc's inequality. 94 Drag c o n ~ p c n s a t i o ~363 ~. Draper Laboratory, 180 Dual-mode guidance, 380 concepts. 381 intcgratcd navigation and guidance system, 381, 384 midcourse plus terminal guidance. 380 Dynamic lead guidance, 372

Earth-fixedlgeographic coordinate ( i l l s ) , 185 transformation (illus.), 185 Earth-to-NAV direction cosine matrix (illus.), 186 ECM. See Electronic countermeasures (ECM) Effective slope and effective noise, 56-57 (ill~ds.),58-59 derived by minimizing mean square error (illus.), 55 Effective navigation ratio (illus.), 64 EKF. See Extended Kalman Filter (EKF) Electronic countermeasures (ECM), 260 Electronic countermeasures (ECM)/ E C C M modeling, 24 Electronic unit (EU), 199 Electro-optical seekers, 7-3 Electro-optical (EO) guided weapons, 33 1 .Electrostatically supported gyro (ESG), 178 Empirical radome slope calculation, 61-62 Environment sensing guidance/ correlation matching, 389-90 EO. See Electro-optical (EO) Error analysis, 146-48 Error analysis model development, 308-10 midcourse guidance system analysis, 308-9 ESG. See Electrostatically supported gyro (ESG) Estimator-controller, combined: optimal filters and optimal guidance law, 476-77 optimal guidance system (OGS), 478-81 Estimator gains, (table) 441 EU. See Electronic unit (EU) Euler attitude estimator (illus.), 163 Expanding memory filter, 415 Extended Kalman Filter (EKF), 133-35, 441-42, 444 continuous dynamics, discrete measurements (table), 135 Extended target detection and segmentation, 412-13

External disturbances, 77 External interface. 195 External navigation aids. 213-29 aided inertial navigation mechanization, 213-15 doppler radar, 225 global positioning system (GPS), 215-16 Kalman filtering, 226-28 performance, 228 long-range navigation (LORAN). 220 star tracker, 225-26 tactical air navigation (TACAN), 217-18, 221 terrain contour matching (TERCOM), 220, 222, 225 External references, 33-34

~

Fading memory filter, 138, 415 Kalman filter, 420 Wiener filter, 420 Fail-operationifail-operationifail-safe (FO-FO-FS), 238 Fan-Tits algorithm, 95, 100 FCS. See Flight control system (FCS) Fiber-optics guidance (FOG) (example), 303 Field of view (FOV), 255, 260 Filter computational requirements, 149-53 Kalman filter design software, 152-53 square-root filter, 151-52 suboptimal filter, 152 Filter divergence, 153-35 and suboptimal design, 155 Filtering: and estimation techniques, modern, 126-28 Filter performance, 149, (illus.) 150, (illus.) 152 First-order a estimator using inertial a rate at center-of-gravity (example), 164 First-order a estimator using inertial a rate at IRU (example), 161 First-order altitude estimator, (example), 163

Index First-order altitude rate Kalman filter (example), 132-33 First-order complementary/Kalman filter, 158-63 altitude rate estimator (example), 161 estimation of center-of-gravity lateral acceleration (example), 160 estimation of center-of-gravity normal acceleration (example), 160 first-order a estimator using inertial a rate at center-of-gravity (example). 162 first-order a estimator using inertial a rate at IRU (example), 161 first-order altitude estimator (example), 163 first-order Euler attitude estimator (example). 163 first-order sideslip estinlator using inertial sideslip rate at cetlter-ofgravity (example), 162 first-order sideslip estin~atorusing inertial sidcslip ratc at IRU (exan~plc).161 first-order velocity estimator (example), 162 n o bias, b = 0, 161-63position bias. 159-60 rate bias, ii = 1 , 161 ratc estin~ator.162 turbulencelgust anglc-of-attack rate estimate (csamplc), 160 turbuletlce/gust t.elocity cstinlator (example), 160 First-order Euler attitude estimator (example), 163 First-order sideslip estimator using incrtial sideslip rate a t C G (example), 162 First-ordcr sidcslip estimator using inertial sideslip rntc at IRU (cxatnplc). 101-02 First-order solution, 515, 517 First-order targct tracker, 43-1-22 First-order velocity cstinlator (cxatnple), 162 Fixed memory filter. 415 Flight control systctn (FCS). 14 and sensing, 22-23 timc constant (ill~cs.),64 Flight phases, 279-80 launch, 270

midcourse, 279-80 terminal, 280 typical laser-guided weapon mission (example), 280 FLIR. s e e ~oiward-lookinginfrared (FLIR) FO-FO~FS. ;see Fail-operationlfailoperationlfail-safe (FO-FO-FS) FOG. See Fiber-optics guidance (FOG) Footprint, 393 Forward-looking infrared (FLIR), 30, 184 Four-gimbal mechanism (ill~rs.),193 Fourth-order optimal C P N G algorithm: with decoupling feature, 496-98 with target acceleration bias estimate, 498 FOV. See Field of view Future guidance processing, 401-13 missile guidance, signal processing, 407-13 technological advances in signal processing. 40.5-6

Gauss-Markov: models. 70, 71 process. 157 GI>OP. See Gconlctric dilution of position (GIIOP) Generalized least squares (GLS), 451 Generalized strapdown mechanization, 199, 201 (ill~ts.),204 (il111.c.).205 Geometric dilution of position (GIIOI'), 33, 220 bias and variancc shrinkage, 454-57 Gimbal anglc tnotion, 56, 60 cffccti\re slope and cffcctive noise, dctcrminants (iil~r$.), 5.5 targct maneuver and n o w causc (1111~s.). 55 Gimballed incrtial navigation system. 179, 186, 192-99 incrtial sensors on stable platform. 193 tncchanism, 192 navigation mechanization and crror model, 195. 197-99 na\.igation nlodc, 194-05. 196-98 platform alignment modes, 193-94

system internal and external interlaces. 105, I97 Gimbal nicchanisrn, 102, 103 (;imbal/strapdown navigation systems, diffcrcncc between, 199 See a l j o Gimbal versus strapdown comparison and analysis Gimbal versus strapdowti comparison arid analysis. 200-13 ieaturc comparisoti, 200 navigation errors comparison. 209-13 Glint noisc. 47. 75. 557 Glint noise uyi,,,,, 548 Global positioning system (GI's), 179, 215-17 acctlracy irnprovcmcnt. 445-47 bias and variance shrinkage, 454-57 position-fix navigation, 446-52 recursive estimation and Kalman filter, 452-53 ridge regression, 5-11-42 inertial navigation systems updating, 217 aided inertial-global positioni~lg system mechanization (illrrs.), 218 measurement principles, 216-17 range, 316 range rate. 216-17 time, 216 overview, 213-16 performance, 217 pseudo-rangc error budget (table), 217 requiremctlts, 215 Globdl positioning systemlinertial navigation system Kalman filter systcm design, 457-64 Global positioning system pseudo-range error budget (table), 217 GLS. See Generalized least squares (GLS) GPS. See Global positioning system (G S-') Gravity anomalies, 177 Guidance, 14 computer function, 14 Guidance algorithm, 252, 310-47 direct guidance methods, 312-47 preset guidance, 310-12 Guidance filter design trade-off, 591-600 missile midcourse guidance system analysis, 593-600

optimal guidance filter design analysis, 592-93 Guidancc filtcringlprocessi~ig,45-51 advanced multivariablc control system design. 49-50 classical versus niodcrn guidance and control, 50-51 guidance kinematic loop. 45 navigation, guidance, and control systeni design rcquirenicnts and design considerations, 48-40 noisc i~iputs.45-48 nottnoisc inputs. 48 simplified pursuer dynamics, 45 Guidance kinematic loop, 45 Guidancc kinematic loop stability analysis (example), 52-53 Guidance law, 347-79 classification (illlrs.), 469 comparisoli of (table), 353 line-of-sight (LOS) angle guidance, 348 line-of-sight (LOS) rate guidance, 348-19. 35 1-73 sensitivity and comparison, 373-79 simultation, 502-5 survcy, 469-74 Guidance loop (illrrs.), 37 Guidance loop transfer function with radomc slope bias effect and without radome slope effect, 533 Guidance mission and performance, 268-98 operation, 280-98 performance, 369-79 phases of flight, 279-80 Guidance performance analysis with inflight radome error calibration, 520-28 adaptive radome estimation design, 520-26 closed-loop analysis for adaptive radome estimator, 524-28 Guidance processing, 252 algorithm, 310-47 defense and offense systems, 392-404 future processing, 404-13 law, 347-79 mission and performance, 268-98 multiple mode guidance modeling, 298-310 processors, 252, 256-68 single-mode, dual-mode, multimode guidance. 379-92

Guidance processors, 252, 254-70 precision requirements and achievements, 260-62 standard missile guidance system development (example), 262-67 Talos guidance system (example), 267-70 Guidance system classification, 21-22 Guidance tracking filter, 252, 427-30 design approach, 428-30 function and requirements, 427-28 Guided interceptor, 609-13 clutter, 610-13 deception repeater jammer, 613 IF seeker in weather condition, 611 jamming and clutter, 607 Guided weapons, 280-81 air-to-air, 281-83 air-to-surface, 283-84 air-to-surface roles and missions, 284-87 surface-to-air, 287 surface-to-surface, 294 Gyros. See Gyroscopes Gyroscopes, 14, 177-78, 192-95 ~ y r sensitivity o gain design (example), 67-68

Handover, definition, 299 HCE. See Homing controller expansion Helicopter integrated inertial navigation system (HIINS), 241-45 accuracy requirements, 241-43 configuration 1 and 2, 243-45 (illus.), 246-49 Hermite matrix for third-order perturbed polynomial, 93 Higher order system analysis, 62-63 High-frequency dynamics (table), 26 High-frequency requirement, 85-86 High-value target (HVT), 261 HIINS. See Helicopter integrated inertial navigation system (HIINS) HOJ. See Home-on-jamming (HOJ) Home-all-the-way guidance, 379-80 Home-on-jamming (HOJ), 331, 337, 344

Homing controller expansion (HCE), ' 537-38 Homing guidance, active, 324-26 Homing missile controller, 16 Homing missile guidance and control analysis (example), 63-65 Homing of aerospace vehicles, 331, 337, 344 modes of homing, 324 Homing problems, 397-401 Homing system performance: desired/actual effective navigation ratio, determinant of, 52 guidance system time constant, 32 HVT. See High-value target (HVT)

IC. See Integrated circuits (IC) IEEE conference on decision and control, 4 IINS. See Integrated inertial navigation system (IINS) IISA. See Integrated inertial sensing assembly (IISA) Illuminator (radar devices), 328 IMU. See Inertial measurement unit (IMU) Inertial guidance, multimode guidance using, 388 Inertial measurement unit (IMU), 195 (illus.). 198 Inertial midcourse guidance, 299, 301 Inertial navigation, 176-81 basic functions, 179 common requirements, 181-84 comparison and analysis of gimballed versus strapdown, 209-13 error analysis, 188-89 error models, 188. 189-92 external navigation aids, 213-29 gimballed inertial navigation system, 192-99 integrated inertial navigation system, 229-51 navigation computation and error modeling, 184-92 strapdown inertial navigation system. 199-209 updating, 217, (illus.) 220

Inertial navigation requirements. 181-84 stand-off weapon systems, 182-84 vehicle and weapon systems, 181-82 visual attack systcms, 184 weapon dclivcry. 182 lncrtial position. 186-87 Inertial reference unit (IRU), 428 primary ti~nction,299 Inertial sensors on stable platforn~,193 alignment nlodes, 193-94 three-gyrolacccleromctcr (illus.), 104 Inertial space, 177 Inertial system nlechanization (illus.), 198 Infrared (IR): detection, 309 sensor, 28 Infrared (IR) guidance. 330 Infrared (1R)-guided missiles, advantages, 281 Infrared homing short-range AAM, 508-10 Infrared (IR) image processor, 310 Infrared (IR) seeker in weather condition, 613 Integrated circuits (IC), 261 Integrated Doppler, 216-17 See also Pseudo-delta-range (PDR) Integrated GPSIIN navigation system, 457-64 data-decoupling algorithm, 462-63 GPSIINS Kalman filter system design, 458-62 tests and results, 463-64 Integrated inertial navigation system (IINS), 229, 231-51 helicopter integrated inertia! navigation system (HIINS), 241-45 (illus.), 246-49 integrated inertial sensing assembly (IISA), 232, 234, 236, 238-43 integrated missile guidance systems, 245, 249-51 integrated sensinglflight control reference system (ISFCRS), 231 (illus.), 232 (illus.), 233 (illus.), 234 integrated sensory subsystem (ISS), 231-32 (illus.), 235

Integrated inertial sensing assembly (IISA). 232. 234. 236-41 configuration of sensor locations, 236, 238 configuration versus attitude1 velocity redundancy (table), 238 effect of gyro dither. 238 error sources, 240-41 acceleration scale factor stability, 240-41 axis alignment errors, 240 data synchronization. 241 input axis bending, 241 noise and filtering, 241 performance, 240 redundancy management, 238-40 Integrated missile guidance systems, 245, 249-51 integrated seeker-guidance navigation system, 249-51 lntegrated modified PNG and optimal controller, 520 with radome compensation, 520 Integrated navigation and guidance system, 381, 384 Integrated sensinglflight control reference system (ISFCRS), 231 (illus.), 233 (illus.), 234 geometry (illus.), 232 MlRA system (illus.), 234 Integrated sensory subsystem (ISS), 231-32 Intelligent knowledge-based systems techniques, 332 Intelligent TVIimage, 330 Intercept guidance, relative geometry and parameters for (illus.), 35 Interface information for gimballed inertial navigation systems, 197 (illus.), 197 Intermediate-frequency requirement or crossover frequency, 85 Internal interface, 195 IR. See Infrared (IR) IRU. See Inertia! reference unit (IRU) ISFCRS. See Integrated sensing/flight control reference system (ISFCRS) ISS. See Integrated sensory subsystem (ISS)

Jamming and clutter effects on guided interceptor, 609-13 clutter, 610-12 deception repeater jammer, 609-10 IR seeker in weather condition, 613 Jam-to-signal ratio USR), 609 Jet Propulsion Laboratory: NASA software contract of, 152 JSR. See Jam-to-signal ratio USR)

Kalman-Bucy filter, 2 See also Recursive minimum variance estimator Kalman filter (KF), 2, 71, 420 analysis, 558-59 continuous, 131-32 design and perforn~anceanalysis, 140-48 design process, 142 error analysis, 146-48 noise intensity matrices, selection of, 142-46 discrete, 130-31, (table) 141 cstirnation techniques. 126-28 operational considcrations. 148-58 absetice of a priori inforn~ation,153 coniputational requirements, 149-53 filtcr divcrgencc, 153-58 filter performance, 149 modeling process noise and biases, 156-58 simplified, 420-21 Kalman filtering, 226-28 approach, 227-28 pcrformancc, 228. (illris.) 230 Kclley-Bryson variational optin~ization procedure, 2 KF. Srr Kalnian filtcr (KF) Kincniatic equations (illrrs.), 37 Kinematic/rclati\~cgeometry, 34-37

Laplacc domain. 162 Largc-scale integrated (LSI), 261

Laser guidance (example), 303 Laser-guided weapon mission, typical, (example), 280 Linearized airframe response (table), 25 Linear discrete/continuous smoother (table), 141 Linear miniinum variance estimation: Kalman filter, 128-33 continuous Kalman Filter, 129, 132-33 discrete Kalman filter, 129 Linearlnonlit~earintercept navigation. guidance, and control processing, 14-18 Linearlnonlinear intercept navigation, guidance, and control system, 13-23 flight control system (FCS) arid sensing, 22-23 guidance system classification. 21-22 modeling and simulation, 18-21 processing, 14 Linear radome miss prediction. 57-59 Line-of-sight (LOS): angular rate sensors, 33 guidance, 312-13, 349 system seeker model, 43 sensors. 32 LOAL. See Lock-on after launch (LOAL) LOUL. Sce Lock-on before launch (LOUL) Lock-on after launch (LOAL). 255 Lock-on bcfore launch (LOBL), 254 Long-range navigation (LORAN), 34, 179, 220 advantages and limitations, 220 characteristics, 220 Look-down Doppler radar, 184 e LORAN. Scr ~ b n ~ - r a t i gnavigation (LORAN) LOS. Scc Line-of-sight (10s) Low-altitude air defense systems. 397-404 air defense homing system, 401-3 further missile dcvclopnicnts, 403-4 homing problems, 397-401 Low-altitude cruise tiiissilc threat. 293 Low-altitude target cngagcmcnt with grazing atiglc control (example). 505-00 Low-frequency rcquirctncnt, 84 LQG tcchniquc, 2, 470, 471, 472, 473 LSI. Src Largc-scalc integrated (LSI)

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Maneuver scale factor. 536 Markov process. 71 Matched models versus minimum MSE models, 456 Maximum-likelihood filter, 415 MDASE. See Modeling, design, analysis, simulation and evaluation (MDASE) Measurement errors. 77 Medium-scale integrated (MSI), 261 Midcourse guidance, 298-302 boost-sustain propulsion, 587-91 computational scheme, 587-90 near-optimal trajectories, 590-91 command, 299 command to line-of-sight, 299 (illus.). 300 inertial, 299, 301 Midcourse euidance for stand-off " tactical weapons (example), 301-2 Midcourse guidance system analysis, 308-9 Midcourse phase navigation, guidance, and control processing (illus.), 308 ...

Midcourse plus terminal guidance, 380-81 Millimeter wave (MMW): sensors, 29 Miss-distance analysis for command guidance, 550-57 asymptotic system performance, 553-54 system time constant and missile acceleration limits, 554, 556-57 Miss distance (MD) and missile lateral acceleration due to launch error,

-5n5 --

Miss distance and missile lateral acceleration due to step-target acceleration, 505 Miss due to radome slope, 54-56 Missile, 178, 262-67 developments, 403-4 range and function, 297-98 Missile attitude, 396 Missile footprint, 396 Missile guidance: comparing neural network classifiers, 411-12

extended target detection and segmentation, 412-13 future signal processing for, 407-13 Missile midcourse guidance system analysis, 593-97 error analysis, 397-98 results, 598-600 Missile nonalignment, 607 Missile range, 396 Missile sizing, 536 Missile speed, 396 MIT Instrumentation Laboratory, 178 MMW. See Millimeter wave (MMW) Modeling, design, analysis, simulation and evaluation (MDASE): cycle, 3 linear/nonlinear intercept navigation, guidance, and control system, 13-23 of navigation, guidance, and control processing, 13 navigation, guidance and control system design and analysis, 45-68 process, 1 target signal processing, 23-45 target tracking state modeling, 68-76 Modeling biases, 156-58 Modeling errors, 76, 146-48 Modeling process noise and biases, 156-58 Modeling and simulation, 18-21 computer simulation block diagram (iil~is.),19 Model with a priori information, 452 Modern filtering and estimation techniques, 126-28 combined complementary/Kalman filter approach to estimator design, 158-75 Kalman filter design and performance analysis, 140-48 linear minimum variance estimation: Kalman filter, 128-33 nonlinear filtering, 133-38 operational considerations, 148-58 absence of a priori information, 153 computational requirements, 149-53 filter divergence, 153-55 filter performance, 149, (illus.) 150-51 modeling process noise and biases, 156-58 prediction and smoothing, 138-40

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Modern multivariable control analysis, 8-9 Modified maximum-likelihood filter, 424-25 definition, 425 Modified P N G with radome compensation, 520 Monopulse homing system, 343 tests, 344 Monte Carlo: CADET and SLAM qualitative comparison, 122 cost comparison for linear and nonlinear systems (illus.), 123 simulations, 142 Monte Carlo analysis, 100-102, 474 acceleration limit study utilizing Monte Carlo approach (illus.), 101 for second-order systems, 95 theoretical confidence intervals for Gaussian distributed random variable (illus.), 101 Monte Carlo technique, 20, 78, 92, 111 algorithm, 94, 99 robustness margins, 94 single-input single-output autopilot, 95 for third-order system, 94 MRSI. See Multiresolution spatial integration (MRSI) MSI. See Medium-scale integrated (MSI) circuits Multiniode guidance: applications, 384-92 developnient of a niultimode/ multiband HOJ system (example), 391-92 environment sensing guidance/ correlation matching, 389-90 radar and infrared, 385-88 STANDARD missile-2 upgrade program (cxamplr). 390 using coniniand guidance, 389 using inertial guidance, 388 Multiniode guidance analysis, 618-19 Multimode guidance systcrtl design, 613-19 analysis, 61 8-1 9 Multimodcln~ultiba~~d HOI- system, . dc\~cloprncntoT (exaniple), 39 1-92 Multiple guidance system using optimal

C P N G algorithm with LOS angle and LOS rate estimator (illus.), 614 Multiple mode guidance modeling, 298 error analysis model development, 308-10 midcourse guidance, 298-310 terminal guidance, 302-8 Multiple model estimation, 137-38 Multiple-rate Kalman filter (table), 131 for radar tracking, 443 Multiresolution spatial integration (MRSI), 405 Multivariable control analysis. modern, 77 adjoint method, 106-11 covariance analysis, 102-6 design requirements, 84-86 general robustness analysis, 89-90 Monte Carlo analysis, 100-102 other performance analysis. 124-25 performance analysis, 78 qualitative comparison, 121-23 robustness analysis, 77-78 robustness of real perturbations, 90- 100 sensitivity and complementary sensitivity functions. 82-83 singular value analysis, 78-87 statistical linearization, 113-21 structured singular value, 86-88 Multivariable Nyquist stability, 79

Navaho missile, 187 Navigation aids, external. See External navigation aids Navigation computation and error modeling, 181-92 coordinate systems. 184-86 data processing, 188 inertial navigation systems error analysis and modeling, 188-92 position and velocity generation. 186-88 Navigation filtering for position and velocity estimate (exaniple), 167 Navigation, guidance, and control (NGC): application to:

aerospace, 2 energy management. 2 industrial nianufacturing, 2 medicine, 2 design overview, 1-8 advanced, systems, 7-8 inlportance of advanced, concepts and their impact, 4-7 history, 1-4 outlitic and scope, 8-1 1 advanced systems design, 10 (ill~rs,),11, 12 design algorithms for advanced systems, 9-10 MDASE of navigation, guidance, and control processing, 8 modern multivariable control analysis, 8-9 processing, 10 theory, 2 Navigation, guidance, and control stability and performance analysis, 52-68 autopilot response characteristics necessary to achieve homing, 65-66 comparison of statistical digital simulation methods (example), 53 effective slope and effective noise, 56-57 empirical radome slope calculation, 61-62 guidance kinematic loop stability analysis (example), 52-53 gyro sensitivity gain design (example), 66-67 higher order system analysis, 62-63 homing missile guidance and control analysis (example), 63-65 linear radome miss prediction, 57-59 miss due to radome slope, 54-56 miss less with periodic radome, 56 noise and target maneuver miss, 60-61 radome error compensation, 66-67 radome error-induced miss-distance predictions (example), 53-54 radome error slope in guidance response time, 66 range-independent noise (RIN) miss, 60 Navigation, guidance, and control system design and analysis,

45-68 guidance tiltcri~iglprocessin~, 45-51 rcquirenictits and design considerations. 48 stability and performance analysis, 52-68 Navigation and guidance filtering design, 414 advanced system design, 445-64 practical filter design, 427-42 radar tracking, 442-44 spacecraft attitude estimation, 444-45 target state estimation. 414-27 Navigation and guidance for position estimate, 430-31 Navigation and guidance for position and velocity estimate, 431-34 complementary filtering, 431-33 differentiation of position information, 431 inertial navigation systems with periodic radar position updates, 433-34 Navigation and, guidance filtering for position, velocity, and acceleration estimate, 434-40 poisson jinking maneuver, 71, 438-39 practical estimator gain design, 439-40 uniformly distributed target maneuver, 434-38 Navigation mechanization and error model, 195, 197-99 error models, 199 (illtrs.), 202-3 (illus.), 204 mechanization, 195, 197-98 performance trade-off analysis, 199 Navigation mode, 194-95 (illus.), 196 (ill~rs.),197 Navigation requirements: precision-guided stand-off weapon (PGSOW), 182-84 visual attack system, 184 weapon delivery, 182 Navigation signal time and range (NAVSTAR), 215 Navigation system, 13 NAVSTAR. See Navigation signal time and range (NAVSTAR) NAV-to-body direction cosine matrix (iiius.), 186

Neutrally stable first-order filter (example), 145-46 NEV. See North, east, and vertical (NEW Nine-state guidance filtering equation, 75-76 N o bias, b = 0, 161-62 Noise: glint, 47, 75 receiver, 47, 75 Noise and target maneuver miss, 60-61 Noise-driven time-varying systems, 100 Noise inputs, 45-48 atmospheric noise, 558 atmospheric noise U.am4,549-50 glint noise, 557 glint noise a,,,,,, 548 , range-dependent noise, 557 range-dependent noise u,,, 548-49 range-independent noise (RIN), 557 range-independent noise (RIN) u,, 548 total angular noise PSD Q,, 550 Noise intensity matrices, selection of, 142-46 neutrally stable first-order filter, (example), 145-46 QR-' -, x , 146 QR-' -, 0, 143 stable, first-order filter (example), 143-45 unstable first-order filter (example), 145 unstable open-loop model dynamics, 145 Nonlinear dynamic systems (illirs.), 18 Nonlinear filtering, 133-38 extended Kalman filter (EKF), 133-35 multiple model estimation, 137-38 statistical linearization technique, 135-37 Nonli~~carities. 77 Nonlinear stochastic systems: analyzing and evaluating tool, 116 Monte Carlo versus CAIIET, 116, 118 steps to apply CAIIET to guidancc systems, 117 Nonnoisc inputs. 48 Normalized radomc slope (illlrs.), 64 North, cast, and vertical (NEV) coordinates. 193

OGS. See Optimal guidance system Operational considerations, 148-58 absence of a priori information, 155 computational requirements, 149-53 filter divergence, 155-57 filter performance, 149 modeling process noise and biases, 156-58 Optimal BPNG algorithm, 487-88 implementation, 488 Optimal control theory, 2 Optimal CPNG algorithm with complementary: LOS angle and LOS rate estimator, 495 LOS rate estimator, 493 Optimal filters and optimal guidance law, analytical solution of, 474-81 analytical three-dimensional optimal guidance law, 474-76 combined estimator-controller: optimal filters and optimal guidance law, 476-77 optimal guidance system (OGS), 378-81 Optimal guidance gain time variation (ill~rs.),477 Optimal guidancc law, 476-77 analytical three-din~ensional, 474-76 Optimal guidance system (OGS), 478 Optimallnonoptin1a1 trajectory, 581 Optimal shaping guidance law structure (ill~rs.),580 Optin~alsolution for air-breathing enginc or thc power-on stagc of a solid rocket engine, 571-73 Optimal solution for the pon'er-off stagc of solid rockct engine, 500 Optimal trajcctory shaping guidancc, 562-63 law, general solutions of, 574-75 mathematically probing analytical solution of, 576-78 problcn~formulation, 3 3 - 6 5 Ordcr reductiot~by truncation, 530-31 Ostrowski matrix. 81 Outputting, INS function, 179

Parasitic fcc~lbacks.43-45 Passive hornins. 330-31 CHAI'AIIRAL guidance (c?tample), 33 1 1'1111. Src I'scudo-delta-rangc (PIIR) PGM. SLY l'rccision-guided munitions [I'GM) Penalty function technique, 99 Pert'orrn'ince analysis, 78 Performance comparison in linear homing systems, 503 Periodic radonic, less miss with, 54 (i//ll~.), 35 Perturbation (true frame) approach, 189 PGSOW. Srr Precision-guided stand-off weapon (PGSOW) Phoney data interpretation, 454 Platform aligiinicnt niodes, 193-94 PNG. S r r Proportional navigation guidnnce (I'NG) Point defense. 293 Poisson jinking maneuver, 71. 438-39 Poisson process, 71 Position and rate estimators with position and acceleration measurements (Formulation I), 163-67 Position and rate estimators with position and rate measurements (Formulation 2), 163, 168-69 Position and rate estimators with rate and acceleration measurements (Formulation 3), 163 Position bias, 159-60 Position error and rate error estimator with position error and acceleration measurements (example), 167-68 Position-fix navigation, 446-52 Position and velocity generation. 186-87 Position sensing, 310 PPS. See Precise positioning service (PPS) Precise positioning service (PPS), 216 Precision-guided munitions (PGM), 252 Precision-guided stand-off weapon (PGSOW), 182-83 mission scenario, 182 (illus.), 183, 183-84 Prediction and smoothing, 138-40

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PRN. See Pseudorandom binary noise "Probing Bocing's Crossed Connections" (Karen Fitzgcrald),

5 Proportional navigation (illtrs.), 23 Proportional navigation guidance (L'NG), 16, 360-63, 468 pursuit plus I'NG, 371 system design and analysis, 363 velocity conipcnsated L'NG (VCPNC;). 371 Proportional na\~igationguidance (PNG) law. 477 Pseudo-dcltd-range (PI>11), 217 Pseudorandom binary noise (PRN), 216 Psi-angle (computer frame) approach, I89 Pulsed rocket control, 600-606 Pursuer Parget tracking processing soft~vareniodulcs (illlrs.), 23 Pursuit guidance, 353-38 attitude, 357-58 plus proportional navigation ~ u i d a n c e (PNG), 360

Radar and infrared, multimode guidance, 385 Radar command to line-of-sight guidance (illus.), 300 Radar cross section (RCS), 396 Radar-guided missiles, 281 Radar search mode, 309 Radar tracking. 442-44 a tracker, a-P-y tracker, 443 discrete Kalman filter, 442-43 extended Kalman filter, 444 multiple-rate Kalman filter for, 443 Radiometer and MMW guidance, 331 Radome boresight error slope on miss distance, effect of (illus.), 55

Radome coupling and compensation loops, 38-41 Radome error calibration and compensation, 517-43 compensation, 66-67, 517-20 design equations for P N G with parasitic feedback, 530-46 guidance performance analysis with in-flight radome error calibration, 520-28 Radome error compensation, 517-20 example, 66-67 Radome error distorts boresight error (illus.), 38 Radome error-induced miss-distance predictions (example), 53-54 Radome error slope in guidance response time (example), 66-67 Radome slope, miss increases with (illus.), 53 Range-dependent noise, 557 or,,548-49 Range-independent noise (RIM), 557 miss, 60 ,'rc 550 Range measurement, 216 Range rate (closing velocity), 33 Range rate measurement, 216-17 Range sensors, 32 Range-tracking servo. 309 Rate bias, h = 1. 161 Rate estimator (example), 162 Rate-sensing feedback, 310 RCS. See Radar cross section (RCS) Real-time implementation and performance, 578-79 comparison between optimal trajectory and nonoptimal trajectory, 581-82 implementation and simulation of three-dimensional flight, 579-80 midcoursc phase. 580 switching phase and terminal phase, 581 vertical launch of tactical nlissiles, 582-83 Receiver noise, 75 Recursive cstinlatiot~and Kalman filtcr, 452-53 Recursi\rc Kalman filtcr. 420 Recursi\*c filter, block diagram (illtrs.), 416

Recursive minimum variance estimator. See Kalman-Bucy filter Relinearization, 134 Repeater jammer, deception, 609-10 Residual data block, 458 RF detection, 309 Riccati equation, 152 Ridge regression, 453-54 RIM. See Range-independent noise (RIM) Ring laser gyro (RLG), 178 RLG. See Ring laser gyro (RLG) RMS miss distances, 56, 57 Robust integrated control law design process (illtrs.), 49 Robustness analysis, 77-78 general framework, 89 Robustness measures, various (table), 90 Robustness of real perturbations, 90 (illtrs.), 97 Root locus, 92 and SSV, 92

SAM. See Surface-to-air missiles (SAM) SAR. See Synthetic aperture radar^ (SAW SDF. &e ~ingle-degree-of-freedom (SDF) sea-skimming missile, 296 Search mode, 309 Search patterns: raster scan, 309 sector scan, 309 spiral scan, 309 Second-order a estiri~ator(example), 169 using inertial a rate at center-ofgravity, 169 using inertial a rate at IRU, 169 Second-ordcr c o n l p l c n i e n t a r y / K a l n ~ a ~ ~ filtcr, 163 for~rlulation1, 163-67 navigation filtering for position and velocity estimate (example), 166 position error and rate error estimator with position error and acceleration measurements (example), 167-68 formulation 2, 168-69

second-order a cstimator (exanlple). 169 second-order sideslip cstimator (example), 169-70 Second-order optinlal C P N G algorithm, 493 Second-order sideslip cstimator (cxaniplc). 169-70 using inertial sideslip rate of centerof-gravity, 169-70 using inertial sideslip rate of IRU, 169 second-order target tracker: a-p tracker, 422-23 -- -Seeker block diaeram. 38 Seeker gyro-loop bandwidth, 538-39 Seeker and optimal guidance system block diagram, (illus.) 480 Semiactive (or semipassive) homing, 326-27 guidance dynamics, 546 loop (forward time) (illus.), 5-17 pros and cons of. 339 Semiactive laser homing, 327-29 Semi-Markov filter, 417, 418 Sensing, INS function, 179 Sensitivity and comparison of guidance law, 373-79 Sensitivity and complementary sensitivity functions, 82 Sequentially correlated noise, 156 Servo angular positions sensor, 309 Short-range tactical missiles, comparison of guidance laws for, (table) 352 SIDEWINDER missiles, 330 Signal and noise curve magnitude bode plot (illus.), 429 Signal processing: autonomous acquisitionlimage processing, 406-7 missile guidance, 407-8 technological advances in, 405-6 Simplified Kalman filter, 420-21 Simplified optimal guidance law, 477 Simplified pursuer dynamics, 45 Simplified suboptimal BPNG algorithm: without pursuer acceleration feedback, 492 with pursuer acceleration feedback, 49 1 Simplified suboptimal C P N G algorithm, 500-501 Singer model, 71

Single-degree-of-freedom (SDF),193 Single-input single-output (SISO), 86 feedback control system, 78 Single-mode guidance, 379 home-all-the-way guidance, 379-80 Singular perturbation for advanced midcoursc guidance, 584 fast dynamic approach, 587 medium dynamic approach, 586 midcoursc guidance algorithm, 587 near-optimal solution using singlepcrturbation technique, 585 problem formulation, 584 slow dynamic approach, 585-86 Singular value analysis, 78-82 maneuver., 433 -. SISO. See Single-input single-output (SISO) SLAM. See Statistical linearization with adjoint method (SLAM) Monte Carlo and CADET qualitative comparison, 121-23 cost comparison for linear and nonlinear systems (illus.), 123 Smoothing, 140 linear discrete1continuous smoother (table), 141 Software: Kalman filter design, 1 2 - 3 3 SOG. See Suboptimal guidance (SOG) SONAR. See Sound navigation and ranging (SONAR) Sound navigation and ranging (SONAR), 294-95 Spacecraft attitude estimation, 444-45 Space-fixed navigation computation, 186 (illus.), 187 SPS. See Standard positioning service (SF'S) square-root' filter, 151-52 SSM. See Surface-to-surface missiles (SSM) Stable, first-order filter (example), 143-45 Stable platform, inertial sensors on, 193 Stable platform phase follow-up system (STAPFUS), 337 Stability margin computation, 98-100 Standard missile guidance system development (example), 262-67 STANDARD missile. 329 ~

STANDARD missile-2 upgrade program (example), 390-92 Standard missile-2 terminal guidance (example), 305 Standard positioning service (SPS), 216 Stand-off weapon systems, 182 STAPFUS. Scc Stable platform phase follow-up system (STAPFUS) Star trackers, 179, 225-26 advantages and limitations, 226 State modeling (mblc), 21 State-space model for additive uncertainty, 91-92 State-space singular values, 92 Statistical linearization, 111-16 applications, 120-21 missile acceleration for target intercept: nonlinear systcm analysis (illlrs.), 121 approsiniation (illris.), 112 statistical linearization with adjoint mcthod (SLAM), 118-19 steps for using SLAM. 118-19 technique. 135 and tools for. 116-17 Statistical lincarization approach, 442 Statistically linearized filter, 135 continuous dynamics, discrete mcasuremcnts (mhlr), 136 Statistical lincarization with adjoint nicthod ISLAM). 78. 118-20 Stochastic inputs. adjoints for, 109. 11 1 Strapdo~vn.dcfinition, 177 StrapdoLvn computcr requirements, 201, 204 Strapdown crror model and analysis, 2U4 computational errors. 207 crror model. 208 (ill~rs.),210 instrument errors, 204, 206 StrapdoLvn inertial navigarion systcm, 170. 180, I')'J-20<) conlputcr. 3 1 1 (~I/IIs.).210 (ill~rs.), 21 1 computer rcquircn~cnts.203. 200 error and ~nodclanalysis. 200. 208-9 gcncralizcd n1echanization. 190, 201 operation flow diagram. 208 (illlts.), 21 3 Strapdown navigation colnputcr, 201 (ill~rs.),208 (ill~rs.).200

requirements, 201, 204 Strapdownlgimbal navigation systems, difference between, 199 See also Gimbal versus strapdown comparison and analysis Structured singular value, 86-88 Suboptimal BPNG algorithm: with pursuer acceleration feedback, 489 without pursuer acceleration feedback, 489-90 Suboptimal C P N G algorithm, 498-500 Suboptimal filter, 152 Suboptimal guidance (SOG), 489-90 Surface-to-air missiles (SAM), 280 defense considerations. 392-93 intercept of an ASM, 392 offensive considerations, 393 target coverage and missile survivability. 393-97 weapons. 287 Surface-to-surface nliss~les(SSM), 294 Surface-to-surface \\.capons, 294 target types, 294 Synthetic aperture radar (SAR). 28. 261 System time constant and nlissilc acceleration limits. 554, 5 3 - 5 7

TACAN. Scc Tactical air navigation (TACAN) Tactical air navigation (TACAN), 33, 179, 217-20 (ill~rx.),221 advantages and limitations, 218-20 Talos: direct-guidance application, 333 first Talos h o n ~ i n gsystcm, 335 terminal guidance, 333. 335-37 Talos beam-riding midcourse guidancc systcm (cxamplc). . . 321 Talos g ~ ~ i d a n csystem c (csamplc). 267-OX Talos homing system, first. 3.33. 330 Target acceleration n~odclitig,71-72 Target ccho, 320 Targeting, 30-31 Targeting sensor dynan~ics.37 Targeting sensors (tahl(,). 31 distributed. Target nlancuvcr, unifort~~ly 69-70. 434-37

Target maneuver co~npcnsation,363 Target noise and target maneuver rnodclittg. 68-60 corrclatcd accclcratiot~process, 70-71 other targct accclcratior~modeling, 71-72 Targct rccog~litionusing neuralnetwork-based algorithms, 400- 1 1 Targct signal processing, 23-45 Kinematic/rclative geometry, 34-37 targeting. 30-34 targeting sensor dytlamics, 37-45 414 Targct state csti~~iation. alpha-beta-garnma filter, 420-23 comparison of target tracking filters, 424-26 Kalman filter, 419 modified maximum-likelihood filter, 423-24 simpliticd Kdrnan filter, 419-20 targct tracking filter, 415-16 summary, 416-18 two-point estrapolator, 424 Wiener filter, 420 Target tracker (seeker) modeling, 41 Target tracking filter, 415 adaptive filtering, 419 comparison of, 423-27 accuracy, 4 5 - 2 6 computer requirements, 25-26 tilrer implementation, 418-19 summary of various types of stochastic filters, 416-18 Target-tracking processing (illus.), 27 Target-tracking sensors, 27-30, 414 Target tracking state modeling, 68-76 target noise and target maneuver modeling, 68-72 three-dimensional intercept state modeling, 74-73 two-dimensional target tracking state modeling, 72-73 TARTAR, 266, 299, 301 Taylor series, 111 Television (TV)/video, 29 TERCOM. See Terrain contour matching (TERCOM) Terminal guidance, 302-8 definition, 302 Terminal guidance laws, 510-17 optimal conditions, 513-14 optimal solution, 514-17

problem formulation, 512-13 terminal guidance law based on a perturbation technique (example), 312 T ~ r t n i n guidance ~~l law based on a perturbation technique (example), 5 I2 Terminal guidancc system analysis, 309- 10, 501 - 10 applicatio~tto infrared homing shortranyc - AAM, 508, 510 augmented proportional navigation (AI'N), 501 conlpariso;l of suboptimal guidance (SOG) and PNG, 505-10 guidancc law simulation, 502-5 performance comparison in linear homing systems, 505-8 Terminally guided submunitions (TGSM), 262 Terminally guided submunitions (TGSM) guidancc (exa~nple), 305-8 (ill~u.),309 Terminal phase navigation, guidance, and control processing (illus.), 309 Terrain contour matching (TERCOM), 179, 220, 222-25, 312 advantages and limitations, 223, 225 TERRIER, 266. 301 TGSM. See Terminally guided submunitions (TGSM) Third-order complementary/Kalman filter, 170-75 formulation 1, 170-74 trajectory estimation (example), 173-74 formulation 2, 174-75 Third-order optimal CPNG algorithm, 495-96 Third-order target tracker, 424 Three-dimensional flight: implementation and simulation of, 579-80 midcourse ~ h a s e ,580 switching phase and terminal phase, 581 Three-dimensional intercept state modeline. -. 74-76 nine-state guidance filtering equation, 75-76 twelve-state guidance law design modeling equation, 74-73 -

Three-gimbal mechanism (illus.), 193 Thrust vector command generator, 308 Thrust vector controller, 308 Time-correlated noise, 156 Time domain concept, 2 Time measurement, 216 Time-to-go estimator, improved, 606-9 algorithm, 606-7 estimation of time to go, 606 missile nonalignment, 607 updating of acceleration estimate, 608-9 Total angular noise PSI Q,, 550 Tracking, - 414 Tracking accuracy, comparison of, 425-26 computer requirements, comparison of, 426-27 Tracking error guidance system seeker model, 42-43 Tracking filter, 308 Tracking mode, 309 Tracking sensorlguidatlce comparisons, 27-30 Tracking via the missile (TVM), 317 Track-while-scan radar, 318 TRG. See Tuned rotor gyro (TRG) Trajectory dynamics model, and equivalent (ill~rs.),46 Trajectory estimation, 173-74 True frame (perturbation) approach, 189 Tuned rotor gyro (TRG), 178 Turbulencelgust anglc-of-attack rate estimation (example), 160 Turbulencelgust velocity estimator (example), 160 TV. Srr Television (TV) T V detcction, 309 TVM. Srr Tracking via the missile (TVM) Twelve-state guidance law dcsign modclitig equation. 74 2DF. Src Two-degrees-of-frcedorn gyros (21lF) Two-degrecs-of-frccdon~ (2DF) gyros, 193 Two-dimetisiotial target tracking state modeling, 72-73 Two-point extrapolator, 425 Two-radar command guidance. 317-19

U-D covariance factorization, 154 U D U T algorithms, 151 UHSIC. See Ultra-high-speed integrated circuit (UHSIC) Ultra-high-speed integrated circuit (UHSIC), 261 Unmodeled dynamics, 77 Unstable first-order filter (example), 147 q-+ 0, 147 r -+ 0, 147 Unstable open-loop model dynamics, 145

VCPNG. See Velocity compensated proportional navigation guidance (VCPNG) Vehicle systems, 181-82 Velocity compensated proportional navigation guidance (VCPNG), 371 Velocity pursuit guidance, 358 Velocity vectors, 186-87 Vertical launch of tactical missiles, 582-83 Vertical plane guidat~ce,567-69 Very hi$h-sped integrated circuits (VHSIC), 261 Very large-scale integrated (VLSI) circuits, 261 VHSIC. SEEVery high-speed integrated circuits (VHSIC) Video-tracking servo, 310 Visual attack systems, 184 VLSI. See Very large-scale integrated (VLSI) circuits -

-

Weapotis, guided: air-to-air. 280, 281-83 air-to-surfacc, 283-84 air-to-surfacc roles and missions, 284-87

close air support. 284 deep tactical strike. 285 defense suppression, 285 tactical interdlction. 284 surface-to-air, 287 (illus.), 288-92 Weapon systems, 181-84 delivery, nav~gationrequirements, 182 breakthroughs in, guidance, 261-62 stand-off, 182-84 White Gaussian state, 156

White excitation (maneuver) noise, 420 White-no~se, 156 sequence, 76 W~cncrfilter, 152, 410

Zero effort miss, 366 Zero-order solution. 514

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