Advanced Techniques for Clearance of Flight Control Laws (Lecture Notes in Control and Information Sciences)

Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari

283

Springer Berlin Heidelberg NewYork Barcelona Hong Kong London Milan Paris Tokyo

Christopher Fielding, Andras Varga, Samir Bennani, Michiel Selier (Eds.)

Advanced Techniques for Clearance of Flight Control Laws With 244 Figures

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Series Advisory Board A. Bensoussan · P. Fleming · M.J. Grimble · P. Kokotovic · A.B. Kurzhanski · H. Kwakernaak · J.N. Tsitsiklis

Editors Christopher Fielding Msc. Aerodynamics (W427D) BAE Systems Warton, Preston PR4 1AX England, UK

Dr.-Ing. Andras Varga Deutsches Zentrum f¨ur Luft- und Raumfahrt German Aerospace Center DLR - Oberpfaffenhofen Institute of Robotics and Mechatronics 82234 Wessling, Germany

Dr. Samir Bennani Delft University of Technology Faculty of Aerospace Engineering Kluyverweg 1 2629 HS Delft, The Netherlands

Michiel Selier Msc. National Aerospace Laboratory (NLR) Flight Mechanics Department Anthony Fokkerweg 2 1059 CM Amsterdam, The Netherlands

Cataloging-in-Publication Data applied for Die Deutsche Bibliothek – CIP-Einheitsaufnahme Advanced techniques for clearance of flight control laws / Christopher Fielding . . . (ed.). - Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 2002 (Lecture notes in control and information sciences ; 283) (Engineering online library) ISBN 3-540-44054-2

ISBN 3-540-44054-2

Springer-Verlag Berlin Heidelberg New York

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Preface

In recent years, a major aim of flight control systems research has been to achieve a high level of performance and safety by improving the design methods. Researchers and academia have concentrated their activities on the synthesis aspects of flight control systems, in particular by demonstrating the applicability and strengths of novel, robust, multivariable synthesis tools. Significantly less research effort has been spent on the global assessment of the achieved designs, which represents a key activity for the certification of today´s aircraft, which are equipped with safety-critical, highly complex flight control systems. Currently, the aeronautical industry is faced with the formidable task of clearance of the flight control laws. Before an aircraft can be tested in flight, it has to be proven to the authorities that the flight control system is safe and reliable, and has the desired performance under all possible operational conditions, and in the presence of failures. This motivated the research presented in this book: an exploration of the benefits of new analysis techniques for the clearance of flight control laws. It is a first step towards a better and deeper understanding of the industrial flight clearance process, with the objective to provide recommendations on how analysis techniques should evolve in order to improve the efficiency and reliability of this process. The Group for Aeronautical Research and Technology in Europe (GARTEUR) provided an ideal framework to bring together research institutes, academia and industry and pursue such a relevant research objective. This book is a result of a research effort performed by GARTEUR Flight Mechanics Action Group 11 FM(AG11). It would not have been possible without all individuals and organisations that have contributed to this group. GARTEUR FM(AG11) is also very thankful to all people from outside the group that have contributed with their constructive comments in the form of reviews or industrial evaluations.

June 2002

The Editors

Table of Contents

Part I Industrial Clearance of Flight Control Laws 1 Introduction Michiel Selier, Udo Korte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Tasks and Needs of the Industrial Clearance Process Udo Korte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Part II Tutorial on Analysis Methods 3 The Structured Singular Value and µ-Analysis Declan G. Bates, Ian Postlethwaite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4 The ν-Gap Metric and the Generalised Stability Margin John Steele, Glenn Vinnicombe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5 A Polynomial-Based Clearance Method Leopoldo Verde, Federico Corraro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6 Bifurcation and Continuation Method Mark Lowenberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7 Optimisation-Based Clearance Andras Varga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Part III The HIRM+RIDE Benchmark 8 The HIRM+ Flight Dynamics Model Dieter Moormann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 9 The RIDE Controller David Bennett . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 10 Selected Clearance Criteria for HIRM+RIDE Federico Corraro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

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Part IV LFT Modelling of Uncertainty Models 11 An Overview of System Modelling in LFT Form Jean-Fran¸cois Magni, Samir Bennani, Jean-Paul Dijkgraaf . . . . . . . . . . 169 12 Physical Approach to LFT Modelling Jean-Paul Dijkgraaf, Samir Bennani, GertJan Looye, Jean-Francois Magni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 13 Uncertainty Bands Approach to LFT Modelling Thomas Mannchen, Klaus H. Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 14 Flatness Approach to LFT Modelling Franck Cazaurang, Lo¨ıc Lavigne, Benoˆıt Bergeon . . . . . . . . . . . . . . . . . . . . 221

Part V Analysis Results 15 Baseline Solution Tobias Wilmes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 16 µ-Analysis of Linear Stability Criteria Declan G. Bates, Ridwan Kureemun, Martin J. Hayes, Ian Postlethwaite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 17 µ-Analysis of Stability Margin Criteria Thomas Mannchen, Klaus H. Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 18 ν-Gap Analysis of Stability Margin Criteria John Steele, Glenn Vinnicombe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 19 Polynomial-Based Clearance of Eigenvalue Criteria Leopoldo Verde, Federico Corraro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 20 Bifurcation-Based Clearance of Linear Stability Criteria Mark Lowenberg, Thomas Richardson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 21 Optimisation-Based Clearance: The Linear Analysis Andras Varga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 22 Optimisation-Based Clearance: The Nonlinear Analysis Lars Forssell, Andreas Sandblom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

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Part VI Conclusions and Recommendations 23 Industrial Evaluation Fredrik Karlsson, Chris Fielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 24 Considerations for Clearance of Civil Transport Aircraft Robert Luckner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 25 Concluding Remarks Michiel Selier, Rick Hyde, Chris Fielding . . . . . . . . . . . . . . . . . . . . . . . . . . 457

A Nomenclature and Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

1 Introduction Michiel Selier1 and Udo Korte2 1

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National Aerospace Laboratory NLR Flight Mechanics Department, Anthony Fokkerweg 2, 1059 CM Amsterdam, The Netherlands. [email protected] EADS Deutschland GmbH, Military Aircraft, MT 62 Flight Dynamics, 81663 M¨ unchen, Germany. [email protected]

Summary. We describe the background and motivation for the research carried out on advanced techniques for the clearance of flight control laws within the GARTEUR Flight Mechanic Action Group 11. This project involved 19 European partners representing research establishments, industry and universities. The core activity of this project was the HIRM+ clearance benchmark, whose main results are presented in this book.

1.1 The Importance of Research on Flight Control Law Clearance 1.1.1 Project Background Aircraft manufacturers have reached a high level of expertise and experience in flight control law design. The current design and analysis techniques applied in industry enable flight control engineers to address virtually any realistic design challenge. However, the development of flight control laws from concept to validation is a very complex, multi-disciplinary task and the many problems that have to be solved make it a costly and lengthy process. Researchers in universities and research institutes have developed new, advanced mathematical methods for design and analysis that have the potential to improve the flight control law development process. In the past decade, the Group for Aeronautical Research and Technology in Europe (GARTEUR) has established action groups to investigate the potential benefits and drawbacks of several of these new synthesis and analysis methods. From 1994 until 1997, the GARTEUR Flight Mechanics Action Group 08, FM(AG08), performed successful research on ”Robust Flight Control”. A design challenge was carried out, in which a set of robust control design methods were applied both to a civil and a military aircraft model. The results produced by this group (reports, two benchmark models and a book ”Robust Flight Control” [1]) are widely appreciated in the aerospace control community. In 1999 the Flight Mechanics Action Group 11, FM(AG11), was C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 3-11, 2002. Springer-Verlag Berlin Heidelberg 2002

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established to address a new and complementary challenge. This resulted in GARTEUR reports, three benchmark models and this book. This new challenge focused on the clearance process of the flight control algorithms. The clearance (or assessment) of control laws can be seen as the last step of the flight control system design, taking place when a mature controller design is available and ready for flight tests. In the clearance process it has to be proven that the flight control laws have been designed such that the aircraft is safe to fly throughout the whole flight envelope, under all parameter variability and failure conditions. The background and motivation for this research are briefly highlighted below. 1.1.2 Flight Control Systems As described in [2], the Flight Control System (FCS) enables the pilot to control the aircraft along a desired trajectory and provides safe and economic operation. Pilot inputs are translated into deflections of the aircraft’s control surfaces, which in turn change the aerodynamic forces and moments acting on the aircraft. In the early days, the FCS was a purely mechanical system, which connected the control devices of the pilot directly to the control surfaces of the aircraft by a system of rods, levers, cables and pulleys. The aerodynamic forces on the control surfaces were limited by the physical capabilities of the pilot. The aircraft in those days usually possessed natural aerodynamic stability. The size and operating speed of aircraft increased as aviation evolved with time. The control force that was needed to overcome the aerodynamic forces on the control surfaces also increased, until a point was reached where the required efforts exceeded the pilots’ physical capabilities. To assist the pilots, the FCS was augmented with hydraulic actuators to provide the required control force. This was the first step that removed the direct connection between the pilot and the control surfaces and the mechanical linkages between the pilots’ inceptors and the actuators now transmitted (mechanical) displacements instead of transmitting force. Eventually, developments in aviation, especially in the area of automation of flight, led to the development of the fly-by-wire FCS, in which electrical signals are transmitted between the pilot and the actuators instead of mechanical signals. Today’s high performance aircraft can no longer be flown directly by the pilot. This is especially true for fighter aircraft, which are often designed to be naturally unstable to improve performance. Redundant electronic flight control systems with sophisticated control algorithms running on digital computers are needed to assure integrity and reliability, and to provide the required stability, performance and handling characteristics. The design process for a modern FCS is a complex, multi-disciplinary activity, which has to be transparent, correct and well-documented in order to allow certification of the aircraft. The design and validation of the flight

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control laws (FCLs) is an important part of the FCS design. The FCL design process can be divided into the following five phases: 1. In the off-line design phase the control system architecture and FCL structure are defined and the control law parameters are tuned to achieve the desired handling qualities and closed-loop performance specifications. The control law design is completed by adding appropriate blocks to the control configuration (e.g., gains, filters, nonlinear functions) to guarantee the correct functioning of control laws within the allowed operational ranges. The stability and performance of the resulting design are assessed mainly by employing linear system analysis techniques and nonlinear simulations. 2. Via pilot-in-the-loop simulation the handling qualities and many operational issues of the augmented aircraft are assessed. 3. In iron-bird tests it is verified that the FCLs operate correctly with the FCS hardware in the loop. 4. In the clearance it is formally proven to the authorities that the designed FCLs fulfil all requirements for safe operation of the aircraft throughout the whole flight envelope, and under all foreseeable parameter variability and failure conditions. 5. Finally, flight tests are executed in which the FCS design is validated with respect to the aircraft specification derived from customer and airworthiness requirements. The FCL design process has a strong iterative nature, especially in the offline design phase. However, deficiencies in control laws found in later stages require iterations as well, since the design engineers have to go back to the first phase to improve the controller design. The cost for such modifications increases significantly with each phase. Once a controller design is considered to be sufficiently mature, the clearance task is started. Although FCL analysis and design clearance takes place during all phases of the design process, a formal clearance (phase 4) is required before flight testing (phase 5) can take place. The clearance process is described in more detail in the next section. 1.1.3 The Clearance Process and Potential Improvements As the safety of the aircraft operation is primarily dependent on the designed flight control laws, it must be proven to the clearance authorities that the flight controller is functioning correctly throughout the whole flight envelope in all normal and various failure conditions, and in the presence of all possible parameter variations. The role of clearance is to demonstrate, via exhaustive analyses, that a catalogue of selected criteria expressing stability and handling requirements is fulfilled. Typically, criteria covering both linear and non-linear stability, as well as various handling and performance requirements are employed for the purpose of clearance.

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The clearance of FCLs is a lengthy and expensive activity, especially for fighter aircraft, where many different store configurations have to be investigated involving large variations of mass, inertia, and centre of gravity location, as well as uncertain structural modes and aerodynamic data (often expressed through highly nonlinear dependencies). In addition, the error tolerances in the aerodynamic data and on air data signals used for control law scheduling, have to be taken into account. The complex aircraft models used for clearance purposes describe the actual aircraft dynamics, but only within given uncertainty bounds. One reason for this is the limited accuracy of the aerodynamic data set determined from theoretical calculations and wind tunnel tests. These parameters can even differ between two aircraft of the same type, due to production tolerances. Especially at high angles of attack, local flow separation effects can be different due to these tolerances. Furthermore, the employed sensor, actuator and hydraulic models are usually only linear approximations, where nonlinear effects are not fully modelled because they are either not known or it would make the model unacceptably complex. To perform the clearance, for each point of the flight envelope, for all possible configurations and for all combinations of parameter variations and uncertainties, violations of clearance criteria and the worst-case result for each criterion have to be found. Based on the clearance results, flight restrictions are derived when necessary. Since flying the aircraft in the presence of failures might involve the use of alternative control laws (e.g., by switching to a backup control law after the loss of a certain sensor or an engine failure), the number of additional cases that has to be investigated can be significant. The huge amount of assessment work, typically on systems of very high order, requires fast, efficient and numerically reliable methods and routines for the calculation and visualisation of results. A major improvement can be expected by increased automation of the tools used for model-based analysis of the aircraft’s behaviour. The objective should not be the faster production of analysis data, because a high degree of automation already exists. New techniques are needed for the faster detection of combinations of parameter values and manoeuvre cases for which flight clearance restrictions are necessary. Such ”worst cases” may be caused by rather obscure combinations of events and flight conditions, which makes it particularly difficult to detect them. Over the past two decades, several mathematical techniques have been developed for the analysis of linear and nonlinear systems with uncertain parameters. Each of these technique has its known strengths and weaknesses. However, at this moment it is still difficult for the aeronautical industry to assess whether their application would improve the efficiency of the FCL clearance process. The main objective of the research activity described in this book was thus to explore the potential benefits of using advanced

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analysis methods for the clearance of flight control laws, by demonstrating some of the most promising techniques on realistic flight clearance problems. Analysis of a complex system produces complex results. Good visualisation is essential to gain a deeper understanding of the clearance results on which important decisions about the airworthiness of an aircraft are based. For this reason, a secondary objective of the research activity was to explore, based on a ”wish list” from industry, new tools that would improve the visualisation of clearance results. It is important to keep in mind that the question addressed here is not a purely technical one, since industry is already technically able to successfully clear flight control laws. The main industrial benefits of new methods should be related to reducing the involved effort and cost, while getting sufficiently reliable results, or increasing the reliability of the analysis results within a reasonable amount of effort.

1.2 Description of GARTEUR FM(AG11) In 1999, GARTEUR FM(AG11) ”New Analysis Techniques for clearance of flight control laws - NEAT” was established to address the research objective described in the previous section. 1.2.1 Project Organisation In this group, 19 organisations from European research establishments, industry and universities participated: Research Establishments 1. Centro Italiano Ricerche Aerospaziali (CIRA, Italy, Capua) 2. Deutsches Zentrum fu ¨r Luft- und Raumfahrt e.V. – DLR-Braunschweig, Germany, Braunschweig – DLR-Oberpfaffenhofen, Germany, Oberpfaffenhofen 3. Totalf¨orsvarets Forskningsinstitut - The Swedish Defence Research Agency (FOI, Sweden, Stockholm) 4. Instituto Nacional de T`ecnica Aeroespacial (INTA, Spain, Madrid) 5. National Aerospace Laboratory (NLR, The Netherlands, Amsterdam) ´ 6. Office National d’Etudes et de Recherches A´erospatiale – CERT-ONERA, France, Toulouse – ONERA-Salon, France, Salon de Provence

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Industrial Members 7. 8. 9. 10. 11. 12. 13.

BAE SYSTEMS (BAE, United Kingdom, Warton) Dassault Aviation (DAv, France, Paris) Airbus Deutschland GmbH (Airbus, Germany, Hamburg) EADS Military Aircraft (EADS-M, Germany, Munich) QinetiQ Group (QG, United Kingdom, Bedford) Saab AB (SAAB, Sweden, Link¨oping) The MathWorks Ltd. (TMW, United Kingdom, Cambridge)

Universities 14. 15. 16. 17. 18. 19.

L’Universit´e Bordeaux (UBOR, France, Bordeaux) University of Bristol (BU, United Kingdom, Bristol) University of Cambridge (UCAM, United Kingdom, Cambridge) Delft University of Technology (DUT, The Netherlands, Delft) University of Leicester (ULES, United Kingdom, Leicester) Universit¨at Stuttgart (UST, Germany, Stuttgart)

The Action Group was chaired by CIRA (Dr. Stefano Scala) from April 1999 until May 2000 and by NLR (Mr. Michiel Selier) from May 2000 until the end of the activity in September 2002. Two workshops, intended to present the results obtained within this Action Group, have been organised: the first by INTA in Madrid (2000) and the second by CIRA in Capua (September 2002). 1.2.2 The HIRM+ Analysis Challenge This book describes the results of an analysis challenge in which seven analysis teams have applied five methods to the same problem. The aim of this design challenge was to describe how these advanced methods can be applied to the clearance process and to demonstrate this on the basis of a benchmark model. Initially, all analysis teams needed to get acquainted with the industrial clearance task. For this purpose a description of the current industrial clearance process of flight control laws was provided by industry, which proved to be of great informational value. In parallel, a clearance benchmark problem was defined. It was decided to use the High Incidence Research Model (HIRM), a generic fighter model with a canard, wing, horizontal tail and vertical tail. This model was available in a mature state from the previous GARTEUR FM(AG08) action group. Within FM(AG11), the flight envelope of HIRM was expanded to suit the needs of the group and parametric uncertainties representing the main variabilities in the model have been defined and included in the model. This updated model, called HIRM+, has been used as the basic aircraft model for nonlinear simulations, trimming and linearisations. The basic aircraft model was

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augmented with control laws from the previous GARTEUR FM(AG08). The controller, called RIDE, is based on a robust inverse dynamics estimation. Finally, for the clearance of closed-loop HIRM+RIDE configuration, analysis criteria have been defined that were representative of industrial practice. In a first stage, the current industrial practice has been demonstrated within a so-called Baseline Solution, which was intended to serve as a basis for comparisons with more advanced techniques. In a second stage of the project, several advanced analysis techniques have been applied to the HIRM+ benchmark problem: – – – – –

µ-analysis ν-gap analysis a polynomial-based analysis method bifurcation analysis optimisation-based worst case search.

The basis for the µ-analysis based approaches is the so-called ”Linear Fractional Transformation” (LFT) based parametric uncertainty model. An LFTmodel represents an approximation of a continuum of linear models, where a special (LFT) representation of parametric dependence is used to account for parametric model uncertainties. A complementary activity to the µ-analysis based approach was the generation and validation of LFT-models. Obtaining good quality LFT-models is time consuming. The order of LFT-models depends on the complexity of the parametric dependencies, the number of parameters, and the employed method for the LFT-model generation. Several approaches have been employed to illustrate the generation of LFT-models for the longitudinal dynamics of HIRM+ with five uncertain parameters. The results obtained by the analysis teams have been described in detailed reports, showing what steps were necessary to apply the method, presenting complete analysis results, and discussing advantages of methods and encountered difficulties. These reports served as basis for the industrial partners to identify and to evaluate, from their point of view, the benefits and limitations of each method. 1.2.3 Other Project Activities Developing visualisation tools for clearance. As already indicated, a very important aspect of the clearance is the presentation of the results of the analysis to the control design engineers, pilots and clearance authorities in a straightforward way. In FM(AG11) specifications for visualisation tools were identified based on input from current industrial practice and conceptual ideas generated by the analysis teams in the group. Setting up more realistic benchmark models. The HIRM+ was used mainly because it was readily available at the beginning of the project in a

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mature state. However, HIRM+ is a generic model and is not based on an actual aircraft. The industrial partners indicated their desire for demonstration of the analysis methods on more realistic and more complex models that are closer to the large and complex models used in industry. Thus two additional models were developed: ADMIRE and HWEM. ADMIRE (Aero Data Model In Research Environment) is a realistic fighter model, with a configuration comparable to a Gripen delta-canard configuration. This model is especially of interest for the clearance of the transonic region. HWEM (Harrier Wide Envelope Model) is a realistic model of an actual Harrier aircraft. An interesting aspect of this model is its use in the clearance of the control laws in the transition phase between normal wing-borne flight and hovering flight. The adjustment of these models to meet the needs of the group took a substantial amount of time and effort. Once the new benchmarks were defined, the analysis teams could choose one of the two models for application of their analysis techniques, in addition to HIRM+. The results of this additional work are not discussed in this book because of publication time constraints, but have been presented at the final workshop, which was held at CIRA in September 2002. A public web site on the FM (AG11) project is available at: http://www.nlr.nl/public/hosted-sites/garteur/rfc.html.

1.3 Objectives and Structure of This Book Due to the large amount of work performed by the members of the group, it was decided to gather the main results into a single book. This book focuses on the HIRM+ analysis activities of the group, since this model has been addressed by all analysis teams. The main objectives of this book are: – to describe the current clearance process for flight control laws for fighter aircraft and the typical flight clearance criteria, and to specify which improvements would increase efficiency; – to demonstrate advanced analysis techniques for the clearance of fighter aircraft flight control laws; – to indicate the advantages and limitations of each method and give directions for further improvements and research. This book consists of six parts. Part one describes the clearance process as it is currently applied in the military aircraft industry. In part two, tutorial sections provide the reader with a brief explanation of the theory behind each method, and references are given for more elaborate descriptions of the analysis techniques. Part three introduces the HIRM+ benchmark problem that the analysis teams have addressed for demonstration of their techniques.

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The basic aircraft model, the controller and a representative set of analysis criteria are described. Part four discusses the generation of LFTs, which is an essential task of µ-analysis. In part five the analysis techniques are applied to the HIRM+ benchmark and the results are described. As a reference, a baseline solution is provided, which used the classical clearance methods. Finally, the industrial view on the project’s achievements is given in part six, together with recommendations for future improvements and research.

References 1. J.F. Magni, S. Bennani, J. Terlouw (eds). Robust Flight Control - A Design Challenge. Springer-Verlag London Limited, United Kingdom, 1997. 2. C. Fielding and R. Luckner. Industrial Considerations for Flight Control. In Flight Control Systems. Co-published by IEE Control Engineering Series, London, UK, 2000 and American Institute of Aeronautics and Astronautics (AIAA), USA, 2000.

2 Tasks and Needs of the Industrial Clearance Process Udo Korte EADS Deutschland GmbH, Military Aircraft MT 62 Flight Dynamics 81663 Mu ¨nchen Germany [email protected]

Summary. The process of clearance of the flight control laws of a fighter aircraft is described. In order to better understand the industrial task and needs in this field, the different steps of the clearance assessment, the methods, procedures and criteria currently applied in industry to derive a clearance are explained. Finally, requirements on evaluation and visualisation tools are addressed.

2.1 Introduction In the introduction to this book it was pointed out that clearance of the flight control laws of a fighter aircraft is a tremendous task because of the many different store configurations, the large number of parameter variations and uncertainties and the large flight envelope for which a clearance has to be provided. This book chapter describes the process, methods and procedures which are currently used in industry to carry out this task. It will thus enable a better understanding of the industrial needs and stimulate research for improvements. The chapter starts with the requirements for the generation of an analysis model and addresses the need for familiarisation with the basic aircraft and the controller. It then demonstrates the effect of important variabilities and uncertainties and describes the methods and criteria applied in linear and nonlinear analysis and simulation to generate a clearance. Finally it addresses the needs for data processing and software tools and how flight limitations – derived from the worst case parameter combinations – can be visualised to give precise information on where the aircraft is allowed to fly.

2.2 Steps of the Analysis Process The clearance of the flight control laws provides information about the flight envelope, angle of attack/load factor limits and the manoeuvres which are C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 13-33, 2002. Springer-Verlag Berlin Heidelberg 2002

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allowed to be flown. In order to derive this information, a number of consecutive steps has to be performed. For the current industrial process these steps can be summarised as follows: Step 1: Generation of an analysis model This involves establishing a full-size nonlinear model which includes all parametric uncertainties. From this model, linear (small perturbation) models are derived (from trimming and linearising). These linear models have to be validated against the nonlinear model. Step 2: Familiarisation with aircraft and controller To provide the designer with a good appreciation of the uncontrolled aircraft, studies are carried out to produce plots of aerodynamic stability and control derivatives (unaugmented aircraft), plots of scheduled feedback gains and the open-loop eigenvalues. Step 3: Trend studies on the effect of uncertainties This involves studying the effects of the various uncertain parameters on the stability and handling of the given aircraft. Step 4: Linear stability analysis This step involves the calculation of open-loop stability margins (Nichols plots) for a narrow grid of flight envelope points and for different uncertainties. It also includes the calculation of closed-loop eigenvalues for all flight envelope grid points and for different uncertainties. It covers the identification of worst case results for all grid points and the visualisation of all results. Step 5: Linear handling analysis This involves the evaluation of appropriate frequency and time domain criteria, the identification of worst cases and the visualisation of the results via plots and tables. Step 6: Nonlinear analysis by off-line and manned simulation This step involves assessing the general flying characteristics with and without uncertainties, the identification of handling and control problems, and the derivation of manoeuvre limitations. Step 7: Clearance report This involves the derivation and visualisation of manoeuvre and flight envelope limitations based on linear and nonlinear analysis results, and the provision of flight test recommendations. Step 8: Improvement of the clearance This is to address some non-compliances and involves special investigations of reduced stability margins, based on reduced tolerances from flight test results. The details of the main steps are described in the following paragraphs.

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2.3 Aircraft Analysis Model The first task of the clearance process is the generation of a representative model of the aircraft and its flight control system (FCS) because this is the basis of all clearance work. 2.3.1 Nonlinear Model For design purposes a reduced model is often used but the clearance analysis work requires a full-size model. Such a model includes the nonlinear, elastified aerodynamic data-set, the nonlinear equations of motion with configuration dependent data, the nonlinear control laws and the hardware model of the actuators, sensors, filters, pilot inceptors, hydraulics, hinge moments, engine, computing delays and data processing. Models for atmospheric turbulence and gusts also have to be included. The nonlinear model will be used extensively both in mathematical simulation (non-realtime) and manned simulation to test stability, handling and control of the aircraft under all conditions, as realistically as possible, in the air and on the ground. Rig tests with real hardware in the loop (i.e. flight control computers with control law software and redundancy management, sensors, actuators and hydraulics) will complement these simulations and will be used to check the transient behaviour when system failures are occurring. 2.3.2 Linear Model The linear, small perturbation model is separated into longitudinal and lateral/directional motion and is derived from the nonlinear model by trimming and linearising the model at a large number of grid points over the entire flight envelope which are dependent on Mach number (M), angle of attack (AoA) and altitude (or dynamic pressure). It is important to choose the grid points narrow enough in order not to miss the critical points of the envelope where large changes of the nonlinear aerodynamics occur within a small AoA or Mach range. If, for instance, the normal gridding is in steps of 10 kft for altitude, 0.2 in Mach number and 2 ◦ in AoA it might be necessary to choose Mach number steps of 0.05 and AoA steps of 1 ◦ for the transonic area. Indications for a suitable choice will become evident from familiarisation with the model. For a complete assessment, thousands of grid points have to be trimmed, and hence fast trim routines are an important factor for time-efficient analysis. In the linear model the hardware (actuators, sensors, time delays etc.) is represented by linear transfer functions which approximate the nonlinear models in the frequency range of interest (up to 5 to 7 Hz). The linear model will be used for calculation of stability margins and eigenvalues and to check that the customer-agreed linear handling criteria are fulfilled. As the linear model may easily reach a dynamical order of 70 or more, numerically stable and efficient algorithms are required for analysis.

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2.3.3 Familiarisation with the Model Before clearance assessment work starts, the engineer should make himself familiar with the aerodynamic characteristics of the unaugmented aircraft and the controller logic and dynamics. Essential stability and control power derivatives should be plotted as a function of Mach number, AoA and control surface position to get an indication of potential problem areas such as reduced control surface effectiveness or sign reversal. For naturally unstable aircraft it is very useful to plot the unstable (positive) real eigenvalue as a function of M and AoA. An example is given in Fig. 2.1 for the pitch axis, where sigma denotes the magnitude of the aircraft’s unstable pole (originating from the short-period mode) and is plotted against AoA for a fixed Mach number.

i n

a

l

4

i t u

d

3

n

g

2

L

o

1

S

i g

m

a

0 - 1 0

2

4

A

6

o

A

8

[ d

e

g

1

0

1

2

1

4

]

Fig. 2.1. Plot of unstable short period eigenvalue

The plot shows a strong peak in instability (positive sigma) followed by a change to stability over a narrow range of AoA. In this area, stability margin problems might be expected due to the high instability level and the wide variation in stability. If the controller were to be scheduled with AoA then tolerances in the airdata system might lead to misadaptation of the controller gains and thus induce over- or undergearing. 2.3.4 Inclusion of Uncertainties in the Model For clearance, it must be demonstrated that the aircraft is safe under all conditions and variations. This means that the assessment must be performed not only for the nominal model but also for all possible deviations, operating conditions and store configurations. Therefore, the model has to be extended to enable the inclusion of such variabilities/uncertainties. Many of the variations are known to a large extent whereas others are uncertain and known only within certain confidence levels. The boundaries of c.g. position, mass

2 Tasks and Needs of the Industrial Clearance Process

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A

/

C

M

a

s

s

[

K

g

]

and inertia, for instance, are known for a given store configuration. For unusual fuel demand, fuel sloshing, fuel system failure or missile firing, the c.g. position will deviate from nominal but not exceed a specified most forward or most aft limit, as indicated in Fig.2.2.

L

o

n

g

i

t

u

d

i

n

a

l

C

G

P

o

s

i

t

i

o

n

[

%

m

.

a

.

c

.

]

Fig. 2.2. Centre of gravity diagram

The aerodynamic data-set is another example. The data-set values which are used in the initial analysis model are derived from wind tunnel measurements and theoretical calculations. They are therefore known only within given tolerance bands. The tolerance values will be largest at the beginning of the aircraft’s development. In-flight parameter identification during flight test will later improve the knowledge of the aerodynamics and allow a reduction in the tolerance levels and thus to delete possible clearance restrictions. For stability and handling investigations the change in pitching, yawing and rolling moments due to changes in AoA, M, control surface position, angular rates etc. are needed. Therefore, the aerodynamic tolerances are defined as deviations of the stability and control derivatives and not of the moment coefficients themselves. In paragraph 2.3.5 the four groups of variabilites / uncertainties are defined and how they can be implemented in the analysis model is described in chapter 8 of this book. In industry it is current practice not to derive separate clearances for each store configuration but to provide block clearances. For this purpose, a grouping of configurations which are aerodynamically equal (within tolerances) is made. For each group, mass/c.g. boundaries as shown in Fig. 2.2 are defined which cover the extreme values of all included configurations. For this analysis, several representative mass/c.g. points on the forward and aft boundary curves are chosen (i.e. maximum mass, minimum mass and medium mass). At each of these operating points (fixed values of mass, c.g., inertia, throttle setting and air data uncertainty) the aircraft is trimmed for all grid points and then linearised. After linearisation the aerodynamic uncertainties with their extreme values (e.g. Cmα tolerance = 0.1) are added to the nominal

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derivatives. This has the advantage that no new trimming is required when type or sign of the tolerance is changed, only the corresponding elements of the state-space matrices have to be re-calculated. This approach saves a significant amount of time and effort. 2.3.5 Type and Effect of Uncertainties Variabilities and uncertainties can be divided into four groups: Configuration dependent variabilities such as mass c.g. and inertia, which differ with stores and fuel Aerodynamic uncertainties on stability derivatives, control power and damping derivatives Hardware dependent variabilities such as changes of the actuator or sensor dynamics or computing delays Air data system dependent tolerances such as measurement errors in signals like AoA, M or dynamic pressure which are used for scheduling of the control laws or control surfaces (to optimise performance) The effect of these uncertainties on stability, handling and performance differs with aircraft type, store configuration, control laws and flight condition. Before actual analysis work is started, the relative importance of the different uncertainties on the clearance results of the aircraft to be assessed should be investigated in trend studies. Parameters with minor effect can thus be excluded from further assessment in order to reduce the amount of necessary calculations. For the most important tolerances the effects are pretty clear - at least for linear analysis. The trends are more straightforward for the longitudinal axis than for the lateral/directional axes, where coupling effects between roll and yaw can lead to different results at different flight conditions. Examples for the effect of variabilities / uncertainties are given below. Configuration dependent variabilities There are several configuration dependent variabilities which are now described. Shift of longitudinal c.g. position The c.g. position is a dominant parameter for the longitudinal characteristics of an aircraft (much less for the lateral characteristics) because it directly influences the stability. When the c.g. is moved aft, the longitudinal stability of the aircraft is reduced and when it is aft of the neutral point the aircraft becomes unstable. This means that the feedback gains of a controller designed for a given c.g. position will be higher than needed for forward c.g. (over-gearing) and less than needed for a more aft c.g. (under-gearing).

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In a Nichols diagram (see explanation in subsection 2.4.1), a more forward situated c.g. will shift the plot of the open-loop frequency response upwards and - in the low frequency range - to the right, because the aircraft becomes more stable. This means that more forward c.g. will decrease the upper gain margin (UGM) and increase the lower gain margin (LGM). Note that a LGM only exists for basically unstable aircraft.

NICHOLS PLOT 12

Pitch Loop Cut

f in [hz]

0.050 0.089

9

0.282

0.158 0.089

6

Gain [db]

3

0.282

0.501 0.891 0.501 0.891

0 -3

0.891 1.585 1.585

-6

1.585 2.818

-9

2.818 2.818

-12 -15 -220

0.501

0.282 0.158

-200

-180

-160 -140 Phase [deg]

-120

-100

-80

-60

nominal c.g. 2% forward c.g. 2% aft

Fig. 2.3. Pitch actuator loop cut: effect of 2% c.g. shift

A more aft c.g. will accordingly shift the frequency plot downwards (increased UGM) and - for low frequency - to the left (decreased LGM). An example is given in Fig. 2.3. An effect of an aft c.g. shift to be seen in the time domain would be a quickened response in a pull-up manoeuvre with the danger of overshooting the allowed limits of AoA or normal load factor. For the c.g. more forward than nominal, the pull-up response would be slowed down (creeping response). Shift of lateral c.g. position The effect of a lateral c.g. shift on linear stability is usually neglected. Investigations will concentrate on the handling aspect and are done by mathematical and manned simulations. A noticeable effect might be seen for strongly asymmetric configurations (i.e. jettison of an underwing

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tank). The roll stick deflection needed for compensation must not be too large (< 70%) and must leave some margin for manoeuvring. The induced sideslip due to lateral g must not lead to departure at high AoA. Shift of vertical c.g. position This effect is of minor importance for a fighter aircraft and can usually be neglected. Changes in inertia A reduction of the pitch inertia (Iy ) would mainly reduce the UGM by

NICHOLS PLOT 12

Pitch Loop Cut

f in [hz] 0.089

9

0.158 0.158 0.282 0.282

6

0.501 0.501

Gain [db]

3

0.891

0

0.891

-3

1.585 1.585

-6 2.818

-9

2.818

-12 -15 -220

-200

-180

-160 -140 Phase [deg]

-120

-100

-80

-60

nominal Iy 20% reduced

Fig. 2.4. Pitch actuator loop cut: effect of 20% reduction in Iy

shifting the high frequency part of the pitch Nichols plot upwards (Fig. 2.4). The effect of the yaw inertia Iz on the rudder loop and the roll inertia Ix on the aileron loop is similar. Changes in mass Mass changes are usually of much less importance than inertia or c.g. changes. This might be different with controllers where the gains are a function of mass.

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Aerodynamic uncertainties There are several aerodynamic uncertainties which are now described. Changes of stability derivative Cmα This derivative defines the static stability around the pitch axis of the

NICHOLS PLOT 12

Pitch Loop Cut

f in [hz]

0.252 0.449

0.142

9

0.252

6

0.449 0.449

3

0.798

Gain [db]

0.798 0.798

0

1.418 1.418 1.418

-3 -6

2.522 2.522 2.522

-9 -12 -15 -220

4.486

-200

-180

-160 -140 Phase [deg]

-120

-100

-80

-60

Nominal +0.1 Cm_alfa Tolerance -0.1 Cm_alfa Tolerance

Fig. 2.5. Pitch actuator loop cut : effect of Cmα tolerance (± 0.1)

aircraft. For positive Cmα the aircraft is usually unstable and a controller is needed to provide artificial stability. A positive Cmα tolerance will increase the basic instability and affect the low frequency part of the Nichols plot: the upper part is shifted left and downwards, thus reducing the LGM. The high frequency range (lower part of the plot) will not be affected. The effect is demonstrated in Fig. 2.5. Changes in weathercock stability Cnβ and dihedral effect Clβ For weathercock stability, Cnβ must be positive. Whether or not a negative value of Cnβtotal (nominal + tolerance value) will produce lateral instability can only be determined by a full lateral analysis. The influence of a Cnβ tolerance on the Nichols plot of the rudder loop cut is comparable to that of a Cmα tolerance on the pitch actuator loop cut: a reduction of the static stability term (negative Cnβ tolerance) will turn the upper (lower frequency) part of the plot to the left and downwards with the high frequency part remaining unaffected. An example is given in Fig.

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2.6, where the tolerance value is large in relation to the nominal Cnβ and therefore, a big effect is observed. Furthermore, the unaugmented aircraft is directionally unstable. The effect of a Clβ tolerance on the plot of the rudder loop cut is similar to that of a Cnβ tolerance. The effect on the aileron loop is small and negligible.

NICHOLS PLOT 12 9

Rudder Loop Cut 0.004

f in [hz]

0.014 0.008 0.025 0.045 0.080 0.142

6

Gain [db]

3

0.008 0.0040.014 0.025 0.045 0.080

0.252

0.003

0

0.142

0.449 0.252 0.449 0.798

-3

0.798

-6

1.418 1.418

-9 -12 -15 -220

2.522 2.522

-200

-180

-160 -140 Phase [deg]

-120

-100

-80

-60

nominal -0.04 Cn_beta Tolerance

Fig. 2.6. Rudder loop cut: effect of Cnβ tolerance (-0.04)

Changes in the pitch control effectiveness CmδT S , CmδCS Tolerances on the effectiveness of symmetric flap or canard have an effect similar to a change of the pitch feedback gains. An increase of the effectiveness (negative CmδT S or positive CmδCS tolerance) will shift the pitch actuator Nichols plot upwards and thus reduce the UGM. A decrease will have the opposite effect. This is shown in Fig. 2.7. Changes in the roll control effectiveness ClδT D , ClδCD An increase in the roll control effectiveness shifts the Nichols plot of the roll loop upwards and thus decreases UGM and possibly the phase margin (PM). Changes in the rudder control effectiveness CnδR Comparable to the pitch loop, an increase in rudder effectiveness decreases the UGM of the rudder loop and possibly the PM. Changes in coupling terms CnδT D and ClδR The effect of changes in these coupling terms is not as predictable as for the other tolerances. It will depend more on flight condition/AoA.

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Changes in damping derivatives Cmq , Clp , Cnr The effect of changes in these coupling terms is not as predictable as for the other tolerances. It will depend more on flight condition/AoA. The effects are usually small and negligible in linear analysis.

NICHOLS PLOT 12

Pitch Loop Cut

f in [hz]

0.089 0.089 0.158

9

0.158

6

0.282 0.282

Gain [db]

3

0.501 0.501 0.891

0

0.891

-3

1.585 1.585

-6 2.818

-9

2.818

-12 -15 -220

-200

-180

-160 -140 Phase [deg]

-120

-100

-80

-60

Nominal 0.04 Cm_dts Tolerance

Fig. 2.7. Pitch actuator loop cut: effect of CmδT S tolerance (0.04)

Hardware tolerances The effect of hardware tolerances is straightforward. The hardware dynamics (actuators, sensors, delays) are represented by transfer functions (filters) whose frequency response will change when the filter time constants change due to tolerances. This will introduce additional phase lag/lead and gain decrease/increase into the feedback loop as a function of frequency and thus influence the stability margins. In the example plot of Fig. 2.8, for simplicity, a single-input system is shown with an actuator transfer function of 1 for the nominal case and a transfer function of 1/(1 + 0.1s) for the tolerance case. Here, the difference in phase and gain between the two transfer functions will directly add to the Nichols plot (at 1.585Hz, we get for example, an additional phase shift of −44.9◦ and a gain decrease of -3dB from the actuator). Additional time delays from hardware tolerances will have a detrimental effect on handling qualities. With increasing delay it becomes more and more difficult for the pilot to predict the response of the aircraft. Too large a time delay will lead to pilot-involved oscillations (PIO). For good handling (Level 1) the ”equivalent time delay” between pilot input and aircraft response should be smaller than 100ms.

24

U. Korte NICHOLS PLOT 12

Pitch Loop Cut

f in [hz] 0.089 0.089

9

0.158 0.158 0.2820.282

6

0.501

Gain [db]

3

0.501

0

0.891 0.891

-3 1.585

-6 1.585

-9

2.818

-12 -15 -220

5.012

-200

-180

-160 -140 Phase [deg]

-120

-100

-80

-60

Nominal Nominal plus Actuator

Fig. 2.8. Effect of actuator change on Nichols plot

Air data system tolerances Measurement errors in AoA, M, altitude or dynamic pressure could have a considerable effect on stability, handling and performance because they could lead to incorrect scheduling of the controller gains or incorrect positions of air data scheduled control surfaces. As already pointed out in chapter 2.3.3, clearance problems might then be found in areas of fast stability changes (Cmα ) over a small range of AoA or in the transonic area where the aerodynamics change considerably with small Mach number changes. An example for the transonic area is given in Fig. 2.9, where a pressure measurement error leads to a Mach error which in turn leads to wrong gain scheduling of the controller. The effect is a violation of the stability margin requirement as defined by the trapeziodal boundary.

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NICHOLS PLOT 12

Pitch Loop Cut

f in [hz] 0.142 0.252

9

0.252

6

0.449

0.449

3 0.798

Gain [db]

0

0.798 1.418

-3

1.418 2.522

-6

2.522

-9

4.486 4.486

-12 -15 -220

-200

-180

-160 -140 Phase [deg]

-120

-100

-80

-60

Nominal 0.085 Mach Tolerance

Fig. 2.9. Pitch actuator loop in the transonic area: effect of Mach tolerance (+0.085)

2.4 Clearance Requirements, Criteria, and Tasks 2.4.1 Stability Analysis The basic aim of all clearance work is to prove that the aircraft is stable over the entire flight envelope with sufficient margin against instability for a given set of uncertainties - just to prove stability (boundedness) is not enough. The first step in demonstrating sufficient stability is the calculation of linear stability margins for the open-loop frequency responses in pitch, roll and yaw at all points of the flight envelope within the required AoA or load factor limits. The open-loop frequency responses are obtained by breaking the loop at the input of each actuator or sensor. The results are plotted in Nichols diagrams where the required gain and phase margins are shown as exclusion zones which must not be violated by the plot. An example is given in Fig. 2.10, where it can be seen that with increasing AoA the UGM decreases and at 16 ◦ and 18◦ the stability margin requirements are no longer fulfilled. Therefore, a flight limitation for an AoA above 14◦ would be necessary unless the problem could be solved by a modification of the controller or a reduction of the tolerance. Such a deterioration of the stability margins would be reflected in decreased damping characteristics of the lateral response of the aircraft. The boundary values of the Nichols exclusion zone are based on experience. The outer boundaries are valid for the nominal case whereas the inner boundaries have to be observed when uncertainties are applied. The stability margin requirements [1] are valid for frequencies between 0.06Hz and the first

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rudder - loop cut

NICHOLS PLOT

12 9

0.014 0.025 0.008 0.045 0.004 0.080 0.003

6

0

0.252

0.142

0.025 0.014 0.045 0.008 0.080 0.004 0.142 0.0030.008 0.025 0.014 0.045 0.080 0.004 0.142 0.003

3

Gain [db]

f in [hz]

0.142

0.252 0.449 0.252 0.252

0.449 0.449 0.798 0.449 0.798

-3

0.798 0.798 1.418 1.418 1.418 1.418

-6 -9

2.522 2.522 2.522 2.522

-12 -15 -220

-200

ident 683 684 685 686

-180

alt.[ft] 40000. 40000. 40000. 40000.

-160 -140 Phase [deg]

ma/kcas[-/kts] 1.50/494. 1.50/494. 1.50/494. 1.50/494.

-120

-100

alpha[deg] 12.00 14.00 16.00 18.00

-80

-60

nz[g]

Fig. 2.10. Rudder loop with Cnβ -tolerance: margin degradation with AoA

elastic mode of the aircraft. The aim of the analysis is to find all violation points in the flight envelope and for each of these points, the worst case, i.e., that uncertainty combination which yields the biggest violation. If no violation is found then the aircraft is cleared without limitations. In addition to the open-loop stability margins, the worst case unstable eigenvalues (those with positive real part) of the augmented closed-loop system must be identified. More details about the stability margin and eigenvalue requirements can be found in chapter 10 of this book. The single-loop analysis is usually supported by a limited multi-loop analysis to check for the effects of multiple loop perturbations. Finally, it must be shown by simulation that the stability/controllability is not destroyed by non-linear effects such as rate and position saturation, inertia coupling etc. 2.4.2 Handling Analysis Apart from the stability requirements, the aircraft must fulfil the requirement of good handling. The clearance assessment must show that the pilot can control the aircraft precisely and easily to accomplish the mission. During the last two decades a number of linear and nonlinear criteria have been developed whose fulfilment will give a high degree of confidence that the aircraft will

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exhibit good handling qualities without pilot-in-the-loop oscillations and that flight testing can be started without large risk. The American military specification F-8785C [2] defines 3 levels of flying qualities: – Level 1: Handling clearly adequate (satisfactory) for the mission flight phase – Level 2: Handling adequate but some increase in pilot workload and/or degradation in mission effectiveness exists – Level 3: Aircraft can still be controlled but pilot workload is excessive and/or mission effectiveness is inadequate For combat aircraft Level 1 handling is required within the operational flight envelope and Level 2 handling within the service flight envelope [2] . Linear Handling Analysis Most of the linear handling criteria which are presently in use are based on frequency domain calculations like: – – – – – – – –

Pitch/bank attitude frequency response Pitch/bank average phase rate (to assess resistance to PIO) Absolute amplitude (to assess resistance to PIO) Frequency and damping of short period mode, Dutch roll and FCS modes [1], complemented by the low-order equivalent system approach for highorder systems Closed-loop pitch axis bandwidth (Neal Smith) Open-loop pitch axis bandwith (Hoh) Phase and gain margin criterion (R¨oger) Ride discomfort index [1]

Time domain criteria [3] in use are, among others: – Pitch rate overshoot/dropback, pitch rate peak time, pitch acceleration peak time, flight path angle time delay (Gibson) – Equivalent CAP (control anticipation parameter) – (Effective) Roll mode time constant, time to bank. 2.4.3 Nonlinear Analysis Linear analysis is complemented by nonlinear, non-realtime simulation and manned simulation which are used for detailed investigation of problem areas found in the linear evaluation, to check for the effect of nonlinearities, such as rate and position saturation, dead zones, inertia coupling etc., on aircraft stability, handling and control, and to finally decide whether the aircraft is fit for purpose.

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Off-line simulation Defined control inputs in pitch, roll and yaw are used to test whether the aircraft response is fast and precise without overshoots of AoA, AoS and load factor limits. Pull/push Modern flight control systems are usually of the manoeuvre demand type. This means that pitch stick deflection is proportional to AoA- or nz demand etc. By using full stick rapid pulls/pushes and ramp inputs of defined duration it is checked whether the aircraft shows the required response and stays within the limits for the nominal case. With uncertainties added, the deviations from the nominal trajectory should not exceed a given limit - the overshoot of the AoA-/nz -limit for instance should be less than 2◦ /0.4g and not lead to departure. Rapid rolling Important features to be checked for the nominal and the uncertainty cases are maximum roll rates/overshoots, maximum sideslip generated during roll, roll angle overshoot when trying to stop the roll, variation of normal load factor during full stick rapid roll and available pitch down control power at high AoA and high roll rate (absence of departure due to gyroscopic effects). Pedal response Pedal inputs will generate sideslip. It must be demonstrated for all uncertainties that the maximum sideslip does not exceed safe limits, defined by loading and aerodynamic considerations (values different for different aircraft). Roll due to pedal which is generated as a side effect should always be in the direction of the input to help with turn co-ordination. The pedal inputs will be step inputs and 3-2-1-1 inputs (consecutive steps of alternating sign and a duration of for instance 3s, 2s, 1s, 1s). The latter input is well suited to show whether dynamic inputs will lead to limit overshoots. Rate/position saturation If saturation is encountered it must not lead to control problems or PIO. Limit cycles The aircraft should be free of limit cycles, i.e. sustained oscillations. Integrator windup A problem connected with position saturation is potential integrator windup in control systems with integrator feedback. If for instance, in a pull-up the flaps and canards reach their limits before the commanded AoA has been reached, then the integrator value - if not stopped - would continue to increase. The aircraft could then not follow a nose down command until the integrator has run down below some threshold value. This could be disastrous.

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Fading/switching The correct functioning of fading and switching in the control laws or between different control modes should be demonstrated. Response after failures Depending on the type of failure, aircraft behaviour might change. Actuator failures for instance, might introduce additional phase lag and thus lead to a deterioration in stability and handling. In aircraft with two hydraulic systems, loss of one system could inhibit full use of the control surfaces due to high hinge moments produced by the aerodynamic forces at high dynamic pressure. It must be demonstrated that in such cases the aircraft is still safe - for all combinations of variabilities/uncertainties. Hinge moment effects are mainly seen in the transonic/supersonic area. Manned simulation/rig tests Real-time manned simulation is used to check for nominal and tolerance cases, whether handling and control are adequate in general and for special tasks such as air-to-air refuelling. The pilot will give the final answer whether the aircraft can be flown safely and allows fulfilment of the mission. Tasks to be addressed in manned simulation are listed below. They partly overlap with non-realtime simulation. Control sensitivity, overcontrol, PIO Checks should be made for tasks requiring tight control such as approach and landing, tracking, air-to-air refuelling. Control of AoA/normal load factor It must be demonstrated that the pilot can precisely control the aircraft and stay within allowed AoA/nz -limits. Local instabilities (pitch-ups) must not lead to exceedance of these limits. Bank angle control Precise control of bank angle must be possible. Rapid Rolling Manned simulation complements the non-realtime simulation. Tracking and gross manoeuvring Pilot opinion is required whether aircraft response is adequate. Steady heading sideslip It shall be possible to maintain constant heading without difficulty when applying up to full pedal. Application should be made in a progressive manner. Take-off and landing It should be possible to take-off and land the aircraft safely and without undue pilot workload for all combinations of uncertainties. This must be demonstrated for dry and wet runways, with and without crosswinds (up to 30kts dry) for calm and turbulent weather (moderate and severe turbulence). How turbulence can be included in the model is described in [2].

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U. Korte

Stick authority with asymmetric configurations After missile firing, underwing tank jettison failure (hang-up) or fuel failures, considerable asymmetries can be produced. It must be possible to compensate for the asymmetries with less than 70% roll stick deflection. Acceptable handling must be demonstrated. Carefree handling For those types of aircraft which have been designed to be carefree meaning that the FCS provides automatic protection of certain limits (i.e. AoA-, AoS-, nz -, roll rate limit) - the correct functionality must be demonstrated for nominal cases and under uncertainty. Crossed controls combined with throttle inputs should be used in such a sequence that would most likely generate departure. Handling after failures Depending on the failure, agile manoeuvring might no longer be required and the envelope to be cleared might be reduced. The criteria to be applied remain the same, but the required handling level however might be reduced by one. The transient behaviour during occurrence of a system component failure is checked in Rig (iron bird) tests with real hardware and redundancy management (multi-channel system) included.

2.5 Requirements on Clearance Methods and Tools 2.5.1 Methods Generally all analysis techniques - current ones and new ones-must be able to find out where the aircraft is safe to fly with the given control system and uncertainties. Therefore, they must give answers to the following questions: – Are there any stability margin violations in the required flight envelope? What is the strongest violation at each point and which uncertainty combination has caused it? – Are there any unstable eigenvalues which are outside the requirement limits? For which uncertainty combination are they largest? – Are there any limit cycles? Where and for what combination? – Is there any rate or position saturation? Which combination gives the longest duration? Does it lead to control problems? – Is there any handling criterion which indicates handling worse than Level 2? Where and for which uncertainty combination? – Are there any limit cycles? Where and for what combination? – Is there any rate or position saturation? Which combination gives the longest duration? Does it lead to control problems? – Are there any exceedances of +ve/-ve limits of AoA, AoS, load factor or roll rate? Which uncertainty combination yields the worst case and for which manoeuvre?

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– Which uncertainty combination yields the largest deviations from nominal manoeuvre response? – Is the aircraft still flyable with sufficient precision and ease in specified turbulence, crosswind and gusts for the given set of uncertainties? Current methods do not allow inclusion of all variabilities and uncertainties simultaneously in one calculation run. For each c.g. position or mass for instance, a new calculation is required, and this increases the assessment effort. The procedure to put a grid on the flight envelope and do all calculations at the grid points does not give an answer about the behaviour between these points. The hope is that with a very narrow grid, all critical points will be found - but there is no guarantee and more grid points lead to more calculations. In current practice in industry, only the maximum/minimum values of the uncertainties are considered because otherwise, the necessary analysis effort would increase tremendously and might become unaffordable. A guarantee that the extreme values will deliver the worst case does not exist. There is a need for methods which are not conservative and free from the above limitations, and they should reduce the computational effort - or at least they should not increase it above the level required by current methods. 2.5.2 Software Tools and Visualisation The control laws clearance assessment of a fighter with so many different configurations and parameter uncertainties requires a huge amount of data processing. Therefore, fast and efficient software tools are needed, for instance, tools for fast trim calculations or calculation of higher order frequency responses. Commercial-off-the-shelf software tools are often not well suited for this purpose. Besides the calculation of results, a difficult and time consuming part of the work is to mentally put together all pieces of information obtained from linear and nonlinear analysis in order to derive necessary clearance limitations. For this task database structures for easy interactive information retrieval in combination with good Visualisation and plotting tools are of high importance. Industry is presently using commercial software packages like MATLAB or MATRIXX combined with their in-house developed facilities. The commercial products provide many useful graphical capabilities but they are not tailored to the specific needs of the clearance task. Adaptation of existing tools in this respect is needed and tailored plotting packages for 2D- and 3Dplots are needed for: Familiarisation with the aircraft and the controller Plots of important aerodynamic parameters or controller gains against AoA, Mach number, dynamic pressure etc. will help to develop a feeling about where the weak points of the system with respect to clearance might be.

32

U. Korte

Display of clearance assessment results Nichols plots, plots of minimum Gain and Phase Margin (see Fig.2.11), plots or tables of unstable Eigenvalues, plots of Handling Qualities metrics

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Fig. 2.11. Minimum phase margin versus Mach number for all AoA

Visualisation of clearance limitations It is extremely important that clear and unmistakable information about necessary clearance restrictions is provided to the operators of the aircraft. The pilot must know exactly where he is not allowed to fly (prohibited areas) and in which area of the flight envelope restrictions in AoA or load factor must be observed. Plots of the flight envelope - altitude versus Mach number - are used to display the prohibited or restricted areas, as indicated in Fig. 2.12.

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A general requirement for all plotting information is that it contains a legend which allows clear identification of the presented results and the basis from

2 Tasks and Needs of the Industrial Clearance Process

33

which they have been be derived (model standard and issue of the aerodynamic data set, hardware, configuration, control laws, flight condition, failure states etc.). This means the documentation must allow exact reproduction at a later time or by other users, if necessary.

2.6 Conclusions The different tasks in the process of clearance of flight control laws have been described and it has been made obvious that this process is time consuming and expensive due to the problem complexity and the huge amount of effort required. It has been pointed out that new methods and tools are needed which could help to reduce the computational effort and/or give a guarantee for the detection of the worst case results with respect to variabilities and uncertainties.

References 1. Military Specification of Flight Control Systems - Design, Installation and Test of Piloted Aircraft MIL-F-9490D (USAF), 6 June 1975 2. Military Specification of Flying Qualities of Piloted Airplanes MIL-F-8785C Department of Defense, USA, November 1980 3. Military Handbook of Flying Qualities of Piloted Airplanes HDBK-1797 Department of Defense, USA, 19 December 1997

3 The Structured Singular Value and µ-Analysis Declan G. Bates and Ian Postlethwaite Control and Instrumentation Research Group, Department of Engineering, University of Leicester, University Road, Leicester, LE1 7RH, UK. [email protected], [email protected]

Summary. We introduce the structured singular value µ and discuss its use as an analysis tool for flight control applications. To apply µ-analysis tools to flight control law clearance problems, linear fractional transformation (LFT) based uncertainty models must first be generated to capture the effect of uncertain aircraft parameters on the closed-loop system dynamics. The clearance criteria which are most easily addressed with µ-analysis are those which are defined in the frequency domain such as the stability margin criterion or the unstable eigenvalue criterion. µ-analysis tools which have been developed to address these specific clearance criteria are discussed.

3.1 Introduction It is generally possible to arrange any linear time invariant (LTI) closedloop system which is subject to some unstructured and/or structured type of norm-bounded uncertainty in the form shown in Fig. 3.1, where P , K1 and K2 denote the plant, pre-filter and feedback controller respectively. With respect to this figure, unstructured uncertainty means that the uncertainty matrix ∆ is fully populated, while structured uncertainty means that it has some (typically diagonal or block diagonal) structure. In the context of a flight control clearance problem, unstructured uncertainty could correspond, for example, to unmodelled high frequency aircraft dynamics, while structured uncertainty is used to represent particular aircraft parameters such as stability derivatives, inertias, etc, which are subject to change, or known only to within a certain tolerance. Techniques for converting standard aircraft models into the form shown in Fig. 3.1 are discussed in Part IV of this book. Given a model in this form, it is then straightforward to rearrange the system into the form shown in Fig. 3.2, where M represents the known part of the system (aircraft model and controller) and ∆ represents the uncertainty present in the system. Partitioning M compatibly with the ∆ matrix, the relationship between the input and output signals of the closed-loop system shown in Fig. 3.2 is then given by the upper LFT: C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 37-55, 2002. Springer-Verlag Berlin Heidelberg 2002

38

D.G. Bates and I. Postlethwaite

∆

w r -

K1

-

- h u6



z y -

P

K2



Fig. 3.1. Interconnection structure of a general uncertain closed-loop system

∆

w

-

r

-



M

z

- y

Fig. 3.2. Upper LFT uncertainty description

y = Fu (M, ∆) r = (M22 + M21 ∆(I − M11 ∆)−1 M12 ) r

(3.1)

Now, assuming that the nominal closed-loop system M (s) in Fig. 3.2 is asymptotically stable and that ∆ is a complex unstructured uncertainty block, the small gain theorem (SGT), [3], gives the following result: The closed-loop system in Fig. 3.2 is stable if and only if σ(∆(jω)) <

1 σ(M11 (jω))

(3.2)

This result defines a test for stability (and thus a robustness measure) for a closed-loop system subject to unstructured uncertainty in terms of the maximum singular value of the matrix M11 . For aerospace systems it is often the case that uncertainty can be related to variations in specific aircraft parameters, such as centre of gravity, inertias, stability derivatives etc. In such cases, it is possible to generate models of uncertainty which have a particular structure, and thus reduce the level of

3 The Structured Singular Value and µ-Analysis

39

conservatism in the robustness analysis. The generation of structured LFTbased uncertainty models which accurately capture the effect of uncertainty on the original non-linear aircraft model is considered in detail in Part IV of this book. In this tutorial, we will assume that such an uncertainty model is available, and concentrate on the tools which can be used to analyse the resulting uncertain closed-loop system. The generation of a structured LFT-based uncertainty model means that we have been able to place all of the uncertainty affecting the system into an uncertainty matrix ∆ which has a diagonal or block diagonal structure, i.e., ∆(jω) = diag(∆1 (jω), ....., ∆p (jω)), σ(∆i (jω)) ≤ k

(3.3)

where k defines an upper bound on the size of the maximum singular value of any uncertainty block ∆i . Now again assume that the nominal closed-loop system is stable, and consider the question: What is the maximum value of k for which the closed-loop system will remain stable? We can still apply the SGT to the above problem, but the result will be conservative, since the structure of the matrix ∆ will not be taken into account. The SGT will in effect assume that all of the elements of the matrix ∆ are allowed to be non-zero, when we know that most of the elements are in fact zero. Thus the SGT will consider a larger set of uncertainty than is in fact possible, and the resulting robustness measure will be conservative, i.e. pessimistic. In order to get a non-conservative solution, Doyle [4, 5], introduced the structured singular value µ: µ∆ (M11 ) =

1 min(k s.t. det(I − M11 ∆) = 0)

(3.4)

The above result defines a test for stability (robustness measure) of a closedloop system subject to structured uncertainty in terms of the maximum structured singular value of the matrix M11 . Note that the structured singular value robustness measure can be derived directly from the multivariable Nyquist stability theorem, [3]: Theorem. For the system of Fig. 3.2, let Pol denote the number of unstable poles in the open-loop transfer function matrix M11 ∆. Then the closed-loop system is stable if and only if the Nyquist plot of det(I − M11 ∆(s)) (i) does not pass through the origin, and (ii) makes Pol anti-clockwise encirclements of the origin. Note that for the problem at hand we are looking for zero encirclements of the origin, since Pol = 0. This arises because M , the nominal closed-loop system is assumed to be stable, and the uncertainty matrix ∆ is also constrained to be stable. Singular value performance requirements can be combined with stability robustness analysis in the µ framework to measure the robust performance properties of the system. Consider a modification to the standard LFT uncertainty structure of Fig. 3.2, which is illustrated in Fig. 3.3. Assume, for

40

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∆1 0 w r

-

˙˙

0  ∆1

0  ˙˙ 0 ∆p+1

∆p

M

z y

⇒ w

-

M

z

∆p+1 

Fig. 3.3. Robust performance requirement as fictitious uncertainty block

example, that the signal y is the error between the reference demands and corresponding (closed-loop) measured outputs. One way of defining a performance specification for these variables is to require that the maximum singular value of the frequency response matrix from r to y lies below some weighting function W (jω). For example, for zero steady-state tracking error we would require σ(M22 (jω)) << 1 at low frequencies, which could be specified by making the magnitude of W (jω) very small at those frequencies. To prove good robust performance characteristics we need to check if the above nominal performance specifications are satisfied for all variations in the uncertainties ∆1 to ∆p . In a µ-analysis framework this can be accomplished as follows: We assume, as usual, that the model uncertainty blocks ∆1 to ∆p have been normalised to 1, so that the uncertainty matrix ∆ is given by ∆(jω) = diag(∆1 (jω), ....., ∆p (jω)), σ(∆i (jω)) ≤ 1 Assume further that the performance weight W (jω) has been absorbed into M , so that we require σ(M22 (jω)) < 1, ∀ ω. The overall transfer matrix M (jω) can therefore be given by · ¸ · ¸ · ¸ z M11 M12 w = (3.5) y M21 M22 r Thus we have that Robust stability ⇔ µ∆ (M11 ) < 1 ∀ ω and

Nominal performance ⇔ σ(M22 ) < 1 ∀ ω

The robust performance requirement is that the gain from r to y be less than 1 for any uncertainty in the set ∆, thus

3 The Structured Singular Value and µ-Analysis

41

Robust performance ⇔ σ(Fu (M, ∆)) = σ(M22 +M21 ∆(I−M11 ∆)−1 M12 ) < 1 where Fu (M, ∆) is the upper LFT of M and ∆. We can replace this robust performance requirement with a robust stability requirement using the SGT, as follows. The key idea is to notice that having the closed-loop gain from r to y be less than one at all frequencies implies that we can feed back y to r through a fictitious uncertainty block ∆p+1 having gain σ(∆p+1 ) < 1, without de-stabilising the system. This is because the loop gain of this ‘fictitious’ loop will be less than one, and the system is nominally stable, thereby meeting the SGT conditions. In turn, if σ(Fu (M, ∆)) > 1 at any frequency for some value of ∆, then the fictitious loop could de-stabilise the system, since the phase of ∆p+1 is completely uncertain. Thus robust stability of the fictitious p + 1 block system is equivalent to robust performance of the original p block system. To test for robust performance, simply define the new uncertainty set ∆P (jω) = diag(∆1 (jω), ....., ∆p+1 (jω)), σ(∆i (jω)) ≤ 1 and then check that

µ∆P (M ) < 1 ∀ ω

as illustrated in Fig. 3.3. Note that the above test is a necessary and sufficient condition for robust performance, and is thus completely nonconservative as long as the actual model uncertainty is well described by the block structure of ∆. The above discussion on performance robustness has considered performance from a frequency domain viewpoint only. Although some specifications on closed-loop performance can naturally be written in the frequency domain (e.g. handling/flying quality specifications given in terms of low order equivalent system transfer functions), other performance specifications for aerospace systems are given in the time domain, e.g. maximum allowable rise times or overshoot for pilot demands. Worst-case performance in terms of µ only has an exact time domain meaning in terms of sinusoids, and thus care must be taken in interpreting µ-analysis results for general time-domain problems.

3.2 Computation of µ The computation of µ is a non polynomial time problem, i.e. the computational burden of the algorithms that compute the exact value of µ is an exponential function of the number of uncertainties. It is consequently impossible to compute the exact value of µ for large dimensional problems associated with real industrial systems. A usual solution in this case is to compute upper and lower bounds on µ. If these are sufficiently tight, then little information is lost. Note that to fully exploit the power of the structured singular value theory, tight upper and lower bounds on µ are required. The upper bound provides a sufficient condition for stability/performance in

42

D.G. Bates and I. Postlethwaite

the presence of a specified level of structured uncertainty. The lower bound provides a sufficient condition for instability, and also returns a worst-case ∆, i.e. a worst-case combination of uncertain parameters for the problem. The degree of difficulty involved in computing good bounds on µ depends on (a) the order of the ∆ matrix, and (b) whether ∆ is complex, real or mixed, as discussed below. In Part IV of this book, it is shown how different types of uncertainty, and different approaches to generating LFT-based uncertainty models, lead to either complex, real or mixed uncertainty blocks. 3.2.1 Computation of Complex and Mixed µ When all the blocks in ∆ are complex, tight bounds on µ may be computed relatively easily. Polynomial time algorithms are available to compute upper and lower bounds on complex µ, [6]. Both bounds converge to exact µ for low order problems and extensive computational experience, [7], has shown that the bounds remain quite tight even for high order problems. For mixed real and complex uncertainty, polynomial time algorithms are also available for calculating both upper and lower bounds on µ. The upper bound algorithms use linear matrix inequality based optimisation, [8, 9], while the lower bound is generated via power algorithms, [10, 11]. The upper bound is generally quite tight, but the quality of the lower bound depends heavily on the amount of complex versus real uncertainty present in ∆. If ∆ is dominated by real uncertainty then the lower bound algorithm may fail to converge, especially for high order problems. 3.2.2 Computation of Real µ For purely real µ problems, examples appear in the literature which show that µ can even be a discontinuous function of the problem data, [12, 13]. For real µ problems with a physical engineering motivation, however, it is shown in [12] that discontinuity problems do not arise, and convergent upper, [14], and lower, [15, 16], bound algorithms for µ exist. Unfortunately both of these algorithms are exponential time, and thus in practice this limits the size of the ∆ matrix to about 11. A fix for the upper bound problem is to apply the polynomial time mixed µ upper bound algorithms - this generally gives good results, even for high order systems. However, the lower bound for real µ obtained from the mixed µ algorithm is generally poor and the algorithm often fails to converge, particularly for high dimensional problems. This represents a particular problem for the application of µ to flight control law analysis problems, since the number of uncertain parameters which needs to be considered can often result in high order real uncertainty matrices, and the computation of a tight lower bound is essential in order to identify the ‘worstcase’ combination of uncertain parameters. One convenient engineering ‘fix’ for the problem is to add small amounts of ‘artificial’ complex uncertainty to

3 The Structured Singular Value and µ-Analysis

43

each real ∆i in order to improve the lower bound derived from the mixed µ algorithm, [6]. While this approach can work reasonably well, it has some disadvantages. One is that the introduction of ‘enough’ complex uncertainty to generate a tight lower bound can significantly increase the associated upper bound, thus making the analysis results more conservative and more difficult to interpret. Secondly, it has been shown that the real part of the worst-case structured ∆ determined using existing µ software is particularly sensitive to the addition of uncertainty in this manner, [13]. This can result in the computation of a combination of uncertain real parameters that may be very different from the real worst-case combination. In [2, 1], a state-space approach for computing the peak values of the lower bound on real µ is proposed. The approach basically consists of extracting the real part from a destabilising mixed uncertainty matrix and increasing it until one of the closed-loop poles migrates through the imaginary axis. While this approach is polynomial time and can thus be applied to high dimensional problems, it only returns the peak value of the lower bound over a specified frequency range, and cannot be used to generate a tight lower bound at each point of a frequency grid. Information on the ‘shape’ of the µ-lower bound as a function of frequency can provide insight into the type of uncertainty which is causing the problem, e.g. narrow peaks on the µ-plot due to aircraft structural modes. In the analysis results presented in Chapter 16, two new methods are used for generating real µ lower bounds as a function of frequency, [17, 18]. The first of these methods seeks to selectively reduce the size of the highorder real ∆ uncertainty matrix until the use of exponential time lower bound algorithms becomes feasible. To ensure that the reduced order lower bound is a tight lower bound for the original problem, we seek to identify those elements ∆i of the ∆ matrix which make little or no contribution to the maximum value of µ by evaluating the µ-sensitivity function, [19], for each uncertainty. Those uncertainties with the lowest µ-sensitivity functions are then discarded before the lower bound is calculated. The second method casts the problem of computing a lower bound for µ as a search for the worst case (i.e. smallest) real diagonal destabilising uncertainty matrix ∆. Denote the vector of ∆i diagonal entries of ∆ by x. Thus, if ∆ ∈ Rp×p x = [∆1 , . . . , ∆p ]T ∈ Rp (3.6) For real scalar uncertainty this search can be formulated as an equivalent constrained minimisation problem, f (x), over a frequency range Ω: min f (x) =

min

∆i ∈R, ω∈Ω

σ ¯ (∆) subject to |det (I − ∆M11 (jω)| ≤ tol (3.7)

tol in the above constraint is a user defined parameter which can be used to trade-off computation time versus tightness of the resulting lower bound. Commercially available optimisation software from the MATLAB Optimisation toolbox, [20, 21], can be used to solve eqn. (3.7). As the search for a worst

44

D.G. Bates and I. Postlethwaite

case destabilising ∆ is non-convex, local minima can still occur using this approach. However, extensive computational experience using this method on the HIRM+ and other aircraft, [22, 18] indicates that tight lower bounds are generally produced. 3.2.3 Computation of µ without a Frequency Gridding In general, the structured singular value µ(ω) is computed at each point of a frequency grid. In the case of purely complex or mixed real-complex perturbations, µ(ω) is a continuous function of ω. However, purely real uncertainties can cause discontinuities in the value of µ with respect to ω. In addition, in the case of flexible systems, it is possible to miss the critical frequency at which narrow and high peaks occur if the grid is not fine enough. Invariably, refining the grid means increasing the computational effort involved in computing the µ bounds. One solution to this problem is to recast the standard frequency dependent µ analysis problem as an augmented skewed-µ problem, where the frequency ω is treated as an additional uncertainty [1]. In this approach it is then possible to compute µ as a continuous function of frequency over some interval [ω, ω]. The main drawback with this approach is that the size of the augmented skewed-µ problem can be much larger than the size of the original problem. In fact, the uncertain frequency ω appears as a repeated real scalar δω In whose size increases with the order (n) of the original system. Where this approach has been used in the results presented in Chapter 16 the computational cost has been minimised by casting the skewed-µ problem as an optimisation procedure around the critical frequency (identified via an initial frequency grid).

3.3 µ-Tools for Clearance of Stability Margin Criterion In this section we describe one way in which the structured singular value robustness measure can be used to address the clearance of the Stability Margin Criterion defined in Chapter 2. As we shall show, µ has a straightforward interpretation in terms of classical gain/phase margin and Nichols exclusion region robustness specifications. In fact, µ-analysis provides the capability to efficiently test for the avoidance of exclusion regions in the Nichols plane, for all combinations of parametric uncertainty, without resorting to the gridding approach traditionally used by industry. Our treatment follows that in [23, 24, 25]. Other approaches for casting Nichols exclusion region robustness specifications in a µ-analysis framework are described in [26, 27]. We begin with the single loop case. In the framework of µ-analysis, it turns out to be convenient to work with alternative Nichols plane exclusion regions of the form shown in Fig. 3.4. These elliptical regions are centred around the critical point (-180,0) and satisfy the equation

3 The Structured Singular Value and µ-Analysis

| L(jω) |2dB (∠L(jω) + 180)2 + =1 Gm2 Pm2

45

(3.8)

where L(jω) is the open-loop frequency response, Gm is the desired gain margin and Pm is the desired phase margin. Thus for example, any feedNichols Plot

10 8 6

Exclusion region A

Open−loop gain (dB)

4 Exclusion region B

2 0 −2 −4 −6 −8 −10 −240

−220

−200

−180 −160 Open−loop phase (degrees)

−140

−120

Fig. 3.4. Elliptical Nichols plane exclusion regions

back system whose open-loop frequency response avoids the regions A and B in Fig. 3.4 provides gain and phase margins of ±6dB/ ± 36.87◦ and ±4.5dB/ ± 28.44◦ respectively (note that these values are very close to those required under the classical exclusion regions defined in Chapter 2). A key point is that for these particular choices of gain and phase margins the corresponding exclusion regions in the Nyquist plane are circles with (centre,radius) given by (-1.25,0.75) for region A, and (-1.14,0.54) for region B - see Fig. 3.5. Now, as shown in [23], another way to interpret the requirement for avoidance of, for example, the circle B in the Nyquist plane by the open-loop frequency response L(jω), is to consider a plant subject to disc uncertainty of (centre,radius) given by (+1.14,0.54) at each frequency. It is then easy to see that avoidance of the (-1,0) critical point in the Nyquist plane by L(jω) for all possible plants in this set is exactly equivalent to avoidance of the exclusion region B by L(jω) for the original plant. The set of possible plants can be represented as P (s) = P1 (s)(1.14 + ∆N )

(3.9)

46

D.G. Bates and I. Postlethwaite Nyquist Plot 1

Exclusion region A 0.5

Im L(jw)

Exclusion region B 0

−0.5

−1 −3

−2.5

−2

−1.5 Re L(jw)

−1

−0.5

0

Fig. 3.5. Circular Nyquist plane exclusion regions

where P1 is the original plant, ∆N is complex and k ∆N k∞ ≤ 0.54. This is of course the same as writing P (s) = 1.14P1 (s)(1 + WN ∆N )

(3.10)

with WN = 0.47 and k ∆N k∞ ≤ 1. In this way we can represent the Nichols exclusion region as a ‘fictitious’ multiplicative input uncertainty for the scaled nominal plant. It now remains for us to represent the actual uncertainty present in the original model of the plant. This uncertainty can be structured (i.e. resulting from uncertainty in actual aircraft parameters) or unstructured (i.e. resulting from unmodelled dynamics in the aircraft model). For simplicity, here we consider an unstructured multiplicative uncertainty on the original plant P1 of the form P1 (s) = P0 (s)(1 + WM (s)∆M ), k ∆M k∞ ≤ 1

(3.11)

We thus end up with the system shown in Fig. 3.6. We can now do some block diagram manipulations to ‘pull-out-the-deltas’ from Fig. 3.6, and thus convert this system into the standard M ∆ form for robustness analysis under the µ framework, as in Fig. 3.7. Now consider the following robustness specification: Robustness Specification 1: For the control system in Fig. 3.7, we require µ∆ (M11 (jω)) ≤ 1, ∀ ω

3 The Structured Singular Value and µ-Analysis

- WN - ∆N

j 6-

- WM - ∆ M ? - j- 1.14

47

P1

? - j- P0

-

¾

K

Fig. 3.6. Single-loop feedback control system with uncertainties - test for exclusion region B

-

∆N 0 0

∆M

M11

¾

Fig. 3.7. Standard M∆ form for robustness analysis using µ

This specification is exactly the same as saying that the open-loop frequency response of every plant in the set of uncertain plants P1 (s) lies outside the Nichols plane exclusion region B of Fig. 3.4. The extension of the above results to the multivariable case is quite straightforward. Robustness (in terms of avoidance of a prescribed Nichols exclusion region) can be measured (a) cutting one loop at a time, with all the other loops closed, and all loops subject (simultaneously) to an LFT-based uncertainty model representing the uncertain aircraft parameters, or (b) cutting all loops simultaneously, with all loops subject (simultaneously) to an LFT-based uncertainty model representing the uncertain aircraft parameters. We illustrate the approach for the former case via a two-input two-output flight control system design for the HIMAT aircraft model, [28, 6], the extension to the case of simultaneous cuts in all loops will be seen to be automatic. The HIMAT model represents the dynamics of a scaled, remotely piloted version of an advanced fighter aircraft, and has been widely used in the robust control literature as a benchmark for evaluation of controller synthesis and analysis techniques. Assuming effective decoupling between the aircraft’s longitudinal and lateral dynamics, a linearised model P0 for the longitudinal rigid body dynamics is given by:

48

D.G. Bates and I. Postlethwaite

x˙ = Ax + Bu;

y = Cx + Du

(3.12)

with state vector x, control inputs u and controlled outputs y given by x = [δV, α, q, θ], u = [δe , δc ], y = [α, θ]

(3.13)

where δV is forward speed, δe is elevon deflection, δc is canard deflection, and the other variables have their usual meanings. Numerical values for the statespace matrices at a given flight condition can be found in [28, 6]. Potential differences between the nominal model P0 and the actual behaviour of the real aircraft P1 due to uncertainty in the actuator dynamics, aircraft stability derivatives, etc, are represented by a simple diagonal, frequency dependent uncertainty model at the plant input, so that P1 = P0 (s)(I2×2 + WM (s)∆M )

(3.14)

where WM =

50s + 100 × I2×2 , s + 10000

k ∆M k∞ ≤ 1

(3.15)

For the above model, a controller was designed to independently control α and θ in order to provide vertical translation, pitch pointing and direct lift manoeuvring capabilities. Consider now the robustness specification that the open-loop frequency response of loop 1 avoids Nichols exclusion region B, with loop 2 closed, and both loops subject to the multiplicative plant uncertainty defined above. The block diagram corresponding to this test is shown in Fig. 3.8, with WN = 0.47 and k ∆N k∞ ≤ 1. Converting this system into the standard M ∆ form for µ-analysis, Fig. 3.9, gives: Robustness Specification 2: For the system in Fig. 3.9, we require that µ∆ (M11 (jω)) ≤ 1, ∀ω The corresponding µ and Nichols plots for the controller are shown in Figs. 3.10 and 3.11, from which we can see that Loop 1 fails the Nichols exclusion region test for the worst case plant in P1 . Finally, we note that the problems of computing the maximum allowable level of plant uncertainty or size of exclusion region before instability occurs can be cast as skewed-µ calculations [6], and thus can be answered exactly in the structured singular value robustness analysis framework.

3.4 µ-Tools for the Worst Case Eigenvalue Criterion In this section we describe how the structured singular value robustness measure can be used to address the Worst Case Eigenvalue Clearance Criterion defined in Chapter 2. From the standard block diagram for µ-analysis, stability of the closed loop system is equivalent to stability of the quantity

3 The Structured Singular Value and µ-Analysis

- WN - ∆N

-

-

P1

- W M 1 - ∆ M1 ? - j- 1.14

49

? - j- j-

P0

-

6 - W M 2 - ∆ M2 K

¾

Fig. 3.8. Multi-loop system with uncertainties - test for loop 1, exclusion region B

-

∆N 0 ∆ M1 0 ∆ M2

M11

¾

Fig. 3.9. Standard M∆ form for robustness analysis using µ

(I − M11 ∆)−1 . By testing the stability of (I − M11 ∆)−1 as the ∆i elements vary, we can find the worst case, or smallest, set of simultaneous changes in ∆i which drive the system unstable. From matrix theory, (I − M11 ∆)−1 = adj(I − M11 ∆)/det(I − M11 ∆) Thus, for a given set of model perturbations ∆, and a given complex number s0 that is not an open loop pole of M11 (s) or ∆(s), s0 is a closed loop pole if and only if det(I − M11 (s0 )∆(s0 )) = 0 Suppose we want to find the smallest set of ∆i elements which places a pole at s0 : km =: min {k ∈ [0, ∞] such that det(I − M11 (s0 )∆(s0 )) = 0} ∆

D.G. Bates and I. Postlethwaite 1.5 Loop1 − exclusion region A test

Loop1 − exclusion region B test

1

Loop2 − exclusion region A test

0.5

0 −3 10

−2

−1

10

0

10

10

1

2

10

3

10

10

Fig. 3.10. Plot of µ bounds for closed-loop system: loop1 (-), loop2 (-.-) Nichols Plot

20

Loop 1

10 Open−loop gain (dB)

50

Exclusion region A 0

Exclusion region B

−10 Loop 2 −20

−30 −200

−180

−160 −140 Open−loop phase (degrees)

−120

−100

Fig. 3.11. Nichols plots for worst-case systems: loop 1 and loop 2

3 The Structured Singular Value and µ-Analysis

51

where ∆ = diag(∆1 ....∆p ) and σ(∆i (s0 )) ≤ k ∀ i Then µ∆ (M11 ) = 1/km Most published work on µ-analysis has assumed that µ must be computed on a frequency sweep along the s = jω axis. However, computing µ away from the imaginary axis can provide a lot of useful information about changes in the closed-loop performance as well. Some possible tests are shown in Fig. 3.12.

constant damping test

6Im(s) stability test

@ @ @

worst-case unstable eigenvalue test

@

@

@

real-axis eigenvalue test

@

-

Re(s)

worst-case stable eigenvalue test Fig. 3.12. Possible real-µ tests in the s-plane

The Worst Case Eigenvalue Criterion is checked by shifting the imaginary axis into the left half plane until an uncertainty combination is found which places a closed loop pole on the axis. By sweeping s0 along a line of constant damping, such as ξ = 0.4, one may find the smallest perturbation which reduces damping below this level. Since km is typically discontinuous as s0 moves from the real axis to neighbouring complex points, it is also useful to check stability along the real axis. Another useful way to present the data is to compute km on a grid in the s-plane around a nominal closed-loop pole, and then make a contour map of km . This shows directly how the closed-loop poles migrate in the s-plane as a result of the uncertainty, i.e., it corresponds to a multi-parameter root locus - see [29] for more on this approach. Another possibility is to study the behaviour of some specific eigenvalues (e.g., those associated with the longitudinal and lateral modes of the aircraft) as the aircraft control law is subjected to increasing percentages of the worstcase uncertainty. In this approach, we first of all plot the eigenvalues of the nominal closed-loop system, (∆=0). We then use µ to calculate the worstcase ∆ matrix, and plot the associated closed-loop eigenvalues. The nature of the movement of the closed-loop eigenvalues from the nominal to the worst case can then be shown by plotting the eigenvalue positions for different per-

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D.G. Bates and I. Postlethwaite

centages of the worst case ∆, and ‘joining the dots’ to make a root-locus type plot. Plots of this type can provide much useful information about the relative movement of the different eigenvalues with respect to increasing uncertainty. In addition, for the case of systems which are de-stabilised by the worst-case δ, this method can help to understand which pole goes unstable first (an additional requirement for the Eigenvalue Criterion) and also at what level of uncertainty this occurs.

3.5 An Analysis Cycle Using µ-Tools We propose the following cycle for the application of µ-analysis methods to flight control law clearance problems: 1. Generate an LFT-based uncertainty description for the closed-loop aircraft model using one or more of the methods described in Part IV of this book. 2. Validate the LFT-based model against the original non-linear and linearised closed-loop aircraft models via frequency-domain analysis and time-domain simulations. 3. Check selected clearance criteria over the required set of flight conditions. For each criterion, the corresponding µ upper bound provides a guarantee that the criteria is satisfied, while the µ lower bound computes the worstcase combination of uncertain parameters. 4. Examine worst-case uncertainties for any failure cases using traditional methods and/or simulations.

3.6 Conclusions Compared to the standard approaches to the clearance of flight control laws currently used by the aerospace industry, the main advantages of µ-analysis techniques can be summarised as follows: • Clearance criteria can be checked, and worst cases found, for all possible combinations of the values of the uncertain aircraft parameters. This provides a stronger guarantee that a criterion is not violated than that provided by traditional gridding approaches, which generally only check that the criterion is not violated for all combinations of the extreme values of the uncertain parameters. • Whole portions of the flight envelope can be cleared by including Mach No. and altitude as uncertain parameters, thus removing the need to grid the flight envelope itself in the analysis. Clearance criteria can also be checked for continuous intervals of for example α and nz.

3 The Structured Singular Value and µ-Analysis

53

• Computation times for traditional gridding approaches increase exponentially with the number of uncertain parameters. Computation times for µ bounds are generally polynomial functions of the number of uncertainties, and thus for clearance problems involving large numbers of uncertain parameters µ-analysis tools can offer significant computational savings. • Since µ-analysis results are given as a function of frequency, they convey more information about the nature of the worst-case uncertainty and how it affects the aircraft dynamics than is generally provided by traditional methods. The main limitations of current µ-analysis techniques for flight control law clearance tasks are: • The generation of LFT-based uncertainty models which accurately capture the effect of uncertainty on strongly non-linear systems can be a difficult process. • In general, µ-analysis techniques are only well developed for uncertain multivariable linear systems. This fact limits their application at present to the linear, frequency domain clearance criteria defined in Chapter 2. Further developments of the basic theory are required in order to produce µ-tools which can be used reliably to address non-linear and/or time-domain clearance criteria. See, for example, [1, 2, 30] for an overview of recent developments in this area. • The computation of tight bounds on µ can sometimes present difficulties, especially for problems involving a large number of purely real uncertainties.

References 1. Ferreres, G., A Practical Approach to Robustness Analysis with Aeronautical Applications, Kluwer Academic, New York, 1999. 2. Doll, C., Ferreres, G., and Magni, J. F., “µ tools for flight control robustness assessment” Aerospace Science and Technology, No. 3, pp. 177-189, 1999. 3. Skogestad, S. and Postlethwaite, I., Multivariable Feedback Control, Wiley, 1996. 4. Doyle, J. C., “Analysis of feedback systems with structured uncertainty”, IEE Proceedings on Control Theory and Applications, Part D, 129(6), pp. 242-250, 1982. 5. Doyle, J. C. Lecture Notes on Advances in Multivariable Control, ONR/Honeywell Workshop, Minneapolis, 1984. 6. Balas, G. J., Doyle, J. C., Glover, K., Packard, A. and Smith, R., µ-Analysis and Synthesis Toolbox User’s Guide, The Mathworks, 1995. 7. Packard, A., and Doyle, J. C., “The complex structured singular value”, Automatica, 29(1), pp. 71-109, 1993. 8. Fan, M., Tits, A. and Doyle, J. C., “Robustness in the presence of mixed parametric uncertainty and unmodelled dynamics”,IEEE Transactions on Automatic Control, AC-36(1), pp. 25-38, 1991.

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9. Safonov, M. and Lee, P., “A multiplier method for computing real multivariable stability margins”, Proceedings of the IFAC World Congress, IFAC, Vol. 1, pp. 275-278, 1993. 10. Packard, A. and Doyle, J. C., “A power method for the structured singular value”, Proceedings of the American Control Conference, IEEE, Vol. 2, pp. 1213-1218, 1988. 11. Young, P. M., Newlin, M. P., and Doyle, J. C., “Computing bounds for the mixed µ problem”, International Journal of Robust and Nonlinear Control, Vol. 5, pp. 573-590, 1995. 12. Packard, A. and Pandey P., “Continuity properties of the real/complex structured singular value”, IEEE Transactions on Automatic Control, Vol. 38, No. 3, pp. 415-428, 1993. 13. Barmish, B. R., Khargonekar, P. P., and Shi, Z. “Robustness margin need not be a continuous function of the problem data”, Systems and Control Letters, Vol. 15, pp. 91-98, 1990. 14. Jones, R., “Structured singular value analysis for real parameter variations”, Proceedings of the AIAA Conference on Guidance, Navigation and Control , AIAA, Vol. 2, pp. 1424-1432, 1987. 15. Dailey, R. “A new algorithm for the real structured singular value”, Proceedings of the American Control Conference, IEEE, Vol. 3, pp. 3036-3040, 1990. 16. DeGaston, R. and Safanov, M., “Exact calculation of the multiloop stability margin”, IEEE Transactions on Automatic Control, AC-33(2), pp. 156-171, 1988. 17. Bates, D. G., Kureemun, R., Hayes, M. J. and Iordanov, P., “Computation and application of the real structured singular value”, Proceedings of the 14th Int. Conf. on Systems Engineering, Coventry, UK, Vol. 1, pp. 60-66, 2000. 18. Hayes, M. J., Bates, D. G. and Postlethwaite, I., “New tools for computing tight bounds on the real structured singular value”, AIAA Journal of Guidance, Control and Dynamics, 24(6), 2001. 19. Braatz, R. D. and Morari, M., “µ-sensitivities as an aid for robust identification”, Proceedings of the American Control Conference, IEEE, Vol. 1, pp. 231-236, 1991. 20. Branch, M. A. and Grace, A., MATLAB Optimization Toolbox User’s Guide, The MathWorks, 1996. 21. Coleman, T. F., and Y. Li. “A reflective newton method for minimising a quadratic function subject to bounds on some of the variables”, SIAM Journal on Optimisation, Vol. 6, pp. 1040-1058, 1996. 22. Bates, D. G., Kureemum, R., Hayes, M. J. and Postlethwaite, I., Clearance of the HIRM+ RIDE Flight Control Law: A µ-Analysis Approach, GARTEUR Technical-Publication TP-119-11, 2001. 23. G. Deodhare and V. V. Patel, 1998, “A ‘modern’ look at gain and phase margins: an H∞ / µ approach”, Proc. of the AIAA Conf. on GNC, Boston, USA, 1998. 24. Bates, D. G. and Postlethwaite, I., “Quantifying the robustness of uncertain feedback control systems using Nichols exclusion regions and the structured singular value”, Proc. of the UKACC International Conference on Control 2000, Cambridge, 2000.

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25. Kureemun, R., Bates, D. G. and Postlethwaite, I., “Quantifying the robustness of flight control systems using Nichols exclusion regions and the structured singular value”, IMechE Journal of Systems and Control Engineering, 215(16), 2001. 26. Aslin, P. P. and Glover, K., “µ-Analysis Technical Report”, Cambridge Control Ltd., 1990. 27. Mannchen, T., Petermann, C., Weinert, B. and Zobelein, T., “Flight control law clearance of the HIRM+ fighter aircraft model using µ-analysis”, GARTEUR Technical Publication, TP-119-12, 2001. 28. Safonov, M. G., Laub, A. J. and Hartman, G. L., “Feedback properties of multivariable systems: the role and use of the return difference matrix”, IEEE Trans. on AC, AC-26(1), 1981. 29. Dailey, R. and Gangsaas, D., “Worst case analysis of flight control systems using the structured singular value ”, AIAA Paper No. A89-49406, 1989. 30. Tierno, J. E., Murray, R. M., Doyle, J. C. and Gregory,I., “Numerically efficient robustness analysis of trajectory tracking for nonlinear systems”, AIAA Journal of Guidance, Control and Dynamics, 20(4), pp. 640-647, 1997.

4 The ν-Gap Metric and the Generalised Stability Margin John Steele and Glenn Vinnicombe University of Cambridge, Department of Engineering Trumpington Street, Cambridge CB2 1PZ,UK [email protected], [email protected]

Summary. In this chapter we introduce the ν-gap metric and a tool from the H∞ loop-shaping controller design method, namely the generalised stability margin. These tools have been designed to exploit the structure of the feedback interconnection, and capture the inherent robustness of feedback systems.

4.1 Introduction Our method is based on the ν-gap metric and ideas from the H∞ loop-shaping controller design method. The first fundamental object is the generalised stability margin ²P,C of a feedback system consisting of a system P with feedback controller C. ²P,C > 0 guarantees that the closed-loop system is stable, and higher values of ²P,C correspond to a greater degree of relative stability (and also some degree of performance). In the single-input singleoutput case ²P,C > 0.3, for example, ensures that the the Nichols diagram of P C avoids an elliptical region centred on −1 guaranteeing a phase margin of 35◦ and gain margin of 5.4dB. In the multi-loop case, closed loop stability is guaranteed in the face of simultaneous and independent gain/phase offsets at each input and output. The second fundamental object is the ν-gap distance δν (P, P∆ ) between a nominal plant P and a perturbed plant P∆ , which measures the importance of any difference between the open-loop systems from a closed-loop perspective. These tools will be used in an effort to exploit the structure of the feedback interconnection , and the inherent robustness of feedback systems, to reduce the complexity of the analysis.

4.2 The ²-Margin, a Generalised Stability Margin Consider the standard linear feedback configuration illustrated in Fig. 4.1 and referred to as [P, C]. The transfer function from the noise signals [ vy vu ]T to [ y u ]T is given by the 2 × 2 block transfer function matrix · ¸ £ ¤ P (I − CP )−1 −C I T [ vy ] → [ y ] = I vu u C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 57-75, 2002. Springer-Verlag Berlin Heidelberg 2002

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J. Steele and G. Vinnicombe

(i.e. Tvu →y = P (I − CP )−1 etc). We define a stability margin in terms of the H∞ norm of this transfer function matrix, the H∞ norm being defined as the worst case gain over frequency. 1 Definition 1 (Generalised stability margin ²P,C ). Given a closed-loop system [P, C], define °" # ° ° h i°−1   ° ° P −1 (I − CP ) if [P, C] is (internally) stable −C I ° ° ²P,C := ° I °  ∞  0 otherwise. (4.1) This measure of stability is large when then the norm of the closed-loop system transfer function T[ vy ]→[ y ] is small (implying small amplification vu

u

of noise). It was shown, in [1], that ²P,C is a number never less than zero or greater than one. In that reference, ² was also first referred to as a stability margin, since it equals the size of the smallest perturbation to the normalised coprime factors of P for which the perturbed closed-loop system is destabilised. The stability margin ²P,C is well-known in the context of H∞ loop-shaping controller design [2, 3]. In this design process, the frequency response of the open-loop plant is shaped using weighting transfer functions to reflect the desired loop shape (e.g. large low-frequency gain for disturbance rejection and sufficiently fast high-frequency roll-off for noise immunity). The loop-shaping weights implicitly define the desired crossover frequency of the system. These weights are usually selected to be diagonal since they then have the direct interpretation as frequency-wise scalings which reflect the relative importance of each signal to closed-loop behaviour. wy

wu u vu

-

-

P

C

y -

vy

Fig. 4.1. Standard feedback configuration for H∞ loop-shaping design

We can also define a frequency-wise version of the stability measure ²P,C . 1

That is, if T (s) is the transfer function of some stable system, then kT k∞ is defined as maxω σ(T (jω)), where σ(T (jω)) is the maximum singular value of T (jω), defined in turn as the square root of the maximum eigenvalue of the matrix T (jω)∗ T (jw) (and T (jw)∗ is the complex conjugate transpose of T (jω)).

4 The ν-Gap Metric and the Generalised Stability Margin

59

Definition 2 (Frequency-wise generalised stability margin ²P,C ). ¡ ¢ ρ P (jω), C(jω) :=

1 ¶ ¸ µ· £ ¤ P (jω) −1 −C(jω) I (I − C(jω)P (jω)) σ I

(4.2)

Comparing (4.1) and (4.2), it should be clear that, provided [P, C] is stable, ¡ ¢ (4.3) ²P,C = min ρ P (jω), C(jω) . ω

This frequency-wise computation of stability margin will be revisited in more detail in Section 4.4.3.

4.3 Using the ²-Margin for Analysis 4.3.1 The Effect of Weighting the ²-Margin In order to assess the ²-margin of a specified plant and controller, we will need to apply weights which scale the closed-loop disturbance signals in a similar manner to those used in H∞ loop-shaping design. Since the plant and controller are fixed, the loop shape must not be altered by our choice of weights. Consequently, we consider applying input and output weights to the plant along with the inverses of these weights to the controller as shown in Fig. 4.2. uˆ vˆu

-

Wi

Wi−1

u

P

C

y

Wo

Wo−1

yˆ -

vˆy

Fig. 4.2. Standard feedback configuration for H∞ loop-shaping analysis

It may be shown that ²Wo P Wi ,W −1 CWo−1 i

°· · −1 ¸ ¸°−1 ° Wo 0 Wo 0 ° ° v T =° y y ° 0 W −1 [ vu ]→[ u ] 0 Wi °∞ i

(4.4)

When faced with the problem of selecting weights for analysis, the user must consider the control system design specifications and capture the essential ideas with the analysis weights. If the controller was designed using H∞ loop-shaping, the same weights can be used for the analysis. When the given

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controller does not come with such weights, the analysis must begin with a selection of appropriate weights. If these weights have diagonal constant gains, then they can be interpreted as a change of engineering units to reflect the relative importance of each signal. However, we shall go further and use frequency-dependent diagonal weights to reflect frequency-wise the relative importance of each signal. 4.3.2 Selecting Weights for Gain/Phase Stability Analysis For a certain class of input and output disturbances perturbations, we shall see that robust stability is equivalent to considering the best case ²-margin over all diagonal frequency-dependent weights. The results in this section summarise those in [4]. That paper considers closed-loop robustness to simultaneous and independent multiplicative complex perturbations to each plant input and output as shown in Fig. 4.3. This concept of robustness is a generalisation of gain and phase margins used in single-loop analysis, and it will later be used to assess the stability margins of the HIRM+ model.

√1+δi12

1+δo1 √ 2

√1+δi22

1+δoi2 √ 2

1−β

P∆

1−β

1−β

1−β

P

.. .

.. .

√1+δiq 2

1+δop √ 2

1−β

1−β

.. .

C

.. .

Fig. 4.3. Input/output perturbations for multi-loop generalised gain/phase margins (|δ| < β)

Proposition 1 (Gain and phase offset robust stability [4]). Let ∆1 and ∆2 be complex diagonal matrices which perturb a nominal plant P to

4 The ν-Gap Metric and the Generalised Stability Margin

61

−1

P∆ = (I + ∆1 ) P (I − ∆2 ) . If ²Wo P Wi ,W −1 CWo−1 ≥ β for any diagonal ini put and output analysis weights, Wi and Wo , then [P∆ , C] is stable for any perturbations satisfying k∆2 k∞ < β and k∆1 k∞ < β. For a given β ∈ [0, 1), the perturbation structure P∆ −1 (I + ∆1 ) P (I − ∆2 ) for diagonal ∆1 and ∆2 can be rewritten as

=

p 1 −1 1 − β2 (I + ∆1 ) P (I − ∆2 ) P∆ = p 2 1−β ½ ¾ ½√ ¾ 1−β 2 Note that the sets √1+δ1 2 : |δ1 | < β and : |δ | < β are iden2 1−δ2 1−β

tical; therefore, the result in Proposition 1 is equivalent to saying that each input and output can be independently and simultaneously multiplica¾ ½ tively perturbed by a term δio ∈

√1+δ

1−β 2

: |δ| < β . This set of complex

gain/phase offsets describes a region of allowable input/output multiplicative perturbations δio satisfying: Ã

1

Re (δio ) − p 1 − β2

!2

¡ ¢2 + Im (δio ) <

β2 . (1 − β 2 )

On a Nichols plot, the resulting region of allowable perturbations is very closely approximated by the following ellipse which lies strictly within the region of allowable perturbations: 2 ¶2 µ ∠δio |δ | 20 log 10 io   q < 1. + 1+β arcsin β 20 log10 1−β 

(4.5)

This elliptical region for an ²-margin of β = 0.27 is illustrated by the solid line in Fig. 4.4. This result shows that when considering stability with respect to maximum allowable input/output perturbations (in terms of the regions described here), the choice of analysis weights reduces to a simple frequency-wise optimisation problem to find the best weighted ²-margin over all diagonal weights. At each frequency, the problem of finding the optimal weights is a convex optimisation which is readily solved using LMI (linear matrix inequality) techniques. We then solve this problem over a grid of frequencies to assess frequency-wise stability margin and consider the worst such optimised frequency-wise margin as a measure of the robustness of the closedloop system to input and output gain/phase offsets. As in the use of ²P,C and ²-margin, we will refer to this input/output stability margin as ²scaled (P, C) and ²scaled -margin. Formal definitions of this measure follow.

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Definition 3 (Frequency-wise optimally scaled input/output stability margin). A frequency-wise measure of closed-loop stability to input/output gain and phase offsets is given by ¢ ¡ ¢ ¡ max ρscaled P (jω), C(jω) := ρ Wo P Wi , Wi−1 CWo−1 (jω) Wi , Wo diagonal Definition 4 (Optimally scaled input/output stability margin). Similar to (4.3), we can define the input/output perturbation stability margin in terms of the frequency-wise optimally scaled input/output stability margin, provided [P, C] is stable, as: ¡ ¢ ²scaled (P, C) := min ρscaled P (jω), C(jω) . ω∈R

8

max. permissible single loop offsets for b(P,C)≥ 0.27 max. permissible multi−loop offsets for b(P,C)≥ 0.27

6

4

offset gain (dB)

2

0

−2

−4

−6

−8 −60

−40

−20

0 offset phase (degrees)

20

40

60

Fig. 4.4. A multi-loop feedback system achieving an ²scaled -margin of at least 0.27 will remain stable when independent complex multiplicative gains (referred to as gain/phase offsets) from within the solid ellipse are inserted onto each of the plant inputs and outputs. If only one input or output is perturbed, the gain/phase offset may be selected from within the dashed larger ellipse.

If we wish to consider the case of a single complex multiplicative perturbation on a single input or output, [4] also shows that the perturbation model in Proposition 1 can be selected as I + ∆1 = (1 + δ1 ) I  1 − δ2  1 + δ1  I − ∆2 =  ..  .

     1 + δ1

4 The ν-Gap Metric and the Generalised Stability Margin

so that

 1+δ1 P∆ = (I + ∆1 ) P (I − ∆2 )

−1

  =P 

1−δ2

63

 1

  . ..  .  1

A similar formulation is possible for consideration of a single output disturbance. With such a selection of perturbations, Proposition 1 allows any single 1+δ1 with |δi | < β. input or output to be perturbed by a term 1−δ 2

P∆

(1+δi1 )2 1−β 2

P

.. .

.. .

C

.. .

.. .

Fig. 4.5. Input/output perturbations for single-loop gain/phase margins

We can find equivalent expressions for the allowable single-loop perturbations for β < 1: p 1 − β2 1 + δ1 1 + δ1 =p 1 − δ2 1 − β 2 1 − δ2 ) ½ ¾ ( 2 1 + δ1 (1 + δ1 ) =⇒ : |δ1 | < β, |δ2 | < β = : |δ1 | < β 1 − δ2 1 − β2 since √1+δ1 2 and 1−β

√

1−β 2 1−δ2

define equivalent sets for |δi | < β. This result is

illustrated in Fig. 4.5. Notice that the allowable single-loop perturbations are the square of the allowable multi-loop perturbations. On a Nichols plot, the

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resulting ellipse which defines allowable perturbations is therefore twice the size of that for the multi-loop case. Such an ellipse is indicated with dashed lines in Fig. 4.4.

4.4 The ν-Gap Metric and H

ÿ Loop-Shaping

The ν-gap metric measures the distance between systems in terms of how their differences can effect closed-loop behaviour. Such a measure of closedloop distance is important since two (stable) plants which may differ widely in the H∞ norm (a measure of open-loop distance) may achieve similar levels of closed-loop performance with the same controller. For example, consider s+2 10 with the lead compensator C = −2 s+10 . The the nominal system P0 = s+0.1 ²-margin of this closed-loop system is quite good since ²P0 ,C = 0.381. If we consider the performance of this compensator with any system of the form 10 with α ∈ [−0.1, 0.1], we find that the ²-margin is always greater Pα := s+α than 0.362. However, kP0 − Pα k∞ becomes unbounded as α approaches 0. When α is negative, Pα is unstable and such an open-loop comparison is impossible using the H∞ norm. However, the ν-gap distance between P0 and Pα is less than 0.02 (which is a small ν-gap distance) for all α ∈ [−0.1, 0.1]. This small ν-gap distance mathematically captures the intuition that any plant Pα should be easily stabilised by the provided lead controller. In general, if the ν-gap distance between two plants is small then any controller which performs well with one plant (in terms of ²-margin) will also perform well with the other. As illustrated by the example above, the ν-gap metric also allows us to consider the distance between plants with different numbers of right half plane poles. 4.4.1 Definitions The ν-gap metric was introduced in [5]. Definition 5 (ν-gap metric). Given P0 , P1 ° −1/2 ° ∗ −1/2 ° ,  (P1 − P0 ) (I + P0∗ P0 ) °(I + P1 P1 ) ∞ δν (P0 , P1 ) := if η[P1 , −P0∗ ] = η[P0 , −P0∗ ]   1, otherwise where η[P, C] denotes the number of open RHP poles of the (positive) feedback system comprising the plant P and controller C. It can be shown that η[P0 , −P0∗ ] equals the degree of P0 , since the closed-loop poles of this (fictitious) feedback system are symmetric about the imaginary axis. We shall refer to the condition η[P1 , −P0∗ ] = η[P0 , −P0∗ ] as the winding number condition. It will be satisfied, for example, if P0 = 1, P1 = (s + 1)/(s + 1.001) but fail if

4 The ν-Gap Metric and the Generalised Stability Margin

65

P0 = 1, P1 = (s − 1)/(s − 1.001). For this second example, the winding number condition captures the fact that, although the frequency responses of P0 and P1 are close, almost any stabilising controller for P0 will fail to stabilise the unstable pole in P1 . As for ²P,C , δν (P0 , P1 ) is always in the range [0, 1]. Efficient state space methods for computing δν (P0 , P1 ) are provided in [6, 7, 8]. The ν-gap metric is a metric on the set of linear time-invariant systems. Importantly, it satisfies the triangle inequality so that if P2 is the result of perturbing P0 to P1 and then P1 to P2 , then δν (P0 , P2 ) ≤ δν (P0 , P1 ) + δν (P1 , P2 ) .

(4.6)

That is, the distance between P0 and P2 is no greater than the sum of the effects of the individual perturbations. 4.4.2 ν-Gap Robustness Results The ν-gap metric induces a topology on the set of systems in which closedloop stability, as measured by ²-margin is a robust property. That is if ²P0 ,C is large and δν (P0 , P1 ) is small, then ²P1 ,C will also be large. This result is formalised in the following proposition. Proposition 2 (Robustness in the ν-gap[7, from Theorem 3.10]). For P0 , P1 ∈ Rp×q and C ∈ Rq×p , then |arcsin ²P0 ,C − arcsin ²P1 ,C | ≤ arcsin δν (P0 , P1 ) Taking just one side of this bound, and ignoring the tighter result provided by the use of the arcsin functions, we can see how the ν-gap metric guarantees robust stability: (4.7) ²P1 ,C ≥ ²P0 ,C − δν (P0 , P1 ) . Moreover, if δν (P0 , P1 ) < max{maxC ²P0 ,C , maxC ²P1 ,C }, then [7, Section 3.1.2] shows that sin |arcsin ²P0 ,C − arcsin ²P1 ,C | . δν (P0 , P1 ) = max q×p C∈R

So δν (P0 , P1 ) can be viewed as a measure of the worst-case difference of two feedback systems over all simultaneously stabilising controllers. 4.4.3 Frequency-Wise Computation of the ν-Gap Similarly to ²P,C , the ν-gap metric can be computed frequency-wise if the ν-gap winding number condition is satisfied.

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Definition 6 (Frequency-wise ν-gap metric [7, pages 122–123]). Given P0 (jω), P1 (jω) ∈ Cp×q , define ´ ³ ¡ ¢ −1/2 −1/2 (jω) κ P0 (jω), P1 (jω) := σ (I + P1 P1∗ ) (P1 − P0 ) (I + P0∗ P0 ) where we use the shorthand σ(G) (jω) to denote σ(G(jω)).

(4.8)

Combined with the frequency-wise version of ²P,C from Definition 2, robustness results similar to those in Proposition 2 and (4.7) hold frequencywise. For example, ¡ ¢ ¡ ¢ ¡ ¢ ρ P1 (jω), C(jω) ≥ ρ P0 (jω), C(jω) −κ P0 (jω), P1 (jω) ∀ω. 4.4.4 Visualising the ν-Gap Metric For single-input, single-output systems the frequency-wise ν-gap metric can be illustrated using ideas first proposed in [9] and extended in [6, 7]. First, realize that for a single-input, single-output system, the frequency-wise ν-gap (4.8) can be written as ¢ ¡ κ P0 (jω), P1 (jω) = p

|P1 (jω) − P0 (jω)| p . 1 + |P0 (jω)|2 1 + |P1 (jω)|2

This expression gives the chordal distance between the stereographic projections of P0 (jω) and P1 (jω) onto the Riemann sphere (a sphere of unit diameter sitting on the origin of the Nyquist plane). This interpretation is illustrated in Fig. 4.6. Notice in Fig. 4.6 that at frequencies where the gain is large (À 1), the frequency responses of P0 and P1 are projected near the north pole of the Riemann sphere. This means that even frequency responses with large gain which are vastly different on the Nyquist plane cluster together on the sphere so that these differences are small in the chordal distance metric. This feature of the ν-gap metric captures the concept that any well-designed controller will still perform well if the plant is altered at frequencies where the gain is large. Similarly, any well-designed closed-loop system is insensitive to changes where both the nominal and perturbed plant gain is small (¿ 1). In such cases the projected frequency responses are then clustered near the south pole of the Riemann sphere. The chordal distance metric is most sensitive to changes in the plant frequency response near crossover (which is the most critical frequency band for determining closed-loop behaviour). The close nature between the ν-gap and ²P,C can be illustrated for singleinput, single-output systems by realizing that in this case ¢ ¡ ¢ ¡ ρ P (jω), C(jω) = κ P (jω), C −1 (jω) . If P1 (jω) = C −1 (jω) for some ω, then 1 − P1 (jω)C(jω) = 0 implying that the closed-loop system has a pole at s = jω and is consequently unstable.

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Fig. 4.6. For single-input, single-output systems, the frequency-wise ν-gap metric measures the chordal distance between the stereographic projections of frequency responses onto the Riemann sphere.

So¡ applying the same ¢ Riemann sphere interpretation used above, we see that κ P (jω), C −1 (jω) is a measure of how much the plant P0 can be perturbed ¡ ¢ in the chordal metric, κ P0 (jω), P1 (jω) , before P1 = C −1 and the loop [P1 , C] is destabilised. 4.4.5 The Weighted ν-Gap Metric All of the preceding results on the ν-gap metric are only useful when the controller achieves a reasonable ²-margin with at least the nominal plant. As with using ²P,C for analysis in Section 4.3, input and output weights can be introduced to implicitly define critical aspects of the closed-loop design. The ν-gap metric can then be used to measure the distance between plants with respect to these weights. Importantly, the ν-gap robustness results still apply with these weighted plants and controllers. For example, we have ¡ ¢ ρ Wo P1 Wi , Wi−1 CWo−1 (jω) ≥ ¡ ¢ ¡ ¢ ρ Wo P0 Wi , Wi−1 CWo−1 (jω) − κ Wo P0 Wi , Wo P1 Wi (jω)∀ω. In order to maintain notational simplicity in the remainder of this work (unless otherwise stated), we will assume that plant and controller models have been already been properly weighted for the analysis.

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4.5 An Approximation of the ν-Gap In this section we are primarily interested in finding a linearised approximation of a (possibly non-linear) model with parametric uncertainty. This linearised uncertainty model should closely approximate the ν-gap distance between the nominal plant and any plant with perturbed uncertainty parameters. We will consider an approximation of the ν-gap metric which is good for small perturbations and which will allow us to obtain a linear fractional approximation of the uncertain plant model. We begin with a mapping proposed in [10] which is defined relative to a nominal model P0 and exhibits many useful properties related to the ν-gap between perturbed systems. Given a nominal plant P0 , we define P∆ 7→ XP∆ := (I + P0 P0∗ )

−1/2

(P0 − P∆ ) (I + P0∗ P∆ )

−1

(I + P0∗ P0 )

1/2

(4.9) Note that the nominal plant P0 maps to XP0 = 0. ¡ ¢ XP∆ can be used to compute the frequency-wise ν-gap κ P0 (jω), P∆ (jω) as shown in [7, Equation 3.11]: ¡ ¢ κ P0 (jω), P∆ (jω) (4.10) σ(XP∆ (jω)) = q ¡ ¢. 1 − κ2 P0 (jω), P∆ (jω) More importantly, use of XP∆ can also provide a good approximation of the ν-gap distance between two plants, P1 and P2 , as formalised in the following result. Proposition 3 ([10, Theorem 3.3]). Given P0 , P1 , and P2 satisfying δν2 (P0 , P1 ) + δν2 (P0 , P2 ) < 1 and the mapping P1 7→ XP1 defined in (4.9), then δν (P1 , P2 ) p δν (P1 , P2 ) ≤ kXP1 − XP2 k∞ ≤ p 2 1 − δν (P0 , P1 ) 1 − δν2 (P0 , P2 ) Remark 1. A similar result holds frequency-wise: ¢ ¡ κ P1 (jω), P2 (jω) ≤ σ(XP1 (jω) − XP2 (jω)) ¡ ¢ κ P1 (jω), P2 (jω) ≤q ¡ ¡ ¢q ¢. 1 − κ2 P0 (jω), P1 (jω) 1 − κ2 P0 (jω), P2 (jω) Proposition 3 suggests that the transformed systems XPi combine to approximate the ν-gap distance between systems as illustrated in Fig. 4.7. In fact, for plants sufficiently close to the nominal model (e.g. δν (P0 , Pi ) . 0.3), the upper bound on the approximation is fairly tight: kXP1 − XP2 k∞ ≤ 1.10δν (P1 , P2 ). Because of this ability to approximate the distance between

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two differently perturbed plants by linearly combining their associated XPi systems, in the next section we will consider approximating the result of simultaneous perturbation of multiple parameters in terms of the parameters’ individual effect upon XP∆ .

XP1 ) , P1

P0

P kX

k∞

≈

( δν

kXP1 − XP2 k∞ ≈ δν (P1 , P2 )

1

XP2

XP0 kXP2 k∞ ≈ δν (P0 , P2 )

Fig. 4.7. Vectors XP1 , XP2 ∈ L∞ combine to permit approximation of δν (P1 , P2 ).

For a single-input, single-output system we can illustrate the result of this frequency-wise mapping XP∆ (jω) relative to the single-input, single-output ν-gap metric analogy to chordal distance on the Riemann sphere discussed in Section 4.4.4. Figure 4.8 shows a cross section of the Riemann sphere taken through a meridian. The values of P0 (jω) and P1 (jω) at a particular frequency are assumed to lie in the cross section of the Nyquist plane for this illustration. The stereographic projections of P0 (jω) and ¡P1 (jω) onto the¢ Riemann sphere along with the chordal distance which is κ P0 (jω), P1 (jω) are also indicated on the figure (the latter is labelled ν-gap). The mapping P1 (jω) 7→ XP1 (jω) is equivalent to making another stereographic projection of the projection of P1 (jω) onto a plane tangent to the Riemann sphere at the projection of P0 (jω).

4.6 Linearising the Approximation of the ν-Gap for Parametric Uncertainty Since the mapping to XP∆ has a linear fractional dependence on the perturbed plant, given any approximation of XP∆ , we can reverse the linear fractional mapping to obtain an approximation of P∆ . In this section we consider such a linearisation of XP∆ for systems with parametric uncertainty. Given a model with uncertainty parameters numbered 1, 2, . . . , n, we will denote a specific set of parameter values as ∆ = {δ1 , δ2 , . . . , δn }. As in the

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proj. of P1 (jω) Tangent Plane XP1 (jω)

δν XP0 (jω) = proj. of P0 (jω) P0 (jω)

P1 (jω)

Nyquist plane

0

Fig. 4.8. For single-input, single-output systems, we can visualise the mapping P1 (jω) 7→ X1 (jω) which approximates the frequency-wise ν-gap at ω between the nominal plant P0 and a perturbed plant P1 . X1 (jω) is a double stereographic projection of P1 (jω) onto a plane tangent to the Riemann sphere at the projection of nominal value P0 (jω). This projection is illustrated in a cross section of the Riemann sphere and the Nyquist plane.

previous section P0 will denote the nominal plant. For ease of notation we consider using normalised parametric uncertainty where the value of each parameter is in the range [−1, 1] with a nominal value of 0. For models with sufficiently smooth parameter dependence we can consider obtaining an approximation of the mapping P∆ 7→ XP∆ via a first-order Taylor expansion of that mapping: XP∆ ≈ Xapp :=

n X

δi Xi

(4.11)

i=1

where the terms in the summation are Xi :=

∂XP∆ ¯¯ . ∂δi ∆=0

(4.12)

We can consider each partial derivative term, Xi , as an approximation representing how the application of the ith parameter at a value of δi = 1 will effect XP∆ . Due to the linearity of this approximation, δi Xi is an approximation of the effect of the application of the ith parameter at a value of δi . This approximation does not require that the parameters enter the plant model linearly—only that the dependence upon parameters which are important to closed-loop behaviour is well-approximated by first-order effects. Once we have this representation of Xapp which is linear in the uncertainty parameters, we can easily invert the mapping from (4.9) to obtain the plant

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model Papp which is associated with Xapp . If the linear approximation of Xapp were accurate, the resulting approximate plant would depend linear fractionally on the uncertainty parameters and well-approximate the ν-gap distance between the nominal plant and the true perturbed plant. The actual plant model, P∆ , and the approximation, Papp , may differ widely in terms of open-loop frequency response or time-domain response, but if the uncertainty parameters had little effect on the closed-loop system, then kXapp k∞ would be small and the approximate plant would be close to the nominal plant in the ν-gap metric. This concept is the basic justification of this method of approximation. Frequency-wise methods of computing these approximations along with graphical interpretations of these methods will be provided in the following two sections.

4.7 Illustration of the Parametric Uncertainty Approximation An example of this ν-gap approximation method is illustrated in Figs. 4.9 to 4.11. This example considers a particular frequency near crossover for the single-input, single-output differential tailplane input disturbance response of the HIRM+ model. For this example the model contains three uncertain parameters whose effect is approximated using the results presented in the previous section. Figure 4.9 shows the Nyquist plane and tangent plane projections as described in Fig. 4.8. Details of the tangent and Nyquist planes are show in Figs. 4.10 and 4.11. In each figure the frequency responses of actual plants taken over a grid of the possible parameter values are shown. On the tangent plane in Fig. 4.10, we show the vectors representing the approximation of the effects of the full-scale application of each individual parameter on XP∆ as determined by the first-order terms in (4.11). The hull of the Minkowski (vector) sum of these approximation vectors is also plotted since it delineates the region of all possible values of Xapp . Figures 4.9 and 4.11 show this region of approximate values of XP∆ projected onto the Nyquist plane for comparison with the system frequency response. Notice that a polytope on the tangent plane can take on a non-convex shape in the Nyquist plane. The overriding concept of this approximation method is that all the systems in these ν-gap-approximation regions are close in a ν-gap sense. The results presented here are typical. Of course, the ν-gap approximation will not necessarily be accurate in all circumstances, but it worked well for the multi-input, multi-output aircraft HIRM+ model.

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Fig. 4.9. For single-input, single-output systems with real parametric uncertainty, we can consider visualising the ν-gap approximation method. The region delineated with a solid line is generated from a linear fractional approximation of the effects of three uncertainty parameters on a single-input, single-output system. Detailed figures of the tangent and Nyquist planes are provided in Figs. 4.10 and 4.11, respectively.

4.8 Estimating Worst-Case Parameter Combinations Details of the application of these tools to clearance problem are presented in a later chapter. However, it is appropriate to present some observations at this point. 4.8.1 Approximate Lower Bound on the Worst-Case Stability Margin

If we were to use Xapp of (4.11), and generate a Papp by inverting (4.9), as an approximation of the combined effects of a number of parameters, the relations between Xapp and ν-gap and the standard ν-gap robustness results from (4.7) would ensure that: ¡ ¢ ¡ ¢ σ(Xapp (jω)) ρ Papp (jω), C(jω) ≥ ρ P0 (jω), C(jω) − q 1 − σ 2 (Xapp (jω)) ¡ ¢ ≈ ρ P0 (jω), C(jω) −σ(Xapp (jω)) .

(4.13)

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nominal plant projected frequency response perturbed plant projected frequency responses ν−gap approximation basis vectors and hull

0.15

0.05

0

−0.05

−0.1

−0.15 −0.2

−0.15

−0.1

−0.05 0 0.05 tangent plane imaginary axis

0.1

0.15

0.2

Fig. 4.10. Detail of the plane tangent to the Riemann sphere from Fig. 4.9 3.2 3 2.8 2.6 Nyquist plane real axis

tangent plane real axis

0.1

2.4 2.2 2 1.8 1.6 nominal plant frequency response perturbed plant frequency responses ν−gap approximation basis vectors and hull

1.4 1.2 −1

−0.5

0 Nyquist plane imaginary axis

0.5

1

Fig. 4.11. Detail of the Nyquist plane from Fig. 4.9

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An upper bound on σ(Xapp (jω)) can be obtained from (4.11) and singular value inequalities which is similar to the ν-gap metric inequality (4.6): σ(Xapp (jω)) ≤

n X

σ(Xi (jω)).

(4.14)

i=1

Using this bound with (4.13) gives the following lower bound on approximate plant stability margin: n ¢ ¡ ¢ X ¡ σ(Xi (jω)). ρ Papp (jω), C(jω) & ρ P0 (jω), C(jω) −

(4.15)

i=1

This result provides a lower bound on worst-case ²-margin in terms of a number of different perturbations. 4.8.2 Neglecting Parameters which Have No Effect We can also use the Xi individual parameter ν-gap effect approximations from (4.12) to determine which parameters have negligible impact on closedloop behaviour. That is, parameters which contribute a negligible amount (e.g. < 0.001) to the worst-case value of σ(Xapp (jω)) in (4.14) are not significant to closed-loop behaviour at that frequency as shown in (4.15). To see this ¡ effect visually¢ for each parameter, δi , we can plot the approximation of κ P0 (jω), Pδi (jω) for the full-scale application of δi = ±1 as approximated by σ(Xi (jω)). This type of analysis can assist in reducing the complexity of the analysis process by identifying parameters which can be excluded from consideration. 4.8.3 A Polynomial-Time Approach Using ν-Gap Information Having generated LFTs, µ-analysis could be used to find the worst case ²-margin. However, given that the lower bound algorithms for computing µ are computationally intensive and not globally optimal, we propose using additional information from the ν-gap approximations to generate a heuristic method for identifying parameters which give poor performance. Similar to the procedure in Section ¢ 4.8.2, we can use the frequency-wise ν-gap val¡ ues of κ P0 (jω), Pδi (jω) to determine which parameters have the greatest potential to affect closed-loop performance. Pn Since the maximisation of σ(Xapp (jω)) = σ( i=1 δi Xi (jω)) over parameter values δi involves a convex optimisation, the parameter values which give the largest frequency-wise ν-gap distance between Papp and P0 occur at vertices of the parameter space. Therefore, we will consider searching for worst-case closed-loop performance over all the vertices of the parameter space (i.e. all combinations of extreme values of the parameters).

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References 1. D. McFarlane and K. Glover. Robust stabilization of normalized coprime factor plant descriptions with H∞ -bounded uncertainty. IEEE Transactions on Automatic Control, 34:821–830, 1989. 2. D. C. McFarlane and K. Glover. Robust Controller Design Using Normalised Coprime Factorisation Plant Descriptions. Lecture Notes in Control and Information Sciences. Springer-Verlag, Berlin, 1990. 3. D. McFarlane and K. Glover. A loop shaping design procedure using H∞ synthesis. IEEE Transactions on Automatic Control, 37(6):759–769, June 1992. 4. K. Glover, G. Vinnicombe, and G. Papageorgiou. Guaranteed multi-loop stability margins and the gap metric. In 38th IEEE Conference on Decision and Control, pages 4084–4085, 2000. Paper number 9007. 5. G. Vinnicombe. Frequency domain uncertainty and the graph topology. 38(9):1371–1383, Sept. 1993. 6. G. Vinnicombe. Measuring the Robustness of Feedback Systems. PhD thesis, University of Cambridge, 1993. 7. G. Vinnicombe. Uncertainty and Feedback: H∞ loop-shaping and the ν-gap metric. Imperial College Press, London, 2000. 8. G. J. Balas, J.C. Doyle, K. Glover, A. Packard, and R. Smith. µ-Analysis and Synthesis Toolbox for use with MATLAB. The MathWorks, Inc., 1998. Version 3. 9. A. K. El-Sakkary. Estimating robustness on the Riemann sphere. International J. Control, 49(2):561–567, 1989. 10. G. Vinnicombe. Approximating uncertainty representations using the ν-gap metric. In Proceedings of the 1999 European Control Conference, Karlsruhe, Germany, 1999.

5 A Polynomial-Based Clearance Method Leopoldo Verde and Federico Corraro Centro Italiano Ricerche Aerospaziali Flight System Department Via Maiorise, 81043, Capua (CE) Italy [email protected],[email protected]

Summary. In this chapter we present an overview on polynomial-based analysis methods for application to robust stability of linear systems subject to uncertain parameters. These methods basically check the robust stability property of a dynamic system by looking at the characteristic polynomial coefficients. A comparison of the most important Kharitonov type approaches proposed in literature and their applicability to the flight control law clearance problem of highly augmented aircraft is also discussed. Finally, a novel algorithm is proposed, which can deal with high-order uncertain dynamic aircraft models within reasonable computation time by introducing some degree of approximation in determining the clearance region’s shape.

5.1 An Overview of Polynomial-Based Methods The proposed clearance analysis technique is mainly based on some theoretical results that allow verification of whether the eigenvalues of an uncertain (linear) dynamic system belong to a pre-determined region D of the complex plane (Robust D-stability problem). This allows direct application of the proposed method for clearance of the unstable eigenvalue criterion described in Chapter 10. The proposed method might also be used for any linear clearance criterion that can, in some way, be mapped to a test on system eigenvalue locations in the complex plane. In the past, a large effort has been expended to address the robust stability problem of linear systems subject to uncertain parameters. A strong impetus to the research has been given by the paper of Kharitonov [1], where a necessary and sufficient condition for robust stability of a family of polynomials with uncertain coefficients has been provided. Although an elegant mathematical result, Kharitonov’s theorem is not suited to engineering applications since it assumes uncorrelated polynomial coefficients. Indeed, in practical situations, the coefficients of the characteristic polynomial of a given system depend on the same physical parameters (e.g. Angle-of-Attack, Mach number) which implies that the coefficients themselves are related to each other. Kharitonov’s result has been introduced in the western literature by Barmish [2]. Since then, many papers have been published on this topic, C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 77-88, 2002. Springer-Verlag Berlin Heidelberg 2002

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trying to extend the original result to cope with more general parameter dependencies and/or to take into account performance as well as stability. We recall the work by Petersen [3], which extends Kharitonov’s theorem to deal with the so-called robust D−stability problem (see definition 1), with D being a given domain in the complex plane (see for example Figure 5.1); the fundamental result by Bartlett et al [4], which states that to check stability of an uncertain polynomial with coefficients ranging into a given polytope, it is necessary and sufficient to check the edges of the polytope; the works by Sariderely and Kern [5], Tesi and Vicino [6], Cavallo et al [7], all dealing with robust stability analysis of uncertain polynomials with coefficients depending affinely on parameters ranging in a given box. These results, together with further insights on the topic can be found in [8]. Unfortunately these results are not useful when: a) the characteristic polynomial depends on parameters in a nonlinear way (this is the case of many flight control applications as shown in [9]), and/or b)we are interested in the more general problem of determining the region shape in the parameter space where the system is robustly D−Stable.

Imaginary Axis

D Real Axis

Fig. 5.1. Typical D-Stability domain in the complex plane

In this context an algorithm will be described which addresses the two issues mentioned above. This algorithm is based on the results provided in [7] and on a method for adaptive grid generation. More precisely, in [7] a necessary and sufficient condition for the D−stability of an uncertain polynomial depending affinely on parameters is given, while here a procedure to approximate a nonlinear vector function with a minimal set of affine ones is proposed. Roughly speaking, this algorithm uses these results to:

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1) determine a set of boxes whose union includes the initial uncertain parameter region and such that, in each box, the uncertain polynomial coefficients can be considered to be affinely dependent on parameters, 2) compute the actual D-stability region in the parameter space by applying the algorithm proposed in [10] to each box (as defined in the above step). In this way the D-stability region is approximated to the desired resolution by the union of the final resulting boxes which satisfy the condition given in [7]. The key point to guarantee that the stability region found via this way converges to the true stability region, is that the errors due to the use of a set of affine functions instead of the actual nonlinear vector function (which gives the characteristic polynomial coefficients of the uncertain system) can be neglected, if the boxes are sufficiently small. This is always true when the mapping of the parameter space into the polynomial coefficient space is continuous.

5.2 D-Stability Problem Statement Let us consider an uncertain linear system described by the differential equations: x(t) ˙ = A(π)x(t) n

k

(5.1)

n×n

, π → A (π), is a continuous matrixwhere x (t) ∈ R and A : R → R valued function of the parameter vector π, Rn is the model state space (of dimension n) and Rk is the uncertain parameter space (of dimension k). In this context, we need the following definition. Definition 1 (Robust D-Stability) Given the compact set Γ ⊂ Rk (i.e. a region in the parameter space) and the open domain D in the complex plane, system (5.1) is said to be robustly D−stable in Γ if λi (A (π)) ∈ D, i = 1, .., n, for all π ∈ Γ , where λi (A)denotes the i-th eigenvalue of the matrix A. In other words, system (5.1) is defined to be robustly D-stable if its poles all belong to a given region D of the complex plane (see Figure 5.1 for an example), for each point π in the uncertain parameter region Γ . Note that, when D coincides with the left half of the complex plane, we simply talk about robust stability. Now let us refer to the system described by eq.(5.1) and let a (.) : Rk → n R , π → a (π), the vector-valued function containing the coefficients of the characteristic polynomial of the matrix A (π). T We denote by L : Rn → P n , a = (a1 , ..., an ) → p (s, a), where p(s, a) = sn + a1 sn−1 + · · · + an ,

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the linear operator mapping a vector of Rn into P n , the set of monic polynomials of degree n. Finally, define the compound operator La := L ◦ a. From a robust D−stability point of view, the complete behaviour of system (5.1) is described by the following family of monic polynomials: La (Γ ) = {p(·, a(π)) | π ∈ Γ }

.

Indeed it is clear that the system described by eq. (5.1) is robustly D−stable within the given set Γ if and only if the roots of all polynomials belonging to the family La (Γ ) are in the domain D. Now consider the following problem. Problem 1 (Determination of the robust D-stability region in the ∗ parameter space Rk ) Determine the region ΠD ⊂ Rk such that system ∗ (5.1) is robustly D−stable in ΠD . As we shall see, the idea behind the polynomial coefficient based approach ∗ proposed here is that of approximating the D-Stability Region ΠD by the union of boxes in the space Rk . To check robustness in the given box, it is necessary to have a procedure to solve the following basic problem. Problem 2 (basic problem) Given a box V ⊂ Rk , determine if system (5.1) is robustly D−stable in V . With the methods currently available in the literature, the above stated Basic Problem can be solved without conservatism when the dependence of the characteristic polynomial on parameters is affine (see [5], [4], [6], [7]). The nonlinear dependence has been considered in [11] and [12] (multiaffine dependence), while in [13] a multivariate dependence has been assumed. In these previous papers, the stability analysis is performed by introducing fictitious parameters which allow the multivariate dependence to be transformed into a multiaffine one. Then the test is performed (at the price of some conservatism) on the fictitious multiaffine characteristic polynomial, by using one of the approaches proposed in the literature. Another algorithm dealing with nonlinear dependency on parameters, implements the idea proposed in [14]. In this approach, known as the polytopic covering approach, the image of the given nonlinear function is “immersed” into that of an affine function. In [15] it is shown that the polytopic covering approach leads to less conservative results than those obtainable with other methods. The main drawback of these ”polytopic set covering” based methods is that the dimension of the parameter space in which the D-stability analysis algorithm needs to be applied, can dramatically increase. In [10] it has been shown that good results can be achieved when the augmented parameter space dimension is at least the same as the polynomial order. For the aeronautical application under investigation and, specifically, in flight control law clearance problems, the order of the closed-loop polynomial

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is too high to allow these methods to work well and to obtain results in a reasonable time.

5.3 The Proposed Algorithm Let us come back to the solution of Problem 1; here we consider a slight variation of the problem, taking into account the fact that, in flight control problems, the range of parameter variations or parameter uncertainties can be estimated. Thus, let us consider that: π1 ≤ π1 ≤ π 1 π2 ≤ π2 ≤ π2 .. .

πk ≤ πk ≤ πk Where the underline is used to indicate the minimum value of a parameter while the bar above stands for the maximum value. Hence we have that π ∈ Π (i.e. a box in the parameter space), where ¤ £ ¤ £ ¤ £ Π = π1 , π1 × π2 , π2 × · · · × πk , πk

.

∗ ∗ Therefore, our goal is to determine the set ΠD := ΠD ∩ Π, where ΠD is the robust D−stability region defined in Problem 1. The nonlinear mapping a (Π) can be approximated by a set of affine mappings, each of them suitably defined on a partition of Π. In other words, let us consider instead of a (Π), the following mapping: N [ i=1

a∗i (Πi ) ,

N S i=1

Πi = Π ∧ Πi ∩ Πj = ∅, ∀i 6= j ∈ 1..N

(5.2)

Here a∗i (Πi ) is an affine approximation of a (Πi ) calculated by linear regression methods and N is the number of subsets into which the initial box, Π, has been divided. ˆ D corresponding to the polyIt is expected that the D-stability region Π nomial coefficient mapping defined in equation 5.2, will approach the true stability region ΠD , provided that the linear regression approximation error tends to zero as the volumes of the boxes tend to zero. In this respect, the following procedure gives an approximate solution to Problem 1. It computes the boundary of the stability region ∂ΠD up to a ˆ D will be evaluated instead of desired resolution (actually an estimation ∂ Π ∂ΠD ). Any dependence of the system matrix on uncertain parameters can be covered. The procedure is made up of two main steps: 1. Compute an optimal partition {Πi } of Π (trying to minimise N , the number of subsets Πi ) where the generic nonlinear map a(.) can be

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approximated by an affine map a ∗i (.) in each Πi , with a maximum estimated error of deps . The algorithm also stops when subsets Πi become smaller than a pre-defined grid size eps 1 . 2. Compute (up to a desired resolution eps 2 ) the D-stable region in the uncertain parameter space for each partition Πi by using the approximated affine vector function a ∗i (.). Specifically, we can schematically describe the first procedure as follows: Procedure 1 (Adaptive Grid Generation) Put the box Π in the List For each box of the List Evaluate coefficients in the box vertices and in the centre Compute an affine function approximation in the box (linear regression fitting) Compute error derr (from linear regression algorithm) If derr < deps or kboxk < eps1 then add box to the final list Else divide box in sub-boxes and update List End End End of Procedure In this procedure and in the second one, given a generic box V,the operation kV k is defined as follows: kV k =

max

i=1,...,2k

li

where li is the i-th side of the box V. In other words the size of the box is given by the length of the longest side of the box. A more sophisticated algorithm for adaptive grid generation (i.e. a grid where the number of partitions is not a priori defined) could be investigated and implemented to increase the reliability of the error fitting, but this work is beyond the scope of this chapter. The above procedure can treat points where the nonlinear vector function a(.) is not defined during the Evaluate statement. It should also be noted that, by using mathematical manipulations, only dot products between matrices and vectors (no matrix pseudo-inversion) are required, thus leading to a very fast algorithm. The only time-consuming task is actually the evaluation of the nonlinear vector function at 2 k +1 points for each examined box. More precisely, because 2k smaller boxes are generated each time we divide a box and the algorithm used for the Evaluate statement does not allow multiple evaluations of the same point in the uncertainty space, the maximum total

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number of polynomial coefficient evaluations after j steps (i.e. the number of evaluations required in the case that no boxes are below the maximum error ¡ ¢k deps ) is: 2(j−1) + 1 + 2(j−1)k . In other words, the maximum number of trimming and linearisations is equivalent to the number of evaluations performed with a grid of 2(j−1) + 1 points for each uncertainty, plus all centre points of boxes generated at step j. Thus, the effectiveness of the proposed technique can be also assessed by comparing it with a grid of the same size. Furthermore, by putting j=1 in the above relation, the minimum number of characteristic polynomial coefficient evaluations is obtained, which is actually equivalent to only evaluating the polynomial coefficients at the vertices of Π (i.e. min/max combinations of the uncertainties), plus its centre point. The output of this first procedure is a list of boxes {Πi } where the initial nonlinear vector function can be considered affinely dependent on the uncertain parameters. The main steps of the second procedure are schematically listed below (see [16] and [10] for details). Procedure 2 (Computation of D-Stable Region) Put the box set {Π} in the List For each box of the List If cond(box ) then Compute eigenvalues of the system in the centre point of box If all the eigenvalues belong to domain D then box is D-stable Else box is D-unstable End Elseif kboxk < eps2 then box is not D-stable Else divide box in sub-boxes and update List End End End of Procedure Given a generic box V , the logical operation cond(V) gives a necessary and sufficient condition that guarantees the box V is entirely included in the D-stable or D-unstable regions of parameter space. For the sake of brevity, we do not detail the procedure here, but only highlight that it is based on the simple knowledge of the polynomials’ coefficients at the vertices of the considered parameter space box (see [7] for details). In this procedure we use the affine vector functions {a ∗i (.)} computed in procedure 1. Evaluation of such vertex polynomial coefficients can be implemented with simple matrix and vector dot products, thus leading to a very fast execution time. Finally, it should be noted that eigenvalues of the system are only computed in the

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centre point of each box when cond(V) is true, so dramatically reducing the number of eigenvalue evaluations.

5.4 An Example Here, we consider the system analysed in [8]. We have: p (s, a (π)) = s3 + a1 (π) · s2 + a2 (π) · s + a3 (π) , where T

π = ({π1 , π2 }) ∈ Π := [−3, 7]

2

and a1 (π) = 1 + π1 + π2 a2 (π) = 3 + π1 + π2 a3 (π) = 1.25 + 6 · (π1 + π2 ) + 2 · π1 · π2 In this case, the region D is given by the open left half of the complex plane (robust stability problem). Just to check the algorithm’s capability to exclude undefined regions in the parameter space and/or to analyse non-rectangular regions, we added the following constraint to the parameters (which will exclude the upperright corner of the rectangular uncertain parameter region defined above): π1 + π2 < 11 . By applying the proposed algorithm, two results can be visualised: the adaptively generated grid with the undefined region, and the D-stability region shape. In Figure 5.2 the resulting grid as a function of π1 (x-axis) and π1 (y-axis) is shown. It can be noted how the algorithm produces a non-uniform grid which is more dense in some areas than others. The resulting robust stability region is instead plotted in Figure 5.3 as a function of π1 (x-axis) and π1 (y-axis). In this figure, the stability (grey) and instability (dark grey) regions are directly plotted in the uncertain parameter space, thus leading to a very intuitive way to show the region(s) where a criterion is cleared. From Figure 5.3, the capability of this approach for detecting the ”hole of instability” (marked dark grey in the figure) contained within the stability region (grey in figure) is also evident. This result perfectly agrees with [8].

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7 6 5 4

π2

3 2 1 0 −1 −2 −3 −3

−2

−1

0

1

2 π1

3

4

5

6

7

Fig. 5.2. Example of an adaptive grid generated by the polynomial-based method 7 6 5 4

π2

3

Stable

2 1 0

−1

Not Stable −2 −3 −3

−2

−1

0

1

2 π

3

4

5

6

7

1

Fig. 5.3. Example of a robust D-stability region generated by the polynomial-based method

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5.5 Conclusions A clearance method has been proposed which allows checking whether eigenvalues of a dynamic uncertain system belong to a predefined domain D of the complex plane (Robust D-stability property). The proposed method basically proceeds in two steps: 1. automatic generation of local affine approximations of the vector valued function a(.), which maps the uncertainties into the characteristic polynomial coefficients; 2. checking of D−stability by using these local affine approximations and theoretical results available from the literature, which can deal with Dstability of polynomial families whose coefficients depend affinely on the uncertainties. The capability to clear whole regions of the uncertain parameter space is the key feature of the proposed polynomial-based clearance method, provided that all local affine approximations are accurate. Furthermore, no limitation on the choice of the uncertain parameters, because there is no assumption made about the kind of dependence of the system dynamic matrix A on the uncertainties. Another important feature of this technique is that it can give an estimate (i.e. an approximation) of the actual D-stable region shape in the uncertainty space. How good this estimation is, depends upon the parameter settings of the proposed adaptive grid generation routine (i.e. eps 1 and deps ), which have to be preliminarily and interactively tuned. A bad parameter tuning can lead to violations being missed or to conservative results. This is because of a bad approximation of the nonlinear vector function a(.) with the affine vector function set {a ∗i (.)} computed by the adaptive grid generation algorithm. Since the proposed method is based on analysis of the closed-loop characteristic polynomial, it is not a suitable approach when dealing with clearance criteria based on nonlinear simulation results. Furthermore, this approach needs to pre-determine the domain D in the complex plane, where the eigenvalues have to belong. Thus, it can be directly used for the eigenvalue criterion selected in Chapter 10 and may, in principle, be used with any linear clearance criterion that can, in some way, be mapped to a test on system eigenvalue locations in the complex plane. Thus, the proposed technique can be directly applied to all clearance criteria which are based on the eigenvalue locations in the complex plane. Nevertheless, some mathematical manipulation can be used to map (at least in an approximate way) other linear criteria to the complex plane, thus extending the applicability and scope of the proposed technique. The consideration reported above can be easily used for a comparison with conventional gridding methods. It is clear that this approach gives a better coverage than gridding methods, as it clears entire regions of the uncertain

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parameter space. With gridding methods, only local information (i.e. in the points of the grid) can be rigorously guaranteed. An aspect that needs more attention is the number of evaluations (i.e. trimming and linearisation) needed to perform an analysis and the required computational burden. In order to make a comparison with gridding methods, we recall here that in the case of a nonlinear vector function a(.), an adaptive grid is generated in order to reduce the computational burden of the successive step. Furthermore, it is only during the generation of the adaptive grid that the function a(.) is evaluated (i.e., trimming and linearisation are performed). Once the grid has been generated, the algorithm uses the computed information for determining the shape of D-stable region(s) in the uncertain parameter space, without performing any further trim or linearisation. In this way, a very fast algorithm can be implemented which takes typically less than 15-20% of the total computing time, depending upon the presence of a D−unstable region and the specified accuracy for determining its shape. This is actually the computational overhead due to clearing entire regions instead of points (as with gridding methods). Furthermore, it is clear that most of the computational effort is spent in performing trimming and linearisations, as is the case for conventional methods. On this matter the proposed algorithm still shows complexity which grows exponentially with the number of uncertain parameters, as with conventional methods. While with conventional gridding methods the number of trimming and linearisations can be determined once the grid size is fixed, in the proposed algorithm the actual number of trimming and linearisations depends upon how ‘regular’ (i.e., mostly linear) the vector function a(.) is. Typically, it is more than 10 times less than gridding methods, if the same resolution is used (see Chapter 19 for some examples). On the other hand, this method typically needs to compute many more trimming and linearisation points, compared with classical clearance methods based only on criteria evaluation in the min/max vertex points of the uncertain parameter space. Future improvement of the proposed algorithm could further reduce the number of evaluated trim and linearisation points. For example, more sophisticated algorithms for automatic grid generation could be investigated, which leave unchanged the grid size of the uncertainties, whose dependence is mostly linear. In this way, the final grid size can be adapted, not only on the basis of the nonlinearity location in the parameter space (as is done in the proposed algorithm), but also to account for the different kinds of dependence of function a(.) on the uncertain parameters.

References 1. Kharitonov V., Asymptotic stability of an equilibrium position of a family of systems of linear differential equations. Differential Equations vol. 14, 1979

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2. Barmish BR., Invariance of the strict Hurwitz property for polynomials with perturbed coefficients. IEEE Transaction on Automatic Control, vol.29, pp.935937, 1984. 3. Petersen R., A new extension to Kharitonov’s theorem. Proc. 26th CDC, Los Angeles (CA), 1987 4. Bartlett C, Hollot CV, Lin H Root locations of an entire polytope of polynomials: it suffices to check the edges. Math. Contr. Signal and Sys., vol.1, pp.61-71, 1988 5. Sariderely MK, Kern FJ., The stability of polynomials under correlated parameter perturbations. Proc. 26th CDC, Los Angeles (CA), 1987 6. Tesi A, Vicino A., Robustness analysis of uncertain dynamical systems with structured perturbations. Proc. 27th CDC, 1988 7. Cavallo A, Celentano G, de Maria G Robust stability analysis of polynomials with linearly dependent coefficient perturbations. IEEE Transaction on Automatic Control, vol. AC-36, pp 380-384, 1991 8. Ackermann J., Robust Control. Springer Verlag, London, 1993. 9. Cavallo A, de Maria G, Verde L Robust analysis of handling qualities in aerospace systems. In Proc. IFAC World Congress, Tallin, 1990 10. Amato F, Verde L., ROBAN: a parameter robustness analysis tool and its flight control applications. 3rd IFAC Symposium ROCOND 2000, Prague, Czech Republic. 11. Petersen R., A collection of results on the stability of families of polynomials with multilinear parameter dependence. Tech. Rep. EE8801, University of South Wales, Australia, 1988 12. De Gaston RRE, Safonov MG., Exact calculation of the multi-loop stability margin. IEEE Transaction on Automatic Control, vol. AC-33, pp 156-171, 1990 13. Pena RSS, Sideris A., A general program to compute the multivariable stability margin for systems with parametric uncertainty. Proceedings of American Control Conference, Atlanta (GA), 1988 14. Amato F, Garofalo F, Glielmo L, Verde L., An algorithm to cover the image of a function with a polytope: application to robust stability problems. Proceedings of the IFAC World Congress, Sydney, 1993. 15. Amato F, Garofalo F, Glielmo L, Pironti A., Robust and quadratic stability via polytopic set covering. Int. J. Robust Nonlinear Control, vol.5, 1995. 16. Cavallo A, De Maria G, Verde L., Robust Flight Control Systems: a parameter space design. Journal of Guidance, control and Dynamics, Vol.15, No.5 pp.12071215, 1992

6 Bifurcation and Continuation Method Mark Lowenberg Department of Aerospace Engineering University of Bristol, Queens Building University Walk, Bristol, BS8 1TR, UK. [email protected]

Summary. The bifurcation and continuation method utilises the nonlinear equations describing the full aircraft system and locates worst-case parameter combinations for all selected flight conditions. It offers benefits over existing analysis techniques mainly in respect of clearance criteria that require checking throughout the incidence range — usually these are linear criteria. The criteria themselves are implemented directly, as in conventional clearance methods, so that there is no conservatism in the results. The method offers a significant reduction in computational effort, a high degree of accuracy in locating angle of attack at which criteria are violated and good visibility of the physics of the nonlinear phenomena governing the behaviour. It does, however, involve some assumptions in the process of selecting worst-case uncertainty combinations. This chapter outlines some of the concepts of bifurcation and continuation methods, the manner in which they are applied to the clearance problem and the analysis cycle.

6.1 Background to Bifurcation Methods Bifurcation analysis is the term given to a technique for studying behaviour of dynamic systems in terms of their geometric structure: the topology of the steady and non-steady solutions of the system in respect of its state and input variables. It is founded upon elements of nonlinear dynamical systems theory and is implemented numerically via algorithms known as “continuation methods”. The nonlinear systems of concern in aircraft stability and control can usually be formulated as a set of n ordinary differential equations, where n is the number of dynamic state variables. They will also include a number of inputs, or parameters, upon which the behaviour depends. In a nonlinear system it is possible for multiple steady state solutions to co-exist at a fixed combination of parameter values; some of these may be stationary (equilibria), others recurrent1 , and some stable and others not. In open-loop flight mechanics, for example, both a spin mode and a stable wings-level trimmed state may exist for the same control surface positions. 1

A state is recurrent if, after sufficient time, the system returns arbitrarily close to the state value; if it returns precisely to its starting point at periodic intervals the state is periodic.

C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 89-106, 2002. Springer-Verlag Berlin Heidelberg 2002

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These multiple solutions endow nonlinear systems with their rich behaviour. As parameters vary, so does the degree of local stability and the relative dominance of the multiple attractors/repellors 2 ; so the parameter variation can be accompanied by abrupt changes in qualitative nature of the response. This switching of dynamic characteristics at critical points in stateparameter space is known as a bifurcation, and the points at which these are triggered are bifurcation points. Thus in an aircraft model the flight states – regular trims and turns, oscillations, spins and other autorotations – can be classified as equilibria, limit cycles (periodic orbits), quasi-periodic orbits or chaos. The transitions from one to another can also be of various types, which may be abrupt or subtle; these are the bifurcations, examples of which are onset of “wing rock” and “departure” into a spin. The principal strength of bifurcation analysis lies in its ability to explain and predict the phenomena underlying such behaviour across global regions of state-parameter space. In flight mechanics and control, the most commonly used nonlinear tool is simulation (generation of time histories); but the responses depend on initial conditions, length of run and the sequence of parameter changes invoked during the run. Therefore, they are unique to these conditions: important phenomena can be missed and little information on the mechanisms governing the behaviour is provided. For example, when a simulated pull-up results in large-scale excursions in the state variables, it may be observed that departure has occurred and perhaps that the resulting behaviour is steady, oscillatory or otherwise. But there is no information on the mechanism responsible for the departure and its dependence on parameters. Therefore the simulation provides inadequate insight into cause and effect for direct use in analysis and design. However, the combination of bifurcation diagrams with simulation time histories and phase portraits forms an extremely powerful and systematic framework for study of nonlinear dynamics. The numerical implementation of bifurcation analysis – via continuation methods – provides a flexible means within which parameter-dependence can be evaluated. Since most controllers can be represented in the form of ordinary differential equations, both open-loop and closed-loop nonlinear systems can be addressed. The technique may be applied to design studies; but also, by including model variabilities and uncertainty parameters and incorporating clearance criteria in the continuation method, robustness analysis for clearance can be performed. 2

An attractor is a steady state to which the system settles down after some transient motion, when left unperturbed; a repellor is a steady state from which it is repelled.

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6.2 Application of Bifurcation Analysis Bifurcation theory itself is beyond the scope of this chapter, and the reader is referred to the many textbooks on this topic (e.g. [1], [2], [3]). Also, a detailed account of flight mechanics applications is not possible here (see [4] or [5]). Rather, a brief explanation of the manner in which the method is implemented is provided. Bifurcation methods applied to continuous (as opposed to discrete) dynamical systems are based on the model being formulated as a set of differential equations. For most existing aircraft models, where time-dependency in the applied forces and moments is not represented, the system is autonomous and can be expressed as a set of 1 st -order ordinary differential equations: x(t) ˙ = f (x(t), δ)

x(t), f ∈ IRn

δ ∈ IRm

(6.1)

where x is a vector of n state variables, δ is a vector of m parameters, t is time, x˙ is the time derivative of x and f is a vector function with n components. In standard open-loop aircraft models, the state vector x typically consists of the translational and rotational motion rate or attitude variables (p, q, r, α, φ, etc.) and the parameter, or input, vector δ may include, besides system parameters, the various control effectors (aileron, elevator, thrust and so on). In a more complete model, such as that of HIRM+, the vector x includes actuator and sensor states. The model is further augmented when control laws are added, the state vector now including the controller states and the vector δ all the inputs. The theory underpinning bifurcation analysis and continuation algorithms requires that functions of state and control variables be smooth (continuously at least twice differentiable). In practice, models are only piecewise continuous, due to: – linear interpolation of data tables; – discrete changes in aerodynamic loads due to scheduling of slats or other surfaces, or as a result of hysteresis; – controller nonlinearities – saturation, dead bands, switching, etc. Nevertheless, bifurcation analysis can be run under such circumstances, although numerical difficulties arise more readily. However, care must be taken in interpretation of the bifurcation output as piecewise continous systems can introduce additional types of bifurcations. The principal numerical task in bifurcation analysis is to generate all the stationary point and periodic solutions to the system described by equation (6.1) within the state-parameter space of interest. When such a model is extended to incorporate variability and uncertainty parameters within it, it corresponds to the ODE-PUM (Ordinary Differential Equation Parameter Uncertainty Model) as defined in the scope report for GARTEUR FM(AG11) [6].

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The most common means of presenting bifurcation analysis information is to depict the solution branches of the n states as a parameter varies, the others remaining fixed ... so-called one-parameter bifurcation diagrams. Denoting this free parameter as λ (a component of δ), stationary point solutions are defined by: x˙ = f (x, λ)

x, f ∈ IRn

λ ∈ IR

(6.2)

and limit cycles (periodic orbits) by: x(t) ˙ = x˙ (t + TL ) = f (x(t), λ)

x, f ∈ IRn

TL , λ ∈ IR

(6.3)

where TL is the period of the orbit. The free parameter λ – referred to as the “continuation parameter” or “bifurcation parameter” – is conventionally chosen to be a pilot input (e.g. control effector position, stick deflection). However, it can alternatively be any parameter of the system that is of interest, such as centre of gravity location, a stability derivative, a control system gain or an uncertainty parameter 3 . The continuation method used to generate the bifurcation diagrams is a path-following algorithm. Given an initial solution to equation (6.2) or (6.3), it makes use of the local smoothness of the system to find a locus of points as λ varies from its initial guess value. It traces out a so-called “solution path” or “solution branch” of the system. At each solution the eigenvalues of the Jacobian matrix of the locally linearised system, fx , are determined. These indicate local stability and yield information on bifurcations of solution paths. The algorithms can usually cope with discontinuities in the slope of the system, especially when looking for stationary point solutions (as opposed to the more computationally intensive periodic orbits). There are several examples of continuation method software generally available, such as “AUTO” [7] which is an effective program for both equilibria points and periodic orbits, and their associated bifurcations. 6.2.1 Visualisation of Results Graphical representation of the n state variable solutions as the parameter varies is not easy for high-dimensional models. Usually, it is sufficient to plot out each state one at a time versus λ (so for an nth -order model there will be n one-parameter bifurcation diagram projections, one for each component of the state). Dynamical systems theory defines local stability and bifurcation characteristics in terms of eigenvalues of the system Jacobian matrix. In order to 3

In this chapter, we shall treat “uncertainty parameters” and “model variabilities” (such as centre of gravity range) in the same manner. For simplicity, we shall use the term “uncertainty parameter” to refer to both.

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infer likely behaviour from the bifurcation diagrams, it is therefore necessary to indicate the relevant information in some way on the plots. Typically, a selection of symbols and line types is adopted to characterise the solutions as equilibria or limit cycles, as stable or unstable, and to indicate bifurcations (here, solid lines represent asymptotically stable equilibria and dashed lines unstable equilibria). Used in conjunction with time histories, phase plane plots and sometimes two-parameter bifurcation diagrams, these one-parameter bifurcation diagrams provide a wealth of information on the nonlinear system behaviour. 6.2.2 Types of Elementary Bifurcation The majority of bifurcations that manifest themselves in aircraft models, particularly in closed-loop systems, are relatively simple in nature. These elementary bifurcations from stable equilibria are folds, branch crossings and the Hopf bifurcation to a limit cycle. Whilst these are common in open-loop flight dynamics, most are not manifested in the clearance analysis of aircraftplus-controller combinations such as the HIRM+RIDE results presented in Chapter 20. Therefore, only the fold is described here. A fold bifurcation from stable equilibria occurs as a real eigenvalue of the system Jacobian matrix, fx , crosses into the right half plane so that its rank is reduced from n to (n − 1). To one side of the bifurcation point on a 1-parameter bifurcation diagram there exist locally two equilibria paths – one stable, one unstable; they coalesce parabolically at the bifurcation point and on the other side no solutions exist. An example of two such folds on a bifurcation diagram appears in Fig. 6.1 (which shows a locus of equilibrium solutions to a system of the form (6.2) over a range of λ). If the system is operating at or near a stable part of the solution path and the parameter, λ, is varied such that the fold is encountered, there will be a dynamic jump at the fold away from the previous state-parameter space region. In the case illustrated, it will be attracted to the other stable solution branch, and it is evident that hysteresis will occur if λ is varied up and down through the two critical points. It should be noted that the actual behaviour of a nonlinear system only follows the bifurcation diagram solution paths if λ is varied quasi-statically. Transient motions arising from rapid input variation are, of course, still influenced by the solution paths but time history information is necessary in order to piece together a clear picture of how the response relates to the underlying dynamical structure. Fig. 6.2 shows an example of a low-order hypothetical aircraft model bifurcation surface, i.e. an extension of the one-parameter bifurcation diagrams to show variation in state with two parameters – elevator and pitch thrust vectoring (over a large deflection range). Although only elementary bifurcations exist here, including folds, it illustrates the manner in which bifurcation dia-

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stable

fold

unstable λ

fold

Fig. 6.1. Pair of fold bifurcations in state-parameter space.

grams provide information on the mechanisms underlying system behaviour over “global” regions of state-parameter space. stable

1 unstable

α (rad)

0.5

stable

0

−0.5 unstable

−1 −40

−20

elevator (deg)

stable

0

20

40

−100

−50

50 0 thrust vectoring (deg)

100

Fig. 6.2. Two-parameter bifurcation surface (hypothetical aircraft model).

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Experience has shown that the depth of mathematical understanding required by dynamics and control engineers in order to successfully implement the methodology for these elementary bifurcations is no greater than that required for many standard dynamics methods.

6.3 Conventional Bifurcation Analysis Implementation Application of bifurcation analysis to a dynamical system described by differential equations involves, firstly, the need to formulate the system as in equation (6.1). The user identifies the state variables (x) and the inputs (δ). Continuation method codes require the right-hand side of (6.1) to be provided (f (x, δ)), as well as all the data (constants, tables, polynomials or other functions). In order to perform a bifurcation run (solution of a branch of steady states), the user must specify: – which of the m members of δ is to be the continuation parameter, λ; – values for the remaining elements of δ; – a starting value of λ and the range of λ over which solutions are to be sought; – an initial guess for x at the starting λ; – a number of code-specific data inputs (such as solution algorithm parameters). The continuation method software should then produce data for the desired bifurcation diagram, including local stability. Problems can arise due to the system becoming stiff or ill conditioned but these can be resolved by standard means. Not all software solves for periodic orbits. Those that do usually utilise as a starting solution a Hopf point found previously whilst generating equilibrium paths. The process of solving for stable and unstable periodic orbits, along with data to identify bifurcations from these limit cycles, is more computationally intensive than solving for equilibria. Fortunately, periodic orbit solutions are often unnecessary for many closed-loop applications. A typical bifurcation analysis implementation on a closed-loop model might involve specifying, say, the pitch rate demand input to be the free parameter, λ, and then solving for the state as λ varies between its minimum and maximum prescribed limits. By providing a number of initial guesses and/or by implementing “branch switching” at bifurcation points, a number of branches can be solved. In principle, therefore, it is possible to find all the paths of stable and unstable equilibria and periodic orbits in the desired state-parameter space.

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6.4 Application to Control Law Clearance In the context of flight systems clearance, the bifurcation analysis/continuation method approach offers the advantage of using the full nonlinear ODE-PUM and the versatility to conduct investigations in respect of any parameter of the system. It is particularly powerful in providing a link between clearance issues and the underlying physics. It produces a direct indication of clearance violations associated with eigenvalues, and helps to explain phenomena exhibited when evaluating nonlinear criteria via time history traces (e.g. incidence or load factor exceedence). Clearance criteria are not usually associated directly with pure eigenvalue stability. In the GARTEUR specifications, for example, the unstable eigenvalues criterion permits a limited excursion into the right-half plane (to allow for slow unstable modes) and frequency response criteria are used extensively. Therefore it is the continuation method that offers the potential in clearance analysis, rather than bifurcation analysis per se. It is well suited to application of linear clearance criteria over a wide operating range. The continuation method finds the trimmed solution path with respect to which the linearisations are to be conducted. By incorporating the clearance criteria evaluations within a continuation framework, the onset of violations can be detected as solutions are found through, for example, the desired angle of attack (AoA or α) range. The “bifurcation diagrams” that are plotted are still steady states (trim points) versus the continuation parameter; but now the line type or symbols used to indicate changes in stability actually denote violation of a clearance criterion. If several criteria are incorporated in the continuation software, then one plot for each criterion can be produced from the run (as can standard bifurcation diagrams, if required). Clearance criteria are often applied to a different form of model from that used to locate the trim points. This is because the trims may in fact not always be true steady states: to obtain “trim” points over a range of α for a conventional aircraft, it may be that the point at the bottom of a pull-up (or top of a push-over) manoeuvre is defined as “trimmed”. Also, it may be necessary to omit the controller command path when seeking these accelerated “quasi-trims” (e.g. at incidences that exceed the α-limiter). The clearance criteria, however, must be applied to the full system. Therefore, the continuation method is implemented in what is referred to as the “dual-model” framework. One model is specified for finding the solution (trim) points as a parameter varies 4 ; using the states thus obtained, a second version of the system model is called at each solution point to evaluate the clearance criteria. (In fact, any number of models can be used, depending on the nature of the various criteria to be tested throughout the run.) 4

This must be equivalent to the trimming specified for a conventional clearance task.

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The implementation of continuation methods for clearance purposes therefore mirrors that of conventional bifurcation analysis except that: – the clearance criteria must be “attached” to the software – essentially in exactly the same form as they are applied in the traditional baseline clearance approach – and represented on the plots for evaluating the output; – more than one version of the system model may be required, if trim solutions require a different formulation from that used to evaluate the criteria. It is likely that both the system model itself, and possibly the clearance criteria, may be specified in a computing language other than that in which the continuation method software is written. In FM(AG11), for example, the models are provided in MATLAB/Simulink, as are the tools provided with the baseline clearance. However, the continuation code is the Parametric Continuation Solver (PCS) which was created by the author in Fortran 77, specifically for flight mechanics problems. It was updated for the GARTEUR project to run within the MATLAB environment to permit specification of the model right-hand side equations (this also allows MATLAB functions and tools to be utilised where appropriate). PCS does not perform limit cycle continuation but this is not a limitation in the project. Therefore, an important factor in the choice of which continuation software to use is its flexibility in being applied not only to high order models but also to models specified in a different language. The bifurcation/continuation method approach to clearance is founded upon the fact that aircraft steady-state solutions are usually continuous (in value, if not slope) as parameters vary 5 . This means that a solution run under nominal conditions (nominal values of variabilities and uncertainties) will show where in state-parameter space violations of clearance criteria occur, or where the system comes closest to such violations. At these critical solution points, the versatility of continuation methods is then exploited by choosing each of the uncertainty parameters as continuation parameter. This allows information to be obtained on the influence of the uncertainties (across their specified range, not just at their extreme values), which can in turn indicate worst-case combinations of uncertainties. Once the worst case is defined, the original continuation parameter is resumed, this time with the uncertainties in place, and the violation point under worst-case conditions obtained. Details of this process, and its strengths and weaknesses, follow. 5

If features such as mode changes, switches or hysteresis do introduce discrete jumps in behaviour, these should be accounted for when searching uncertainty parameter space – whatever analysis technique is adopted.

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6.5 Description of the Analysis Cycle The analysis cycle as used for clearance in this GARTEUR project is illustrated in Fig. 6.3. Shaded blocks in this flow diagram refer to actions involving running of the continuation method. Broadly, the approach involves two stages: firstly producing results for each clearance criterion in the nominal case; and secondly, based on information from the nominal results, a procedure to choose and analyse the worst-case combination of uncertainty parameters. In order to understand Fig. 6.3 fully, the numbered blocks in the flow diagram are described in more detail. (1) The model as provided is linked in to the software to be used for implementing bifurcation/continuation (e.g. PCS). The interface must include provision for exchange of state variables and their time derivatives, continuation parameter, uncertainty parameters and other relevant data. When solving for equilibrium solutions, the order of the model must reflect the required trim conditions (e.g. accelerated quasi-trims, command path excluded, etc.). If the model from which the clearance criteria are to be determined is different, the dual-model form of the continuation framework must be invoked, where two (or more) models are called for each solution point. The software and plotting must be set up to facilitate the clearance criteria that are to be investigated. Values of the system constants and inputs must match the required flight conditions. The software should be configured to permit changes in choice of continuation parameter for the various stages in the analysis cycle. (2) The continuation method is run for the nominal system, using an appropriate continuation parameter over its allowable range (e.g. pitch rate demand, qdem ). This generates the path of stationary point solutions for the system as continuation parameter is varied. The solutions must correspond to the trim conditions specified for the clearance analysis, and the required α range must be covered by the continuation parameter variation. At each point the clearance criteria are evaluated. The results can be plotted as bifurcation diagrams (each state variable versus qdem ) and also as clearance measure versus, say, α. The former provide information on the behaviour characteristics of the dynamical system, while the latter are a direct output of the clearance measure of interest. Root loci 6 and frequency responses can also be plotted. At one level, this step is conventional clearance implemented “continuously” through the α range. However, it offers the additional benefits of finding exact violation points at which to investigate parameter uncer6

Root loci in this instance refer to loci of roots as λ varies (rather than a gain varying).

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Fig. 6.3. Clearance analysis cycle.

tainties, providing insight into the system behaviour as λ varies and of revealing any singularities that may arise. (3) Points on the nominal bifurcation diagrams where clearance criteria are violated are located. These nominal critical points identify neighbourhoods where the uncertainty parameters are likely to further limit the clearance envelope.

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If no nominal critical point exists on the bifurcation diagrams for a particular criterion, locate where the clearance measures come closest to their allowable bounds (nominal worst-case points). It is possible that the nominal worst-case point for a particular criterion is well away from the allowable bounds. However, in the absence of substantial working experience with the model, no assumptions should be made and such points must be studied further as with nominal critical points. (4) The continuation method is now run again, this time starting at a nominal critical or worst-case point and using one uncertainty parameter at a time as the continuation parameter. Each uncertainty is run through its full range (−1 to +1 in terms of normalised values, where the nominal value is 0) so that the path of equilibria versus uncertainty is generated in the neighbourhood of the starting point. There is one important difference for these nonlinear sensitivity runs in terms of model set-up. In a regular run, starting from the nominal point and varying an uncertainty parameter, the value of α would change (along with other states)7 ; the results would then include not just the uncertainty parameter effect but also that of the incidence change. What we really want is to find the worst-case uncertainty parameter combination at the critical point – i.e. we wish α to remain constant at the critical point value. Therefore, we modify the coding of the method to hold α constant and, instead of solving the α˙ equation for α, we solve it for what was the continuation parameter in nominal runs (e.g. qdem ). (5) The worst-case combination of uncertainties for a particular criterion at a particular critical point is estimated as follows. First, we observe from the nonlinear sensitivity plots created in step (4) the value of uncertainty parameter that in each case gives rise to the maximum adverse displacement of the criterion under consideration. So for example, in the sketch in Fig. 6.4(a), the normalised value of uncertainty parameter that gives the largest positive increment in maximum real eigenvalue is −1. This approach is taken for all k uncertainty parameters of interest, so that each would usually take on the value of either −1 or +1, depending on which gives rise to the largest degradation in clearance criterion. We assume next that this worst-case value for each of the k uncertainties in isolation also holds for the k uncertainties in combination. Then, knowing both the value of each uncertainty parameter and its effect on the clearance criterion (“∆(Max. real eigenvalue)” in Fig. 6.4), the reduction factors can be applied to aerodynamic uncertainties in order to choose the worst-case uncertainty combination. In this way, the method shows 7

α changes significantly with parameters such as Xcg U nc (variation of fore/aft centre of gravity position) but negligibly with lateral-directional parameters.

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Fig. 6.4. Possible nonlinear sensitivity scenarios. ∆(Max. real eigenvalue) is the change in (real part of) the most positive eigenvalue relative to its nominal value.

explicitly whether using all k parameters gives the actual worst case or whether, due to reduction factors, a subset (fewer than k) is worse. Note that this method can account for situations such as illustrated Fig. 6.4(b) and (c). In the case of (b), the maximum sensitivity does not occur at −1 or +1 but at an intermediate value. The sensitivity variation in Fig. 6.4(c) shows a fold in the curve: to the left of the fold there are two possible trimmed solutions for each value of uncertainty parameter; to the right there are no defined trim points. The information provided by the nonlinear sensitivities is vital to correct implementation should such a situation arise; i.e. where the nominal point is close to a fold bifurcation induced by varying an uncertainty parameter. Situations such as in Fig. 6.4(c) can arise via, for example, a discontinuity (such as saturation) or via a smooth nonlinearity inducing a fold bifurcation. Either way, there then exists a range of uncertainty parameter values where there is no solution and another with two possible solutions — one

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being a worse case than the other (larger ∆(Max. real eigenvalue)). The power of this analysis technique is that if the nominal point lies on the non-worst case branch, the continuation method will reveal the “new” worst case condition. (6) Once the most severe combination of uncertainties is found in step (5), this is applied to the model during a “local” continuation run, again starting at the nominal critical point and with the continuation parameter as in the nominal runs (e.g. qdem ). The run is strictly only valid as a worstcase in a small region because the selection of uncertainty parameters is based on the nominal critical point (and also because the definition of uncertainties for stability derivatives is only meaningful nearby the selected trim value). Such runs will reveal the value of α to which the critical point (clearance violation) moves in the presence of uncertainties. Fig. 6.5 illustrates the principle behind these local continuation runs. It shows part of a nominal bifurcation diagram, together with the corresponding solution paths for the minimum and maximum values of one uncertainty parameter, Xcg U nc . The clearance criterion is the maximum real eigenvalue; solid lines denote cleared equilibria and dashed lines represent solutions that violate the criterion. We see immediately that the worst case is Xcg U nc = −0.15 (α at violation decreases from 25◦ to 24◦ ). All three paths shown in Fig. 6.5 are valid over the entire qdem range. In step (6), however, a combination of uncertainties is used and these were selected in step (5) as the worst-case combination in the vicinity of the nominal critical point (α = 25◦ in this example). Thus we can assume this combination to be the worst-case during a local continuation run – but not globally8 . The analysis cycle described above and depicted in Fig. 6.3 can be partially automated to improve efficiency in performing the clearance task. There is, however, some advantage to retaining a degree of human intervention in the process (helps relate results to the physics of the problem and to identify any anomalies that may arise from model discontinuities).

6.6 Assumptions Implicit in the Analysis Cycle The process outlined in step (5) of the analysis cycle assumes that each uncertainty parameter value corresponding to maximum clearance measure 8

In fact, it has been found that by running the local bifurcation diagram over an extended α range, a very good indication of other possible violation regions is obtained. Once located, a new selection of worst case uncertainties can be performed at such points, to ensure that the local worst case is used (i.e. repeat steps (4) to (6) locally).

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degradation is the same irrespective of the values of the other k − 1 uncertainties. This “uncertainty parameter decoupling” has two consequences: 1. Prior to considering reduction factors, the implication is not too severe: the actual magnitude of degradation due to an uncertainty when other uncertainties are non-zero is accounted for in step (6); however, the value of the uncertainty at which this is a maximum remains fixed. (In other words: if it is indicated that an uncertainty should take on a normalised value of, say, −1 for maximum influence on a criterion with all the other uncertainties set to zero, then the assumption is that the −1 remains valid even if the other uncertainties are non-zero.) This implies that the multi-dimensional surface – in clearance criterion and uncertainty parameter space – has no saddle-type features in the neighbourhood of the nominal point. More specifically: the behaviour with respect to single parameters is monotonic over the full uncertainty parameters range. 2. When deciding upon how many aerodynamic uncertainties to include in the worst-case combination, the test utilises the actual magnitudes by which each uncertainty degrades the clearance measure. In general,

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the choice may then be incorrect as the magnitudes will vary when the uncertainties are used in combination. In the HIRM+RIDE clearance, checks indicated that the worst-case combination of uncertainties found in this manner were correct. In general, however, it is a fairly strong assumption and an improved approach is described in section 6.7.

6.7 Conclusions and Recommendations It has been shown that the bifurcation/continuation framework for control law clearance offers the advantage of effectively linking the clearance results with the underlying physics of the system. The actual clearance analysis at a particular critical point uses the same implementation of criteria as a traditional (baseline) clearance, so there is no conservatism in the technique. The real strength of the approach is the time saving that can be achieved by pinpointing precisely which regions in the flight envelope violate or come close to violating clearance criteria. Efforts at determining the worst-case scenario are then concentrated in these violation zones only. This applies in terms of AoA regions in the above descriptions – but the versatility of the method means that alternative implementations, such as across a Mach number range, can be carried out in a similar way. A related benefit is that the “continuous” variation in α allows the violation point to be located rather accurately, and yields information between the sample points analysed in a conventional gridding technique. The premise upon which it becomes acceptable to define these violation regions based on nominal analysis assumes that other regions do not move markedly closer to violation when uncertainty parameters are applied. As already mentioned, this requires the system characteristics to vary smoothly as parameters change (but does not require their rate-of-change to vary smoothly). When there are mode changes or other forms of discontinuity in the system that cause some sort of jump in the dependence of the system on parameters, then the premise is no longer valid. However, this assumption is equally implicit in conventional clearance analysis: by looking only at discrete values of incidence and flight conditions, the implication is that there are no dramatic phenomena occurring between these points. Thus the bifurcation/continuation methodology is considered to be of practical use. If there are known points of discontinuity (associated, for example, with a switch in the system) these should be identified beforehand, so that they can be accounted for in the clearance process. The continuation method framework is best suited to clearance criteria that require checking over a range of state-parameter space – e.g. a range of AoA. This includes, in principle, nonlinear criteria. However, as these are based on simulations (time histories) and the models are complex, they add

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a significant run-time penalty if computed throughout the incidence range. In practice, such criteria tend to be applied only from straight and level trim points (not performed across the α range) and so part of the benefit of the technique is lost on such criteria. Nevertheless, by taking the straight-andlevel trim as the critical point, the nonlinear sensitivity analysis to identify worst-case uncertainty combinations could be useful – although the benefit over conventional clearance is much reduced. As previously described, the bifurcation/continuation technique offers an explicit means of finding “true” worst-case uncertainty parameter combinations for each criterion – but within the constraints of an assumption on the system topology. This means that the parameter combination found in this manner cannot be guaranteed to be the actual worst case. An improvement to the analysis cycle would be to perform a further nonlinear sensitivity study step: once the worst-case combination has been found using each uncertainty in isolation (others at nominal values), the process is repeated, this time with the fixed uncertainties adopting those values, rather than the nominal ones. Thus nonlinear coupling between the uncertainty parameters will be incorporated in the sensitivity evaluation. In many practical cases this will confirm the worst-case choice of parameters; if not, the process can be iterated, using the newly-found uncertainty combination at the latest step. For a full guarantee that the worst-case has been found, the conventional clearance approach (trying all possible combinations and permutations) could even be contemplated. Since the critical regions have already been detected by the continuation process, the number of times that this time-consuming iterative analysis is required is substantially less than in a full baseline/conventional implementation. A more sensible recommendation, however, lies in the combination of the bifurcation/continuation technique with an optimisation method. In this hybrid approach an optimisation algorithm would be integrated within the continuation framework and applied for each criterion at each nominal critical point to find the worst-case combination of uncertainty parameters (possibly using the nonlinear sensitivity analysis as described above as an initial guess). This would “free up” the continuation method to be extended to solve also for worst-case paths through the flight envelope, i.e. locating a worst-case boundary in terms of altitude and Mach number. Such a hybrid method could indeed prove a significant benefit in terms of efficient clearance analysis.

6.8 Acknowledgements The author is grateful to Tom Richardson (postgraduate student, Dept. of Aerospace Engineering) for Fig. 6.5 and his help in developing the continuation method framework, and to Guy Charles (postgraduate student, Dept. of Mechanical Engineering) for generating Fig. 6.2.

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References 1. S. H. Strogatz. Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry and Engineering, Addison-Wesley, 1994. 2. Y. A. Kuznetsov. Elements of Applied Bifurcation Theory, 2nd ed., Springer, 1998. 3. J. M. T. Thompson and S. Bishop. Nonlinearity and Chaos in Engineering Dynamics, John Wiley & Sons, 1994. 4. J. M. T. Thompson and F. B. J. Macmillen. Nonlinear Flight Dynamics of HighPerformance Aircraft, Philosophical Transactions of the Royal Society London A, 356, 2167-2180, 1998. 5. M. G. Goman, G. Zagainov and A. Khramtsovsky. Application of Bifurcation Methods to Nonlinear Flight Mechanics Problems, Prog. Aerospace Sci, 33, 539-586, 1997. 6. J. Terlouw, U. Korte, L. Forsell, C. Fielding, A. Varga, D. Littleboy, J. Kos and A. Martnez. Scope of a new GARTEUR Flight Mechanics Action Group on ”New Analysis Techniques for Clearance of Flight Control Laws”, GARTEUR/TP-119-1, 1999. 7. E. Doedel, X. Wang and T. Fairgrieve. AUTO94: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations, Applied Mathematics Report, California Institute of Technology, 1994.

7 Optimisation-Based Clearance Andras Varga German Aerospace Center, DLR - Oberpfaffenhofen Institute of Robotics and Mechatronics D-82234 Wessling, Germany. [email protected]

Summary. The basic feature of the optimisation-based clearance approach is to reformulate the clearance problems as equivalent minimum distance problems for which ”anti”-optimisation is performed to determine the worst-case parameter combination/flight condition leading to worst performance. The basic requirements for the applicability of the optimisation-based approach are the availability of suitable parametric models describing the overall nonlinear dynamics of the augmented aircraft and of accompanying efficient and reliable trimming, linearisation and optimisation software tools. The optimisation-based approach has no limitations with respect to clearance criteria, being able to address all kind of clearance requirements which are expressible as mathematical criteria.

7.1 Classical versus Optimisation-Based Approach Let c(p, F C) be a given clearance criterion, depending on the uncertain parameters grouped in a q-dimensional vector p and flight condition vector F C usually having up to three components (e.g., Mach-number M , altitude h, angle of attack α). p is generally unknown, but it is assumed that all its components lie in known intervals, defining a hyper-box P in the q-dimensional Euclidean space. The variation of flight condition F C is determined by the defined flight envelope where the aircraft is required to operate. We can easily formulate the clearance problem for a given performance criterion c(p, F C) as a distance minimisation problem. Let c0 be the limiting acceptable value of c(p, F C), as defined in the clearance documents. Then, the difference d(p, F C) = c(p, F C) − c0 (7.1) can be interpreted as a signed distance function to the limiting acceptable performance c0 . If for a fixed F C, d(p, F C) is positive for all parameter values p ∈ P, then the clearance requirement is fulfilled in F C and the point F C is cleared. The minimum distance d(F C) = min d(p, F C) p∈P

can be interpreted as the robustness measure of how far the system is from the limiting acceptable performance c0 . A negative value of d(F C) can be C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 107-117, 2002. Springer-Verlag Berlin Heidelberg 2002

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interpreted as a measure of the lack of robustness and the corresponding point F C is not cleared. The current industrial clearance approach relies on an exhaustive search on a grid for both flight conditions and uncertain parameters. Typically, one chooses N flight conditions F Ci , i = 1, . . . , N and in each flight condition F Ci , c(p, F Ci ) is evaluated only in ν(P), the set of vertex points of the q-dimensional hyper-box P. Thus, e Ci ) := min d(p, F C) ≥ d(F Ci ) d(F p∈ν(P)

e Ci ) is only an approximation (upper bound) of the true minimum and d(F e Ci ) is used to decide if F Ci is cleared or distance d(F Ci ). The value of d(F not. Clearing N flight conditions, each in 2q vertex points, requires N · 2q evaluations of c(p, F C). Thus, the required computation time increases exponentially with the dimension q. To have a feeling what exponential computational complexity means, assume that 1 second is necessary for one function evaluation. Then for q = 5, 9 and 15, the time needed to check only one flight condition in the 2q vertices is 32 seconds, 512 seconds, and 9.1 hours, respectively. Note that typical values of N are of order 50001 . Two main difficulties of the classical approach are evident. First, there are tremendous costs involved when simultaneously checking the robustness for many uncertain parameters. Since the evaluation of each robustness measure increases exponentially with the number of parameters q, obviously the computational costs for large problems are too high to be affordable in industrial practice. Second, there is no guarantee that the ”cleared” flight conditions should have been cleared, since for each parameter only the extreme points (maximum and minimum) are checked. Thus, if the minimum occurs in an intermediate point which is not cleared, then the clearance results could be false. The same applies when considering the finite set of flight conditions {F C1 , . . . , F CN } which certainly can not cover all points of the physical flight envelope. The optimisation-based approach offers immediate improvements for both of these aspects. The first improvement is a reasonable computational cost of clearance in case of many parameters. This occurs because the number of function evaluations necessary to compute the worst-case parameter combination is usually much lower than that corresponding to evaluating the function in all vertex points, even in the case when only two values for each parameter are used. The second improvement is achieved by allowing a continuous variation of parameters within the given parameter space P. In this way, the clearance results cover all points of P and therefore are more reliable. A straightforward way of enhancing the classical approach is to perform, in each of the selected N points F Ci , i = 1, . . . , N of the flight envelope, an optimisation-driven worst-case search to determine the minimum distance 1

U. Korte, Private communication

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d(p, F Ci ). This approach can be seen as a combination of the gridding-based classical search in a discrete set of flight conditions with the optimisationbased continuous search for worst-case parameter combinations in the complete parameter space. Note that this approach can be very effective, even when discontinuities of derivatives are present in the mathematical model due to the use of linear interpolation formulas to evaluate aerodynamic coefficients defined by look-up tables. By ”freezing” the flight condition during optimisation, the variations within these tables no longer play any role, and therefore, optimisation methods based on gradient search techniques can be readily employed to locate worst-case parameter combinations in a very effective way. The mathematical optimisation problems to be solved belong to the class of nonlinear programming problems (NLPs) with simple bound constraints on variables for which both gradient-based and gradient-free techniques can be employed (see Section 7.3). An enhancement of this approach can be achieved by explicitly addressing the continuous variation for the flight condition. A possible approach is to define a coarse set of flight conditions FC = {F C1 , . . . , F CK }, where K ¿ N , and associate to each flight condition F Ci a box F Ci ± ∆F Ci centered around F Ci . The sizes of these boxes are chosen such that their union covers the whole physical flight envelope. Then, by including F C among the optimisation variables, solve for all F Ci , i = 1, . . . , K the distance minimisation problems min d(p, F C) d(F Ci ) = p∈P F C∈F Ci ±∆F Ci

If d(F Ci ) > 0, then each flight condition in the hyperbox around F Ci is cleared. Otherwise, a locally finer grid can be considered if necessary and the clearance can be repeated on this finer grid. The main advantage of this approach is the complete and continuous coverage of both the flight envelope and the parameter space, and thus a higher confidence in the clearance results. Another advantage is the potentially lower total costs, by using a reduced set of only K ¿ N flight conditions. The mathematical optimisation problems to be solved is a NLP with only simple bounds on variables and linear constraints (see Section 7.3). It is important to note that robustness analysis problems are essentially global optimisation problems. When qualifying flight conditions as cleared or not cleared on basis of a local search, only the not cleared points are guaranteed. For a rigourous analysis, only the computationally very expensive global search approaches with guaranteed convergence are able to assess cleared points. Often, a restricted preliminary sensitivity analysis with only a few parameters can indicate the probable lack of multiple local minima. In such cases, the cheaper and more efficient local search methods can be used for solving clearance problems practically without any loss of reliability of the results.

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7.2 Description of the Analysis Cycle Let c(p, F C) be a given clearance criterion depending on the uncertain parameters grouped in a parameter vector p and the flight condition vector F C. The analysis cycle used for the clearance of a control configuration for the given clearance criterion c(p, F C) is illustrated by the flow diagram in Figure 7.1. Here we assume that a continuous search is performed only in the parameter space P for a finite set of flight conditions F Ci , i = 1, . . . , N .

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According to the flow diagram in Fig. 7.1, the following main steps have to be performed in an optimization-based clearance procedure: Step 0 is the initialisation step for the optimisation-based clearance procedure and usually involves choosing the flight conditions F Ci , i = 1, . . . , N where the worst-case parameter combinations are to be determined, the definition of vector p (e.g., most relevant or full set, longitudinal or lateral), setting of appropriate options for trimming, setting the values of criterion specific variables (e.g., frequency-grid, time-grid), or choice of optimisation method (see Section 7.3) and corresponding options (e.g., stopping tolerances, maximum number of iterations etc.) Step 1 is necessary to eliminate from the analysis those points where the clearance requirements are not fulfilled for the nominal values of parameters. Furthermore, here we can also check if the normal acceleration nz is within an allowed range of values (e.g., −3 g ≤ nz ≤ 7 g for HIRM+) or the control surface deflection saturation limits for δT S , δT D , and δR are reached. Points where such violations occur are not cleared and are automatically eliminated from the analysis. The neat effect of this check is a reduction of the overall computational effort. Step 2 is the basic optimisation step performed for each selected flight condition F Ci . The results of this step are the worst-case parameter combination pworst , F Ci ). The and the corresponding criterion value c(pworst i i performed number of function evaluations is an indication of the efficiency of the optimisation-based search in comparison with the classical grid-based approach. Step 3 is similar to Step 1 and the performed check is necessary because the worst-case parameter combination can lead to the same possible violations of some conditions as those occurring in the nominal case (e.g., violation of condition −3 g ≤ nz ≤ 7 g or of the deflection saturation limits). Note however, that such points are found only incidentally by the optimiser, and may exist in a particular flight condition F Ci even in the case when the determined worst-case parameter combination does not violate the above conditions. Step 4 is the outputting of computed data to a database. For each flight condition F Ci , the stored information contains typically the computed worst-case parameter combination pworst , the corresponding minimum i distance d(piworst , F Ci ), the number of performed function evaluations, cleared/not cleared status information, etc. Step 5 performs the graphical evaluation of obtained results by producing plots necessary for assessing and documenting the clearance results.

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7.3 Optimisation Algorithms Suitable for Clearance Each clearance analysis problem defined in Chapter 10 can be formulated as a standard nonlinear programming problem (NLP) of the form min f (x) subject to cj (x) ≥ 0, j = 1, . . . , m li ≤ xi ≤ ui , i = 1, . . . , n

(7.2)

to be solved for x ∈ IRn . Here the components of x includes, in general, variables defining the flight condition (e.g., M , h, and/or α) and components of the vector p representing the uncertain parameters of the model. Each component xi of x must lie between the corresponding lower bound li and upper bound ui . The lower and upper bounds are defined by restricting the flight conditions to lie within the admissible region defined by the flight envelope, while the bounds on uncertain parameters are defined on basis of their physical significance. The scalar constraints cj (x) may correspond, for example, to restricting the search to a typical polygonal region, whose boundary is defined by several line segments. Thus, in the most general case, the NLP (7.2) corresponding to a particular clearance problem is still a particular NLP subject only to simple bounds on variables and linear constraints. If the flight condition (i.e., M , h and α) is not part of x, then the clearance problem can be formulated as an even simpler NLP with only simple bounds on variables. The NLPs arising in clearance problems have several particular features: Low order. Since the optimisation variables are the uncertain parameters and possibly some components of the flight condition vector, the dimension of the optimisation problem is relatively small, satisfying n ≤ 25. Multiple local minima. The functions expressing clearance criteria exhibit very complex dependencies of parameters. It follows, that we can always expect that these functions have several local minima. Expensive function evaluation. The evaluation of criteria based on linearised models, involves trimming, linearisation and frequency response or eigenvalue computation of relatively high order systems (up to 60 state vector components). The evaluation of criteria based on nonlinear models usually involves simulations, preceded by trimming. Thus typically, the evaluation of clearance criteria is very time consuming. Fast and reliable trimming (e.g. via inverse models) is a prerequisite to increase the efficiency of function evaluations. Model reduction techniques can be efficiently used to reduce the order of linear models used to evaluate frequency-response based criteria. Discontinuous derivatives. Discontinuities in derivatives of functions arise from several sources. Naive implementation of criteria by defining distance functions to regions with polygonal boundaries will certainly lead to functions with discontinuous derivatives. By approximating boundaries

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by polynomials (e.g., spline functions) we can get rid of such discontinuities, but the clearance problem can be falsified by such an approximation. Other sources of discontinuities can lie in the model itself, if table-driven linear interpolations are present. Finally, failures to accurately evaluate the function (e.g., because of inaccurate trimming) lead to discontinuities even in the functions themselves. Handling trim failures, for example by setting the function values to a very large number, can rise severe problems for certain solvers. Noisy function. Noise in function values originates from various truncation errors made in intermediary computations such as trimming, linearisation, order reduction, numerical evaluation of gradients, simulation, as well as from the round-off errors associated with difficult numerical computations like eigenvalue computation. To handle such functions, the usage of more robust, derivative-free optimisation methods could be necessary (e.g., pattern search) or enhancements of gradient-search techniques are necessary (e.g., usage of central difference approximation of gradients, usage of gradually increased accuracy in gradient computations, etc.). For additional aspects of optimisation with noisy function see [1]. In the following paragraphs we present brief information on several optimisation algorithms which are suitable for solving the NLPs appearing in the clearance problems. For most algorithms software implementations are freely available on the Internet [2]. 7.3.1 Gradient-Based Local Search Methods Gradient-based minimisation methods use local information on the function through its gradient to achieve fast convergence rates. This is why, when applicable, many gradient-based search methods allow highest computational performance in solving general or particular NLPs. For the usage of most gradient-based techniques a basic requirement is the continuity of gradient with respect to the optimisation variables. Furthermore, for a satisfactory performance, the availability of an analytic expression of gradient is highly desirable. However, for complex functions like those typically arising in clearance problems, usually no analytic gradients are available. Therefore, numerical approximations of gradients have to be computed resultsing in a slower and less reliable execution, especially when function evaluations are noisy. The sequential quadratic programming (SQP) method to solve the general NLP with equality and inequality constraints can be used to solve the particular NLP of the form (7.2) which arises in clearance problems. The SQP method can be seen as a generalisation of Newton’s method for unconstrained optimisation in that it finds a step away from the current point by minimising a quadratic approximation of the problem function f (x). Under mild conditions this method has a fast, so-called superlinear convergence [3]. An alternative approach for problems with only simple bounds constraints on the vari-

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ables is the limited memory BFGS with bound constraints (L-BFGSB) described in [4] together with accompanying Fortran 77 software. This approach extends the standard Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method to handle NLPs with simple bounds, by using a gradient projection approach. The algorithm has a superlinear convergence and no violations of bound constraints on variables occur during optimisation. For a more detailed discussion of both approaches see [3]. 7.3.2 Gradient-Free Local Search Methods Derivative-free methods using only function evaluations are a real alternative to gradient-based methods, especially when function evaluations are noisy and/or discontinuities in the gradient are present. Two classes of derivativefree methods are known: direct-search methods which include the popular simplex and pattern search methods, and trust-region methods relying on linear or quadratic interpolation models. Derivative-free methods are useful when the function f (x) is not smooth (e.g., ”noisy” function) or when accurate derivatives are difficult to determine numerically. For more details on derivative-free methods see [5] and for performance comparisons see [6]. Pattern search (PS) algorithms are a class of direct search methods initially proposed for unconstrained minimisation which has a rigourous global convergence theory. The PS techniques has been recently extended to solve NLPs with simple bounds [7]. PS methods use a simple decrease criterion to accept a step as opposed to the sufficient decrease criterion used by gradientbased search. This is why, PS methods usually have a slower convergence rate than a gradient-based search. On the other hand, PS methods are often numerically more robust than gradient-based methods in avoiding local minima as well as tackling with noisy functions. PS methods may require a relatively large number of function evaluations, hence they tend to be effective primarily for problems of relatively small dimensions and low accuracy situations. Model-based trust region methods exploit the smoothness of the objective function and attempt to preserve the convergence properties of their gradient-based counterparts. The constrained optimisation by linear approximations (COBYLA) approach employs linear approximations to the objective and constraint functions [8]. The approximations are formed by linear interpolation at n + 1 points in the space of the variables (regarded as vertices of a simplex) and the size of the simplex is reduced as the optimisation advances. The main advantage of COBYLA over many of its competitors, is that it treats each constraint individually when calculating a change to the variables, instead of lumping the constraints together into a single penalty function. Therefore, COBYLA usually has better convergence than the pattern search method. One disadvantage of the COBYLA software, is that it does not address simple bounds explicitly and these must be transformed to 2n general constraints in the NLP (7.2) of the form cj = xj − uj ,

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cj+n = lj −xj , for i = 1, . . . , n. Unfortunately, this leads to frequent violations of bound constraints during the computation. The derivative-free optimisation (DFO) trust-region method uses a quadratic approximation of the objective function (see [6] and references therein). The quadratic model approximates the function well within a certain ”trust”-region of a given radius and serves to determine new points by minimising the current approximation instead of the function itself. The new points generated by the algorithm are used both to advance the optimisation and to update the approximation. Since the DFO algorithm needs only a relatively few function evaluations, this method is well-suited to minimise expensive functions which depend on few (some hundred at most) variables. 7.3.3 Global Search Methods For functions with many minima, the use of global optimisation techniques is the only alternative for successful computations. In this section we discuss three global optimisation approaches which can be employed for solving optimisation-based clearance problems with simple bounds on the parameters. Typically, these methods require a very large number of function evaluations and therefore they are primarily intended either to determine good starting points for local search based methods, or to address difficult clearance problems with many local minima. The simulated annealing (SA) algorithm is essentially an iterative random search procedure with adaptive moves along the coordinate directions [9]. It permits uphill moves under the control of a probabilistic criterion, thus tending to avoid the first local minima encountered. It has been proved that the sequence of points sampled by the SA algorithm form a Boltzmann distribution and converges to a global minimum with a probability of one as the annealing ”temperature” goes to zero. The genetic algorithm (GA) is a global optimisation approach based on evolution strategies which guarantee the survival of the fittest individual in each population [10]. The GA can easily handle problems with simple bounds on the variables, and even general constraints by using penalty function techniques. There are several selection schemes which can be combined with a shuffling technique for choosing random pairs for mating. The GAs based on binary coding, use mutations (e.g., jump or creep mutations), crossover (single-point, uniform, etc.), niching and various other strategies to produce successive populations. The use of GA for function optimisation is quite costly in terms of the required number of function evaluations, but usually its cost can be predicted in advance by choosing the population size and the number of successive generations. To find the global extremum with high accuracy, this method typically requires a very large number of function evaluations. The global optimisation using multilevel coordinate search (MCS) attempts to find the global minimiser of the bound constrained optimisation problem using function values only, based on a multilevel coordinate search

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that balances global and local search [11]. The local search is done via SQP. The search is not exhaustive, so occasionally the global minimum may be missed. However, a comparison to other global optimisation algorithms shows excellent performance of the MCS method in many cases, especially in low dimensions.

7.4 Conclusions The main benefits of the optimisation-based search are the lower costs in the case of many simultaneous parameters and an increased reliability of the results because of the continuous exploration of parameter space. Prerequisites for the applicability of this approach are appropriate parameterised models, fast and reliable trimming and linearisation procedures (necessary for efficient function evaluation) and robust optimisation software capable of addressing the challenge of solving NLP problems with possibly non-smooth, expensive to evaluate and noisy functions. Taking into account all these aspects, the best suited approach appears to be the trust-region DFO method. For functions with only a few variables, DFO typically requires relatively few evaluations of the problem function. For more difficult problems with many local minima, the MCS method combining local and global search appears to be a viable alternative to more expensive GA and SA methods. The acceptance of the optimisation-based clearance approach by the industry depends on several aspects. Since the optimisation-based clearance can be seen just as a straightforward (more powerful) extension of the classical approach, the effort to learn this method is almost negligible. In fact, the classical gridding-based approach can always be used as a standard option even in an optimisation-based clearance methodology. This is why, the first time setting up of the method is not much different than for the grid based approach. However, the usage of sophisticated optimisation tools requires special care when defining suitable smooth distance functions on basis of standard clearance criteria. Further, the implementation of fast and reliable procedures to evaluate these functions is of crucial importance for the success of the optimisation-based worst-case search. The reusability of software to cope with new aircraft models and control laws can be enforced by performing the optimisation-based clearance within a dedicated software environment which supports the interchange of different models and criteria. Within such an environment, the effort for a new analysis setup with different models and control laws is expected to be easily affordable. For maximum flexibility, such an environment has to provide additional facilities for experimenting with various optimisation techniques, different parameter sets, different criteria, different optimisation options etc. A clearance software environment satisfying all above requirements will be implemented as an add-on to the optimization based design environment MOPS of DLR (Multi-Objective Parameter Synthesis) [12].

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References 1. C. T. Kelley. Iterative Methods for Optimisation. SIAM, Philadelphia, 1999. 2. H.D. Mittelmann and P. Spellucci. Decision tree for optimization software. World Wide Web, http://plato.la.asu.edu/guide.html, 2001. 3. J. Nocedal and S. J. Wright. Numerical Optimization. Springer Series in Operations Research. Springer-Verlag, New York, 1999. 4. C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal. Algorithm 778. L-BFGS-B: Fortran subroutines for Large-Scale bound constrained optimization. ACM Transactions on Mathematical Software, 23:550–560, 1997. 5. M.J.D. Powell. Direct search algorithms for optimization calculations. In A. Iserles, editor, Acta Numerica, Vol. 7, pages 287–336. Cambridge University Press, 1998. 6. A. R. Conn, K. Scheinberg, and Ph. L. Toint. A derivative free optimization algorithm in practice. In Proc. of AIAA St Louis Conference, 1998. (http://www.fundp.ac.be/∼phtoint/pht/publications.html). 7. R. M. Lewis and V. Torczon. Pattern search methods for bound constrained minimization. SIAM Journal on Optimization, 9:1082–1099, 1999. 8. M.J.D. Powell. A direct search optimization method that models the objective and constraint functions by linear interpolation. In S. Gomez and J.P. Hennart, editor, Advances in optimization and numerical analysis, pages 51–677. Kluwer Academic Publishers, 1994. 9. P. Laarhoven and E. Aarts. Simulated Annealing: Theory and Applications. D. Reidel Publishing, 1987. 10. D. E. Goldberg. Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, Reading, MA, 1989. 11. W. Huyer and A. Neumaier. Global optimization by multilevel coordinate search. J. Global Optimizationn, 14:331–355, 1999. 12. H.-D. Joos, J. Bals, G. Looye, K. Schnepper, and A. Varga. A multi-objective optimisation-based software environment for control systems design. In Proc. IEEE CADCS Symposium, Glasgow, UK, 2002.

8 The HIRM+ Flight Dynamics Model Dieter Moormann EADS Deutschland GmbH?? , Military Aircraft MT62 Flight Dynamcis, 81663 M¨ unchen, Germany [email protected]

Summary. The major objective of the GARTEUR Action Group on Analysis Techniques for Clearance of Flight Control Laws FM(AG-11) is the improvement of the flight clearance process by increased automation of the tools used for modelbased analysis of the aircraft’s dynamical behaviour. What is finally needed are techniques for faster detection of the worst case combination of parameter values and manoeuvre cases, from which the flight clearance restrictions are be derived. The basis for such an analysis are accurate mathematical models of the controlled aircraft. In this chapter the HIRM+ flight dynamics model is described as one of the benchmark military aircraft models used within FM(AG-11). HIRM+ originates from the HIRM (High Incidence Research Model) developed within the GARTEUR Action Group on Robust Flight Control FM(AG-08). In building the HIRM+, additional emphasis has been put on realistic modelling of parametric uncertainties.

8.1 Introduction The HIRM+ has been developed from the HIRM, a mathematical model of a generic fighter aircraft originally developed by the Defence and Evaluation Research Agency (DERA, Bedford). The HIRM is based on aerodynamic data obtained from wind tunnel tests and flight testing of an unpowered, scaled drop model. The model was set up to investigate flights at high angles of attack (-50◦ ≤ α ≤ 120◦ ) and over a wide sideslip range (- 50 ◦ ≤ β ≤ +50◦ ), but does not include compressibility effects resulting from high subsonic speeds. The origin of the model explains the unconventional configuration with both canard and tailplane, plus an elongated nose (see Fig. 8.1). The aircraft is basically stable. However, there are combinations of angle of attack and control surface deflections, which cause the aircraft to become unstable longitudinally and/or laterally. Engine, actuator and sensor dynamics models have been added within FM-AG-08 to create a representative, nonlinear simulation model of a twin-engined, modern fighter aircraft. The model building was done by using the object-oriented equation-based modelling environment Dymola [4]. ??

The work on this project was conducted while the author was employed at DLR, Institute of Robotics and Mechatronics, Oberpfaffenhofen, 82234 Wessling, Germany

C . Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 121-139, 2002. Springer-Verlag Berlin Heidelberg 2002

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Fig. 8.1. High Incidence Research Model [1]

In building the HIRM+, the emphasis has been put on realistic modelling of parametric uncertainties. Parameters have been defined to specify uncertainties in mass, inertial data, position of the centre of gravity, aerodynamic control power derivatives, stability derivatives, and some coefficients in the actuator and engine dynamics. In spite of these changes, the nominal models of HIRM+ and HIRM (i.e., with all uncertain parameters set to zero) are essentially the same. Although variations of the uncertainty parameters affect the trim values of states and control surface deflections, due to the HIRM’s fairly linear aerodynamic derivatives over the specified flight envelope, the stability properties remain essentially unchanged Another aspect arising from the current industrial clearance practice is to allow the use of expected tolerance ranges of typical uncertain parameters (e.g., stability and control power derivatives) to be directly accessible in the nonlinear model. This allows the HIRM+ to mimic the industrial clearance approach, which is heavily based on both linear and nonlinear aircraft models. Usually, individual entries of the state-space matrices, with known physical meaning, are considered as uncertain and varied within the expected ranges.

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123

automatic code generation

aerodynamic controls & engine controls & gust inputs measurements & evaluation outputs

Fig. 8.2. The HIRM+ aircraft dynamics model

8.2 The HIRM+ Object Model The HIRM+ aircraft dynamics model in the upper part of Figure 8.2 consists of four basic blocks denoted as: actuator dynamics, engine dynamics, flight dynamics and sensor dynamics. Zooming into the flight dynamics model displays its internal structure, as given in the lower half of Fig. 8.2: The flight dynamics block incorporates the mass properties including equations of motion and the models of aerodynamics, variations in thrust, gravity, atmosphere, and gust disturbances [2, 3].

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In contrast to the (input-output) block-oriented description of the aircraft dynamics model, the flight dynamics model itself is specified using an acausal model formulation [4]. Due to the acausal approach, interconnections between components are not limited to signal flows but represent physical system interactions, like energy flows, or kinematic constraints. Automatic code generation is used to import the flight dynamics into the overall model. The outputs of the flight dynamics model, which are used as measurements for control and evaluation outputs are specified in Table 8.1. Table 8.1. Measurements and evaluation outputs of HIRM+ Name

Description

Unit

measurements p

y(1)

Body-axis roll rate

rad/s

q

y(2)

Body-axis pitch rate

rad/s rad/s

r

y(3)

Body-axis yaw rate

θ

y(4)

Pitch angle

rad

φ

y(5)

Bank angle

rad

ψ

y(6)

Heading angle

ax

y(7)

Body-axis x-acceleration

m/s2

ay

y(8)

Body-axis y-acceleration

m/s2

az

y(9)

Body-axis z-acceleration

m/s2 m/s

VA

y(10)

Airspeed

M

y(11)

Mach number

rad

-

h

y(12)

Altitude

α

y(13)

Angle of attack

rad

m

β

y(14)

Angle of sideslip

rad

Flight path angle

rad m/s

evaluation γ

y(15)

VG

y(16)

Ground speed (magnitude)

x

y(17)

Earth-axes x-position (north)

m

y

y(18)

Earth-axes y-position (east)

m

Fp1

y(19)

Thrust of engine 1 (left engine)

N

Fp2

y(19)

Thrust of engine 2 (right engine)

N

The inputs of the aircraft dynamics model (aerodynamic controls, engine controls, and gust inputs) are specified in Table 8.2.

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Table 8.2. Controls and gust inputs of HIRM+ Name δT S

u(1)

Description

Unit

Symmetric tailplane deflection

rad

δT D

u(2)

Differential tailplane deflection

rad

δCS

u(3)

Symmetric canard deflection

rad

δCD

u(4)

Differential canard deflection

rad rad

δR

u(5)

Rudder deflection

suction

u(6)

Nose suction

-

δT H1

u(7)

Throttle of engine 1 (left engine)

-

δT H2

u(8)

Throttle of engine 2 (right engine)

-

WXB

u(9)

Body-axes head wind

m/s

W YB

u(10)

Body-axes cross wind

m/s

WZB

u(11)

Body-axes vertical wind

m/s

The uncertain parameters of the HIRM+, their formulation, nominal values, upper and lower bounds, units and descriptions are given in sections 8.2.1 to 8.2.5. 8.2.1 Mass Characteristics and Geometric Data The body-object of Fig. 8.2 specifies the mass characteristics and the rigid body differential equations of motion with 6 degrees of freedom. For a derivation of these equations a reference such as [5] should be consulted. The HIRM+ mass characteristics are specified in Table 8.3 Variations in mass and moment of inertia are given by the following equations. For convenience, the uncertain parameters of the HIRM+ are denoted with an asterisk and parameters without, as their nominal values. The uncertainty itself is expressed by the subscript U nc: m∗ = (mU nc + 1) m 

0 −Ixz (1 + Ixz U nc ) Ix (1 + IxU nc ) 0 Iy (1 + IyU nc ) 0 I∗ =  0 Iz (1 + IzU nc ) Ixz (1 + IxzU nc )

 

The centre of gravity varies with respect to its nominal value which is defined as body geometric reference BGR, see Fig. 8.2): ∗ = Xcg + XcgU nc Xcg ∗ Ycg = Ycg + YcgU nc ∗ Zcg = Zcg + ZcgU nc

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D. Moormann Table 8.3. Inertial parameters Name

Nominal value

m

15296.0

Unit kg

Description Aircraft total mass

2

Ix

24549.0

kg m

Iy

163280.0

kg m2

y body axis moment of inertia

x body moment of inertia

Iz

183110.0

kg m2

z body moment of inertia

2

Ixz

-3124.0

Xcg

0

m

Centre of gravity location along x-axis

Ycg

0

m

Centre of gravity location along y-axis

Zcg

0

m

Centre of gravity location along z-axis

kg m

x-z body axis product of inertia w.r.t. body geometric reference BGR w.r.t. body geometric reference BGR w.r.t. body geometric reference BGR

Table 8.4. Inertial uncertain parameters Name

Nominal

[min; max]

Unit

Description

value mU nc

0

[-0.2; 0.2]

-

Uncertainty level of aircraft mass

XcgU nc

0

[-0.15; 0.15]

m

Centre of Gravity offset along x-axis from nominal Xcg , positive toward nose

YcgU nc

0

[-0.10; 0.10]

m

Centre of Gravity offset along y-axis from nominal Ycg , positive toward starboard

ZcgU nc

0

[-0.04; 0.04]

m

Centre of Gravity offset along z-axis from nominal Zcg , positive down

IxU nc

0

[-0.2; 0.2]

-

Uncertainty level of Ix

IyU nc

0

[-0.05; 0.05]

-

Uncertainty level of Iy

IzU nc

0

[-0.08; 0.08]

-

Uncertainty level of Iz

IxzU nc

0

[-0.2; 0.2]

-

Uncertainty level of Ixz

The parametric uncertainties in the HIRM+ mass characteristics are defined using the parameters given in Table 8.4 in terms of their nominal values (see Table 8.3) and their set of uncertain parameters.

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In some cases (e.g. mass) physical units are not shown, because the uncertainties are expressed in terms of percentages (±20% for mass) of the nominal value. 8.2.2 Aerodynamics The aerodyn-object of Fig. 8.2 describes the aerodynamic forces and moments. The aerodynamic force and moment coefficients for HIRM+ are given by the summation of several components [1]. Most components have the form Cab (c, d). The derivative for a force or a moment a with respect to b is determined by linearly interpolating between the values given in a look-up table as a function of the variables c and d. The basic aerodynamic parameters are specified in Table 8.5 Table 8.5. Aerodynamic parameters Name

Nominal value

Unit

Description

c¯

3.511

m

Mean aerodynamic chord

S

37.16

m2

Wing planform area

b

11.4

m

Wingspan

To allow a physically meaningful interpretation of parametric variations with a direct influence on the stability and control power derivatives, the uncertain parameters in the HIRM+ have been defined such that they can be directly recovered in the linearised models. This has the undesired effect that trim values are explicitly used in the definition of uncertain parameters, which means, that the nonlinear simulations are now trim point dependent through initial state components (e.g.,αtrim ) and initial control surfaces (e.g., δCStrim ). Thus, strictly speaking, even for the nonlinear model this approach permits only small manoeuvres close to the trim point. This approach is convenient, in that it allows model upgrades to be made at the level of the nonlinear model, prior to linearisation. In what follows the expressions of the uncertain aerodynamic moment coefficients are given, where trim values of various parameters are specified with the subscript trim (e.g., αtrim ). Uncertain pitching moment coefficient: ∗ = Cm + Cm0 U nc + CmδCS U nc (δCS − δCStrim ) Cm + CmδT S U nc (δT S − δT Strim ) + Cmα U nc (α − αtrim ) c¯ + Cmq U nc (q − qtrim ) VA 2

with Cm as the nominal pitching moment coefficient of HIRM, depending on δT S , δCS , etc.

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Uncertain rolling moment coefficient: Cl∗ = Cl + Cl0 U nc + ClδCD U nc (δCD − δCDtrim ) + ClδT D U nc (δT D − δT Dtrim ) + ClδR U nc (δR − δRtrim ) b b + Clp U nc p + Clβ U nc (β − βtrim ) + Clr U nc r 2 VA 2 VA with Cl as the nominal rolling moment coefficient of HIRM. Uncertain yawing moment coefficient: Cn∗ = Cn + Cn0 U nc + CnδCD U nc (δCD − δCDtrim ) + CnδT D U nc (δT D − δT Dtrim ) + CnδR U nc (δR − δRtrim ) b b + Cnp U nc p + Cnβ U nc (β − βtrim ) + Cnr U nc r 2 VA 2 VA with Cn as the nominal yawing moment coefficient of HIRM. Table 8.6. Uncertain parameters of aerodynamic stability derivatives Name

Nom.

[min; max]

Unit

[0 ; 0]

-

Description

value Cl0 U nc

0

Uncertainty in rolling moment

Cm0 U nc

0

[0 ; 0]

-

Uncertainty in pitching moment

Cn0 U nc

0

[0 ; 0]

-

Uncertainty in yawing moment

Cmα U nc

0

[-0.1; 0.1]

1/rad

Uncertainty in Cmα stability derivative

Clβ U nc

0

[-0.04; 0.04]

1/rad

Uncertainty in Clβ stability derivative, where: k = 1 for α < 12◦ , k = 2 for α > 20◦ , and k is linearly interpolated for 12◦ ≤ α ≤ 20◦ between 1 and 2.

Cnβ U nc

0

[-0.04; 0.04]

1/rad

Uncertainty in Cnβ stability derivative

Cmq U nc

0

[-0.1; 0.1]

-

Uncertainty in pitching moment derivative due to normalised pitch rate

Clp U nc

0

[-0.1; 0.1]

-

Uncertainty in rolling moment derivative due to normalised roll rate

Clr U nc

0

[-0.03; 0.03]

-

Uncertainty in rolling moment derivative due to normalised yaw rate

Cnp U nc

0

[-0.1; 0.1]

-

Uncertainty in yawing moment derivative due to normalised roll rate

Cnr U nc

0

[-0.05; 0.05]

-

Uncertainty in yawing moment derivative due to normalised yaw rate

In Tables 8.6 and 8.7 the ranges of the uncertain aerodynamic stability derivatives and control power derivatives are given. For some parameters, no

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129

value of uncertainty has been defined. These terms have been included to allow for future applications of the model. Table 8.7. Uncertain parameters of aerodynamic control power derivatives Name

Nom.

[min; max]

Unit

Description

CmδT S U nc 0

[-0.04; 0.04]

1/rad

Uncertainty in pitching moment derivative due to symmetrical tailplane deflection

CmδCS U nc 0

[-0.02; 0.02]

1/rad

Unc. in pitching moment derivative due to symmetrical canard deflection

ClδT D U nc

0

[-0.04; 0.04]

1/rad

Unc. in rolling moment derivative due to differential tailplane deflection

ClδCD U nc

0

[-0.02; 0.02]

1/rad

Unc. in rolling moment derivative due to differential canard deflection

ClδR U nc

0

[-0.006; 0.006]

1/rad

Uncertainty in rolling moment derivative due to rudder deflection

CnδT D U nc 0

[-0.02; 0.02]

1/rad

Unc. in yawing moment derivative due to differential tailplane deflection

CnδCD U nc 0

[-0.01; 0.01]

1/rad

Unc. in yawing moment derivative due to differential canard deflection

CnδR U nc

[-0.02; 0.02]

1/rad

Uncertainty in yawing moment derivative due to rudder deflection

value

0

8.2.3 Engine Dynamics Each engine-object of Fig. 8.2 is modelled as shown in Fig. 8.3. A throttle

Fig. 8.3. Engine dynamics model

demand of 0 selects idle which is 10 kN of thrust at sea level. A throttle demand of 1 corresponds to maximum dry thrust of 47 kN. Full reheat is selected when the throttle demand equals 2 and corresponds to a thrust

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D. Moormann

of 72 kN. The rate at which the thrust changes depends on whether the engine is in dry thrust or reheat. For dry thrust, the maximum rate of change is 12 kN/s whereas in reheat it is 25 kN/s. The sea level engine thrust is scaled with the relation of local density ρ to sea level density ρ0 . The engine setting angles are zero and so the thrust acts parallel to aircraft x-body axis. The engine positions with respect to the body geometric reference BGR are given in Table 8.8. Table 8.8. Engine parameters Name

Nom.

Unit

Description

value XT AP

-6.0

m

Body-axes x-position of thrust application point

YT AP

± 0.56

m

Body-axes y-position of thrust application point

0.35

m

Body-axes z-position of thrust application point

thridle

10 000

N

Idle thrust (at sea level)

thrdrymax

47 000

N

Maximum dry thrust (at sea level)

thrreheatmax

72 000

N

Maximum reheat thrust (at sea level)

ZT AP

thrdryrL

±12 000

N/s

Ratelimit at dry thrust

thrreheatrL

±25 000

N/s

Ratelimit at reheat thrust

Variations due to parametric uncertainties in engine rate limiters for dry thrust and reheat thrust are given by the following equations: thrdryrL ∗ = thrdryrL (1 + EngrLU nc ) thrreheatrL ∗ = thrreheatrL (1 + EngrLU nc ) The uncertainty level of the engine rate limits is given in Table 8.9. Table 8.9. Engine uncertain parameter Name

Nominal value

EngrLU nc

0

[min;

Unit

Description

max] [0 ; 0]

-

Uncertainty level of engine rate limits

8.2.4 Actuator Dynamics The actuator dynamics block of Fig. 8.2 is specified by the following transfer functions:

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131

Taileron actuator transfer function: T ∗ (s) =

1 (1 + 0.026 (1 + δTbw U nc ) s) (1 + 0.007692 s + 0.00005917 s2 )

with an uncertain rate limit defined as ± 80 (1 + δTrL U nc )◦ /s Canard actuator transfer function: T ∗ (s) =

1 (1 + 0.0157333 (1 + δCbw U nc )s + 0.00017778s2 )

with an uncertain rate limit defined as ±80 (1 + δCrL U nc )◦ /s Rudder actuator transfer function: T ∗ (s) =

1 (1 + 0.0191401 (1 + δRbw U nc ) s + 0.000192367s2 )

with an uncertain rate limit defined as ± 80 (1 + δRrL U nc )◦ /s For the actuator dynamics block, currently no values of uncertainty has been defined. These terms have been included for future applications of the model. Table 8.10. Actuation uncertain parameters Name

Nom. value

δTrL U nc

0

[min;

Unit

Description

max] [0 ; 0]

-

Uncertainty level of tailplane rate limit

δTbw U nc

0

[0 ; 0]

-

Uncertainty level of tailplane bandwidth

δCrL U nc

0

[0 ; 0]

-

Uncertainty level of canard rate limit

δCbw U nc

0

[0 ; 0]

-

Uncertainty level of canard bandwidth

δRrL U nc

0

[0 ; 0]

-

Uncertainty level of rudder rate limit

δRbw U nc

0

[0 ; 0]

-

Uncertainty level of rudder bandwidth

8.2.5 Sensor Dynamics To reduce the complexity of the overall model, and thus the computation times required by simulations, the sensor dynamics for HIRM are replaced by lower order approximated sensor models for the HIRM+, described by transfer functions. The HIRM+ sensor dynamics for body axis angular rates [p, q, r] and body axis accelerations [ax , ay , az ]: T ∗ (s) =

1 − 0.005346 s + 0.0001903 s2 1 + 0.03082 s + 0.0004942 s2

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D. Moormann

The HIRM+ sensor dynamics for airspeed, Mach-number, altitude, angle of attack and angle of sideslip [VA , M a, h, α, β]: T ∗ (s) =

1 1 + 0.02 s

The HIRM+ sensor dynamics for body axis attitudes and heading angle [ϕ, θ, ψ]: 1 T ∗ (s) = 1 + 0.0323 s + 0.00104 s2 A measurement error signal is added to the signal of α and β. These errors are assumed to be constant during the period of simulation: α∗ = α + αU nc β∗ = β + βU nc For the HIRM+ sensor dynamics block, currently no value of uncertainty for α- and β-measurement errors have been defined. These terms have been included for compatibility with the HIRM, in which these uncertainties had been used. Table 8.11. Sensor uncertain parameters Name

nom

[min; max]

Unit

Description

αU nc

0

[0 ; 0]

[rad]

Uncertainty in sensed angle of attack (added to the α-measurement signal)

βU nc

0

[0 ; 0]

[rad]

Uncertainty in sensed sideslip angle (added to the β-measurement signal)

8.3 Automated Model Generation for Parametric Time Simulations and Trim Computations The object model in Fig. 8.2 is graphically specified using components from the Flight Dynamics Library [3], that are instantiated with HIRM+ specific system model parameters. From this object model, simulation and analysis models of the aircraft system dynamics and documentation can be generated automatically (see Fig. 8.4). In the mathematical model building process, the equation handler of Dymola solves the equations according to the inputs and outputs of the complete HIRM+ model. Equations that are formulated in an object, but that

8 The HIRM+ Flight Dynamics Model physical model composition

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modelling

physical system model

3.511 15296.0

systemparameter

component libraries

y

dts dtd dr

u

V alpha beta

V alpha beta q gamma

y

8

u

8

specification of model inputs and outputs

dts dtd dr throttle1 throttle2

mathematical model building

(interactive)

(automatic) sorted & solved equations for simulation

sorted & solved equations for trim calculation

codegeneration

(automatic) simulation model e.g. Matlab/Simulink(cmex-) S-function

trim code e.g. Matlab/Simulink(cmex-) S-function

Fig. 8.4. Model building process

are superfluous for capturing the behaviour of the particular model, are automatically removed. The result is a nonlinear symbolic state-space description with a minimum number of equations for this task x˙ = f (x, u, p) y = h(x, u, p) From the symbolic description, numerical simulation code for different simulation environments is generated automatically. In this way, it is possible to generate, for example, a Matlab-Simulink m-file or cmex-code, or CCode according to the neutral DSblock standard [7], which can be used in any simulation environment, being capable of importing C-Code. From the sorted and solved equations for simulation, symbolic analysis code can be generated, describing a parameterised state-space model. This

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code can be used to extract in an automated procedure the so-called Linear Fractional Transformation (LFT) standard form, that may serve as the basis for µ robustness analysis. A detailed description of the generation of an LFT representation from an object model as depicted in Fig. 8.4 can be found in [8]. One of the key aspects of the successful usage of an optimisation-based clearance methodology is an efficient trimming approach. The trimming of HIRM+ is a very challenging computational task, involving the numerical solution of a system of 60 nonlinear equations for the stationary values of state and control variables appearing in the HIRM+ state model. The difficulties mainly arise because of the lack of differentiability of the functions due to the presence of various look-up tables used for linear interpolations. Severe nonlinearities in the engine model and in the aerodynamics, as well as the presence of control surface deflection limiters make the numerical solution of this high order system of equations very difficult. To manage the trimming problem, an highly accurate and efficient approach has been employed in [2]. The facilities of an equation based modelling environment as Dymola [4] allows the generation of C-code for an inverse model to serve for trimming. Such a model has as inputs the desired trim conditions (such as Va , α, . . . ) and as outputs the corresponding equilibrium values of trimmed state (x) and control vectors (such as δT S , δCS ,. . . ). Dymola generates essentially explicit equations for the inverse model by trying to solve the 60th order nonlinear equation symbolically. Even if a symbolic solution cannot be determined, Dymola is still able to reduce the burden of solving numerically a 60th order system of nonlinear equations to the solution of a small core system of 13 nonlinear equations which ultimately must be solved numerically. Thus, the trimming procedure based on such an inverse model is very fast and very accurate.

8.4 Flight Conditions and Envelope Limits The analysis of HIRM+ is restricted to the flight conditions defined in Table 8.12. Depending on the clearance problem, the equilibrium conditions in these Table 8.12. Set of flight conditions for clearance analysis FC No.

F C1

F C2

F C3

F C4

F C5

F C6

F C7

F C8

M

0.2

0.3

0.5

0.5

0.6

0.7

0.8

0.8

h [ft]

5,000

25,000

40,000

15,000

30,0000

20,000

5,000

40,000

points are defined by the trimming conditions for straight and level flight for given γ, M and h or pull-up manoeuvres for given α, M and h. For the

8 The HIRM+ Flight Dynamics Model

135

variation of the α the interval [−15◦ , 35◦ ] has been chosen, and for gridding a step size ∆α = 2◦ has been suggested. When several aerodynamic uncertainties are simultaneously used in the analysis, reduction factors must be applied on their absolute values as specified in Tables 8.6 and 8.7. The values of reduction factors for different numbers of aerodynamic uncertainties are given in Table 8.13. Table 8.13. Reduction factors for simultaneous aerodynamic uncertainties Number of aerodynamic uncertainties Reduction factor

2

3

4

≥5

0.62

0.46

0.37

0.31

Due to load factor limitations (section 8.4.1) and control surface deflection limits (section 8.4.2) it is not possible to trim all flight conditions of Table 8.12 for all angles of attack between −15◦ and 35◦ . This is already true for the nominal model, for which all uncertainty parameters are set to zero. The number of not trimmable points in the flight envelope increases with more uncertainty parameters being used. This fact must be accounted for during the assessment. 8.4.1 Load Factor Limits The clearance task is restricted to a ”true” flight envelope, where additional restrictions on variables must be satisfied. The first condition is to restrict the load factor to meet −3 [g] ≤ nz ≤ 7 [g] All flight conditions, where this condition is violated can not be cleared. A preliminary check involving only nominal cases has been performed. In Fig. 8.5 the values of load factors versus α for the eight flight conditions are presented. It can be seen that, because of violation of load-factor limit, F C6 is defined only for α ∈ [−9◦ , 29◦ ] and F C7 is defined only for α ∈ [−2◦ , 12◦ ]. It is helpful to have the dependence of nz on various parameters in mind. In general, nz can be expressed as nz = −

ρ VA 2 SCZ 2mg

and thus depends on the Z-force aerodynamic coefficient CZ , altitude (via air density ρ), airspeed VA , and mass of the aircraft m. For HIRM+, CZ is given by [1] CZ = CZδT S (α, δT S ) + CZδCS (α, δT S )δCS + 1.7555 CZq (α, δCS )

q VA

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15

10

Load factors for nominal cases FC 1 FC2 FC3 FC4 FC5 FC6 FC 7 FC 8

n

z

7 5

0 −3 −5

−10 −15

−10

−5

0

5

10 AoA [deg]

15

20

25

30

35

Fig. 8.5. Nominal load factors for HIRM+

Because δCS = 0◦ (the canards are not used) and the term 1.7555 CZq (α, δCS ) VqA being much smaller than CZδT S , CZ can be approximated by the single term CZ ≈ CZδT S (α, δT S ), where the dependence on δT S , being not significant, can be dropped. Thus, if we neglect the pitching motion, nz for straight and level flight can be expressed as nz ≈ −

ρ VA 2 SCZδT S (α) 2mg

and depends finally only on α, altitude (influence on air density), the airspeed, and the mass of the aircraft. The uncertain parameters, with exception of the mass, do not have any influence on the values of nz . A remarkable property of HIRM+ is that, independently of any values of uncertain model parameters, nz ≈ 0 for α close to 2◦ , because CZδT S (2◦ ) ≈ 0. This particular feature of HIRM+ can be observed in Fig. 8.5. 8.4.2 Control Surface Deflection Limits A second set of conditions originate from the deflection limits on taileron and rudder actuators:

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−40◦ ≤ δT S + δT D ≤ 10◦ −40◦ ≤ δT S − δT D ≤ 10◦ −30◦ ≤ δR ≤ 30◦

Nominal actuator deflections for δTS+δTD

20

10

Actuator deflections

0

−10

−20

−30

−40 −15

FC 1 FC2 FC3 FC4 FC5 FC6 FC7 FC8 −10

−5

0

5

10 AoA [deg]

15

20

25

30

35

Fig. 8.6. Summation of symmetrical and differential tailplane deflection

Nominal actuator deflections for δTS−δTD

20

10

Actuator deflections

0

−10

−20

−30

FC1 FC2 FC3 FC4 FC5 FC6 FC7 FC 8

−40 −15

−10

−5

0

5

10 AoA [deg]

15

20

25

30

35

Fig. 8.7. Difference between symmetrical and differential tailplane deflection

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All flight conditions, where the above conditions are violated, lead to saturation of control surfaces, and thus are automatically not cleared. For the nominal cases, the variations of δT S + δT D and δT S − δT D for the rigid body equations of HIRM+ can be seen in Figures 8.6 and 8.7 1 . The values computed in these figures have been determined with the inverse trim routine where these limits are not present, and therefore the trimming is always possible. This is intentionally done, in order to make trimming numerically easier and to be able to study points also slightly outside of the limits for the control surface actuators. It follows that the trimming results are valid only if the above bounds are fulfilled. As a practical consequence, the above conditions must be checked after each trim computation. Ignoring these conditions leads to strange (but expected) effects, as for example, zero columns in the input matrix B of the linearised HIRM+ in F C1 for α ∈ [−15◦ , −10◦ ] because of saturation of inputs. This further leads to identically zero transfer function, when breaking the symmetric taileron loop. According to these plots, for the nominal parameters, F C1 is defined only for α ∈ [−9◦ , 35◦ ] because of violation for α ∈ [−15◦ , −10◦ ] of the conditions δT S ± δT D ≤ 10◦ . The variation of δR is within the allowed limits and is not shown here. Based on nominal case analysis results, the ”true” set of flight conditions to serve for analysis purposes must be restricted.

References 1. Ewan Muir. The HIRM design challenge problem description. In J. F. Magni, S. Bennani and J. Terlouw, editors, Robust Flight Control, A Design Challenge, Lecture Notes in Control and Information Sciences, vol. 224, pp. 419–443, Springer Verlag, Berlin, 1997. 2. D. Moormann. Automatisierte Modellbildung der Flugsystemdynamik (Automated Modeling of Flight-System Dynamics). Dissertation, RWTH Aachen. VDI Fortschrittsberichte, Mess-, Steuerungs- und Regelungstechnik, Reihe 8, Nr. 931, ISBN: 3-18-393108-7, 2002. 3. D. Moormann and G. Looye. The Modelica Flight Dynamics Library. Modelica 2002, Proceedings of the 2nd International Modelica Conference. Oberpfaffenhofen, Germany, March 18-19, 2002. 4. H. Elmqvist. Object-Oriented Modeling and Automatic Formula Manipulation in Dymola. In Scandinavian Simulation Society SIMS’93, Kongsberg, Norway, June 1993. 5. R. Brockhaus. Flugregelung. Springer Verlag, Berlin, 1994. 6. J. F. Magni, S. Bennani and J. Terlouw. Robust Flight Control, A Design Challenge. Lecture Notes in Control and Information Sciences, vol. 224, Springer Verlag, Berlin, 1997. 7. M. Otter and H. Elmqvist. The DSblock Model Interface for Exchanging Model Components. Simulation, 71:7–22, 1998. 1

Figures 8.6 and 8.7 are the same for α ≤ 20◦ , because δT D is zero for a trimmed straight-and-level-flight within this α-limit. δT D becomes different from zero due to a lateral asymmetry in the aerodynamic model for α > 20◦ .

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8. A. Varga, G. Looye, D. Moormann, and G. Gr¨ ubel. Automated generation of LFT-based parametric uncertainty descriptions from generic aircraft models. Mathematical and Computer Modelling of Dynamical Systems, 4:249–274, 1998.

9 The RIDE Controller David Bennett BAE SYSTEMS Aerodynamics (W427D) Warton, UK [email protected]

Summary. The Robust Inverse Dynamic Estimation (RIDE) control laws and HIRM+ aircraft model provide a suitable basis for engineering research within the Matlab / Simulink design environment. With the implementation of a dynamic pressure scheduler, the control laws provide consistent handling qualities across an extended flight envelope over which flight control analysis techniques can be applied. The model offers reliable trimming, linearisation, simulation and analysis capabilities through the functionality of the model and Matlab.

9.1 Introduction The following section describes the integration of the Robust Inverse Dynamic Estimation (RIDE) control laws with the HIRM+ aircraft model . The RIDE control laws and the HIRM+ were developed separately, and therefore the task was to integrate the two and ensure that the combined model was suitable for robustness analysis research by the GARTEUR Action Group, AG11. The RIDE control laws were developed for the HIRM model by DERA Bedford, for the GARTEUR Group FM-AG-08. A full description of the design process and implementation of RIDE is provided in [1] and therefore, it will not be repeated in this chapter. Replacing the HIRM model with that of the HIRM+ did not require any modifications to the controller design for the nominal model, i.e. with all the uncertainties set to zero. The only significant difference between the HIRM and HIRM+ that may have influenced the performance of the control system, is the reduced complexity of the sensor models. However, the modified sensor models are an approximation of the original models, i.e. there is no significant difference in low frequency (< 5 Hertz) dynamics, and therefore, no significant difference in control system performance. Modifications have been made to the control laws, to improve their performance, and to extend their Matlab/Simulink functionality for release to the GARTEUR Action Group. To summarise, the changes include: – – – –

improved trimming, implementation of a dynamic pressure gain scheduler, an improved process for controller initialisation, and improved linearisation.

C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 141-150, 2002. Springer-Verlag Berlin Heidelberg 2002

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9.2 Augmenting HIRM+ with the RIDE Control Laws The complete HIRM+RIDE model was constructed by integrating the HIRM+ model described in the previous chapter and the RIDE control laws, extracted from the existing RIDE/HIRM model, which was provided by DERA. As both were readily available in Simulink, only minor modifications were required to integrate them. The resulting Simulink model, hirmpride.mdl, is shown in Fig. 9.1, which provides consistent notation between its individual components. In addition to the aircraft model and control laws, the model shown in Fig. 9.1 provides functionality for: – specifying pilot demands: roll rate, pitch rate, sideslip and airspeed, – specifying wind inputs: x, y and z axis components, and – forcing the actuator inputs directly. The block trim inputs provide the actuator inputs with the correct trim settings, calculated by the Matlab file trimhirmplus.m.

Fig. 9.1. The augmented HIRM+RIDE Simulink model, hirmpride.mdl

9.3 Trimming the RIDE Control Laws The HIRM+RIDE results presented in [1] showed slight mis-trims at the start of time responses. To remove the mis-trims a two-part trimming strategy was developed and implemented. Firstly, the sensor outputs are initialized in order to set the schedulers to obtain the correct trimmed state, and secondly, the model is initialised to force the control surfaces to their correct positions. The second part is achieved by summing the trimmed control surface deflections

9 The RIDE Controller

143

(block: trim inputs) with the controller outputs. This requires the controller outputs to be zero for a perfect trim. The RIDE control laws contain an inverse dynamics loop that outputs non-zero values for all trimmed flight conditions. As this inner-loop is summed with the integrators to form the output of the control laws, the integrators are initialised so that the total control law output is zero on all paths, as shown in Fig. 9.2.

Fig. 9.2. Initialisation of the integrators

To demonstrate the trimming process, the model was trimmed at Mach=0.4, h=10000 ft and a small amplitude, 1 deg/sec, pitch rate demand was applied. The results for the pitch rate response, before and after the modifications to the trimming process, can be seen in Figs. 9.3 and 9.4. It can be seen that the pitch rate trimming error has been eliminated.

Fig. 9.3. Pitch rate demand showing mis-trimmed response

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Fig. 9.4. Pitch rate demand showing correct trim

9.4 Dynamic Pressure Scheduler Implementation The recommended design envelope for the HIRM control laws as defined in [2] is Mach 0.15 to 0.5 and height from 100 to 20000 ft, with an AoA range of -10 degs to +30 degs, and a sideslip range of 10 degs. To accomplish this, a dynamic pressure scheduler was implemented to provide consistent handling qualities across the required flight envelope. The dynamic pressure scheduler is implemented using one equation and one look-up table, designed for the flight condition Mach=0.5, h=15000 ft. The method is best described with the aid of the following equation, which defines dynamic pressure: 1 2 (9.1) ρv 2 Fig. 9.5 shows the characteristics of the dynamic pressure gain scheduler. The gain scheduler will traverse a single curve as Mach number varies, and will change to a different curve as the altitude varies, thereby providing the gain value that produces the desired handling qualities for the current flight condition. The dynamic pressure scheduler is implemented using the equation: ¶ µ q5 (9.2) Gain = g5 qbar dynamic pressure =

Where:

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145

– g5 is the constant gain value from the original RIDE control laws (at Mach=0.5, h=15000 ft), – q5 is the dynamic pressure (at Mach=0.5, h=15000 ft, which remains constant), and – qbar is the dynamic pressure at the current flight condition.

Fig. 9.5. Dynamic pressure scheduler

The method works by calculating the ratio of dynamic pressure at Mach=0.5, h=15000 ft, to that at the current flight condition, and then modifying the gain at Mach=0.5, h=15000 ft by this ratio. This approach works well where air compressibility effects are not significant (or have not been modelled) and will give reliable results at low Mach numbers. The benefits of the modified scheduler are demonstrated by comparing Figs. 9.6 and 9.7, which show a comparison of the aircraft pitch responses over the flight envelope, before and after the addition of the dynamic pressure scheduler. Both Figures show the flight envelopes and pitch rate responses to pilot step demands at the envelope corner points. A pitch rate step demand of 1 deg/sec was commanded at 1 second, and removed at 4 seconds. It can be clearly seen that by designing a controller scheduler that takes into account both Mach number and height (in comparison to one that uses Mach number only), significantly improved and consistent responses, and therefore consistent handling qualities are obtained. It can also be seen that the flight envelope has been significantly increased.

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Fig. 9.6. Flight envelope with the Mach number scheduler

9 The RIDE Controller

Fig. 9.7. Flight envelope with the dynamic pressure scheduler

147

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9.5 Linearisation of the HIRM+RIDE The following presents an automated procedure for linearising the HIRM+ and RIDE control laws, that have been shown to produce excellent robustness. The procedure is contained completely within the trimming program trimhirmplus.m, which is executed when trimming the HIRM+. The linearisation process is performed in three stages. Initially a linear model is obtained for the HIRM+ using the results from the trimming routine and Matlab linearisation commands. Secondly, the RIDE control laws are linearised, and thirdly, the two linear models are integrated to form a state-space representation of the HIRM+RIDE model as described below, where the input and output vectors are shown in Fig. 9.8, and are defined as: ym yr ye uref uc ud ref mes

– – – – – – – –

sensor output (fed back to the controller) controller-generated actuator demands evaluation outputs (not fed back to the control laws) pilot demands actuator (control) inputs, gust (disturbance) inputs, controller reference input (trim actuator setting) controller measurement input

Fig. 9.8. Connectivity of the HIRM+ and RIDE models

The linearised HIRM+ is represented by the following state and output equations ¸ · uc (t) (9.3) x(t) ˙ = Ax(t) + [ B1 B2 ] ud (t) · ¸ · ¸ ¸ · ¸· ym (t) C1 0 D1 uc (t) = (9.4) x(t) + C2 0 D2 ye (t) ud (t)

9 The RIDE Controller

Similarly, for the linearised RIDE control laws,

149

¸ ref (t) (9.5) mes(t) ¸ · ref (t) (9.6) yr (t) = Cride xride (t) + [Dride,1 Dride,2 ] mes(t) The two sets of state-space equations are then combined using the following equations: ·

x˙ ride (t) = Aride xride (t) + [Bride,1 Bride,2 ]

mes = ym uc = uref + yr

(9.7) (9.8)

Combining equations (9.3) and (9.4), and (9.5) and (9.6) by using equations (9.7) and (9.8), gives the following state and output equations for the linearised HIRM+ and RIDE. ¸ · ¸· ¸ · A + B1 Dride2 C1 B1 Cride x(t) x(t) ˙ = + Bride2 C1 Aride x (t) x˙ ride (t)  ride  ¸ ref (t) · (9.9) B1 Dride1 B1 Dride2 D1 B1  ud (t)  Bride1 Bride2 D1 0 uref (t)   · ¸ · ¸ · ¸· ¸ ref (t) ym (t) C1 0 x(t) 0 D1 0  ud (t)  = + (9.10) ye (t) C2 0 0 D2 0 xride (t) uref (t) Equations (9.9) and (9.10) are coded into the file trimhirmplus.m, resulting in the linearisation being performed automatically when the HIRM+ is trimmed.

9.6 Verification of the Linearised Model Fig. 9.9 shows the results of the linearisation process. The results from small amplitude non-linear and linear responses are overplotted to compare the differences in the two representations of the model for the flight condition Mach=0.4, h=10,000 feet. The figure shows the results of a 1 deg/sec, step demand in pitch rate, initiated after 1 second and held for a duration of 3 seconds. The results show a very accurate match in pitch rate, but the Mach number plot shows a slight difference. Engine rate limiting due to the speed hold function will contribute to this this difference, since rate limiting effects will not be captured by the linearisation routine, and hence the non-linear Mach number response is expected to reduce at a slightly lower rate. These results and those obtained for the other flight conditions were considered to be satisfactory in terms of the verification of the linearisation process and the resulting models.

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Fig. 9.9. Response to a small amplitude pitch rate demand

References 1. E.A. Muir. HIRM Design Challenge Presentation Document: The Robust Inverse Dynamics Estimation Approach. Technical Publication TP088-28, Group for Aeronautical Research and Technology in Europe, Technical Report, GARTEUR-FM(AG-08), 1997. 2. E.A. Muir. Robust Flight Control Design Challenge, Problem Formulation and Manual: The High Incidence Research Model (HIRM). Technical Publication TP088-4, Group for Aeronautical Research and Technology in Europe, Technical Report, GARTEUR-FM(AG-08), 1997.

10 Selected Clearance Criteria for HIRM+RIDE Federico Corraro? Centro Italiano Ricerche Aerospaziali Flight System Department Via Maiorise, 81043, Capua (CE) Italy [email protected]

Summary. In this chapter the main requirements for the HIRM+RIDE clearance problem are given. The HIRM+ flight envelope is firstly introduced and the points where clearance criteria have to be checked are defined. The uncertainty parameters to be considered for the analysis are listed and categorised. Finally, each clearance criterion selected for the HIRM+ is described and mathematically defined.

10.1 Introduction The industrial flight control law clearance is an extensive verification process based on several test criteria which can be grouped into four classes: I. II. III. IV.

Linear stability criteria Aircraft handling/PIO criteria Non-linear stability criteria Non-linear response criteria

In order to define a clear HIRM+RIDE [1] benchmark problem for verifying the capabilities of analysis techniques, a selection of criteria from the above set has been performed and their definition has been mathematically formulated in order to avoid interpretation ambiguities. The selected criteria are: 1. 2. 3. 4.

Stability margin criterion (class I) Unstable eigenvalues criterion (class I) Average phase rate and absolute amplitude criteria (class II) Angle of attack/normal load factor limit exceedance criterion (class IV)

The benchmark definition is based on the description of the unaugmented HIRM model [2] and on the description of the original RIDE (Robust Inverse ?

The content of this chapter is, to a large extent, based on the GARTEUR report [4] TP-119-2-A1v2 edited by S. Scala, F. Karlsson (SAAB) and U. Korte (EADSM). The author whishes to thank the above mentioned people for their support in revising the report [4] for the scope of this book.

C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 151-165, 2002. Springer-Verlag Berlin Heidelberg 2002

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Dynamics Estimation) flight control laws for the HIRM [3]. Some degree of freedom has been still left on how to define worst cases and robustness measures.

10.2 Flight Envelope and Model Uncertainties Taking into account the limited scope of this benchmark problem, the analysis has been restricted to the discrete set of flight conditions defined in Table 10.1 and shown in Fig. 10.1. For each flight condition in Table 10.1, several flight cases must be considered. These flight cases are equilibrium conditions, in straight and level flight and in pull-up manoeuvres, characterised by different values of AoA and load factors, up to the maximum AoA or load factor nz (which respectively range [-15˚,35˚] and [-3g,7g], as reported in [1]).

Fig. 10.1. Flight envelope of the HIRM+ with RIDE control laws

It should be noted that in reality, industry needs to clear the flight control laws across the whole flight envelope, and that reducing the analysis to a limited number of selected flight conditions is only done for the HIRM+ benchmark problem to. As an example, for an aircraft with a flight envelope similar to the HIRM, a typical clearance analysis would be performed on a grid in the flight envelope (M, h), with typical steps of ∆M =0.2, ∆h=5,000ft.

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Table 10.1. Set of points in the flight envelope for the HIRM benchmark problem

FC No. FC1 M 0.2 h [ft] 5,000

FC2 0.3 25,000

FC3 0.5 40,000

FC4 0.5 15,000

FC5 0.6 30,000

FC6 0.7 20,000

FC7 0.8 5,000

FC8 0.8 40,000

In areas of the flight envelope where problems are expected, a more dense grid is used. Methods which can potentially perform ”global” analysis are desired because they allow clearance of whole sets of flight conditions (the entire envelope in the limit) in one shot, instead of single points one at a time. If a model linearisation is performed for the analysis, the linearised models should be those obtained in straight and level flight conditions and in pull-up manoeuvres 1 (with a given AoA grid spacing), for the selected points in the flight envelope. The HIRM+ model contains several uncertain parameters which are defined in Chapter 8 and [1], together with their nominal values and their uncertainty ranges. For the scope of this benchmark problem, the variation of the aircraft dynamics due to the model uncertainties listed in Table 10.2, has to be considered. Furthermore, separate analyses for longitudinal and lateral-directional uncertainties should be performed. In Table 10.2, a rating of the importance of the uncertain parameters of the HIRM+ model has also been given, both for longitudinal and for lateraldirectional axes, see Chapter 2. The intention is to reduce the number of parameters to be taken into account during a preliminary analysis of limited time length. Parameters rated as category 1 are the most relevant for clearance, and therefore they must be taken into account in the analysis from the beginning. Parameters rated as category 2 are less relevant and their variation can be ignored during a preliminary analysis. It should also be noted that the reduction factors in Table 10.3 have to be applied to the ranges of aerodynamic uncertainties only when several of them are applied simultaneously for the clearance analysis. This assumption is made to avoid unduly pessimistic assumptions from being made and is based on a probability argument. Regarding the figures in Table 10.2 and the classification of the uncertainties into two categories of decreasing relevance to clearance, the following should be noted: a problem in itself is to establish which uncertainties should be in which category for an aircraft. Aircraft designers usually have a good idea which parameters matter, firstly from their experience with earlier projects, and secondly, from their understanding of the aircraft’s linearised equations of motion and associated transfer functions. It is common practice to support this knowledge with a preliminary analysis with only one uncertainty at a time, in order to confirm the degree of importance of every uncertainty, and then to repeat the analysis with different sets of simultane1

Usually, trimmed banked turns for different aircraft load factors also considered.

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Table 10.2. Uncertain parameters for the HIRM+RIDE benchmark problem Longitudinal axis Lateral axis Category 1 Category 2 Category 1 Category 2 (most relevant) (less relevant) (most relevant) (less relevant) Xcg [-0.15;0.15] m [-0.2;0.2] Ycg [-0.1;0.1] ClδT D [-0.04;0.04] Iy [-0.05;0.05] Zcg [-0.04;0.04] Ix [-0.2;0.2] ClδCD [-0.02;0.02] Cm α [-0.1;0.1] Ixz [-0.2;0.2] Iz [-0.08;0.08] ClδR [-0.006;0.006] CmδT S [-0.04;0.04] CmδCS [-0.02;0.02] Clβ K [-0.04;0.04]∗ CnδT D [-0.02;0.02] Cm q [-0.1;0.1] Cnβ [-0.04;0.04] CnδCD [-0.01;0.01] Cnr [-0.05;0.05] CnδR [-0.02;0.02] C lp [-0.01;0.01] C lr [-0.03;0.03] Cn p [-0.1;0.1] ∗ K = 1 for α < 12◦ , K = 2 for α > 20◦ and K is linearly interpolated between 1 and 2 for 12◦ ≤ α ≤ 20◦ . Table 10.3. Weights on simultaneous aerodynamic tolerances Simultaneous number of aerodynamic uncertainties 2 3 4 ≥5

w = Reduction factor on uncertainty range 0.62 0.46 0.37 0.31

ous uncertainties, to which the reduction factors of Table 10.3 are applied. Within this analysis it is reasonable to assume that no more than five uncertainties at a time should be considered. Indeed, increasing the number of simultaneous uncertainties would lead to smaller values of the reduction factor in Table 10.3, such that the uncertainty set around the nominal point would become very small, and therefore, the analysis would give results not significantly different from those obtained for the nominal condition. In order to clearly define the selected clearance criteria for this benchmark problem, the following generic definition of the uncertainties is needed. Let k be the number of uncertain parameters to be taken into account in some criterion. For the i-th uncertain parameter, define Π i as the interval in which the parameter can vary according to Table 10.2 and the definition in Chapter 8 and [1]. The actual uncertainty range to be used in the analysis for the i-th parameter is defined as ½ Πi if the i-th uncertainty is not an aerodynamic parameter Πi,w = wΠi if the i-th uncertainty is an aerodynamic parameter where w is the reduction factor given in Table 10.3, when several aerodynamic uncertainty parameter are employed simultaneously in the analysis. For ex-

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ample, if the i-th uncertain parameter is cmαU nc and three aerodynamic uncertain parameters are considered simultaneously, then Πi = [−0.1, 0.1], w = 0.46 and Πi,w = wΠi = [−0.046, 0.046]. The complete uncertainty set is Π = Π 1,w ×. . . ×Π k,w ⊂Rk , i.e. the hyper-rectangle in which the vector of k uncertain parameters varies. In what follows, we denote by π ∈Π a particular value of the vector of uncertain parameters in the uncertainty set Π.

10.3 Stability Margin Criterion (Class I) This criterion requires identification of all flight cases (in terms of M , altitude and AoA) where the Nichols plot stability margin boundaries of Figs. 10.2 and 10.3 (see also [2]) are violated. It is also necessary to identify, for each flight condition, which uncertainty parameter values lead to the biggest violation - i.e. to define the worst-case tolerance combination. 6

4.5

4

2

Gain [dB]

1.5

−180

0

−145

−1.5

−2

−4

−6 −210

−4.5

−200

−190

−180

−170 −160 Phase [°]

−150

−140

−130

−120

Fig. 10.2. Nichols plot exclusion zones for single-loop analysis with uncertainties

Note that for clearance purposes, only violations of the boundaries are of interest, but for analysis and understanding purposes, the worst-case combination is of interest, even if the boundaries are not violated. This knowledge becomes important when considering possible future developments of an aircraft, as it gives an indication of what changes might be possible.

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F. Corraro 4

3

3

Gain offset [dB]

2 1

1

0

0

30

−1

−1

−2 −3

−3

−4 −30

−20

−10

0

10 20 Phase offset [°]

30

40

50

60

Fig. 10.3. Gain and phase offsets for multi-loop analysis with uncertainties

This criterion requires two analyses: a single-loop analysis and a multiloop one. Note that at least the single-loop analysis must be performed for this benchmark problem. In the following, details on both analyses are given. a) Single-loop analysis. The open-loop Nichols plots of the frequency response is obtained by breaking the loop at the input of each actuator, one at a time, while leaving the other loops closed. The frequency response should avoid the region shown in Fig. 10.2. This test is carried out mainly to assess the sensitivity of the system to changes in the dynamics of each actuation system, and to ensure that the system maintains adequate stability margins. It also gives a good indication of the sensitivity of the system to changes in aerodynamic control power. Note that when performing the frequency response, a gain of -1 needs to be included on the input or the output, to obtain the correct phase response. This requirement should be satisfied for each control loop. Note also that the RIDE control laws of the HIRM+ only use symmetrical thrust, symmetrical and differential tailplane, and rudder for control, while symmetrical and differential canards, nose suction and differential thrust are not used. b) Multi-loop analysis. The closed-loop system should be able to withstand the application of simultaneous and independent gain and phase offsets at the input of each of the actuators, without becoming unstable. This test is mainly carried out to check the system sensitivity to simulta-

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neous changes in the dynamics of the actuation systems and is aimed at identifying problem cases that might be missed by the single-loop tests. Simulataneous changes in actuation dynamics can occur, for example, due to a reduction in hydraulic pressure. The corresponding perturbation matrix P will be of the form: P =diag(K1 e−jϕ1 ,. . . ,K 4 e−jϕ4 ) Where Ki and ϕi are gain and phase offsets respectively, taking values in the region shown in Fig. 10.3. Note that P should be placed in series in front of the actuators, giving an uncertain gain scaling in the range [0.7, 1.4] and a maximum phase lag of 30deg on the input to the actuators. The matrix P is of dimension 4 by 4, since four controls are used in the RIDE control laws: symmetrical thrust, symmetrical tailplanes, differential tailplanes and rudder. For the scope of this benchmark, the minimum set of points to be tested can be the four corners of the region in Fig. 10.3. This criterion is not only a pass/fail test. It is further required to give the combination of uncertain parameters that lead to the worst case stability margin. Therefore, a suitable definition of the stability degree, say ρ, is required to identify the worst case.

6

ρ=1

4

Gain [dB]

2

O

0

B

A

D

C

−2 ρ = 0.57

−4

−6 −210

−200

−190

−180

−170 −160 Phase [°]

−150

−140

−130

−120

Fig. 10.4. Definition of a possible stability degree by scaling the exclusion region

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Several different possibilities for the definition of ρ exist [4]. For example, one could be to assign a normalised stability degree of unity to the regions shown in Figs. 10.2 and 10.3. and to scale the region of perturbation by preserving its aspect ratio. Smaller regions will have a stability degree less than one and greater regions will have a stability degree greater than one. The stability degree attained will be defined by the boundary of the greatest region around the critical point in the Nichols diagram, that is not crossed by the set of Nichols plots of the open-loop system (for the single-loop test) or the boundary of the greatest region for which all the internal perturbations do not destabilise the system (for the multi-loop test). An example is presented in Fig. 10.4 for the single-loop test. An hypothetical transfer function (continuous line) is plotted in the Nichols diagram against the stability margin boundary defined in the criterion, which is drawn with a bold line. The example transfer function crosses the boundary and therefore violates the requirements. A second region, of the same shape as the criterion boundary, is plotted in the figure. This is the greatest region having that shape that is not crossed by the transfer function line. The stability degree at the criterion region (with bold borders) is taken as unity. Thus, following the definition above, the stability degree at the inner region is defined by the ratio: ρS =

OB ∼ 20◦ ∼ = ◦ = 0.57 35 OA

10.4 Unstable Eigenvalues Criterion (Class I) For this criterion it is required to identify the flight cases (in terms of M , altitude and AoA) where unstable closed-loop eigenvalues (i.e. those with positive real part) occur, and for what tolerance combination these eigenvalues have the largest value of their real part. This test is to determine the most severe cases of divergent modes in the closed-loop system, to allow an assessment of their acceptability in terms of their influence on aircraft handling. A minimal requirement is to consider category 1 uncertainties, as described in section 10.2. In what follows, we define the precise bound on the real part of the unstable eigenvalues. Let λ = σ + j ω be an eigenvalue of the state matrix of the closed-loop linearised state space model. The real part σ must satisfy (see also Fig. 10.5) the following bounding condition:  for ω ∈ Ω1 = {ω : |ω| ≥ 0.15 rad/s}  σ1 = 0, σ < σ2 = (ln 2)/20, for ω ∈ Ω2 = {ω : 0 < |ω| < 0.15 rad/s}  σ3 = (ln 2)/7, for ω ∈ Ω3 = {0}

(10.1)

The eigenvalues of the closed-loop system depend generally of the uncertain parameters. Let λ(π) = σ(π) + jω(π) be such an eigenvalue depending on

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parameter vector π ∈ Π. For each domain Ω i , i = 1, 2, 3 of the imaginary parts, we can define the set Λ+ i (π) = {λ (π) : σ (π) ≥ σi , ω (π) ∈ Ωi , } representing the eigenvalues whose real parts violates the condition (10.1). Let σi,max the maximum real part of eigenvalues in Λ+ i (π). Then, for each domain Ω i , the maximum unstable eigenvalue criterion can be defined as σi,wc = max σi,max (π) Π

and the corresponding worst-case parameter combination is πi,wc = arg max σi,max (π) Π

The criterion is defined for all flight conditions in Table 10.1, for which Λ+ i (π) is not empty, i.e. for which at least one unstable eigenvalue exists. Note that for real eigenvalues, the definition of worst case as the maximum real part among the positive eigenvalues is quite straightforward. For complex eigenvalues, different definitions of worst cases could have been chosen, such as the magnitude of the complex eigenvalue. Here, the maximum positive real part has been suggested because it can be directly linked to the existing handling qualities requirement on the minimum time to double amplitude of unstable modes. 1

ω

For ω ≥ 0.15 rad/s σ<0

0

Eigenvalue: λ = σ ± jω

For ω < 0.15 rad/s σ < ln(2)/20 For ω ≥ 0 (real eig.) σ < ln(2)/7

−1 −0.05

0

0.05 σ

0.1

Fig. 10.5. Boundary for unstable eigenvalues

0.15

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10.5 Average Phase Rate and Absolute Amplitude Criteria (Class II) These criteria are related to pilot in-the-loop oscillation (PIO) detection. To show that there are no PIO tendencies in pitch and roll the following requirements should be fulfilled for the given set of uncertainties: a) the average phase rate values of the frequency response of pitch/bank attitude to stick force should demonstrate at least Level 2 handling and therefore not lie outside the Level 2 boundaries of the phase rate criterion (see Fig. 10.8). b)only for the pitch axis, add the following requirement as the second part of the criterion. The absolute amplitude of the frequency response of pitch attitude to stick force at -180 degrees should be less than -16dB (if measured in [deg/lbs]) or -29dB (if measured in [deg/N]). All cases where these requirements are violated must be identified. The purpose of this test is to assess the closed-loop response in terms of what the pilot will experience, when he is flying the aircraft. The test (a) is aimed at ensuring satisfactory dynamic response characteristics and the test (b) is aimed at checking that the system is not sensitive to ”pilot gain” and will provide satisfactory handling for a wide range of pilots. In order to understand the above definition, the average phase rate and the criterion boundaries are explained below. The criterion parameters are defined from the transfer function of pitch attitude to longitudinal stick force, TF long , and of bank angle to lateral stick force, TF lat , for the roll axis. The transfer functions have to be computed from the linearised system at flight conditions of steady-state straight and level flight (with nz = 1g). Note that the input variable for both transfer functions is the pilot stick force. The ratio of stick force to stick deflection, in millimetres, assumed for the HIRM+ is given in Table 10.4, taken from [2]. Note also that in the HIRM+ model, the pilot commands in pitch and roll axes are defined respectively, as pitch rate demand and velocity vector roll rate demand, both in degrees per second; therefore, a gain should be applied between the pilot input in millimeters of the stick and the demand in degrees per second of angular rate. The last column of Table 10.4 contains the proposed gain for roll commands. The gain in the lateral axis has been chosen as the ratio of the maximum required roll rate, 70˚/s, to the maximum lateral stick deflection, 80mm, both defined in [2]. For longitudinal pilot commands, gain scheduling with dynamic pressure is proposed. This gain scheduling, presented in Fig. 10.6, is based on the values contained in Table 10.5, computed such that the step response to a full pilot stick command in pitch, gives a satisfactory response for the flight conditions of Table 10.1.

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Table 10.4. Stick characteristics Inceptor Longitudinal stick Lateral stick

Deflection amplitude -72 to +120 mm (positive aft) -80 to +80 mm (positive right)

Stick forces 1.2 N/mm 0.5 N/mm

FCS gain See Table 10.5 and Fig. 10.6 0.875 ˚/s/mm

Table 10.5. FCS gain for longitudinal stick command at specified flight conditions Flight condition

Dynamic pressure [kg/m2 ] 2.4313e+02 2.4704e+02 3.5611e+02 8.4532e+02 1.0027e+03 1.0858e+03 1.8380e+03 4.5077e+03

FC1 FC2 FC3 FC5 FC8 FC4 FC6 FC7

FCS gain for longitudinal stick [˚/s/mm] 0.03 0.03 0.065 0.15 0.15 0.2 0.2 0.2

0.22 0.2

Pilot command gain [°/s/mm] (pitch)

0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02

0

500

1000

1500

2000

2500

3000

Dynamic Pressure [Kg/m2]

3500

4000

4500

5000

Fig. 10.6. Gain scheduling of longitudinal stick command with dynamic pressure

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Gain:degrees altitude per pound stick force [dB]

5

0

-5

fbw

-10

-15

-20 fc -25 2fc -30 -220

-200

-180

-160 Phase [degrees]

-140

-120

-100

Fig. 10.7. Parameters of the average phase rate and absolute amplitude criterion

The magnitude of the average phase rate, AP R , is defined as: AP R = −(Φ2fc − Φfc )/fc = −(Φ2fc + 180o )/fc where fc is the frequency in Hertz at -180˚ phase, Φf c = -180˚, and Φ2f c is the phase angle in degrees at double fc . The example plot shown in Fig. 10.7 meets these criteria. The gain at -180˚ of phase is -21 dB, thus meeting the second requirement for the pitch axis, and the average phase rate is: AP R = - (-204 + 180 )/1 = 24˚/Hz (assuming that fc is 1 Hz and that the phase at 2 Hz is -204˚ in the example shown). Plotting this values of fc and AP R on Fig. 10.8 shows that this system meets the Level 1 handling requirements. For each flight condition FCi defined in Table 10.1, the clearance task is to find all parameter combinations within the uncertainty set Π defined in section 10.2, where TF long violates either requirements a) or b), or TF lat violates requirement a).

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250

200

Average Phase Rate [°/Hz]

Level 3

150 Level 2

100 Level 1 50

0

0

0.2

0.4

0.6

0.8 1 1.2 1.4 Frequency, fc[Hz], at −180° phase

1.6

1.8

2

Fig. 10.8. Average Phase Rate. Criterion boundaries

10.6 AoA/nÿ -Limit Exceedance Criteria (Class IV) For this test it is required to identify all flight cases where in the pull-up manoeuvres defined below (Fig. 10.9), the positive AoA/nz limits (respectively 29˚ - from the limiter - and 7g) are exceeded. This must be done for the nominal case and for the uncertainty case. The combination of uncertainties that yields the largest exceedance must be identified. This test is aimed at assessing the effectiveness of the incidence limiting and g-limiting capabilities of the flight control system, in terms of the attained maximum values. Specifically, two pitch stick commands should be assessed (see also Fig. 10.9): a full stick rapid pull and a pull in three seconds. For the full rapid pull a 1000 mm/s stick rate is applied on the longitudinal stick that brings the stick from the initial position to the maximum amplitude in the aft direction. The pull in 3 seconds is a ramp command that brings the stick from the initial position to full aft longitudinal stick in 3 seconds. Different rates of pitch stick application will result in diffrent aircraft decelerations, compared to the rapid pull, which can lead to different mamixmum AoA values, due to gain scheduling and aerodynamic response. Both commands must be applied from a trim condition of straight and level flight, and the simulation should be run for 10 seconds. For the above commands, a nominal trajectory must first be

164

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generated for each flight condition of Table 10.1. The nominal trajectory is the response to the above pilot commands when all uncertainties are set to zero. The criterion requires identification of the largest exceedance - if there is any - of the positive AoA/n z limit, for the nominal and the uncertainty cases. Full stick rapid pull

150

long. stick [mm]

120 100

slope = +1000 mm/s

50

0 −1

0

1

2

3

4

5

6

7

8

9

10

6

7

8

9

10

Full stick pull in 3s

150

Long. stick [mm]

120 100

50

0 −1

0

1

2

3

4

time [s]

5

Fig. 10.9. Pilot commands for testing largest exceedance of AoA and nz limits

For a mathematical definition of the AoA-limit exceedance criterion, let us define the uncertainty set Π as in section 10.2. Let π 0 denote the nominal case for parameters (i.e., all uncertainties set to zero). Let α(t,π) be the angle of attack response to the pilot command, which depends on the uncertainty vector π. Finally, let: αD (t,π) = α(t,π) - 29˚, and αEXC (π) = maxt≤10s αD (t,π), the difference between AoA and its upper limit and the maximum value of this difference, respectively. A negative αEXC (π) indicates that the AoA limit is satisfied, while a positive one indicates exceedance of the limit.

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Use of this criterion involves solving, for each flight condition in section 10.2, the following optimisation problem: αEXC,wc = max αEXC (π) Π

πwc = arg max αEXC (π) Π

The criterion is satisfied, at the given flight condition, if αEXC,wc is negative. It is also of interest to establish, for each flight condition, if at least in the nominal case (uncertainties set to zero), the criterion is satisfied, i.e. to check if: αEXC (π0 ) < 0 A similar definition can be used for the nz -limit exceedance criterion.

References 1. Moormann D., Bennet D. The HIRM+ Aircraft Model and Control Laws Development. GARTEUR/TP-119-2, 1999 2. Muir E. et al. Robust Flight Control Design Challenge Problem Formulation and Manual: the High Incidence Research Model (HIRM). GARTEUR/TP-088-27, 1997 3. Muir E. HIRM Design Challenge Presentation Document: The Robust Inverse Dynamics Estimation Approach. GARTEUR/TP-088-28, 1997 4. Scala S., Karlsson F., Korte U. Selected Criteria for Clearance of the HIRM+ Flight Control Laws. GARTEUR/TP-119-2-A1, V.2., 2000

11 An Overview of System Modelling in LFT Form Jean-Fran¸cois Magni1 , Samir Bennani2 , and Jean-Paul Dijkgraaf2 1 2

ONERA-Toulouse, CERT-DCSD, BP 4025, F31055 Toulouse Cedex 04, France [email protected] Delft University of Technology, Department of Control and Simulation, Kluyverweg 1, 2629 HS Delft, The Netherlands [email protected] and [email protected]

Summary. This paper presents an overview of system modelling in LFT (Linear Fractional Transformation) form. An LFT form serves often to replace a bank of linearized models by a continuum of linear models to be employed to solve parametric analysis and design problems. In other cases, LFT models are special multidimensional realizations of parametric rational matrices. We discuss several approaches to generate LFT models, addressing both exact as well as approximate generation of LFT models. Many of existing generation methods tend to produce large order LFT realizations which are difficult to be handled when solving analysis and design problems. Therefore, a main focus of our presentation is on obtaining low order LFT models, by using both exact as well as approximate LFT models reduction techniques.

11.1 Introduction This paper presents an overview of system modelling in LFT (Linear Fractional Transformation) form. For a more detailed presentation of this topic see [1]. The toolbox described in [1] provides MATLAB functions implementing all LFT generation and reduction techniques presented in this chapter. LFT models are mostly useful for analysis like µ-analysis (stability, performance, describing function [2]) and IQC-based analysis (non-linear or time varying components). These models can also be used for robust control design like µ-synthesis or multi-model control design ([3]), and for direct scheduled feedback gain design (LPV [4] or LTI [3] approaches). Several papers present techniques for LFT minimal order realization, see for example [5], [6], [7]. But such “minimality” is reached provided that uncertain parameters do not commute (that is not true for uncertain system modelling). It turns out that two main steps are to be considered. First, realization must be done carefully, especially taking advantage of parameter commutativity (details in Section 11.3). Then, some techniques reminiscent of “minimal realization” can be applied for further order reduction (details in Section 11.4). C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 169-195, 2002. Springer-Verlag Berlin Heidelberg 2002

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Section 11.2 introduces LFTs and related vocabulary. Section 11.3 is devoted to low order LFT realization. Three techniques are presented: the object-oriented technique, Morton’s technique ([8]) and the tree decomposition ([9],[10],[11]). The object-oriented approach is the most natural one but the tricks that reduce order must be done manually. The technique of [8] is limited to a special dependency with respect to uncertain parameters (affine dependency). The tree decomposition is the most general technique, it is recommended to use it when a parameter dependent system is given in a symbolic form. Other alternative order reduction techniques are just cited in the introduction of §11.3. Section 11.4 treats the generalization to LFTs of the “minimal realization” problem of standard dynamic systems. Considering LFTs, the size of what is improperly called a “minimal realization” depends on the initial realization considered for oder reduction. The reason for that, is that parameter commutativity is not taken into account, and in fact, minimality is truly attained only if parameters do not commute. First, it is shown how standard system “minimal realization” techniques can be applied to LFTs (1–D technique [12]). Then the generalized Kalman decomposition (or n–D technique) ([7],[13],[6]) is briefly evoked. We conclude this section by considering interpolation and approximations. A technique that permits the designer to evaluate precisely approximation errors, and, by the way to model approximation errors is proposed. Section 11.5 presents alternative techniques based on gridding (therefore to be avoided if there is a risk of missing a worst case). In order to simplify notations, ∆ denotes either a block-diagonal matrix in which all blocks are of the form δi I (context: realized LFTs) either the symbolic vector the entries of which are the δi ’s (context: symbolic form before realization).

11.2 Definitions and Basic Manipulations 11.2.1 Definition of LFTs LFT-objects are objects which can be represented as a matrix (interconnection matrix) connected by feedback through elements of different natures (I/s in which s is the Laplace variable, constant real parameters, parameters depending on time, non-linearities, linear systems). For example, in Fig. 11.1, the interconnection matrix is the bottom one. There are two connections of feedback type, the one through ∆ and the other one through I/s. The uncertain linear systems of Fig. 11.1 represents a dynamic gain (u → y) of the form: x˙ = Ax +B1 w +B2 u z = C1 x +D11 w +D12 u (11.1) y = C2 x +D21 w +D22 u The artificial feedback defining the model variations is

11 An Overview of System Modelling in LFT Form

w = ∆z

171

(11.2)

The derivation can also be considered as an artificial feedback: x=

x w u

In x˙ s

∆

¾

In s

¾

- A B1 B2 - C1 D11 D12 - C2 D21 D22

x˙ z y

-

Fig. 11.1. LFT representation of a linear uncertain system

The LFT-objects considered in this paper are – linear systems (as above), for which the coefficients depend rationally on the parameters. – matrices depending rationally on parameters (as in Fig. 11.1 but without the I/s-loop). Classical linear systems (as in Fig. 11.1 but without the ∆-loop) and fixed matrices are special cases. These objects are said to be LTI (Linear Time Invariant1 ). More generally, parameters might depend on time (LPV systems - Linear Parameter Varying) or might be replaced by non-linearities. We shall not treat these cases, but it is interesting to evoke them because they allow us to introduce the notion of commutativity. Indeed, if two parameters δi and δj are constant they commute because δi δj = δj δi . Furthermore, with 1s playing a role rather similar to that of parameters, it is interesting to note that 1s and δi commute also. On the other hand if δi (t) is a function of time, it is no 1

In practice, it will be possible to consider LPV (Linear Parameter Varying) as LTI systems provided that parameters vary slowly.

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J.-F. Magni, S. Bennani, and J.-P. Dijkgraaf

longer true that 1s δi (t) = δi (t) 1s . If one supposes that δi is a non-linearity (a threshold for example), we lose in addition the commutativity between δi ’s: δi δj 6= δj δi . We shall return later to the commutativity problem which we shall treat only in the LTI case. 11.2.2 Illustrative Examples To illustrate the LFT form of a matrix dependent on parameters, let us take two simple examples. b + δ1 (bc − ad) 1 − aδ1

(11.3)

K2 (δ2 ) = Iq2 (aδ22 + bδ2 + c)

(11.4)

K1 (δ1 ) = Iq1

where δ1 and δ2 are uncertain scalars, Iq1 and Iq2 are identity matrices of respective dimensions q1 , q2 . K1 and K2 can be represented with the artificial feedbacks of Fig. 11.2.

¾

δ 2 I q2 δ 1 I q1

u

-

¾

aIq1

bIq1

cIq1

dIq1

δ 2 I q2 ¾

- 0 -

y

u

-

0

I q2 0 0

Iq 2

aIq2 bIq2 cIq2

-

y

Fig. 11.2. Representation of K1 (δ1 ) : y = K1 (δ1 )u and of K2 (δ2 ) : y = K2 (δ2 )u

11.2.3 Special Cases The system of Fig. 11.1 is often considered in two alternative forms that are illustrated in Figs. 11.3 and 11.4. M − ∆ form. In the context of the µ-analysis, the I/s loop is considered as being closed. After closing this loop, we obtain Fig. 11.3 in which Mij can be written as:

11 An Overview of System Modelling in LFT Form

M11 (s) = C1 (sI M12 (s) = C1 (sI M21 (s) = C2 (sI M22 (s) = C2 (sI

− A)−1 B1 + D11 − A)−1 B2 + D12 − A)−1 B1 + D21 − A)−1 B2 + D22

173

(11.5)

For µ-analysis, these matrices will be considered on a frequency gridding i.e. s = jωi , i = 1, . . ..

∆

w

- M11 (s) M12 (s)

u M21 (s) M22 (s)

z y -

Fig. 11.3. M − ∆ form

In s

x

- A(∆) B(∆)

u C(∆) D(∆)

x˙ y -

Fig. 11.4. Parameter dependent state-space representation

Parameter dependent state-space representation derivation. The loop ∆ of Fig. 11.1 can also be considered as being closed. In that case we obtain the matrices (A(∆), B(∆), C(∆), D(∆)) defining linear systems. After closing this loop, we obtain Fig. 11.4 in which the uncertain matrices of Fig. 11.4 can be written:

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A(∆) = A + B1 ∆(I − D11 ∆)−1 C1 B(∆) = B2 + B1 ∆(I − D11 ∆)−1 D12 C(∆) = C2 + D21 ∆(I − D11 ∆)−1 C1 D(∆) = D22 + D21 ∆(I − D11 ∆)−1 D12 The parameter dependent matrices (A(∆), B(∆), C(∆), D(∆)) (these matrices are LFTs) can be evaluated for given values of ∆, for example the nominal case corresponds to ∆ = 0. These matrices are in particular useful, after evaluation at some given points ∆ = ∆i , i = 1, . . . for validating the LFT model (as in Fig. 11.1) by comparison with linearized models obtained directly by trimming/linearization of the original (possibly nonlinear) model. Moreover, this special form is in fact used by all the realization techniques of Section 11.4 (see in particular §11.3.1). 11.2.4 Star Product and Feedback Some standard notation for LFT systems is recalled first. Definition of Fu (M, ∆). Let us consider the M −∆ form of Fig. 11.3. The transfer between u and y that is obtained after closing the loop ∆ is denoted Fu (M, ∆) : Fu (M, ∆) = M21 ∆(I − M11 ∆)−1 M12 + M22 (11.6) Fu (): upper Linear Fractional Representation where ¸ · M11 M12 M= M21 M22 Definition of Fl (M, K). Let us consider the feedback loop of Fig. 11.3. After closing the loop y = Ku, the transfer “seen from ∆” is denoted Fl (M, K) : Fl (M, K) = M12 K(I − M22 K)−1 M21 + M11

(11.7)

Fl (): lower Linear Fractional Representation The star product. This product consists of replacing the ∆ block by an LFT (normalization treated in the next paragraph is an illustrative example). Let us consider two LFT-objects as in Fig. 11.3 denoted respectively Fu (M, ∆) and Fu (Q, ∆0 ) where ¸ · ¸ · Q11 Q12 M11 M12 and Q = M= M21 M22 Q21 Q22 The star product between M and ∆ is equal to Fu (R, ∆0 ) where the submatrices of R are in a natural partitioned form are: R11 R12 R21 R22

= Q11 + Q12 M11 (I − Q22 M11 )−1 Q21 ; = Q12 (I − M11 Q22 )−1 M12 ; = M21 (I − Q22 M11 )−1 Q21 ; = M22 + M21 Q22 (I − M11 Q22 )−1 M12 ;

11 An Overview of System Modelling in LFT Form

175

∆

¾

∆0

- Q11 -

Q21

Q12 Q22

aa

! aa !!! a ! !! aaa a !! - M11 u

-

M21

M12 M22

y-

Fig. 11.5. The star product

11.2.5 Normalization Normalizing the uncertain parameters means that we replace them by other parameters that vary between −1 and +1. For example, if δi belongs to the interval [δi− δi+ ], after normalization we use δ 0 i instead of δi defined by δi =

δ + − δi− δi+ + δi− + δ0 i i 2 2

with δ 0 i ∈ [−1 + 1]. For computing the new LFT after parameter normalization, it suffices to write ∆ as an LFT depending on the δ 0 i ’s and to apply the star product. Normalization is recommended before using µ-analysis, because it is commonly accepted to consider the values of the µ “measure” with respect to the unity. However, normalization must be done as late as possible (just before µ-analysis). To justify this assertion we consider an example: For example consider two parameters, let us assume that δ1 and δ2 have been replaced by normalized values: δ1 → a1 + b1 δ 0 1 δ2 → a2 + b2 δ 0 2 the simple product δ1 δ2 becomes

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δ1 δ2 → (a1 + b1 δ 0 1 )(a2 + b2 δ 0 2 ) = a1 a2 + a1 b2 δ 0 2 + b1 δ 0 1 a2 + b1 δ 0 1 b2 δ 0 2 The order is 4 instead of 2. After the product is performed, the realization techniques that will be presented in Section 11.3 are not able to return to the initial factorization. This recommendation does not hold if 1/δ appears somewhere (δ denote some uncertain parameter). In this case, it is necessary to give a nonzero nominal value to δ at an earlier stage. An alternative approach would be to consider a descriptor approach.

11.3 Low Order Realization The realization of a linear system given in transfer matrix form (symbolic form) consists of extracting all integrations in order to obtain four matrices (those usually denoted (A, B, C, D)). Drawing a parallel, building the realization of an uncertain system consists of extracting all the uncertainties and the integration symbols of its symbolic representation. It means, in other words, that realizing a system M (s, ∆) reduces to identifying the matrices (A, B1 , B2 , C1 , C2 , D11 , D12 , D21 , D22 ) of Fig. 11.1. This section is devoted to low order LFT realization. Three techniques are presented: the object-oriented approach, the technique of Morton and the tree decomposition. There exist alternative techniques that are not detailed here, the reasons for that are given now. – The graph-based approaches (that can be implemented using Simulink in some cases), require much efforts for order reduction (for example, see [14, 15]). Nevertheless, if only a small part of the flight domain is of interest, the simple special case of §11.6 can be used. – The Horner form conversion based approach of ([16],[17]) is limited to single-input single-output systems and looks like a special case of the tree decomposition. – The matrix-based approach proposed in ([18],[19],[20],[21],[22]) is a priori very attractive, but the final step (that consists of augmenting some linear systems until there exists a solution) is almost impossible to treat automatically in a computer program. 11.3.1 From State-Space Realization to Input/Output Realization Certain authors speak about direct and indirect approaches for system modelling in LFT form. The “direct approach” means that the transfer M (s, ∆) is directly realized. y = M (s, ∆)u (11.8)

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177

The “indirect approach” consists of realizing the matrices A(∆), B(∆), C(∆) and D(∆) as a first step, it is often more efficient to realize globally the matrix S(∆) ¸ · A(∆) B(∆) (11.9) S(∆) = C(∆) D(∆) corresponding to

¸· ¸ · ¸ · x A(∆) B(∆) x˙ = u C(∆) D(∆) y

(11.10)

S(∆) is realized means that a matrix P satisfying: S(∆) = Fu (P, ∆) has been found. In partitioned form:    · ¸ · ¸ P11 P12 P13 x˙ x = Fu  P21 P22 P23  , ∆ y u P31 P32 P33

(11.11)

(11.12)

If we replace x by (1/s)x, ˙ after a simple matrix manipulation we obtain:    ·I ¸ P22 P21 P23 0  u (11.13) y = Fu  P12 P11 P13  , s 0∆ P32 P31 P33 Therefore we have (see (11.1) or Fig. 11.1):     P22 P21 P23 A B 1 B2  C1 D11 D12  =  P12 P11 P13  P32 P31 P33 C2 D21 D22

(11.14)

Summary: realizing the matrix S(∆) is equivalent to compute an input/output realization, it suffices to reorder the sub-matrices (compare (11.12) and (11.13)). The size of the ∆-block, i.e. the LFT order, is the same in both cases. Form (11.12), realizing S(∆) is also equivalent to compute the form of Fig. 11.1 provided that sub-matrices are reordered. 11.3.2 Object-Oriented Realization Within a software environment such as MATLAB, operations on linear systems are performed using standard matrix algebra. Considering the similarity between LFT-objects and linear systems, naturally, it will also be possible to treat LFT-objects as matrices. In particular, it is possible to add (put in parallel), substract, multiply (put in series), transpose, invert, compute a feedback, and concatenate LFT-objects. The first contribution in this direction is in the toolbox for MATLAB described in Terlouw and Lambrechts

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[23], [12], see also d’Andrea [24]. Similar tools were also developed for Maple in Varga and Looye [16],[17]. The mechanism behind these manipulations is rather simple. We are going to clarify it by considering the sum and multiplication of two LFTs. Let us consider two objects identical to that of Fig. 11.3. They are denoted Fu (M 0 , ∆0 ) and Fu (M 00 , ∆00 ) where ¸ ¸ · 0 · 00 0 00 M11 M12 M11 M12 00 ; M = M0 = 0 0 00 00 M21 M22 M21 M22 The transfers between the input u and output y are respectively 0 0 0 0 ∆0 (I − M11 ∆0 )−1 M12 + M22 Fu (M 0 , ∆0 ) = M21

and

00 00 00 00 + M22 ∆00 )−1 M12 ∆00 (I − M11 Fu (M 00 , ∆00 ) = M21

Example. The sum of Fu (M 0 , ∆0 ) and of Fu (M 00 , ∆00 ) can be written £

0 00 M21 M21

· ¸µ · 0 ¸· 0 ¸¶−1 · 0 ¸ ¤ ∆0 0 M11 0 ∆ 0 M12 0 00 I − + M22 + M22 00 00 0 ∆00 0 M11 0 ∆00 M12

so,   0 0 · 0 ¸ M11 M12 0 0 ∆ 00 00 ,  M12 Fu (M 0 , ∆0 ) + Fu (M 00 , ∆00 ) = Fu  0 M11 0 ∆00 0 00 0 00 M21 M21 M22 + M22 

It remains to reorder the δi ’s so that they become contiguous (not scattered in two diagonal blocks). Reordering the uncertain parameters consists of permuting in a similar way the columns and rows of both above matrices. It is very easy to automate such manipulations. The set of required formulae for other operations (multiplication, inversion, concatenation, and so on) is given in [25]. It remains to create the basic elements which will be treated as above. As explained above, the object-oriented approach consists of building big LFTs by combining smaller ones. For example, the simplest objects that we shall have to consider are : integrator 1/s, real or complex scalars, full complex blocks. The third case will not be detailed here because its physical meaning is related to µ-analysis (it corresponds to transfer function matrices of linear systems frozen at a given frequency). The objects 1/s can be represented as follows (easy to check using (11.6)): · ¸ 1 1 01 = Fu ( , ) 1 0 s s an uncertain parameter, for example δ = a0 + a1 δ 0 can be modelled as

11 An Overview of System Modelling in LFT Form

δ = a0 + a1 δ 0 = Fu (

·

¸

0 1 , δ0) a1 a 0

179

(11.15)

It is important to note that using the object-oriented approach, the designer has a direct control on the order of the LFT being built. For example if δ1 and δ2 are uncertain parameters (say δ1 = a0 + a1 δ10 and δ2 = b0 + b1 δ20 ). Suppose we want to realize the uncertain matrix X ¸ · δ1 δ2 X= δ1 δ2 The first step will be to define two elementary LFTs for δ1 and δ2 using (11.15), say lftδ1 and lftδ2 . The designer can build the LFT corresponding to X as follows (we use object-concatenation and object-multiplication) ¸ · lftδ1 ∗ lftδ2 lftX1 = lftδ1 ∗ lftδ2 or, alternatively lftX2 = (lftδ1 ∗ lftδ2 ) ∗

· ¸ 1 1

We have not detailed LFT-concatenation, but it is clear that the order of lftX1 is 4 (δ10 and δ20 repeated twice) and the order of lftX2 is 2 (δ10 and δ20 “repeated” only once). We can say that the order of an LFT built using the object-oriented approach (from elementary object) is equal to the number of elementary objects appearing in the used formula. Therefore, the designer must use factorizations in order to reduce the occurrences of elementary objects i.e., to reduce the order of the final LFT. In the remainder of this chapter, we shall not as above, make further a distinction between the symbolic notation (δ1 , δ2 , . . .) and the corresponding elementary object realizations (lftδ1 , lftδ2 , . . .). 11.3.3 Morton’s Approach Morton’s realization technique ([8]) is very interesting because, despite the fact that S(∆) must have a special form as recalled below (see (11.16)), it leads to a (true) minimal2 order realization. Let us assume that matrix S(∆) of Equation (11.9) admits an affine (symbolic) development of the form ¸ · ¸ · ¸ · ¸ · Aq B q A1 B 1 A(∆) B(∆) A 0 B0 + . . . + δq + δ1 S(∆) = = C1 D 1 Cq Dq C0 D 0 C(∆) D(∆) (11.16) 2

Therefore, it will be useless to apply as a second step, the order reduction techniques of §11.3.4 (these techniques lead to “relative minimality”). This discussion will be clarified in §11.4.1.

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To reduce the realization order, every term (i ≥ 1) is decomposed using the SVD (Singular Value Decomposition) that is · ¸ ¸ · ¸· ¸· Ai B i Ui11 Ui12 Si 0 Vi11 Vi12 = Vi21 Vi22 C i Di Ui21 Ui22 0 0 (Only the non-negligible singular values are retained in Si ) leading to a decomposition · ¸ · ¸ ¤ Ai B i Ui1 £ Vi1 Vi2 = C i Di Ui2 Therefore, ¸ · ¸ · ¸ · ¤ ¤ U11 £ Uq1 £ A0 B 0 V11 V12 + . . . + δq Vq1 Vq2 (11.17) + δ1 S(∆) = C0 D 0 U12 Uq2 The final representation of S(∆) can be written S(∆) = Fu (P, ∆) where   A0 U11 . . . Uq1 B0  V11 0 . . . 0 V12     ..  P =  ... .     Vq1 0 . . . 0 Vq2  C0 U12 . . . Uq2 D0 and

∆ = diag{I/s, δ1 In1 , . . . , δq Inq }

Proof. To check this result, let us consider the case of two uncertain parameters (the generalization is straightforward). Writing S(∆) = Fu (P, ∆) with P and ∆ as above means that      x˙ x A0 U11 U21 B0 · ¸ · ¸  z1   V11 0 0 V12   w1   =    with w1 = δ1 z1  z2   V21 0 0 V22   w2  δ2 z2 w2 C0 U12 Uq2 D0 u y but so or

·

z1 z2

¸

· =

V11 V12 V21 V22

¸· ¸ ¸· ¸ · ¸ · x x w1 δ V δ V → = 1 11 1 12 δ2 V21 δ2 V22 u u w2

¸· ¸· ¸ ¸· ¸ · · ¸ · δ1 V11 δ1 V12 x U11 U21 x A 0 B0 x˙ + = C0 D 0 U12 U22 δ2 V21 δ2 V22 u u y · ¸ µ· ¶· ¸ · ¸ ¸ · ¸ ¤ ¤ x˙ A 0 B0 x U21 £ U11 £ V21 V22 V11 V12 + δ2 = + δ1 y u U22 C0 D 0 U12

which in the form of the development of the equation (11.17).

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181

11.3.4 Tree Decomposition The tree decomposition relies on factoring out parameters which appear as common factors in rows and/or columns in combination with additive decompositions intended to generate terms containing common factors. By using this approach, we obtain a factorised representation of the original matrix, which leads to a lower order LFT realization, when the factors are individually realized. The “tree decomposition” naming comes from the fact that factorizations and decompositions alternate and are repeatedly applied in producing a representation of the original matrix which can be associated with a tree decomposition. This technique was introduced in [11]. In what follows, we assume that S(∆) in (11.9) is given in a symbolic form, having only polynomials entries. The case of rational entries can be also handled by this approach, by applying it to the polynomial factors of a coprime polynomial factorization of S(∆). The details of this approach can be found in [1] or [18]. We illustrate by means of simple examples, the two basic techniques used to determine tree decompositions: factorization and additive decomposition. Factorization. Consider the following example · 2 ¸ 4δ1 δ3 3δ1 0 S(∆) = δ3 δ5 5δ22 δ4 δ2 δ42

(11.18)

which leads to an LFT model of order 12, when all entries are independently realised. By factoring out the common factors δ1 from the first row and δ3 from the first column, we obtain the following factorised representation   ¸· ¸ δ3 0 0 · 4δ1 3 0 δ1 0  0 1 0 S(∆) = 0 1 δ5 5δ22 δ4 δ2 δ42 0 01 which leads to an LFT realisation of lower order 10, if the factors are separately realised. Note that generally such a factorization is non unique and it is a priori not known which factorization leads to the least possible order. Direct sum additive decomposition. This decomposition, as a matter of fact, has as objective to make factorizations possible. The simplest decomposition is just separating a matrix in two terms, in which complementary subsets of parameters (“direct sum”) appear. For example, the decomposition ¸ · ¸ · ¸ · 0 0 0 0 4δ1 3 0 4δ1 3 + = δ5 5δ22 δ4 δ2 δ42 δ5 0 0 0 5δ22 δ4 δ2 δ42 allows us to factor out δ2 δ4 from the second term to obtain ¸ · ¸ · ¸· ¸ · 0 4δ1 3 0 1 0 0 0 0 4δ1 3 = + δ5 5δ22 δ4 δ2 δ42 δ5 0 0 0 δ2 δ4 0 5δ2 δ4

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Thus, by combining additive decompositions and factorizations, we have   ¸ · ¸¶ δ3 0 0 ¸· ¸ µ· · 1 0 0 0 0 4δ1 3 0 δ1 0  0 1 0 + S(∆) = 0 δ2 δ4 0 5δ2 δ4 δ5 0 0 0 1 0 01 which leads now to an LFT realisation of order 8. Alternative additive decomposition. As a matter of fact, direct sum decomposition is not always possible. But it is always possible 3 either to perform a factorization, or to decompose into two parts in the following way: 1. Select one (or more) parameter(s). 2. Isolate a term of the decomposition as that one containing entries with the selected parameter(s) (or at least one of them). 3. Define the other term of the additive decomposition, as the part which is independent of the selected parameter(s). For example, consider

·

δ δ + δ3 S(∆) = 1 2 δ1 δ2 δ3

¸

With δ3 selected, we obtain successively ¸ · ¸ · ¸ ¸ · · δ1 δ2 1 δ3 δ1 δ2 = + δ + S(∆) = δ1 δ2 δ3 0 δ1 δ2 3 0 Alternatively, we can select δ1 and δ2 , to obtain ¸ · ¸ · ¸ · ¸ · δ1 δ2 δ 1 δ = 3 + δ δ S(∆) = 3 + 0 δ1 δ2 δ3 0 δ3 1 2 In both cases, one can factor out the selected parameter, as shown above. In the first case, factoring δ3 out leads to a 5th order LFT model. In the second case, factoring δ1 and δ2 out leads to a 4th order LFT model. Combination of factorizations and decompositions. The tree decomposition method consists in combining all above techniques. It is clear from the simple examples, that the achieved final order depends on various choices of parameters to be factored out, and the order in which factorization and decompositions are performed. Furthermore, all these transformations are not unique (see for example the decompositions of the last example). 3

Except when all entries of sub-matrices are monomials depending on a single parameter, that is, at the end of algorithm.

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183

11.4 Order Reduction and Approximations It is assumed that a realization is available. This section presents first a notion of similarity of LFT realizations. When two realizations satisfy the evoked similarity condition, both realizations represent the same LFT. For order reduction, the key point consists of modifying by similarity a given realization in order to obtain an other one in which order reduction becomes conspicuous (i.e., so that reduction consists of removing rows or columns without modifying the input / output gain). After this discussion on similarity, two techniques (1–D and n–D) are presented. We shall ignore the very popular Gramian-based approach ([5], [26], [27], [28], [29]) because in practice it is not efficient. The main trouble is that this technique is based on LMI (Linear Matrix Inequality) system resolution. But LMI systems that can be handled by LMI solvers are limited in size. Therefore, we have a paradoxical conclusion that only low order LFTs can be reduced using this approach. In addition, it is shown in [30] that this technique is equivalent to the n–D approach that is very reliable, even for very large LFTs. Finally, we treat LFT approximation in view of reducing further the order. We still ignore the Gramian-based approach because, even if it provides a bound of the approximation error, this bound is so conservative that it becomes better to use an algebraic reduction technique with a large value of tolerances when rank conditions are considered. Of course, such approximations must be validated (or invalidated) by computing a measure of the distance between the original and approximated LFTs (this problem is considered in §11.4.4). 11.4.1 Minimality and Similarity Transformation Several concepts are defined in this paragraph such as minimality, similarity, relative minimality. First, ’minimality’ (or ’true minimality’) that is based on ’input / output equivalence’, second, ’relative minimality’ that is based on ’similarity’. In this section, notations are simplified as we do not distiguish between uncertain parameters (δi ’s) and 1/s: the matrix ∆ is assumed to contain the I/s block. Therefore, (A, B1 , B2 , C1 , C2 , D11 , D12 , D21 , D22 ) becomes (M11 , M12 , M21 , M22 ) with obvious correspondance. Definition of equivalence. Two LFT realizations are equivalent if for all values of the parameters δi , the transfer function matrix from inputs to outputs are equal. Definition of minimality. An LFT representation is said to be minimal if there is no equivalent representation having a block ∆ of lower dimension.

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There P is no uniqueness of minimal representations. In particular we might have i=1,...,q ni = n for n minimal but with more or less variable values of the ni ’s. All the references of literature which refer to the notion of equivalence use in fact another much stricter notion. It is a kind of ’equivalence’ defined with a particular class of transformations of the form: T = Diag {T1 , . . . , Tq } where Ti ∈ IR ni ×ni and det(Ti ) 6= 0

(11.19)

where T commutes with ∆, so, T ∆ = ∆T and ∆T −1 = T −1 ∆ Applying this transformation to Fu (M, ∆) (that will be denoted Fu (T (M ), ∆)) leads to ¸ · −1 T M11 T T −1 M12 (11.20) T (M ) = M21 T M22 Lemma 1. Fu (M, ∆) = Fu (T (M ), ∆). It is interesting to give the proof of this result because the fact that commutativity of uncertain parameters is nowhere considered explain the strictness of the concept of similarity. By the way, a strange conclusion will be made, for example: the respective object-oriented realizations of δ1 δ2 and of δ2 δ1 are not similar and δ1 δ2 − δ2 δ1 cannot be reduced to zero. Proof. We have: Fu (T (M ), ∆) = M21 T ∆(I − T −1 M11 T ∆)−1 T −1 M12 + M22 Using the commutativity of T and ∆ Fu (T (M ), ∆) = M21 ∆T (I − T −1 M11 T ∆)−1 T −1 M12 + M22 moving T and T −1 into the parenthesis, Fu (T (M ), ∆) = M21 ∆(I − T T −1 M11 T ∆T −1 )−1 M12 + M22 But the commutativity of T and ∆ implying the commutativity of T −1 and ∆, we have Fu (T (M ), ∆) = M21 ∆(I − T T −1 M11 T T −1 ∆)−1 T −1 M12 + M22 = Fu (M, ∆) Definition of similarity. Considering the above result, two realizations (M 0 , ∆0 ) and and (M 00 , ∆00 ) will be said to be similar if and only if there is a non-singular matrix T satisfying (11.19) s.t. T (M 0 ) = M 00 .

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185

Use of similarity ’to minimize’ order. As stated in the introduction, similarity transformations are used to make conspicuous the fact that the order of an LFT can be reduced by removing a set of rows and columns. To illustrate this fact we shall consider a simple case with only two uncertain parameters. Assuming that T as in (11.20) leads to a matrix 

∗∗∗ 0 ∗ ∗  0 0 ∗  T (M ) =  ∗ ∗ ∗ ∗ ∗ ∗  0 0 ∗ ∗∗∗

∗∗∗ ∗∗∗ 00∗ ∗∗∗ 0∗∗ 00∗ ∗∗∗

 ∗ 0  · 0 0 0 ¸  M11 M12 ∗ := 0 0  M21 M22 0  0 ∗

in which ’∗’ denote non-zero sub-matrices. (The above form is typically what might be obtained using the controllability part of the n–D algorithm that will be briefly evoked in §11.4.3, see [7] for an example leading to this matrix). 0 ∆(I − As we have to compute Fu (T (M ), ∆) that is of the form M21 0 −1 0 0 M11 ∆) M12 + M22 , assume that we permute the rows and columns of the 0 matrix M11 in the following way (1, 2, 3, 4, 5, 6) → (1, 2, 4, 5, 3, 6), the corre0 and sponding transformations are also applied to ∆ and to the rows of M12 0 columns of M21 , we obtain:   ∗∗∗∗∗∗ ∗ 0 ∗ ∗ ∗ ∗ ∗ 0   ∗ ∗ ∗ ∗ ∗ ∗ ∗ · 00 00 ¸   M11 M12   ∗ ∗ 0 ∗ ∗ ∗ 0 T (M ) ≈   := M 00 M 00 0 0 0 0 ∗ ∗ 0 21 22   0 0 0 0 ∗ ∗ 0 ∗∗∗∗∗∗ ∗ With this block triangular form of the matrix to be inverted, it becomes clear 000 0 000 0 ∆(I − M11 ∆)−1 M12 + M22 we can remove the that when we compute M21 two rows and lines that were numbered 3 and 6 in T (M ). Therefore, we have shown that   ∗∗∗∗ ∗ 0 ∗ ∗ ∗ 0    T (M ) =  ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 ∗ 0 ∗∗∗∗ ∗ that is a realization of lower order than the initial one. Definition of ’relative minimality’. An LFT realization will be said to satisfy the relative minimality property if there is no similar representation having a block ∆ of lower dimension.

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Minimality versus relative minimality. It is not really necessary to consider difficult LFT theory to explain the difference between the minimality and the relative-minimality. In case of non commutativity, we have · ¸ · ¸ δ2 δ1 δ = 2 δ1 (11.21) δ1 1 and

·

δ1 δ2 δ1

¸

·

δ 0 = 1 0 δ1

¸·

δ2 1

¸ (11.22)

Realization using the object-oriented approach (section 11.3.2) clearly leads to LFT-objects of order equal to the number of times the parameters occur in the symbolic representation. For simple transfers, it is easy to see if minimality is reached (i.e. the symbolic expression cannot be written with fewer parameters). The object-oriented realization of Equation (11.21) is of order 2 because δ1 and δ2 occur only once. The object-oriented realization of Equation (11.22) is of order 3 because δ1 appears twice and δ2 once. It is also clear that without commutativity, it is not possible to reduce further the transfer of Equation (11.22), so in both cases, the relative minimality property is satisfied, but with respective minimum order 2 and 3. More generally, the fact that commutativity is ignored might lead to extremely different relative minimal orders for the same symbolic object. For example assume that the following symbolic objects are object-oriented realized.     1 δ1 δ2 δ3  δ2 δ3 δ1  and  1  δ1 δ2 δ3 1 δ3 δ1 δ2 Both realizations are clearly equivalent but lead to relative minimal orders respectively equal to 9 and 3. Conclusion. The conclusion of this paragraph is that relative minimality depends on the realization that is considered. Therefore, it is recommended to compute low order realizations before using the similarity-based reduction/approximation techniques (i.e., the techniques presented in the next paragraphs). Note that, for standard linear systems, the minimal realization can be obtained independently of the original realization. 11.4.2 The 1–D Approach The 1–D approach consists of isolating alternately each parameter δi considering it as the 1/s operator. Then, at each step, a standard 1–D order reduction (based on Kalman decomposition) is performed. In order to illustrate more precisely this procedure, let us consider a simple example with only two uncertain parameters δ1 and δ2 . We have



11 An Overview of System Modelling in LFT Form





A x˙  z1   C1a  =  z2   C1b y C2

B1a D11aa D11ba D21a

B1b D11ab D11bb D21b





187

B2 x  w1  D21a    D21b   w2  D22 u

For the first 1–D reduction, the natural quadruple is considered:   ¤ w1 £ B1a B1b B2  w2  sx = Ax +     u   z1 w1 C1a D11aa D11ab D21a  z2  =  C1b  x +  D11ba D11bb D21b   w2  y C2 D21a D21b D22 u Having eliminated non controllable and non observable states (1–D reduction) certain matrices are modified. For simplicity, the corresponding notations are not modified. As a second step, permutation between x and w1 and between x˙ and z1 are performed, then, the following quadruple is obtained:   ¤ x £ (1/δ1 )z1 = D11aa w1 + C1a D11ab D21a  w2  u       x B1a x˙ A B1b B2  z2  =  D11ba  w1 +  C1b D11bb D21b   z2  D21a y C2 D21b D22 u The same reduction is applied as it was done at the first step (x = (1/s)x˙ is replaced formally by w1 = δ1 z1 ). The notions of controllability and observability does not make sense any more, but formally, the reduction is possible. It remains to proceed in a similar way with regard to z2 and w2 to end the procedure of reduction. 11.4.3 The n–D Approach Instead of considering successively 1/s and then every δi seprately as with the previous method, the n–D approach treats simultaneously all the parameters of the block ∆, so, this technique is less conservative. It is shown in [7] that it leads to the ’minimal order’ for a given realization (that should be understood as ’relative-minimality’). As in the case of standard linear systems, it is necessary to apply two steps, the first one consisting of identifying the ’controllable part’, the second one, of identifying the ’observable part’. Minimal form corresponds to the subsystem that is both ’controllable’ and ’observable’. Details are not given here. The reader is referred to [7]. The difference between both technique can be illustrated by comparing the kinds of structures that are obtained after controllability subspaces computation (ignoring

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the observability dual part). For example, the n–D approach might lead to (see §11.4.1)   ∗∗∗∗∗∗ ∗ 0 ∗ ∗ ∗ ∗ ∗ 0   0 0 ∗ 0 0 ∗ 0   T (M ) =   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 ∗ ∗ 0 00∗00∗ 0 using the 1–D approach for each uncertain parameter the obtained structure would be of the form:   ∗∗∗∗ ∗ 0 ∗ 0 0 0  T (M ) =  ∗ ∗ ∗ ∗ ∗ 00∗0 0 The first structure is more powerful because it permits us to reduce the order (see §11.4.1) even if it is not possible to obtain a structure similar to the second one (with a single non-zero block per row that can be eliminated). Illustrative example of the advantages of the n–D approach with regard to the 1–D one. A very simple academic example can be built in order to put in evidence the interest of the n–D approach. Let us consider the symbolic gain · ¸ S(∆) =

1 1+δ1 +δ2 1 1+δ1 +δ2

(11.23)

S(∆) is realized using the object-oriented approach, the obtained realization is of order 4. It is clear that the order of the minimal form is 2 because · ¸ 1 1 S(∆) = 1 1 + δ1 + δ2 This minimal form is found using the n–D technique but not found using the 1–D approach (see [1]). This fact can be simply explained. By considering a single variable at a time, it is impossible to find any factorizations. On the other hand, by dealing simultaneously with δ1 and δ2 , it is clear that factorization becomes possible. 11.4.4 Interpolation and Other Approximations There are several ways to compute approximate LFT models. The Gramianbased approach ([5],[26],[27],[28],[29]) has a good mathematical justification. However, in practice, it appears that heuristic approaches are more efficient. The 1–D and n–D techniques are based on controllable (or observable) state-space decomposition. Therefore, some rank conditions are involved and, by the way, some tolerance parameters are used as limits for rank deficiency.

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If these tolerance parameters are augmented, these techniques become heuristic approximation tools. However, there is no a priori guarantee that these approximations are valid. More heuristic techniques can be considered, for example using engineering knowledge. For instance, some parameter dependencies arising from physical equations might be simplified or even ignored. Such simplifications might also result from Montecarlo analysis rather than from engineering knowledge. Here again, there is no a priori guarantee that these approximations are valid. These remarks justify the necessity of having a tight upper bounds of approximation errors. This problem is addressed now. Evaluation of the interval of variation of a SISO LFT. For that purpose we can use artificial µ-analysis. We just state the main result, the proof can be found in [1]. Lemma 2. Let us consider a (1 × 1) realization with normalized real uncertain parameters (−1 < δi < 1). This object is denoted ¸ · M11 M12 , ∆) M (∆) = Fu ( M21 M22 An artificial system S(λ), with λ ∈ IR , is defined: S(λ) = M11 + M12 (λ − M22 )−1 M21 For λ varying from −∞ to +∞: – The minimum value of M (∆) over the unit ball is the first value of λ at which µ(S(λ)) = 1. – The maximum value of M (∆) over the unit ball is the last value of λ at which µ(S(λ)) = 1. Using lower and upper bounds of µ we can obtain intervals (usually tight) that contains the variations bounds of M (∆). For example if M − < M (∆) < M + Lemma 2 permits us to compute M −− , M −+ , M +− , M ++ such that M −− < M − < M −+

and

M +− < M + < M ++

which means that we have the following bounds M −− < M (∆) < M ++ Use of Lemma 2 for modelling. The above result is interesting in the following situation. Assume that we have two SISO LFT realizations M (∆) f(∆) in which M f(∆) is assumed to be an approximation of M (∆). A and M f realization of M (∆) is given:

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# f12 f11 M M e f(∆) = Fu ( , ∆) M f21 M f22 M "

First, we can compute the approximation error by applying Lemma 2 to f(∆) in order to validate or invalidate the approximation. Second, M (∆) − M the result of Lemma 2 can be used for modelling the approximation error. f(∆) < M ++ , therefore, we Assume that we have found M −− < M (∆) − M can define a new uncertain parameter ε that satisfies M −− < ε < M ++ and replace M (∆) by the following LFT: f(∆) + ε M 0 (∆, ε) = M It is not difficult to check that (use of the sum of two LFTs and of (11.15))  f12 f11 0 M · ¸ M e0 ∆ ) M 0 (∆, ε) = Fu ( 0 0 1  , 0 ε f21 1 M f22 M 

(11.24)

is a realization of M 0 (∆, ε). It is worth noting that using this trick, we can reduce quite a lot the order of the original LFT, and keep a trace of the approximation in a single parameter that is not repeated (ε). The same idea can be used for MIMO LFTs by treating separately each entries. Modelling by interpolation. Interpolation is often required in aeronautics because some derivative coefficients are given as tables. In that cases, it suffices to choose an interpolation formula and to solve a least squares problem to get the desired coefficients. Once the least squares problem is solved, it is worth computing the maximum interpolation error and to add it as a new uncertainty. Here again, we must apply Equation (11.24). Interpolation might also be viewed as a direct approximate modelling approach. For example a parameter dependent nonlinear system is linearized on a gridding of trim points. Interpolation as above can be used. The “trend and bands” technique (see §11.5.2 ) is a special case in which interpolation formulas are affine with respect to the gridded parameters). 11.4.5 Analytical Approaches to LFT Modelling By ’analytic approaches’ we mean, approaches that use the equations defining the system to be modelled. For simple systems, we recommand to use the object-oriented technique (§11.3.2) possibly followed by the (n–D)-order reduction technique (§11.4.3). For more complex cases, all the order reduction/approximation techniques evoked in §11.3 and §11.4 might be useful. For example considering a nonlinear system, the first task might consist of replacing data tables (if any) by interpolations. Then, a symbolic form of the linearized models can be derived as in Equation (11.10). Then use the tree

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decomposition (§11.3.4) followed by the (n–D) order reduction technique. In this last step, tolerance arguments might be augmented for approximation. Then, modelling error should be analyzed using for example the technique of §11.4.4 and new uncertain parameters representing approximation errors should be added to the model.

11.5 Gridding-Based LFT-Modelling Techniques In this book, two chapters consider µ-analysis relative to LFT-models obtained after gridding. When the system is sufficiently well known or the flight domain sufficiently small so that there is no risk to miss a worst case, gridding-based method might be used to replace anlytical approaches. 11.5.1 Min-Max Method A bank of linearized models of the HIRM+ is computed directly from the global Simulink model. Therefore, a bank of quadruples (A, B, C, D) is available. For each entry of these matrices, the minimum and maximum values are computed (all entries of these matrices are assumed to vary independently), say + − + − + − + a− ij < aij < aij ; bij < bij < bij ; cij < cij < cij ; dij < dij < dij

(11.25)

Therefore S(∆) (see (11.9)) can be written     0 1 ... 1 0 ... · ¸     .. A(∆) B(∆)  + ...  + a12  0 . . . . 0 = a11      C(∆) D(∆) .. .. . . Here, ∆ = Diag{a11 , a12 , . . . , b11 , b12 , . . .}. The corresponding LFT realization can be computed using Morton’s method (detailed in §11.3.3). It remains, considering the bounds evoked in (11.25), to normalize the parameter variations between −1 and +1 by using the star product (see §11.2.5). The advantage of this technique is that it is systematic, it leads to nonrepeated uncertainties (there are problems with µ-analysis when some uncertainties are repeated a large number of times, say more than 25 times). The disadvantage is that worst cases found using such an LFT will be expressed in terms of the elements of the matrices A, B, C, D, so, the flight points that induce problems cannot be identified. In addition, the worst case found in that way might be far form actual models (because the entries of (A, B, C, D) are assumed to vary independently).

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11.5.2 Trends and Bands Method This techniques is still based on the use of a bank of linearized models. But here, each coefficient is computed by least squares optimization. For example, let us consider the following interpolation with respect to δ1 . . . δq in which aδijk are the coefficients to be identified and εaij is the interpolation error to be minimized. aij = aδij1 δ1 + aδij2 δ2 + . . . + εaij ¿From a given set of values (extracted from a bank of linearized models) of (aij , δ1 , . . . , δq ), the coefficients (aδij1 , aδij2 , . . .) are computed by least squares. Then, the variations bounds of εij (bands) are easily identified by comparing the interpolated values with respect to the original set of values. We obtain again an expansion of the matrix S(∆). It has the following form  δq δq  δ1 δ1   a11 a12 . . . a11 a12 . . . · ¸   δ ..   A(∆) B(∆)   + . . . + δq  aδq . . . = δ1  a211 . 21     C(∆) D(∆) .. .. . .    0 1 ... 1 0 ...    ..   + ...  + εa  0 . . . . 0 + εa11  12     .. .. . . 

Here, ∆ = Diag{δ1 , . . . , δq , εa11 , εa12 , . . .}. As in the previous case, the corresponding LFT realization can be computed using Morton’s method detailed in §11.3.3. Normalization is performed as in the previous case. Approximations can be based on physical knowledge in the selection of interpolation variable (the δi ’s), but it can also be treated more systematically by using non too small tolerances in the SVDs (see §11.3.3) that are involved in Morton’s approach. The second method has the advantage of keeping the physical meaning of the uncertain (or varying) parameters because the non-structured part is now (εa11 , εa12 , . . .) that corresponds to small variations instead of to the coefficients aij themselves. Nevertheless, this technique does not permit us to find worst cases that are “hidden” between the gridding points.

11.6 Simulink Model Based LFT Generation If the (possibly non-linear) system model is in pure Simulink form, the generation of LFT models (as proposed for example in [31]) is very easy. The idea consists of adding in the Simulink diagram one artificial input and one artificial output at both sides of each parameter that varies or that is uncertain.

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Then, a linearization routine leads to a realization (see Fig. 11.1). In fact, Equation (11.1) and the matrices therin is obtained in the following form: · ¸ w x˙ = Ax + [B1 B2 ] u ¸ · ¸· · ¸ · ¸ (11.26) D11 D12 w C1 z x+ = C2 D21 D22 u y As long as we can ignore parameter dependencies at equilibrium this technique is quite efficient. For covering a large domain of the flight envelope, a bank of models obtained in that way is necessary.

11.7 Conclusion This paper proposes an overview of LFT modelling, however the whole domain is not covered. We preferred to present a specific overview of low order modelling of real parameter dependent systems because, up to our knowledge, such a survey is not available in literature. In addition, this choice is also justified by the fact that aircraft modelling in LFT form is so complex (analytical approach) that, without order reduction tools, it would lead to LFTs of huge orders (larger than 1500 even after some approximations) that would be useless. We did not cover the various ways an LFT model should be augmented in order to treat performance problems or describing function approach analysis. We also ignored the descriptor point of view and so on. Section 11.5 shows that there is not a single approach for LFT-modelling but a large number of possibilities. For selecting the modelling procedure to be used, we must take objectives into account. For example if we want to consider a set of operating points in which there is no risk to have an unexpected behavior, gridding-based approaches might be used. These approaches lead to simple models, so, are also very interesting for control system design. On the other hand, for a clearance problem concerning the limits of the flight domain, if some behavior might not be predictable, it is necessary to use the analytical way. In any cases, it is better to consider a limited part of the flight envelope, rather than trying to find a very general model. For clearance problems, if LFTs are used instead of classical systems, there is a need to validate these models. For gridding based approaches, after checking that the model is valid at the gridding points it is worth validating it between these points. Using the analytic approach, some mistakes might appear during the transcription of equations. In both cases, there is no systematic tool for such a validation unless comparison of simulations.

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References 1. J.F. Magni. Linear Fractional Representations with a Toolbox for Use with MATLAB, Version 1. Technical Report TR 1/05664 DCSD, ONERACERT: http://www.cert.fr/dcsd/idco/perso/Magni/booksandtb.html, December 2001. 2. G. Fererres and V. Fromion. Nonlinear analysis in the presence of parametric uncertainties. Int. J. Control, 69(5):695–716, 1998. 3. J.F. Magni, Y. Le Gorrec, and C. Chiappa. A multimodel-based approach to robust and self-scheduled control design. In Proc. 37th I.E.E.E. Conf. Decision Contr., Tampa, Florida, pages 3009–3014, 1998. 4. J.M. Biannic and P. Apkarian. Missile autopilot design via a modified LPV synthesis technique. Aerospace Science and Technology, 3(3):153–160, 1999. 5. C. Beck. Minimality for uncertain systems and IQCs. In Proc. of 33th IEEE Conference on Decision and Control, Lake Buena Vista, Florida, pages 1233– 1238, December 1994. 6. C. Beck and J. Doyle. A necessary and sufficient minimality condition for uncertain systems. IEEE Transactions on Automatic Control, AC-44:1802– 1813, october 1999. 7. R. D’Andrea and S. Khatri. Kalman decomposition of linear fractional transformation representations and minimality. In Proc. of the American Control Conference, Albulquerque, New Mexico, pages 3557–3561, June 1997. 8. B. Morton. New applications of mu to real-parameter variation problems. In Proc. of 24th IEEE Conference on Decision and Control, Fort Lauderdale, Florida, pages 233–238, December 1985. 9. J.C. Cockburn. Linear fractional representation of systems with rational uncertainty. In Proc. of the American Control Conference, Philadelphia, Pennsylvania, pages 1008–1012, June 1998. 10. J.C. Cockburn and B.G. Morton. On linear fractional representations of systems with parametric uncertainty. In Proc. of the 13th IFAC Triennial World Congress, San Francisco, California, pages 315–320. Pergamon, July 1996. 11. J.C. Cockburn and B.G. Morton. Linear fractional representations of uncertain systems. Automatica, 33(7):1263–1271, 1997. 12. P. Lambrechts, J. Terlouw, S. Bennani, and M. Steinbuch. Parametric uncertainty modeling using LFTs. In Proc. American Control Conference, San Francisco, CA, pages 267–272, 1993. 13. C. Beck and R. D’Andrea. Minimality, controllability and observability for uncertain systems. In Proc. of the American Control Conference, Philadelphia, Pennsylvania, pages 3130–3135, June 1997. 14. C. D¨ oll. La robustesse de lois de commande pour des structures flexibles en a´eronautique et espace. Th`ese pr´esent´ee ` a l’Ecole Nationale Sup´erieure de l’A´eronautique et de l’Espace (SUPAERO), Toulouse, France, 2001. 15. S. Font. M´ethodologie pour prendre en compte la robustesse des syst`eme asservis: Optimisation H∞ et approche symbolique de la forme standard. Th`ese pr´esent´ee ` a l’Universit´e de Paris-Sud, Orsay, France, 1995. 16. A. Varga, G. Looye, D. Moormann, and Gru ¨bel G. Automated generation of LFT-based parametric uncertainty descriptions from generic aircraft models. Mathematical Modelling of Dynamical Systems, 4:249–274, 1998.

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17. A. Varga and G. Looye. Symbolic and numerical software tools for lft-based low order uncertainty modeling. In Proc. of the IEEE International Symposium on Computed Aided Control System Design, Kohala Coast, Hawai’i, USA, pages 176–181, August 1999. 18. C.M. Belcastro. Uncertainty modeling od real parameter variations for robust control applications. In PhD Thesis, University of Drexel, US, pages 1–304, December 1994. 19. C.M. Belcastro. Parametric uncertainty modeling: An overview. In Proc. of the American Control Conference, Philadelphia, Pennsylvania, pages 992–996, June 1998. 20. C.M. Belcastro and B.C. Chang. LFT formulation for multivariate polynomial problems. In Proc. of the American Control Conference, Philadelphia, Pennsylvania, pages 1002–1007, June 1998. 21. C.M. Belcastro, B.C. Chang, and R. Fischl. A matrix approach to low-order uncertainty modeling of real parameters. In Proc. of the 13th IFAC Triennial World Congress, San Francisco, California, pages 297–302. Pergamon, July 1996. 22. C.M. Belcastro, K.B. Lim, and E.A. Morelli. Computer-aided uncertainty modeling of nonlinear parameter-dependent systems, part ii: F-16 example. In Proc. of the IEEE International Symposium on Computed Aided Control System Design, Hawai’i, USA, pages 16–23, August 1999. 23. J.C. Terlouw and P.F. Lambrechts. A MATLAB Toolbox for Parametric Uncertainty Modelling. Technical Report CR-93455-L, National Aerospace Laboratory, NLR, Amsterdam, 1993. 24. R. D’Andrea. Software for modeling, analysis, and control design for multidimensional systems. In Proc. of the IEEE International Symposium on Computed Aided Control System Design, Hawai’i, USA, pages 24–27, August 1999. 25. J.M. Biannic. Commande robuste des syst`emes ` a param`etres variables. application en a´eronautique. Th`ese pr´esent´ee ` a l’Ecole Nationale Sup´erieure de l’A´eronautique et de l’Espace (SUPAERO), Toulouse, France, 1996. 26. C. Beck, R. D’Andrea, F. Paganini, W.M. Lu, and J.C. Doyle. A statespace theory of uncertain systems. In Proc. of the 13th IFAC Triennial World Congress, San Francisco, California, pages 291–296. Pergamon, July 1996. 27. C. Beck and J.C. Doyle. Mixed µ upper bound computation. In Proc. 31st IEEE Conference on Decision and Control, pages 3187–3192, Tucson, Arizona, USA, December 1992. 28. C. Beck and J.C. Doyle. Reducing uncertain systems and behaviors. In Proc. of 35th IEEE Conference on Decision and Control, Kobe, Japan, pages 712–714, December 1996. 29. W. Wang, J. Doyle, C. Beck, and K. Glover. Model reduction of LFT systems. In Proc. of 30th IEEE Conference on Decision and Control, Brighton, England, pages 1233–1238, December 1991. 30. C. Beck and R. D’Andrea. Computational study and comparison of LFT reducibility methods. In Proc. of the American Control Conference, Philadelphia, Pennsylvania, pages 1013–1017, June 1998. 31. P. Gahinet, A. Nemirovski, A. Laub, and M. Chilali. LMI Control Toolbox User’s Guide. The MarthWorks, Inc., Natick, Mass., 1995.

12 Physical Approach to LFT Modelling Jean-Paul Dijkgraaf1,2 , Samir Bennani1 , GertJan Looye3 , and Jean-Francois Magni2 1

2 3

Delft University of Technology, Department of Control and Simulation, Kluyverweg 1, 2629 HS Delft, The Netherlands [email protected] and [email protected] ONERA-Toulouse, CERT, BP 4025, F–31055 Toulouse Cedex, France [email protected] DLR-Oberpfaffenhofen, Institute of Robotics and Mechatronics, D-82234 Wessling, Germany, [email protected]

Summary. The development of an LFT representation for the nonlinear HIRMplus model is presented. The structured singular value µ is applied on the so developed LFT in order to clear critical areas in the flight envelope. µ-Analysis allows to determine the combination of uncertain parameters within their respective bounds, for which a performance criterion or stability margin is worst. For a sensible worstcase analysis, it is important that the uncertain parameters are directly related to the physical uncertain/varying parameters in the nonlinear model. First a symbolic nonlinear model of the HIRMplus is developed, which depends on the physical parameters of interest in a rational way. Then the model is linearised symbolically. A low order LFT model is generated and compared with one obtained using an affine modelling approach. Although both model representations are aimed to cover the same flight conditions, the affine model is a function of a large number of artificial parameters. Both models are compared regarding their ease of generation, conservatism, accuracy, and applicability.

12.1 Introduction This research work is motivated by the flight control law clearance task. Within this task, the concept of µ enables us to ensure Robust Stability (RS) and/or Robust Performance (RP) for an infinite number of combinations of different parameter values, which are bounded. In order to calculate µ it is necessary to describe the nonlinear aircraft model in the form of an LFT, where the nominal or known part of the model is separated from the perturbation/unknown part of the model via a feedback connection, see Fig. 12.1. With N structured as ¸ · N11 N12 N= N21 N22 (12.1) it represents the generalised flight dynamics of the aircraft. The closed loop can be described as C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 197-210, 2002. Springer-Verlag Berlin Heidelberg 2002

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D wD

zD

N d

e Fig. 12.1. Feedback structure

with

e = Fu (∆, N )d

(12.2)

Fu (∆, N ) = N22 + N21 ∆(I − N11 ∆)−1 N12

(12.3)

and with d representing the disturbance and/or reference inputs, e the performance outputs and ∆ a structured uncertainty block. The aim of the presented work is to develop and test a method which allows efficient and accurate µ-analysis for the clearance task. Given a nonlinear model in C-code, with aerodynamic look-up tables, a valid LFT model must be constructed in order to obtain reliable robustness analysis results. The suggested approach consists of the following two steps: 1. Analyse the whole flight envelope, area by area, using affine uncertainty representations of the model to construct the LFT model. This can be done in a quite straightforward fashion (see also chapter 16). The method possibly results into conservative robustness predictions. In Fig. 12.2 an example gridding is depicted. 2. Analyse problem areas of point 1 (dark areas in Fig. 12.2) using more exact LFT models, generated with the aid of the LFR-toolbox, according to Fig. 12.3. LFT models of small (with limited amount of complexity) dynamic systems can be generated by hand or by representing the generalised plant in a Simulink diagram. For models, which involve more parameters and dynamics one is forced to use a structured and preferably automated approach. In this direction LFT models have been generated out of (affine) parameter representations, obtained by repeatedly performed linearisations, with each linearisation belonging to another uncertain parameter combination.

Altitude

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Not cleared Cleared

Air speed

Fig. 12.2. Example Flight Envelope

Nonlinear Matlab/Simulink model, including C-code and aerodynamic tables

Derivation and validation of a simplified, symbolic nonlinear model with rational parameter dependencies

Generation of low order LFT model, including reduction and validation

Robustness analysis using m

Clearance / no clearance of flight envelope area

Comparison of the robustness analysis results with another LFT model

Fig. 12.3. Steps to perform LFT generation and assessment

The analysis of chapter 16 uses this idea by introducing artificial parameters in order to model variations/uncertainty of the system matrix elements. The approach of chapter 13 also uses many linearisations, but in a physical way. Another possibility is to realise 1 a symbolically linearised uncertain aircraft model. Research in this direction has already been performed for several years (see for example [6]), but there continued to be a problem: a drawback 1

One will say that building the realisation of an uncertain system consists of extracting all the parameters and the integration symbols of its symbolic representation.

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of using symbolic models, in which the uncertainties appear in a rational way, is that there is a risk of high order LFT’s. This chapter will give an idea how to use the symbolic approach and how to cope with the problem of possible high order. Hereby we aim at reduction of conservatism (compared with the approaches from chapters 13 and 16) and the ability to trace back worst case perturbations in a physical way. In order to largely automate LFT generation, the LFR-toolbox (see [7]), operating within the Matlab software package, is used. Tools for realisation, normalisation and reducing order are incorporated in this toolbox. The proposed procedure can be applied to every part of the flight envelope, but for simplicity the procedure is demonstrated for two flight conditions, one unaccelerated horizontal flight and one push-over trim condition. The two conditions can be found in Table 12.1. The chapter is structured as follows. In section 12.2 the procedure and results are presented concerning the construction and validation of the symbolic nonlinear equations of motion of the HIRMplus aircraft model (longitudinal motion). This corresponds to the first step of Fig. 12.3. Subsequently, two different LFT realisation approaches are presented in section 12.3, resulting in LFT models with different order (the second step of Fig. 12.3). The use of structure in parameter dependencies and the reduction of already realised LFT models is discussed. After that, the LFT with lowest order is used for robustness analysis. Worst case performance and worst case gain and phase margins are calculated. In section 12.4 the margin results are compared with those obtained with another LFT model, generated with repeatedly performed linearisations, and with the original model. It will give more insight regarding accuracy and conservatism. The chapter finalizes with the most important conclusions in section 12.5. Table 12.1. Flight conditions for analysis Parameter Mach number Angle of attack α Altitude Trim option Flight path angle γ Pitch rate q

FC1 0.2 6 5000 Push-over trim (3) -0.10

FC2 0.5 [-] 5000 Unacc. γ trim (1) 0 0

Unit [-] [deg] [ft] [-] [deg] [rad/sec]

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12.2 Construction of the Symbolic Nonlinear Flight Dynamics Model The goal of this section is twofold. The first is the transformation of the HIRMplus model into a symbolical, nonlinear parametric uncertainty model (PUM), in which the parameters appear in a rational form. The symbolic form serves well for localizing the physical dependencies of the uncertainties. The second objective has to do with the restriction that the final LFT should be kept limited order. One can achieve a good basis for that only by means of simplifying the model equations. As a first step the nonlinear equations are derived and simplified. After this, the aerodynamic tables are fitted by means of a mean square fitting routine or are approximated by an LFT. Subsequently, the sinus and cosinus terms are approximated by a taylor expansion and the model is programmed in a simulink block diagram. This diagram can easily be used to validate and tune the model in two simultaneous ways, i.e. the time responses can be compared with those from the original nonlinear model and the linearisations can be compared with the original ones (from the standard trim and linearisation routines of the HIRMplus model). It is decided to investigate the uncertainty set from Table 12.2 (Iyinv is just a transformation of variable for Iy). These uncertainties will appear in a symbolic way and the system matrix S(p) can be written as: S(p) = f (Cmq , Cma , Cmdts , Xcg , Iyinv )

Table 12.2. Nominal, minimum and maximum values of the uncertainties Uncertainty

Nom value

Cm q Cm a Cmδts Xcg Iy Iyinv

0 0 0 0 163280 ( Iy 1 − min

+ Iy 1

max

1

Including reduction factors

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Min value 1 −0.046 −0.046 −0.0184 −0.15 163280 ∗ 0.95 )/2+ 1/Iymax

Max value1 0.046 0.046 0.0184 0.15 163280 ∗ 1.05 1/Iymin

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12.2.1 Derivation of Nonlinear Equations The states, inputs and outputs of the flight dynamics are: £ ¤0 States: VT AS α β p q r φ θ ψ x y z £ ¤0 Inputs: δts δtd δcs δcd δr suction Fslvl1 Fslvl2 ugust vgust wgust £ ¤0 Outputs: p q r θ φ ψ anx any anz VT AS M a h α β For more background information on the model the reader is referred to [5] and the accompanying software. A couple of choices can now be made: – Only the longitudinal model is considered – Gust is not taken into account – Due to extreme nonlinearity in the aerodynamic tables, the range of angle of attack is limited to [0,20] deg – The states are the 4 standard states: true airspeed VT AS , angle of attack α, pitch rate q and pitch attitude θ The differential equations are derived according to the classical approach, using the standard force, moment and kinematic equations. A description of this approach can be found in [4]. For the force and moment equations we start with ( F = ma (12.4) M = I Ω˙ + Ω × IΩ with F the force vector, incorporating aerodynamic force, thrust force and gravitational force and with M the moment vector, incorporating aerodynamic and thrust moment. The equations of motion are expressions for acceleration in each respective direction, so we rewrite equation 12.4: ( F a = m (12.5) Ω˙ = (M − Ω × IΩ) I −1 All the force, moment and kinematic equations in their general form are derived (in body axes) and the implicit coefficients are investigated in order to determine their importance. The following coefficients can be neglected (either because they are zero or because they have a neglectible influence): Clv = 0, Clrnt = 0, Clpnt = 0, Clpnc = 0, Cnrnc = 0, Cnpnt = 0, Cnpnc = 0, Cnnc = 0 For the longitudinal dynamics the four equations of motion are:

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vb1 v˙ b1 + vb3 v˙ b3 V˙ T AS = = VT AS = cos α[ACxh + B + C + D] + sin α[A(Czh + Czq J) + +E + F ] α˙ =

(12.6a)

sin α cos α [A(Czh + Czq J) + E + F ] − [ACxh + B + VT AS VT AS +C + D]

(12.6b)

q˙ = [−Am3.51(Cmh + Cmq J + G + H + I) + 0.35D + +Xcgunc Am(Czh + Czq J)]

1 Iy

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(12.6c) (12.6d)

with ρVT2AS S 2m −qVT AS sin α −g sin θ (Fslvl1 + Fslvl2 ) ρ ( ) m ρ0 qVT AS cos α

A=−

(12.7a)

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(12.7b)

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(12.7c) (12.7d) (12.7e)

F = g cos θ

(12.7f)

G=

(12.7g)

H= I= J=

1.76 (q − qtrim ) Cmq unc VT AS Cmα unc(α − αtrim ) Cmδts unc(δts − δtstrim ) 1.76q VT AS

(12.7h) (12.7i)

(12.7j)

12.2.2 Fitting of Aerodynamic Tables and Tuning of the Model The equations of motion from the last section contain aerodynamic stability derivatives, which are functions of other parameters. Here they only depend on angle of attack α and/or taileron deflection δts . In general, of course, they

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may depend on more variables. This section describes the procedure to obtain the polynomial fitting of the aerodynamic tables. CXH plot as a function of ALPHA and DTS CXH plot as a function of ALPHA and DTS

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Fig. 12.4. Aerodynamic fittings, (a) Plot of aerodynamic table for Cxh , (b) Reduced α range, (c) First fitting result, (d) Improved fitting result

The development of Cxh for FC1 is taken as an example. The aerodynamic data over the whole α-range is depicted in Fig. 12.4(a). Because of high nonlinearity the angle of attack range is reduced to [0,20] deg, depicted in Fig. 12.4(b). The polynomial expression in (12.8) is taken as an initial polynomial to fit the data (coefficients to be determined): Cxh = a0 + a1 α + a2 δts + a3 α2 + a4 δt2s

(12.8)

The coefficient values are determined by means of a least square routine. The polynomial fitting result is plotted over the original data in Fig. 12.4(c). The similarity between the models turned out to be not good enough. Therefore, the polynomial order of the fitting is raised and, as an extra measure, the

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number of points in the aerodynamic table is increased by interpolation. The new Cxh expression is: Cxh = b0 + b1 α + b2 δts + b3 αδts + b4 α2 + b5 δt2s + b6 α3 + b7 δt3s

(12.9)

This function is depicted in Fig. 12.4(d). For the other stability derivatives this fitting procedure is used as well. Nonlinear pitchrate response on δ input, FC2, gamma−trim

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Fig. 12.5. Time responses to block input on the taileron

In order to judge the quality of the symbolic nonlinear model it is linearised. The linear system matrices are compared with the directly linearised original model. With the information about discrepancies at specific places in the model (matrix elements), it is possible by tuning to correct for aerodynamic fitting errors. This is done together with an adaptation of the trim condition for the symbolic model (due to approximations this trim condition slightly differs from the one for the original model). Fig. 12.5 shows the time responses (FC2) of both models, used for final validation. As can be seen, small discrepancies still exist, which should later be covered by a compensation parameter.

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12.3 From PUM to LFT The construction of the symbolic nonlinear flight dynamics model from the last section mainly involved the derivation of the equations of motion, the polynomial fitting of aerodynamic data tables and the validation of the model. After the symbolically performed partial derivation at the two flight conditions, we have at our disposal the following model structure: ¯ ¯ ¯ ∂F (δx,δu,p) ¯ A(p) = ∂F (δx,δu,p) ¯δx = 0, B(p) = ¯δx = 0, ∂(δx) ∂(δu) δu = 0

¯ ¯ C(p) = ∂G(δx,δu,p) ¯δx = 0, ∂(δx) δu = 0

δu = 0

¯ ¯ D(p) = ∂G(δx,δu,p) ¯δx = 0, ∂(δu) δu = 0

The use of structure before the realisation phase is crucial for arriving at minimal order uncertainty blocks. [3] However, n-D reduction/approximation routines for decreasing order after realisation continue to be necessary in most cases, because structure cannot always fully be exploited. The routines used are based on the removal of uncontrollable/unobservable subspaces according to the proposed procedure in [1]. Some results are depicted in Table 12.3. Table 12.3. Sizes of the LFT sub blocks: direct realisation approach and symbolic structured tree decomposition, with and without reduction afterwards Parameter Cmq U nc Cma U nc Cmδts U nc Iy inv Xcg U nc

Direct realisation no reduction n-D reduced 1 1 1 1 2 1 6 5 53 10

symbolic STD no reduction n-D reduced 1 1 1 1 1 1 1 1 10 9

Validation of the models is performed by means of simulation and matrix element comparison between the original nonlinear model (linearised) and the LFT model for different perturbation sets (not shown here). It can be remarked that within the effort of modelling with the least conservatism possible, one has to account for the following important issue: the polynomial fittings of aerodynamic tables may cause discrepancies between the original model and the LFT. Measures to avoid this will be discussed in the following section.

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12.4 Robustness Analysis The approach of the last section has provided us with a low order LFT model, containing physical uncertainties/variations. This model enables to perform robustness and worst case analysis (in this case only performance). The objective of this section is threefold: • Presentation of worst case performance results 2 and comparison with original linear model. With these results it is possible to judge about conservatism • Proposition for compensating modelling errors • Comparison of the same analysis results with those, obtained by a rather straightforward LFT generation approach

Table 12.4. Worst case gain margins [dB], FC1 (δts : elevator deflection) Loop broken δts q

GM symb method 9.65 9.64

GM orig model 7.86 7.84

GM ’min-max’ model 7.84 7.82

Table 12.5. Worst case gain margins [dB], FC2 (δts : elevator deflection) Loop broken δts q

GM symb method 8.18 8.14

GM orig model 8.03 8.00

GM ’min-max’ model 7.99 7.95

Tables 12.4 and 12.5 show an example of the worst case gain margin results. Analysis of the original model reveals that the margin for e.g. the δts -channel is 8.03 dB (FC2), meaning that the symbolic approach prediction is slightly optimistic. The same holds for the pitch rate channel. Regarding FC1 the situation is worse. When comparing the results with the original model, we see a quite dangerous margin prediction. Both deviations are due to the same cause: one introduces errors when fitting aerodynamic data into functions. Especially in the case of the HIRMplus, these data tables show highly nonlinear curvatures. The situation is deteriorated by the necessary substitution of equilibrium functions (which describe the trim state and trim input as a function of the set of parameters, 2

The robustness analyses are performed analogous to the setup presented in [8]

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which defines the trim). In the validation process of the LFT this problem could be seen by some deviations in matrix elements. There are two approaches to deal with the optimistic predictions. The first approach comes down to a compensation after the LFT is generated. Investigation of the proper variation of matrix elements should indicate the amount of extra variation to be added. This can easily be done by means of an artificial parameter. The introduction of a small artificial compensation parameter, in order to cover polynomial fitting discrepancies, shows that the gain margin for FC2 changes into 8.00 dB (δts -channel) and 7.97 dB (qchannel), respectively. The second way is a more structured idea, which could very well be used in general, just to be sure to be at the ’safe side’. Already in the modelling phase one can introduce some more variation on the implicit functions. Fig. 12.6 shows the idea for an example curvature of Cmh . Doing this for all aerodynamic functions and equilibrium surfaces will cause the LFT model to be more conservative, but this amount of conservatism is very limited compared to possible conservatism with the ’min-max’ method of chapter 16.

Cmh

X

e

X X X

-d1

X

X

0

d1

X

Fig. 12.6. Example aerodynamic function with compensation parameter

Tables 12.4 and 12.5 also show the gain margin results for the LFT model, obtained with another approach. It is based on an affine description of a ’multi-model’, which is the set of system matrices generated by means of many linearisations. The method is also called ’min-max’ approach (see [2] and chapter 16). It is obvious that these margins are very close to what they should be. We could ask ourselves: is the more time consuming symbolic method worthwhile

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doing with respect conservatism? The answer is yes. Although the results for the ’min-max’ method appear to be not conservative at all, the method has a general potential risk for conservatism. The reason is the fact that the artificial parameters are able to form worst case combinations, which are not physically possible. They do not take into account possible parameter dependencies in the matrices (apparently, in the HIRMplus case they did form a physically possible combination). An aproach which also adopted the idea of a multi-model is explained in chapter 13. Contrary to the last paragraph, the ’bands-based’ method uses physical parameters to describe the parameter dependencies. The disadvantage is of course the fact that many linearisations are needed in order to be able to construct the LFT. This means that every time something changes in the model (refined aerodynamic mode, change in control system), the engineer has to do all the linearisations all over again.

12.5 Conclusions A natural setup is presented to perform uncertainty modelling, LFT generation and validation. The first step in the process is the derivation of the nonlinear equations of motion of the aircraft, which is done in section 12.2. This is a time consuming step, especially if one does not really know the model beforehand. The aerodynamic tables were fitted using a least square fitting routine. It turned out to be necessary to tune the obtained polynomials for different flight conditions. The possibly introduced fitting errors are of considerable influence on the quality of the final robustness analysis results. The amount of simplification in the modelling phase already determines the size of the final LFT, later on in the process. For example, the order of the fittings and the approximations of sinus and cosinus have their influence. Once the symbolical nonlinear equations are known, it is possible to start the realisation of the LFT model, see section 12.3. Herefore it is necessary to partially differentiate the symbolic equations around the trim condition. For the standard uncertainty set, two LFT models were now realised, one by a direct realisation and one by the ’structured tree algorithm’. The use of structure leads to a lower order LFT model. For the actual process of ’pulling out the delta’ as well as for reducing order we the LFR-toolbox is used. By means of the symbolic toolbox within Matlab it is possible to use Maple routines as well. The validation of the obtained LFT is performed by simulation and by comparing with the original model linearisations. Introduced errors in the modelling phase can be investigated at this stage and covered by means of compensation parameters. In section 12.4 the analysis results of the symbolically obtained LFT model are assessed by means of comparison with the ’min-max LFT’ and

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with the original model linearisations. The worst case analysis has not really substantiated the advantage of the exact symbolic modelling approach regarding conservatism. The set of uncertain parameters, as defined within GARTEUR, and the structure of the model are the most important reasons for that. The artificial worst case parameter sets of the min-max approach appear to create systems, which are physically possible. However, the symbolic approach enables us to find worst case perturbation sets, which consist of physical parameters. In problem areas in the flight envelope this is almost obligatory. Besides, no repeated linearisations are needed to obtain the LFT, which is a great advantage.

References 1. R. D’Andrea and S. Khatri. Kalman Decomposition of Linear Fractional Transformation Representations and Minimality. In Proceedings of American Control Conference, 1997. 2. D.G. Bates and R. Kureemun and M.J. Hayes and I. Postlethwaite. Clearance of the HIRMplus RIDE Flight Control Law: A µ-analysis approach. GARTEUR FM-AG11/TP-119-11, 2000. 3. J.C. Cockburn and B. Morton. Linear Fractional Representations of Uncertain Systems. In Automatica 33 number 7, 1263–1271, 1997. 4. M.V. Cook. Flight Dynamics Principles. Arnold, London, UK, 1997. 5. Garteur FM-AG08. Robust Flight Control Design Challenge Problem Formulation and Manual: the High Incidence Research Model (HIRM). GARTEUR FM-AG08/TP-088-4, 1997. 6. P. Lambrechts and J. Terlouw and S. Bennani and M. Steinbuch. Parametric Uncertainty Modeling using LFTs. In Proceedings of the American Control Conference, 267–272, San Francisco, California, 1993. 7. J.F. Magni. Linear Fractional Representation: a Toolbox for Use with Matlab. GARTEUR FM-AG11/TP-119-04, 2001. 8. J. Shin and G.J. Balas. Worst-Case Analysis of the X-38 Crew Return Vehicle Flight Control System. In Journal of Guidance, Control, and Dynamics. 24 number 2, 261–269, 2001.

13 Uncertainty Bands Approach to LFT Modelling Thomas Mannchen and Klaus H. Well University of Stuttgart Institute of Flight Mechanics and Control Pfaffenwaldring 7 a 70550 Stuttgart Germany [email protected] [email protected]

Summary. This chapter discusses numerical techniques for generating linear fractional transformation based uncertainty models for use in the clearance procedure of flight control systems using the structured singular value µ. The numerical techniques do not require closed-form linear expressions for the aircraft dynamics – only a non-linear software model of the closed loop aircraft, which can be efficiently trimmed and linearised numerically, is required. A refined approach for generating linear fractional transformation based uncertainty models is presented. The resulting parametric models allow to identify worst-case uncertain parameter combinations.

13.1 Introduction In order to apply µ-analysis techniques to the flight control law certification problem, a parametric uncertainty model which allows a particular linear fractional transformation (LFT) representation of the uncertain closed loop system must first be generated, see Fig. 13.1. M represents the known part of the system (plant and controller) and ∆ represents the uncertainty present in the system. In effect, extra inputs and outputs are introduced so that the system uncertainty can be considered as part of an “external feedback loop”. µ defines a stability-test for a closed loop system subject to structured uncertainty ∆ in terms of the maximum structured singular value [1]. In recent years much attention has been paid to the issue of how to efficiently generate accurate (and ideally minimal) LFT-based uncertainty models for complex uncertain systems – see [2] for an overview. A common assumption among almost all of the approaches suggested is that closed form analytical expressions relating the aircraft dynamics to the uncertain parameters of interest are available, from which LFT-based uncertainty models may be derived. Such expressions usually take the form of non-linear equations of motion involving the uncertain parameters, which when linearised C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 211-220, 2002. Springer-Verlag Berlin Heidelberg 2002

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∆ w r

M

z y

Fig. 13.1. Upper LFT uncertainty description

symbolically using dedicated software tools [3], [4], [5], result in linear statespace models whose coefficients depend explicitly on the uncertain parameters. Given state-space models in this form, the generation of accurate, if not always minimal, LFT-based uncertainty models is then relatively straightforward – see [6], [7] and [8] for some flight control examples. The main drawbacks of the above approach can be identified as the substantial modelling effort required to accurately relate all the uncertain parameters to the non-linear aircraft dynamics, and the fact that the symbolically linearised state-space models are generally valid only at and around the relevant operating point in the flight envelope. Here, an alternative approach for generating LFT-based uncertainty models is presented which does not require the availability of analytical expressions relating the aircraft dynamics to the uncertain parameters – only a non-linear software model of the closed loop aircraft, which can be efficiently trimmed and linearised numerically for different values of the uncertain parameters, is required. The proposed approach thus allows a significant reduction in the modelling effort required, and in general can be applied to complex systems which cannot be satisfactorily described using simple differential equation based symbolic models.

13.2 LFT Modelling Using Trends and Bands 13.2.1 Uncertainty Modelling As shown in [9], LFT-based parametric uncertainty models may be conveniently derived from a linear state space representation of the uncertain system of the form x˙ = (A0 + A1 δ1 + · · · + An δn )x + (B0 + B1 δ1 + · · · + Bn δn )u (13.1) y = (C0 + C1 δ1 + · · · + Cn δn )x + (D0 + D1 δ1 + · · · + Dn δn )u The matrices A0 , B0 , C0 , D0 describe the nominal system, while Ak , Bk , Ck , Dk , with k = 1, . . . , n describe deviations from the nominal system depending on

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the normalized physical uncertain real value parameter δk with −1 ≤ δk ≤ 1. In this section, we focus primarily on the problem of generating state-space representations of the type given in (13.1) from the original non-linear simulation model of the aircraft. Various methods can be used to tackle this problem. The first possibility is to use a Taylor-series approximation around a chosen equilibrium point. For example, the A matrix depending on the k th parameter can be written as ¯ ∂A ¯¯ δk (13.2) A = A (δk = 0) + | {z } ∂δk ¯δk =0 | {z } A0 Ak

where the partial derivatives in the above expression can be computed numerically using finite differences. The advantage of this method is that it uses a standard procedure and the partial derivatives can be calculated easily. This formulation, however, is valid only for regions in the flight envelope where uncertainty results in small or linear variations in the aircraft model. In the case of highly nonlinear dependency of the state-space representation on the uncertain parameters, linearisation intervals become so small that the parameter envelope must be split into many small regions. Moreover, without exact knowledge of the system, the determination of the linearisation intervals, the computation of the derivatives and the choice of appropriate equilibrium points is often difficult. A second approach to the problem of generating LFT-based uncertainty models in the form given by (13.1) is the so-called “min-max” technique [5], [10], used by the University of Leicester in their analysis, Chapter 16. Here, the values of the system-matrix elements are evaluated so that the minimum and the maximum value within the given parameter range are identified element-wise. For example, an element aij of the A matrix can be written as aij = aij0 + aijmin max δaij

(13.3)

with aij0 =

¯ aijmax − aijmin ¯¯ aijmax + aijmin , aijmin max = , δaij ¯ ≤ 1 2 2

(13.4)

Note that for each varying element of the state-space matrices A, B, C, D, an individual δ is needed. Note further that these δ’s are “fictitious” and have no physical meaning. The method is straightforward to implement, fast and can be fully automated, but it can lead to conservatism since possible joint parametric dependencies in the state-space model are ignored. Furthermore, since the “fictitious” δ’s do not directly represent the physical uncertain parameters, it is not possible to identify the worst-case combination of these parameters in the resulting µ-analysis.

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13.2.2 Trends and Bands Based Uncertainty Modelling In this chapter we propose a redefined method for generating LFT-based uncertainty models which is less conservative than the “min-max” approach and moreover allows the computation of worst-case uncertainty combinations in terms of physical uncertain parameters. As with the “min-max” approach, however, no closed form expressions (typically linearised equations of motion) involving the physical parameters are required. The key idea of the proposed method is to model the uncertainties using a curve fitting technique in a least squares sense. Consider, for example, the element aij = f (δk ) of the matrix A depending on one parameter. The formulation aij = aij0 + aijk δk

(13.5)

then represents a linear approximation of the dependency of this element on the uncertain parameter δk , assuming the coefficients ¢ ¡ aij¢0 and a¡ijk aremderived using a least squares fit based on m data-pairs a1ij , δk1 , . . . , am ij , δk . Figure 13.2 illustrates both the actual dependency aij = f¡(δk ) and ¢ the approximation aij = aij0 + aijk δk and shows the rth data-pair arij , δkr , r = 1 . . . m. For k = 1 . . . n uncertain parameters the equation for the coefficients is then given by    1    aij 1 δ11 · · · δn1 aij0  1 δ12 · · · δn2   aij1   a2ij        (13.6)  .. .. . . ..  ·  ..  =  ..  . . .     . . . .  aijn 1 δ1m · · · δnm am ij {z } | {z } | {z } | X

y

z

This usually overdetermined system of equations can be solved for the coefficients in a least squares sense by solving ¢−1 T ¡ X z. y = XT X

(13.7)

In the case of two uncertain parameters, the curve which was approximated by a line, as in Fig. 13.2, can then be interpreted as a surface which is approximated by a plane. In the case of n uncertain parameters, the approximation can be interpreted as a multidimensional regression plane. Note that the implementation of higher order polynomial fits under this approach is also straightforward. For the work presented in this paper, however, a linear fit was found to be adequate. Repeating the above procedure for all elements of the state-space matrices A, B, C, D yields a set of parameter dependent system equations x˙ =

(A0 + A1 δ1 + ... + An δn )x + (B0 + B1 δ1 + ... + Bn δn )u y = (C0 + C1 δ1 + ... + Cn δn )x + (D0 + D1 δ1 + ... + Dn δn )u

(13.8)

13 Uncertainty Bands Approach to LFT Modelling

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Note that the δ’s are now all related to the physical uncertain parameters. For use in a µ-analysis, these fits must be transformed into an LFT-based uncertainty model. To do so, a state-space representation is created, using the approach proposed in [9]. In a matrix representation, the M -block for the linear fit can be assembled in the following manner:   A0 [A1 B1 ] · · · [An Bn ] B0 · ¸     · ¸ x x˙ 0   I 0 ··· 0    z1   0 I    w1        ..   .  .. .. ..  · .. .. (13.9)  ..   . = . . . . .       · ¸ · ¸    zn   wn 0   I 0 ··· 0   y u I   0 C0 [C1 D1 ] · · · [Cn Dn ] D0 T

T

By closing the loop between [w1 ...wn ] and [z1 ...zn ] with the ∆-block, which is of the form ¸ · ¸¶ µ · I 0 I 0 , ..., δn (13.10) ∆ = diag δ1 0 I 0 I it can be shown that the equations (13.9) and (13.10) are an LFT of the system given in equation (13.8), see [9]. The dimension of zi and wi , i = 1 . . . n, is given by: dim(zi ) = dim(wi ) = dim(x) + dim(u) .

(13.11)

13.2.3 Nonlinearity Compensation The method described above is a linear approximation of the uncertain system. Therefore, it does not cover nonlinearities in the dependency of state-

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space matrices on the uncertain parameters. Hence, some unstable combinations could be left out during the µ-analysis, which would make the whole analysis invalid. For instance, assume that a matrix has a varying element aij . As shown in Fig. 13.2, the unstable region could be left out by the test, if only a linear approximation is used. To include deviations from the linear approximations, additional real nonlinearity compensation parameters (CP’s) are added to the existing δ-set [5]. These take the form of aijCP δaijCP , where aijCP represents the maximum deviation from the linear representation and δaijCP is an additional normalized δ. The compensation parameters add an uncertainty “band”-structure to the “trend” established by the dependency of the state-space elements on the physical uncertain parameter. Each compensation parameter independently acts on one state-space matrix element. To keep the variation range or the size of the band as small as possible, the least squares fit is the preferred method for the linear approximation, since it minimizes the size of the required band. The equation for one matrix element of the matrix can now be written as aij = aij0 + aij1 δ1 + ... + aijn δn + aijCP δaijCP .

(13.12)

The size of the band is chosen to be the maximum error between the linear approximation and the actual value of a matrix element. Figure 13.2 illustrates this for the rth data point. Note that the size of the band depends on the number m and locations of the data-points. A strategy for choosing an appropriate number of data-points is given in Sect. 13.2.4. The following equation yields the size of the band for the general case of n parameters. ¯¢ ¡¯ (13.13) aijCP = max ¯aij (δ1r , ..., δnr ) − arij ¯ r=1...m

In the proposed technique, we therefore try to find trends, which represent the parameter variations, but allow for nonlinear variations by introducing compensation parameters. In the general case these trends are modelled by a multi-dimensional regression plane, which describes the linear variation, and a band-structure (compensation parameters), that is limited by planes parallel to the regression plane, above and below, to include nonlinear deviations. The actual value of the uncertain state-space element is then assumed to lie somewhere within this band-structure. Compared with the element-wise “min-max” approach, which also works with bands, the size of the bands is now reduced, thus decreasing the conservatism of the analysis results (see Fig. 13.3 and Fig. 13.4). Furthermore, the trends introduce δ’s with a physical interpretation, which can be used to identify the actual worst case combination of the physical uncertain parameters. The nonlinearity compensation parameters δCP are introduced into the LFT by adding four submatrices E, F, G, H to the M -block representation

13 Uncertainty Bands Approach to LFT Modelling

217

aij

+aijminmax -aijminmax δk Fig. 13.3. Element-wise “min-max” approach

aij +aijCP -aijCP

δk Fig. 13.4. Trends and Bands approach

£ ¤ £ ¤£ ¤ A0 A1 B1 · · · An Bn E 0 · ¸  ·    I 0 ··· 0 0 x  0  z1    .. .. .. ..   ..  .   . .¸ . . . ·  ..    = I   z   0 ··· 0 0  n   0 z  · ¸ CP  G  y 0 ··· 0 0  0 ¤ £ ¤£ ¤ £ C 0 C1 D 1 · · · Cn D n 0 F 

 B0 · ¸  0   x I     ..   w1    .  ·.¸  ..   ·   (13.14) 0    w  n   I  · ¸   wCP  0  u  H  D0

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with the corresponding ∆-block ¸ ¸ · ¶ µ · I 0 I 0 , δCP1 , ..., δCPt . , ..., δn ∆ = diag δ1 0 I 0 I

(13.15)

The total number of compensation parameters t is the number of varying elements in the state-space matrices A, B, C, D. The submatrices E, F, G, H are assembled in the following manner: E and F consist of as many columns as compensation parameters. In each column, the maximum deviation of a matrix element, e.g. for matrix A: aijCP , is placed in the position which corresponds to the row index i of the matrix element, all other entries are set to naught. E contains all aijCP ’s and bijCP ’s, whereas F contains all cijCP ’s and dijCP ’s. In a similar manner, G and H are assembled with rows for each compensation parameter, but now the matrix element with the column index j is set to unity, the other elements equal naught. To illustrate this, an example state-space system with two states, one input and one output is considered. Assume the matrix elements a11 and d11 are varying and there is only one physical uncertain parameter. The LFT would then look like ¸· ¸ ·   a11CP 0 a111 0 0 B0   A0 0 00 0 0           10 0  x˙ x   0 1 0  z1   0 0 w1     =  ·   zCP   0 0  wCP  . (13.16) 1   · ¸ · ¸   y u  10 0    0 0  00 1  £ ¤ £ ¤ 0 0 d111 0 d11CP D0 C0 Note: the all-zeros column (4th column) and the corresponding 4 th row can be deleted but were left in here for clarity. The corresponding ∆ looks like     100 (13.17) ∆ = diag δ1  0 1 0  , δCP1 , δCP2  . 001 13.2.4 Number of Data Points Required To determine the number of data points required for the linear fit in the Trends and Bands LFT generation, the following procedure can be used: – Produce an initial LFT-based uncertainty model using the vertices, e.g. the minimum and maximum value of the uncertain parameters, only. This gives an uncertainty model characterized by trends only.

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– Perform an initial µ-analysis using the initial LFT model and plot the resulting µ upper bound. – Compare this µ upper bound with the results of a µ calculation based on a “min-max” type LFT. – Increase the number of data points, recalculate and plot the corresponding µ upper bound. Repeat the last step until no further increase in the resulting µ upper bounds towards the “min-max” based µ upper bound result is observed. The number of data points used for the final calculation can then be regarded as being sufficient to cover possible nonlinearities in the dependency of the state-space matrix elements on the uncertainty parameters. In Chapter 5 this procedure is used, see Fig. 17.5.

13.3 Conclusions A new technique was introduced for the generation of linear fractional transformation based uncertainty models which are required as inputs for the analysis of flight control systems using the structured singular value µ. In contrast to standard approaches which use symbolic linearised equations of motion, the proposed approach requires only a nonlinear software model of the aircraft which can be efficiently trimmed and linearised. The proposed approach can therefore considered to be a valid alternative proceduce, which may be applied when it is not possible to use an exact approach based on symbolic equations. This might be the case when the model is hybrid in structure, e.g. including look-up tables and fortran or C code. In the proposed technique, linear dependencies (“trends”) of the aircraft dynamics on the uncertain parameters are modelled by a multi-dimensional regression plane. Additional non-linear dependencies are modelled using a “band”-structure defined by nonlinearity compensation parameters. The major limitation of the proposed approach is that it is prone to be at least somewhat conservative, especially for systems with a strong nonlinear dependency of the state-space representation on the uncertain parameters. Its major advantage is that it allows, in contrast to the “min-max” approach, to identify the worst-case combination of uncertain parameters.

References 1. Doyle, J. C. Analysis of feedback systems with structured uncertainties. In: IEE Proceedings, Vol. 129, Part D, no. 6, pp. 242–250, 1992. 2. Ferreres, G. A Practical Approach to Robustness Analysis with Aeronautical Applications, Kluwer Academic, ISBN 0-306-46283-4, 1999.

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3. Terlouw, J. and Lambrechts, P. F. A MATLAB Toolbox for Parametric Uncertainty Modelling, Technical Report CR-93455-L, National Aerospace Laboratory NLR, Amsterdam, 1993. 4. Varga, A. Computational Challenge in Flight Control Design. In: Proc. of the IEEE International Symposium on Computer Aided Control System Design, Hawaii, USA, pp. 1–6, 1999. 5. Varga, A., Looye, G., Moormann, D. and Gru ¨bel, G. Automated Generation of LFT-Based Parametric Uncertainty Descriptions from Generic Aircraft Models, GARTEUR Technical Report TP-088-36, 1996. 6. Belcastro, C. M. Uncertainty Modelling of Real Parameter Variations for Robust Control Applications, Ph.D. Thesis, University of Drexel, USA, 1994. 7. Lambrechts, P., Terlouw, J., Bennani, S. and Steinbuch, M. Parametric Uncertainty Modelling using LFT’s, Proc. of the American Control Conference, San Francisco, 1993 USA, pp. 267–272 8. Idan, M. and Shaviv, G. E. Robust Control Design Strategy with Parameter Dominated Uncertainty, AIAA Journal of Guidance, Control and Dynamics, 19(3), pp. 605–611, 1996. 9. Morton, B. and McAfoos, R. A µ test for real parameter variations. In: Proceedings of the ACC, pp. 135–138, 1985. 10. Hayes, M. J., Bates, D. G. and Postlethwaite, I. New tools for computing tight bounds on the real structured singular value, AIAA Journal of Guidance and Control, 24(6), pp. 1204–1213, 2001.

14 Flatness Approach to LFT Modelling Franck Cazaurang, Lo¨ıc Lavigne, and Benoˆıt Bergeon LAP - University of Bordeaux I –ENSEIRB – CNRS UMR 5131 [email protected]

Summary. In this chapter, an LFT model representing the longitudinal dynamic of the gap between the open loop nominal HIRM+ model and the perturbed model on a trajectory is given. Five of the most relevant uncertainties on the pitch axis are considered. This model describes the gap along a trajectory passing through flight condition FC1, with an angle of attack equal to 6 ◦ . The proposed approach is based on a simplified longitudinal model used to determine a nominal trajectory and corresponding input. The specific outputs are angle of attack α and pitch angle θ. These outputs allow a parametrisation of state-space trajectory. First the LFT generation by flatness approach is described. Then an LFT model of the open loop HIRM+ model for the above flight condition is given. Thereafter an LFT model of the closed loop HIRM+RIDE is proposed to allow the identification of worst case stability margin.

14.1 Introduction In this chapter we apply the flatness property of nonlinear systems to generate Linear Fractional Transformations (LFTs) for systems with uncertain parameters. This approach formalises a concept proposed in 1992 by Balas [1] concerning uncertainty modelling and analysis of feedback linearised systems. The modelling procedure outlined is demonstrated on the nonlinear HIRM+ longitudinal dynamics defined in [2]. Measurements on the so-called flat outputs, used in the linearising feedback, are corrupted by noise and effects of unmodelled dynamics or parametric errors (model simplifications) and the state-space trajectory of the actual plant is not identical to the reference state-space trajectory. Therefore an exact linearisation can not be achieved. But, as long as the tracking error remains small enough, the dynamic of a nonlinear system can be described as a set of perturbed linear systems. Section 2 explains how to obtain a simplified model (aerodynamics data analysis) from a nonlinear system with the property to be input-output linearisable without an unobservable nonlinear dynamic. This property is demonstrated with the flatness approach in section 3. The main contribution, namely the generation of the compact set of models, is given in detail in section 4. Finally we give in section 5 along a desired trajectory, the LFT form of the HIRM+ longitudinal model together with the RIDE controller [3].

C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 221-243, 2002. Springer-Verlag Berlin Heidelberg 2002

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14.2 The Simplified Aerodynamic Model The model used in this chapter is a full six-degree-of-freedom (aerodynamic, gravitational and propulsive (thrust)) reads as: ξ = [α, β, p, q, r, VT , θ, φ, ψ, z, x, y]

T

where the components of the state vector are: x, y, z, co-ordinates of the centre of mass in the Earth axes; VT , α, β, velocity in the body axes, angle of attack and sideslip angle; p, q, r components of the angular velocity in the body axes and θ, φ, ψ the Euler angles. The aircraft is actuated by seven independent controls: the thrust per engine F , the positions of the deflection surface (δT S , δT D , δCS , δCD , δR ) and quantity of the nose suction. Projected on the body axes, the sum of the external forces (aerodynamic, gravitational and propulsive (thrust)) reads         −CX − sin θ 2 X  + F 0  Y  = 1 ρSVT2  CY  + mg  0 2 −CZ cos θ cos φ 0 Z The aerodynamic coefficients CX , CY , CZ were experimentally determined in a wind tunnel and flight tests. Similarly, the sum of the moments about the centre of mass reads in the body axes as:       L 0 bCl 1  M  =  MT  − ρSVT2  cCm  2 N bCn 0 where Cl , Cm , Cn , are roll, pitch and yaw moment coefficients respectively, and L, M , N are the projections of the sum of the moments in the body axes. For the HIRM+ model, these coefficients are defined by the following expressions: – – – – – –

CX is a function of α, δT S , and δCS . CY is a function of α, β, r, p, VT , δT S , δT D , δR , δCD , δCS and suction. CZ is a function of α, q, VT , δT S , and δCS . Cl is a function of α, β, r, p, q, VT , δT S , δT D , δR , δCD , δCS and suction. Cm is a function of α, q, VT , δT S , δCS and suction. Cn is a function of α , β, r, p, VT , δT S , δT D , δR , δCD , δCS and suction.

The aerodynamic forces and moments depend on the air density ρ and the standard atmosphere model allows expressing this quantity as functions depending only on the altitude −z. Projected in the wind and body axis, the state-space model dynamics can be written as: X cos α cos β Y sin β Z sin α cos β V˙ = + + m m m

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X sin α Z cos α + (14.1) mVT cos β mVT cos β X cos α sin β Z sin α sin β Y cos β − − β˙ = −r cos α + p sin α + mVT mVT mVT

α˙ = q − r sin α tan β − p cos α tan β −

x˙ = VT cos β cos α cos ψ cos θ + VT sin β (cos ψ sin θ sin φ − sin ψ cos φ) +VT cos β sin α (sin ψ sin φ + cos ψ sin θ cos φ) y˙ = VT cos β cos α sin ψ cos θ + VT sin β (sin ψ sin θ sin φ + cos ψ cos φ) +VT cos β sin α (sin ψ sin θ cos φ − cos ψ sin φ)

Iy   z Ixz qr Ix −1− IIx L+(Iy −Iz )qr Ixz p˙ = + Ix pq + + ··· I2 Ix Iz − Ixz x

2 ··· +

pq Ix −Iy +

Ixz Ix

+ IIxz L+N x

I2 Iz − Ixz x



M + (Iz − Iy )pr − Ixz (p2 − r2 ) Iy    I2 + − 1 − IIxz + pq Ix − Iy + Ixz x

q˙ =  Ixz qr r˙ =

Iy Ix

Iz −

Ixz Ix L

+N

2 Ixz Ix

φ˙ = p + q sin φ tan θ + r cos φ tan θ θ˙ = q cos φ − r sin φ cos φ sin φ +r ψ˙ = q cos θ cos θ where Ix , Iy , Iz and Ixz denote the considered elements of the aircraft inertia tensor I. To obtain a flat model, the following assumptions are used. CX = f1 (α) qc CZ = a1 α + b1 + f2 (α) 2V qc Cm = a2 α + a3 δT S + b2 + f3 (α) 2V f1 , f2 and f3 are polynomial functions and a1 , a2 , a3 ,b1 , b2 are coefficients calculated by polynomial interpolation on the air data look-up tables in [3]. The only longitudinal control is δT S since δCS is not used in FM(AG11).  f1 = −13.6191α5 + 15.2716α4 − 0.4433α3     −2.9289α2 + 0.035310α + 0.06561     f2 = −120.1437α4 + 43.9568α3 + 36.0977α2 + 1.1089α + 6.848  f3 = 23.1430α5 + 166.9087α4 − 85.2654α3   −19.0724α2 + 4.1305α − 5.4507     a1 = 4.05158, a2 = −0.22065, a3 = −0.61471,    b1 = −0.5191, b2 = −0.0517 Figures 14.1, 14.2 and 14.3 represent the results of these interpolations and the comparison with the actual aerodynamics coefficients.

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Fig. 14.1. CX aerodynamic data and CX interpolation.

Remark 1 The CX interpolation is independent of δT S . This simplification will be taken into account as an uncertain parameter during the analysis procedure.

Fig. 14.2. CZ aerodynamic data and CZ interpolation.

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Fig. 14.3. Cm aerodynamic data and Cm interpolation.

Then the equations of longitudinal dynamics of the HIRM+ model (14.2, 14.3) are obtained using the assumptions that, β = 0, p = 0, r = 0, φ = 0, and α < 21◦ , projected in the body axis. For α < 21◦ the model is symmetric, so longitudinal motions are decoupled from lateral directional motions. X cos α Z sin α V˙ T = + m m Z cos α α˙ = q + mVT M q˙ = Iy θ˙ = q

(14.2)

with ρV 2 Sf1 (α) X = −mg sin θ + 2F − T 2   qc 2 ρVT S a1 α + b1 + f2 (α) 2V T Z = mg cos θ − 2   qc ρVT2 Sc a2 α + a3 δts + b2 + f3 (α) 2V T M = MT − 2

(14.3)

As the study was effected at an angle of attack equal to 6 ◦ , in the second equation of 14.2 the term in sin α (see 14.1) has been simplified in order to make the calculation of the flat output easier. At high angles of attack the term cos α is set to unity. This kind of simplification is not necessary in the first equation. It is clear that such an assumption will introduce modelling errors in the final LFT model.

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14.3 Flatness of the Nominal Model 14.3.1 Introduction Flat systems define a class of nonlinear systems, which are equivalent to linear ones via a special type of dynamic feedback. This dynamic feedback, (called endogenous feedback), is defined as a real-analytic function of state, input and a finite number of its derivatives. For a flat system with m inputs there are m scalar functions yni of state xn , input un and a finite number of its derivatives such that the dynamic with input un and outputs yn1 ,....,ynm can be feedback linearised from an input to state point of view. Outputs yn1 ,....,ynm , which might be regarded as fictitious outputs, are called linearising or flat outputs. The major property of a flat system is that the variables denoting the state xn and the input un variables can be explicitly expressed, without integrating a differential equation, in terms of flat output yn :=(yn1 ,...., ynm ) and a finite number of its derivatives. Flatness turn out to be a useful system property for the analysis of trajectories [4]. From the desired trajectories on the flat output yn ; xn and un trajectories are immediately deduced. 14.3.2 Definition: Flat Outputs[5] Consider the following state-space representation (Σ1 ): x˙ n = fn (xn , un ) with xn = (xn1 , ..., xnn ) ∈ Rn , un = (un1 , ..., unm ) ∈ Rm (14.4) , ..., f ) is a regular function such that f (0, 0) = 0 and where fn = (fn1 nm n

dfn rank dun (0, 0) = m. This system is differentially flat if there is an output vector yn , named flat output, composed of m fictitious outputs (yn1 ,...., ynm ) and three functions A, B and C where yn (k) is the k-th derivative of yn such that: (14.5) xn = A(yn , y˙ n , ..., yn(j) ) un = B(yn , y˙ n , ..., yn(k) ) yn =

C(xn , un , u˙ n , ..., u(l) n )

(14.6) (14.7)

where k > j. The equations (14.5), (14.6) and (14.7) immediately reveal that given a flat (k) outputs trajectory (yn (t),. . . ,yn (t)) for t ∈ [ti , tf ] satisfying both the initial and the final conditions on the state vector xn (t), directly determines the whole state vector trajectory xn (t) with corresponding input vector un (t) for t ∈ [ti , tf ].

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14.3.3 Nominal Dynamic Feedback Once the flat output is determined (yn1 ,...., ynm ), a particular state feedback and a change of base transform the system into a chain of integrators based on the measurement of (yn1 ,...., ynm ). The dynamic feedback is said to be endogenous if, and only if, the converse holds, i.e., if, and only if, any component of yn can be expressed as a real-analytic function of xn , un and a finite number of its time derivatives. Let us give the endogenous state feedback as:  ξ˙n = Xn (xn , ξn , vn ) (14.8) ξn ∈ Rq , vn ∈ Rm un = Un (xn , ξn , vn ) where vn are the new inputs of the system. The change of base Ξ on extended state-space xen is given by: xen = Ξ (xn , ξn )

xen ∈ Rn+q

Then, the extended system is linearisable by static feedback.     x˙ n fn (xn , Un (xn , ξn , vn )) , = fe,n (xn , ξn , vn ) = x˙ en = ˙ Xn (xn , ξn , vn ) ξn

xen ∈ Rn+q

The extended linearised system can be represented as a canonical Brunovski form:  (k1 )    yn1 = vn1 .. .    y (km ) = v nm

nm

where (k1 , . . . , km ) are the controllability indices relating to (yn1 , . . . , ynm ) and also the derivative orders relating to (yn1 , . . . , ynm ). The knowledge of flat outputs is sufficient to determine the input vector un . For flat trajectories yn the equation 14.5 gives the state space trajectories. Then the equation 14.9 gives the nominal input vector un .   vn1  ..  −1  .  = ∆0 (xn ) + ∆ (xn ) un ⇒ un = ∆ (xn ) (−∆0 (xn ) + vn ) (14.9) vnm Fig.14.4 illustrates this approach. The nominal input un of the nominal nonlinear system (N.L System) is built from the nominal flat input reference yn and these derivatives. The sk block is used to build these derivatives of yn .

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Fig. 14.4. Nominal feedback linearisation.

14.3.4 Example Consider the following system:       0 x˙ n1 1 x˙ n =  x˙ n2  =  xn1  un1 +  0  un2 x˙ n3 xn2 0 The flat outputs of this system are:       yn1 = xn2 un1 y˙ n1 = xn1 un1 y˙ n1 =∆ ⇒ ⇒ yn2 = xn3 y˙ n2 = xn2 un1 y˙ n2 un2   xn1 0 is not invertible then the equation (14.9) cannot As the matrix ∆ = xn2 0 be computed, so that the system is not linearisable by static feedback and change of base. A solution consists of introducing a new state variable u˙ n1 = u ˜n1 to delay un1 action in comparison with un2 . The new system becomes:       0 01  x˙ n1   x˙ n2   xn1 un1   0 0  u = +  ˜n1 x˙ en =   x˙ n3   xn2 un1   0 0  un2 u˙ n1 0 10 So



    ˜n1 + un1 un2 y¨n1 = xn1 u y¨n1 u ˜n1 = ∆0 + ∆ ⇔ y¨n2 = xn1 u2n1 + xn2 u ˜n1 y¨n2 un2     xn1 un1 0 , ∆0 = ⇒∆= xn2 0 xn1 u2n1

The linearisation is now possible if xn2 = 0 and un1 = 0. 14.3.5 Flatness of Simplified HIRM+ Longitudinal Model Flat output determination and input-output linearisation by an endogenous state feedback allow transforming the nominal longitudinal model into an integrator chain. Because state vector and output can be expressed as a

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229

function of the angle of attack α and the pitch angle θ, then these outputs are the flat outputs. Indeed, equations (14.2, 14.3) can be changed into: ˙ q = θ˙ = h1 (θ)

VT =

−



˙ ρSf2 (α)θc 4

 +

m(α− ˙ θ˙ ) cos α

1

+ ∆12

ρS(a1 α + b1 )

˙ ˙ θ, θ) = h2 (α, α,

mV˙ T − Z sin α mg sin θ ρVT2 Sf1 (α) ˙ θ) ¨ ˙ α ¨ , θ, θ, + + = h4 (α, α, 2 cos α 2 4 ˙   θc a α + b2 + f3 (α) 2V MT 2Iy T ¨ − 2 ˙ θ) ¨ = − θ = h3 (α, α, ˙ α ¨ , θ, θ, ρV 2 Sca3 Iy a3

F = δts



with ∆1 =

ρSf2 (α) qc m(α˙ − q)2 + 4 cos α

 + 2ρ(a1 α + b1 )mg cos θ

The extended input to state linearised system can be expressed as:  ˙ sin α)VT −Z cos αV˙ T (Z˙ cos α−αZ  v01n = α ¨ = q˙ + mVT2  v02n = θ¨ = M Iy

So inputs v1 and v2 can be expressed as a function of process inputs δT S and F :     δT S v01n = ∆0 (xn ) + ∆ (xn ) un and un = v02n F If ∆−1 exists the knowledge of the output vector yn is sufficient to compute the input vector un ; indeed:    y¨n1 −1 − ∆0 (xn ) un = ∆ (xn ) y¨n2 Then equations (14.2, 14.3) are transformed into the following linear expression:  1  1    ¨1    Y1n Y =α 0100 Y1n 00      Y1n 2 2  2   0 0 0 0   Y1n  Y¨1n  1 0  v01n ˙ 1n = α =  1  +    with 1 1  Y¨   0 0 0 1   Y2n   0 0  v02n Y = θ  2n   2n 2 2 2 0000 01 Y2n = θ˙ Y2n Y¨2n 14.3.6 Flatness of the RIDE Controller The RIDE controller is locally linear. Linear systems are naturally flat as soon as they are controllable and a possible choice of flat outputs is to take the state variables. As the RIDE controllability matrix rank is equal to 4, it

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is necessary to reduce the order of RIDE to have an order four model and the result of this reduction is given by the following state-space representation:   −14.17 −15.086 9.3026 0.081394  −15.084 −31.190 −5.4325 −0.048476   Ar =   8.568 −5.004 −36.923 0.57998  55.269 −32.278 19.417 −7.7175   −1.1410−6 0 −3.7410−3 −1.1510−4 5.6310−6 0  2.1310−4 0 −6.4810−3 2.1110−1 1.0810−2 0   Br =   −1.2729 0 −1.9810−2 3.4210−1 −6.4168 0  83.87 0 −5.0910−2 5.1910−1 −16.637 0   −10.485 6.053 38.150 −14.071 Cr = 0 0 0 0   0 0 0.0393 −0.716 12.732 0 Dr = 0 0.39195 −0.39195 0 0 0 Then the RIDE controller can be represented by the following diagram (Fig.14.5):

Fig. 14.5. RIDE controller.

where:

 qd  Vr   ur =   αr  qr 



 x1r  x2r   and y =   x3r  x4r

For a flat system, the number of flat outputs is equal to the dimension of ur . Choose the flat outputs to be the whole state vector (x1r , x2r , x3r , x4r )T , then for this system an inversion of the state-space equation directly gives an expression of inputs depending on the flat outputs and their derivatives.     qd h5  Vr   h6    =   (x1r , x˙ 1r , ..., x4r , x˙ 4r )  αr   h7  h8 qr

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231

Fig. 14.6. HIRM+ and RIDE closed loop.

14.3.7 Flatness of HIRM+RIDE Fig.14.6 shows the feedback effected between HIRM+ and RIDE for the longitudinal study. In the previous paragraph, the flatness property of the HIRM+ model and RIDE controller are proved separately. These properties are used to demonstrate that the closed loop formed by these two systems is a flat system. The flat outputs are α, θ, x1r , x2r and x3r , which lead to the following expression: F = h4 = C11 x1r + C12 x2r + C13 x3r + C14 x4r − D13 VT − D14 α − D15 q   ˙ x1r , x2r , x3r ˙ θ, θ, (14.10) ⇒ x4r = g1 α, α, Vd =

  h3 − D23 .VT ˙ θ, θ˙ = h3 + VT = g2 α, α, D22

qd = h5   ˙ x1r , x2r , x3r , x4r , x˙ 4r ˙ θ, θ, εVT = h6 − h2 = g3 α, α,   ˙ x1r , x2r , x3r , x4r , x˙ 4r

α = h7 − α = g4 α, α, ˙ θ, θ,   ˙ x1r , x2r , x3r , x4r , x˙ 4r ˙ θ, θ,

q = h8 − h1 = g5 α, α,

(14.11)

The substitution of x4r in g3 , g4 and g5 directly gives the inputs and the states as a function of flat outputs and their derivatives. The extended input to state linearised system can be expressed as:    v1n = x˙ 1r v1n     v2n    v2n = x˙ 2r    v3n = x˙ 3r ⇒   v3n  = ∆0 (xn ) + ∆ (xn ) un    v4n = α ¨ v4n     (3) v5n v5n = θ

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Then the extended state-space equation is transformed into the following linear expression:        Y˙ 1n 00000000 Y1n 10000  Y˙         2n   0 0 0 0 0 0 0 0    Y2n   0 1 0 0 0  v1n  Y˙         3n   0 0 0 0 0 0 0 0   Y3n   0 0 1 0 0     ˙1   1   Y4n  0 0 0 0 0   v2n   Y4n   0 0 0 0 1 0 0 0          ˙2 = 2 +  Y4n  v3n  (14.12) 0 0 0 1 0  Y4n   0 0 0 0 0 0 0 0          ˙1    1   0 0 0 0 0  v4n  Y5n   0 0 0 0 0 0 1 0    Y5n    v  2   5n 2  0 0 0 0 0 0 0 0 0 0 0 0 1   Y5n  Y˙ 5n  3 3 Y 0 0 0 0 0 0 0 0 00001 5n Y˙ 5n

with

 Y1n = x1r     Y2n = x2r     Y3n = x3r    Y1 =α 4n 2 Y4n = α˙    1  Y  5n = θ   2  = θ˙ Y    5n Y 3 = θ¨ 5n

14.3.8 Path Planning The nominal path planning approach determines input vector un (t) on a finite time (T ) so that the state vector path xn goes from the initial point xn (0) to the final point xn (T ). In the flat system case, this is achieved without the resolution of differential equations. It simply requires finding a curve in (k) the space (yn ,. . . ,yn ) which verifies final and initial conditions of the state vector path xn . It is preferable to take into account the structure and the physics of the system to obtain a curve adapted to the problem. The input un that brings the system from the state xn (0) to the state xn (T ) is then obtained from expressions (14.10) to (14.11). In the HIRM+ study case, the movement of flat outputs is determined to pass through the flight condition FC1 :Mach number Ma = 0.2 , altitude 5000 ft, angle of attack α = 6◦ and a pitch angle θ = 6◦ at t1 = 0.1s (see Fig.14.7 and Fig.14.8). Time polynomial expressions are used to describe both movements. Using time polynomial expressions of α(t) and θ(t), the nominal input v1n , v2n , v3n , v4n and v5n are computed. These inputs are used by the feedback linearisation loop to determine nominal process inputs δT S and Fn .

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7

6.8

6.6

6.4

6.2

6

5.8

5.6

5.4

5.2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Fig. 14.7. Time evolution of pitch angle θ (degrees). 7

6.8

6.6

6.4

6.2

6

5.8

5.6

5.4

5.2

5

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Fig. 14.8. Time evolution of angle of attack α (degrees).

14.4 Perturbed Flat System 14.4.1 Definitions The endogenous feedback is computed from the measured flat outputs, without considering the effects of disturbances. The actual trajectories of the flat outputs do not coincide with the specified ones, so that the actual plant, including the linearising feedback, is not exactly the desired linear one. The definition of a compact set of models certainly aims at containing a good description of the perturbed linearised plant. The approach is based on differential geometric theory of jets and prolongation of infinite order developed in particular by Vinogradov [6]. This approach is also used by Fliess et al. [4] to generalise the differentially flat nonlinear systems to orbital flat systems. Consider the state-space representation (Σ1 ) defined in (14.4). The endogenous feedback determined for the nominal flat system (Σ1 ) is described

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by (14.8). This particular feedback leads to the following relations between the flat outputs yn,i and the inputs vn,i : (k )

yn,ii = vn,i ∀i ∈ {1, · · · , m} Denote:

  (r) U n (xn , ξn , vn ) = un = un , ..., un , ...   (k) v n = vn , ..., vn , ...

The nominal system and the particular feedback are represented by:     x˙ n f (xn , Un (xn , ξn , vn )) = fe,n (xn , ξn , vn ) = Xn (xn , ξn , vn ) ξ˙n yn = hn (xn , U n (xn , ξn , vn )) = hn (xn , ξn , v n ) The global dynamic system fe,n can be extended to the infinite dimensional state-space and the global dynamic system definition Fe,n is: ∞

¨n , · · · ) = fe,n (xn , ξn , vn ) Fe,n (xn , ξn , un , u˙ n , u

 ∂ ∂ uµ+1 + ∂n x µ=0 n ∂u(µ) n

Consider now a perturbed plant. As a first consequence, the extended perturbed field Fe is different from the nominal one Fe,n . Secondly the actual state-space trajectory is in a vicinity of the nominal one. The global perturbed model is described by:     x˙ f (x, Un (x, ξ, v)) = f (x, ξ, v) = e Xn (x, ξ, v) ξ˙ y = hn (x, U n (x, ξ, v)) = hn (x, ξ, v) The global perturbed dynamic system fe can be extended to the infinite dimensional state-space ξn . Then: ∞

˙ u ¨, · · · ) = fe (x, ξ, u) Fe (x, ξ, u, u,

 ∂ ∂ uµ+1 + ∂x µ=0 n ∂u(µ)

Denote the distances between the nominal xn , ξn and perturbed x, ξ statespace co-ordinates. δx = x − xn δξ = ξ − ξn δv = (v − vn , v˙ − v˙ n , ...) Then, the flat output yn is governed by the new input vn and yn(k) = vn . The compact set of models is given by [7]:

14 Flatness Approach to LFT Modelling

y (k)

235

    (k) (k) (k)
= vn + δ LFe hn (xn , ξn , v) + DLFm,e hn + δ(DLFe hn ) · · ·   δx   · · · (xn , ξn , v)  δξ  + o |δx|2 + |δξ|2 δv

Where LF e hn is the Lie derivative of hn on the Fe field and:   (k)
(k) (k) δ LFe hn (xn , ξn , v) = LFe hn (xn , ξn , v) − LFe,n hn (xn , ξn , v)


DLFe,n hn (xn , ξn , v) = LFe,n hn (xn , ξn , v) − LFe,n hn (x, ξ, v)  
(k)
(k) (k) δ DLFe hn (xn , ξn , v) = DLFe,n hn (xn , ξn , v) − DLFe hn (xn , ξn , v) (k)

(k)




(k)

DLFe hn (xn , ξn , v) = LFe hn (xn , ξn , v) − LFe hn (x, ξ, v) (k)

(k)

(k)

The dynamic due to the distance between the nominal extended field Fe,n and disturbance extended field Fe (second term on the right hand side) must be stable and satisfy the criterion proposed in FM(AG11) group ([3]). Furthermore the gap between the nominal path and the perturbed path (third term on the right hand side) should be analysed. 14.4.2 HIRM+ Perturbed Flat Model Model errors and the parametric uncertainties are included by introducing the following disturbing terms in the nonlinear state-space equations (14.2, 14.3).  ˜ X = X0 + X    ˜ Iy = Iy0 + Iy ˜1 + M ˜2  M = M0 + M   Z = Z0 + Z˜ ˜ and Z˜ take the aerodynamic In these uncertainty expressions the terms X coefficients CX and CZ modelling errors and the simplification effected in ˜ the aerodynamic coefficient equations (14.2, 14.3) into account, in the term M ˜ 1 ) but another term appears with Cm modelling error is also introduced ( M Xcg uncertainty, this term is defined as: 2

˜ 2 = −xcgunc CZ ρVT S M 2 Then, disturbing terms modify equation (14.12)

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



       Y˙ 1n 0 00000000 Y1n 10000  Y˙         0   2n   0 0 0 0 0 0 0 0     Y2n   0 1 0 0 0  v1n   Y˙    0        3n   0 0 0 0 0 0 0 0   Y3n   0 0 1 0 0      v  ˙1    1   0 0 0 0 0   2n   pert1   Y4n   0 0 0 0 1 0 0 0      Y4n      v + +  ˙2 =  2   0 0 0 1 0   3n   pert2   Y4n   0 0 0 0 0 0 0 0     Y4n       v  ˙1   4n  pert3   1  0 0 0 0 0  Y5n   0 0 0 0 0 0 1 0     Y5n     v  2   5n 2   pert4  0 0 0 0 0 0 0 0 0 0 0 0 1   Y5n  Y˙ 5n  3 Y5n 00000000 pert5 00001 Y˙ 3 5n

(14.13) with the following expressions: ˜ IY 0 − M0 I˜Y M q˜˙ = IY 0 (IY 0 + I˜Y ) ˜ ˜ ˜˙ = − (X0 + X) sin α + Z cos α α mVT mVT ˜ cos α Z˜ sin α X V˜˙ = + m m ˜ ˙˜f (α) = ∂f2 (α) α˙ 2 ∂t α˙

 ˜ ˜˙ ˜ 1 2 ˙ ˜˙ + ˙f2(α)qc+f2 (α)qc Z = − 2 ρVT S a1 α − 2VT   qc · · · − ρV˜˙T VT S a1 α + b1 + f2 (α) 2V T

f2 (α)qcV˜˙T 2VT2

 ···

pert1 , pert2 , pert3 , pert4 and pert5 are determined by: ˜˙ pert1 = α   ˜ sin α + α˙ Z˜ sin α mVT · · · ˜˙ 0 + Z) Z˜˙ cos α + α(Z pert2 = q˜˙ + m21V 2 T  · · · − Z˜ cos α(V˙ + V˜˙ )m − Z V˜˙ cos(α)m 0

T

T

pert3 = 0 pert4 = q˜˙ pert5 =

˜˙ I − M˙ I˜ M Y0 Y ˜ IY 0 (IY 0 + IY )

14.5 Linear Fractional Representation 14.5.1 Introduction Here, a classical way is used to represent a linear system with varying parameters. This method from Steinbuch et al 1991 [8] is a systematic procedure to transform uncertainties in a model into the LFT format.

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14.5.2 LFT Definition Suppose M is a complex matrix partitioned as a 2×2 block matrix:   M11 M12 ∈ C (p1 +p2 )×(q1 +q2 ) M= M21 M22 and let ∆u ∈ C q1 ×p1 and ∆l ∈ C q2 ×p2 be arbitrary. We will then define the upper and lower LFTs as operators on ∆u and ∆l respectively: −1

Fu (M, ∆u ) = M22 + M21 (I − ∆u M11 ) ∆u M12 −1 Fl (M, ∆l ) = M11 + M12 (I − ∆l M22 ) ∆l M21

(14.14)

Either LFT will be called well defined if the relevant inverse exists: det(I − ∆u M11 ) = 0 and det(I − ∆l M22 ) = 0. LFTs can be seen as operations resulting from feedback structures as given in Fig.14.9; equations (14.14) then define closed loop transfer functions from w to z in both cases. An

Fig. 14.9. Upper and lower LFT as feedback structure.

important reason for using the concept of LFTs in linear system theory is that linear interconnections of LFTs can be rewritten as one single LFT. A complete description and some examples on LFTs are given in [9]. 14.5.3 Transformation of a State-Space Model to a LFT   A (θ) B (θ) Consider S (θ) = with θ=(θ1 ,...,θn ) an uncertain system state C (θ) D (θ) representation G(s, θ) and G(s, θ) = Fu (S(θ), 1/s)).   Anom Bnom . Let us introThe nominal system is defined as S (θnom ) = Cnom Dnom duce the following theorem [8]: Consider   Anom B1 Bnom P (s) =  C1 D11 D12  and ∆ (δ) = blockdiag (δ1 Ir1 , ..., δn Irn ). Cnom D21 Dnom So the uncertain system G(s, θ) can be written as the following P (s) and ∆(δ) Linear Fractional Transformation:

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G(s, θnom+δ ) = Fu (P (s), ∆(δ)) if and only if a real matrix exists defined as:     D11 C1 D12 A∆ B ∆ =  B1 0 0  S∆ = C∆ 0 D21 0 0

(14.15)

such that Fu (S∆ , ∆ (δ)) = S (θnom + δ) − S (θnom )

(14.16)

14.5.4 Academic Example Consider a second order system with damping ratio and natural frequency uncertainties defined as:      0 0 1  x+ u x˙ = 2 −2ζω 1 −ω n  n  y= 10 x 

where θ=

ωn = ω0 + δ1 ζ = ζ0 + δ2

Then the state-space representation of the uncertain system is given by:   0 1 0 S (θ) =  −ωn2 −2ζωn 1  1 0 0 If ∆S = S (θnom + δ) − S (θnom ) then ∆S can be written:   0 0 0 ∆S =  −2ω0 δ1 − δ12 −2ω0 δ2 − 2ζ0 δ1 − 2δ1 δ2 0  0 0 0 Let uf and yf be fictitious input and output vectors such that:      0 0 0 u21 y1 yf = ∆S uf =  y2  =  −2ω0 δ1 − δ12 −2ω0 δ2 − 2ζ0 δ1 − 2δ1 δ2 0   u22  y3 u23 0 0 0 Fig. 14.10 represents the second line of ∆S such that: 

y2 = −2ω0 δ1 −

δ12

−2ω0 δ2 − 2ζ0 δ1 − 2δ1 δ2





u21 u22



14 Flatness Approach to LFT Modelling

239

Fig. 14.10. Diagrammatic representation of y2 in academic example.

Fig. 14.11. LFT representation of ∆s .

Including the trivial equation between (y1 , y3 ) and (u21 , u22 , u23 ) and the above diagram, the global system can be transformed on an LFT representation (Fig. 14.11).    δ1 0 0   y2 = Fu S∆ ,  0 δ1 0  u21 u22 0 0 δ2 with



   y11 u11  y12   u12       y13   u13   =   y1   u21  ,      y2   u22  y3 u23



0 1  0 S∆ =  0  0 0

0 0 0 1 0 0 0 0 −1 −ω0 0 0

1 2ω0 0 0 0 0

0 2ζ0 2 0 0 0

 0 0  0  0  0 0

Now application of the above theorem directly gives the LFT form of augmented system P (s) by replacing S∆ Matrix components in P (s).

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

0 1 0 0 0  −ω02 −2ζ0 ω0 0 −1 −ω0   1 0 0 0 0 P =  2ω0 2ζ0 1 0 1 0   0 2 0 0 0 1 0 0 0 0

 0   1  Anom B1 Bnom 0  =  C1 D11 D12    Cnom D21 Dnom 0 0

Finally the uncertain system can be represented by a standard diagram (Fig 14.12).

Fig. 14.12. LFT representation of uncertain system.

14.5.5 LFT Model of Perturbed HIRM+ Longitudinal Model The linear system with parametric disturbances comes from a first order linearisation of the perturbed equations along the above flat output movement. This method depicted in figure 14.13 gives an LFT model of the HIRM+ longitudinal model with uncertainties.

Fig. 14.13. LFT analysis diagram.

In order to check the stability of a closed loop system with the RIDE controller we use the gap between the nominal model (14.12) and the set of perturbed models (14.13) in the vicinity of a nominal trajectory. The corresponding dynamic is given by the following nonlinear equation:

14 Flatness Approach to LFT Modelling





241





ε˙z1 pert1 (Y1 , ..., Y35 , v1 , ..., v5 , εz1 , ..., εz5 , εv1 , ..., εv5 )  ε˙z2   pert2 (Y1 , ..., Y35 , v1 , ..., v5 , εz1 , ..., εz5 , εv1 , ..., εv5 )       ε˙z3  =  pert3 (Y1 , ..., Y35 , v1 , ..., v5 , εz1 , ..., εz5 , εv1 , ..., εv5 )       ε˙z4   pert4 (Y1 , ..., Y35 , v1 , ..., v5 , εz1 , ..., εz5 , εv1 , ..., εv5 )  ε˙z5 pert5 (Y1 , ..., Y35 , v1 , ..., v5 , εz1 , ..., εz5 , εv1 , ..., εv5 ) According to the first Lyapunov theorem, analysis uses the set of models composed of first order linearisation of the disturbance items along on the vicinity of a nominal trajectory. This set of linear models is given by:        ε˙z1 a11 a12 a13 a14 a15 εz1 b14 b15  ε˙z2   a21 a22 a23 a24 a25   εz2   b24 b25            ε˙z3  =  a31 a32 a33 a34 a35   εz3  +  b34 b35  εv4         ε˙z4   a41 a42 a43 a44 a45   εz4   b44 b45  εv5 ε˙z5 a51 a52 a53 a54 a55 εz5 b54 b55 Parameters aij and bij evolve along the trajectory and are defined by: aij =

∂P erti ∂Zj

and

bij =

∂P erti ∂vj

The mean values of all parameters constitute a median value parameter named θm . The components of vector θm are (a11m , . . . , a55m , b14m , . . . , b55m ). Scaling terms (paij or pbij ) allow the description of all parameters in the following expression: aij = aijm + paij δaij

and bij = bijm + pbij δbij   A m Bm with If the global system is defined by P = Cm D m     a11m a12m a13m a14m a15m b14m b15m  a21m a22m a23m a24m a25m   b24m b25m         Am =  a31m a32m a33m a34m a35m  , Bm =   b34m b35m   a41m a42m a43m a44m a45m   b44m b45m  a51m a52m a53m a54m a55m b54m b55m   10000 0 0 0 0 0    Cm =   0 0 1 0 0  , Dm = 0 0 0 0 0 0 00000 Then the augmented model of the gap between nominal and perturbed model along a movement passing through the flight condition FC1 is given by:   Am B 1 B m Paug =  C1 D11 D12  Cm D21 Dm

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and



δa11 0 · · ·  . .  0 .. ..   .. . .  . . δb15   .. ∆= .    δa51   . ..  .. . 0 ··· ···

···

.. ..

.

 0 ..  .         ..  .    0 

. 0 δb55

with 

pa11  0  B1 =   0  0 0

 · · · pb45 0 · · · ··· 0 ··· 0  · · · 0 pa21 · · · pb25 0 · · ·  ··· 0  ··· · · · 0 pa31 · · · pb35 0 · · ·  ··· · · · 0 pa41 · · · pb45 0 · · · 0  ··· · · · 0 pa51 · · · pb55 

   10000 00   0 1 0 0 0 0 0 C0     0 0 1 0 0  C0  0 0            C0 =   0 0 0 1 0  , C1 =  C0  , D0 =  0 0  0 0 0 0 1 0 0  C0      0 0 0 0 0 1 0 C0 00000 01   D0  D0     D12 =   D0  , D11 = 0, D21 = 0  D0  D0 This model is available along a defined trajectory and the µ-analysis checks only this trajectory. To obtain a complete analysis of the model over all the flight area it is necessary to determine a set of available movements and to generate an LFT including all these movements.

14.6 Conclusions Representation tools for uncertainty modelling on input to state linearised flat systems have been presented and applied to the HIRM+ model. The process consists of firstly constructing the nominal path for the flat system. For such a system, this operation is easier within the geometric differential framework. Therefore, the fictitious flat outputs are used to obtain an input

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to state linearisation with an endogenous feedback. Thus, there is no hidden state and the stabilisation problem is simple. However, the endogenous feedback is sensitive to parametric variations and exogenous disturbances. So the perturbed system and endogenous feedback is equivalent to a linear nominal model troubled by nonlinear disturbance items. These items are linearised along the nominal path and put in an LFT form. This representation can be used to check the robustness stability of the HIRM+ model, by using standard µ-analysis. Future work involves the comparison with LFT results given by the other proposed approaches. One difficulty for the comparison comes from the choice of flat outputs not necessarily being the usual choice. Finally, as the coupled nonlinear model is flat, it should be interesting to build the LFT form in this more realistic model.

References 1. J. Reiner and and W. Garrard G. Balas. Design of a flight control system for a highly maneuverable aircraft using robust dynamic inversion. In AIAA Guidance, Navigation and Control Conference, pages 1270–1280, Washington, DC,, 1994. AIAA. 2. D. Moormann and D. Bennett. TP-119-02 The HIRMplus Aircraft Model and Control Law Development. GARTEUR Action Group on Analysis Techniques and Visualisation Tools for Clearance of Flight Control Laws (FM-AG-11), 1999. 3. F. Karlsson, U. Korte, and S.Scala. TP-119-02A Selected Criteria for Clearance of the HIRMplus Flight Control Laws (Addendum to TP-119-02). GARTEUR Action Group on Analysis Techniques and Visualisation Tools for Clearance of Flight Control Laws (FM-AG-11), 2000. 4. M. Fliess, J. L´evine, Ph. Martin, and P. Rouchon. A l¨ıe-b¨ acklund approach to equivalence and flatness of nonlinear systems. IEEE Transaction on Automatic Control, 44(5):922–937, 1999. 5. M. Fliess, J. L´evine, Ph. Martin, and P. Rouchon. Flatness and defect of non-linear systems: introduction theory and examples. Int. Journal of Control, 61:1327–1361, 1995. 6. A.M. Vinogradov. Symetries of partial Differential Equations. Kluwer, Dordrecht, 1989. 7. F. Cazaurang, B. Bergeon, and S. Ygorra. Robust control of flat nonlinear system. In IFAC Workshop on lagrangian and hamiltonian methods for nonlinear control, pages 167–169, Princeton U.S.A, 2000. IFAC. 8. M. Steinbuch, J.C. Terlouw, and O.H. Bosgra. Robustness analysis for real and complex perturbation applied to electro-mechanical system. In Proceedings of of ACC’1991, pages 556–561, Boston, 1991. 9. K. Zhou, J.C. Doyle, and K. Glover. Robust and Optimal Control. The Math Works Inc., 1992.

15 Baseline Solution Tobias Wilmes German Aerospace Center DLR - Braunschweig Institute of Flight Systems D-38108 Braunschweig, Germany [email protected]

Summary. Techniques used currently in the aeronautical industry are applied to solve several flight control law clearance tasks. The obtained results for the clearance of the HIRM+RIDE control configuration are intended to serve as basis for comparisons with several more advanced approaches presented in this book.

15.1 Introduction The clearance of flight control laws prior to flight test and certification is addressed by the numerical evaluation of clearance criteria. The clearance criteria to be used for the so-called ’baseline solution’ and the new analysis methods have been discussed in Chapter 10. These are typical criteria as used in today’s aeronautical industry and have proven applicability and reliability. For the HIRM+ application, criteria were defined in three different sub-groups, i.e. linear stability criteria, pilot in-the-loop oscillation (PIO) criteria and nonlinear response criteria. For every criterion the longitudinal parameters (Xcg ,Iy ,Cmα ,Cmdts , Cmq ) and the lateral parameters (Ycg , Ix , Iz , Clβ , Cnβ , Cnr ), have been varied across the flight envelope within given uncertainty bands. Usually, for the uncertainty parameters, only the nominal condition and the extreme values are evaluated to reduce the number of cases to be analysed. When problems are expected in a certain area of the flight envelope, a more dense grid can be used. This was not done during the HIRM+ baseline analysis because of the limited time and effort available for the work. In order to evaluate the linear criteria, aircraft model trimming and linearisation routines were employed. It was agreed to restrict the analysis of the baseline solution to the same discrete set of eight flight conditions that have been used for the new methods, in order to allow direct comparison of the results and to keep the analysis effort low. For all flight conditions, several flight cases were taken into account. These flight cases are aircraft equilibrium conditions, in straight and level flight or in pull-up manoeuvres, characterised by angle-of-attack values within the range -15 ◦ to 35◦ (∆α = 2◦ ) and within the load factor range from -3g to 7g. The FCS has been designed to limit the C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 247-263, 2002. Springer-Verlag Berlin Heidelberg 2002

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angle-of-attack to 29◦ , but it was decided to examine the different criteria beyond this limitation, to safeguard against peak overshoots outside of the angle-of-attack limits. Taking all this into account, the baseline solution required 5504 longitudinal and 11008 lateral function evaluations for each linear criterion. Additionally, for the nonlinear criteria, 256 function evaluations per criterion were evaluated.

15.2 Model Limitations Due to the load factor limitation of -3g to 7g, certain angle-of-attack/flight condition combinations were not evaluated because the load factor was outside of these limits. In particular, the angle-of-attack range of FC6 was restricted to between -9◦ and 29◦ , and for FC7, the angle-of-attack was limited to between -1◦ and 11◦ for the analysis. A set of deflection limits on symmetric/differential tailplane and rudder actuators are defined for the HIRM+ model: −40◦ 6 δT S ± δT D 6 10◦ , −30◦ 6 δr 6 30◦

(15.1)

It follows that the trimming results are valid, only if the above bounds are fulfilled. The above conditions were checked in order to extract the valid flight envelope for the evaluation. As a result, the angle-of-attack range of flight condition 1 (FC1) is restricted to −9◦ 6 α 6 35◦ due to the violation of the condition δT S 6 10◦ (δT D = 0◦ ) at low angle-of-attack. The rudder authority limits were not violated and stayed within the allowed limits. At high angle-of-attack (> 20◦ ), depending on the flight condition (M, h), the unaugmented HIRM+ model is asymmetric and experiences a rolling/yawing moment which has to be counteracted by a differential tail and rudder deflection to obtain level flight.

15.3 Initial Considerations The HIRM+ aircraft model was originally augmented by using pitch rate command as an input variable, thereby neglecting the command path shaping which transfers physical pilot stick deflections into a pitch rate command signal. Therefore, command path gains had to be defined to enable evaluation of the PIO criteria and to provide the desired PIO resistance. The command path gains were scheduled to provide a consistent steady-state step response but detailed handling qualities have not been taken into account. For the broader application of general handling and performance criteria for fighter aircraft (neglected in this assessment) the command path gains selected would probably be inadequate due to insufficient performance and high stick force per load factor.

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15.4 Method The intention of the baseline solution is to demonstrate the application of current industrial methods in finding problem areas of the HIRM+ model with its flight control system at the given eight flight conditions. The results obtained can then be used as a basis to assess the results from other methods. When an aircraft is cleared and deemed to be safe for flight test (or service), hard limitations must be provided to divide between cleared and not-cleared. For the PIO criteria, it is often difficult to defend a not-cleared decision if there are only minor violations while other criteria are cleared. In practice, a not-cleared condition with only slight violations of one criterion can be turned into a cleared configuration after thorough simulation assessments. Current research is therefore trying to combine the probability of loss-of-control of an aircraft, with the amount of violation of the stability margins in the Nichols plot [1]. The aim is to get a better understanding of the empirically-based stability margins and the influence of a quantified violation on safe operation of an aircraft. All configurations (not only the worst case) which fail a certain requirement have to be further analysed in order to identify what sort of violation occurred and, in a practical application, to determine a solution. This is not included in this chapter due to space restrictions but all cases where criteria have been failed for the HIRM+ have been illustrated in a separate report [2]. One goal of the project was to get information about the measures of merit for different clearance methods, including their computational efficiency. This, in particular, was omitted from considerations in the benchmark analysis due to two reasons. Firstly, the computational time is highly dependent on the hardware used for the evaluation. Every analysis team used a different computer set-up, thus gaining completely different time measures for an equal evaluation. Secondly, the computational time is highly dependent on the software code used for the evaluation. MATLAB is optimised for certain methods of calculation. Time-consuming code can be due to unnecessary computation (cascaded instead of integral calculation, etc), costly function calls (not built-in), re-computation of variables, etc. To optimise the code, a profiler has to be applied to identify bottlenecks, if speed is an issue.

15.5 Linear Stability Analysis One of the main issues in feedback control is stability. If the feedback gain is too large, then the controller may overact and the closed-loop system will become unstable. Two methods are commonly used to determine closed-loop stability, i.e. frequency response and closed-loop eigenvalues analysis. Stability margin criterion. The Nichols plot can be used for stability evaluations by reading the value of the gain, when the phase is minus 180 ◦ .

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This method is based on the frequency response and has the advantage of a nice graphical interpretation to provide useful measures of relative stability. The Nichols stability margin evaluation in this chapter was conducted according to the boundary specifications (see Chapter 10). Single-loop analysis was performed for all three control surfaces. The boundary in the Nichols plot represents a stability margin relative to the critical point (-180 ◦ , 0 dB). The minimum distance to the critical point is therefore a direct measure for the stability of the augmented aircraft. This stability margin parameter will be named ’stability degree’. The stability margin criterion is violated for a stability degree of less than unity. For unstable closed-loop behaviour, the stability degree parameter is set to zero, since there is no margin left with respect to the critical point. Figs. 15.1 to 15.3 illustrate the Nichols plot evaluations for the different control surfaces. It can be observed that for some flight conditions at high angle of attack, parameter variations lead to a violation of the Nichols plot boundaries (ρ < 1) and hence the aircraft cannot be cleared.

Stability margin criterion − symmetric tailplane

1.5

Stability degree [−]

1

CLEARED NOT CLEARED

0.5

0 −15

FC1 FC2 FC3 FC4 FC5 FC6 FC7 FC8 −10

−5

0

5 10 15 Angle of attack [deg]

20

25

30

35

Fig. 15.1. Stability margin criterion - symmetric tailplane

15 Baseline Solution Stability margin criterion − differential tailplane

Stability degree [−]

2

1.5

1

0.5

0 −15

CLEARED NOT CLEARED

FC1 FC2 FC3 FC4 FC5 FC6 FC7 FC8 −10

−5

0

5 10 15 Angle of attack [deg]

20

25

30

35

Fig. 15.2. Stability margin criterion - differential tailplane

Stability margin criterion − rudder

1.6

1.4

Stability degree [−]

1.2

1

CLEARED NOT CLEARED

0.8

0.6

0.4

0.2

0 −15

FC1 FC2 FC3 FC4 FC5 FC6 FC7 FC8 −10

−5

0

5 10 15 Angle of attack [deg]

20

25

30

Fig. 15.3. Stability margin criterion - rudder

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Unstable eigenvalues criterion. The pole locations of the closed-loop system are evaluated and the system is stable if all closed-loop poles are in the left-half plane. The poles are determined numerically by calculating the eigenvalues of the linearised model Figs. 15.4 and 15.5 illustrate the variations of the real part of the longitudinal and lateral worst case eigenvalues in the form of the ’time-to-double amplitude’, which is a practical measure of the speed of response divergence. Figs. 15.6 and 15.7 illustrate the variations of the imaginary part of the longitudinal and lateral worst case eigenvalues. It can be observed that some flight conditions, parameter variations lead to a violation of the eigenvalue requirement and hence the aircraft cannot be cleared.

30

CLEARED if Im(λ)<0.15

Time to double amplitude T2 [sec]

25

Unstable eigenvalues criterion FC1 FC2 FC3 FC4 FC5 FC6 FC7 FC8

20

15 CLEARED if Im(λ)=0

10

5

0 −15

NOT CLEARED

−10

−5

0

5 10 15 Angle of attack [deg]

20

25

30

35

Fig. 15.4. Longitudinal eigenvalue evaluations - time to double amplitude

15 Baseline Solution 3

2.5

Unstable eigenvalues criterion FC1 FC2 FC3 FC4 FC5 FC6 FC7 FC8

Im(λ)max

2

1.5

1

0.5 Im(λ) = 0.15 0 −15

−10

−5

0

5 10 15 Angle of attack [deg]

20

25

30

35

Fig. 15.5. Longitudinal eigenvalue evaluations - imaginary part

30

CLEARED if Im(λ)<0.15

Time to double amplitude T2 [sec]

25

Unstable eigenvalues criterion FC1 FC2 FC3 FC4 FC5 FC6 FC7 FC8

20

15 CLEARED if Im(λ)=0

10

5

0 −15

NOT CLEARED

−10

−5

0

5 10 15 Angle of attack [deg]

20

25

30

35

Fig. 15.6. Lateral eigenvalue evaluations - time to double amplitude

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2.5

Unstable eigenvalues criterion FC1 FC2 FC3 FC4 FC5 FC6 FC7 FC8

Im(λ)max

2

1.5

1

0.5 Im(λ) = 0.15 0 −15

−10

−5

0

5 10 15 Angle of attack [deg]

20

25

30

35

Fig. 15.7. Lateral eigenvalue evaluations - imaginary part

15.6 Pilot In-the-Loop Oscillation (PIO) Analysis The introduction of electronic flight control technology has increased the susceptibility to a special type of man-machine problems called pilot in-theloop oscillation (PIO). The phenomenon, in essence, is a destabilisation of the aircraft-pilot closed-loop system. In chapter 10, two linear PIO criteria were defined for the HIRM+ analysis: the average phase rate and the absolute amplitude criteria. For the evaluation of the absolute amplitude criterion for the HIRM+, the pilot command path gain has been adjusted in order to obtain Level 1 handling qualities across the flight envelope. The evaluated command gain values are so low that the typical pilot’s aggressiveness (pilot gain) will not destabilise the aircraft-pilot system. On the other hand, the pilot will not feel like being in a fighter, due to low command authority or high stick forces per achieved load factor and piloted simulation would be necessary to assess whether such low gains gave acceptable aircraft performance. For the HIRM+ analysis a MATLAB toolbox called HAREM is used for the evaluation of the PIO criteria. This toolbox was developed at DLR in order to provide a comprehensive set of handling qualities requirements based on transfer function and frequency response analyses [3]. Many of these criteria consist of more than one output variable and hence, multi-objective functions have to be developed to find the worst-case [4]. However, for the phase rate

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criterion, which has two output variables (phase rate, PIO frequency), only the phase rate was used to find the worst case. Figs. 15.8 and 15.9 illustrate the average phase rate evaluations for the longitudinal and lateral cases respectively. It can be observed that nearly all flight conditions are Level 1 and can be cleared, and only 2 angle-of-attack conditions (for FC1) lead to a minor violation of the Level 1 phase rate requirement and hence cannot be cleared.

level boundaries

300

250

phase rate [deg/Hz]

Level>3 200 Level 3 150 Level 2 100 Level 1 50 Level 1* 0

0

0.2

0.4

0.6

0.8 1 1.2 frequency [Hz]

1.4

1.6

1.8

2

Fig. 15.8. Average phase rate criterion, longitudinal

Fig. 15.10 illustrates the absolute amplitude criterion evaluation for the longitudinal case. It can be observed that every parameter and angle-ofattack variation results in level 1 handling qualities, i.e. the gain at -180 ◦ phase lag is less than -29◦ /N [dB]. Due to the low pilot gain the criterion exhibits Level 1 behaviour. It is noted that this will not guarantee overall desired handling qualities, which has been discussed previously, since the PIO criterion will be level 1 even if the gain is set to zero! Some angle-of-attack values for certain conditions are missing. This is because the pitch stick to pitch attitude transfer function provided by the linearisation routine did not reach -180◦ phase lag at gain values less than 0 dB and further investigation is required.

T. Wilmes

level boundaries

300

250

phase rate [deg/Hz]

Level>3 200 Level 3 150 Level 2 100 Level 1 50 Level 1* 0

0

0.2

0.4

0.6

0.8 1 1.2 frequency [Hz]

1.4

1.6

1.8

2

Fig. 15.9. Average phase rate criterion, lateral

Absolute amplitude criterion

0

FC1 FC2 FC3 FC4 FC5 FC6 FC7 FC8

−5 −10 −15 Absolute amplitude [deg/N]

256

−20 LEVEL 3 −25 LEVEL 2 −30

LEVEL 1

−35 −40 −45 −50 −15

−10

−5

0

5 10 15 Angle of attack [deg]

20

25

30

35

Fig. 15.10. Absolute amplitude evaluation (longitudinal)

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15.7 Nonlinear Response Analysis Two types of manoeuvres are applied to the HIRM+ aircraft and its flight control system. A maximum deflection ramp input with a steep slope (1000 mm/sec) and a gradual slope (40 mm/sec). The results for the ramp input analysis are illustrated Figs. 15.11 to 15.14. Flight conditions 3, 4, 5 and 7 cannot be cleared due to violations of the angleof-attack or load factor limit. One reason for the increase in AoA beyond the clearance limit of 35◦ is the rapid decay of airspeed in the pull-up manoeuvres. The nose-down functionality of the AoA limiter does not operate properly in such circumstances. A work-around that is used in industry is to simulate the pull-up manoeuvres with fixed speed instead, but that may hide possible problem areas of the aircraft. An example time-domain response for a 40 mm/sec ramp input is illustrated in Fig. 15.15. After leaving the flight envelope (airspeed decay) the AoA and sideslip increase rapidly beyond their allowable limits.

AoA limit exceedance criterion − Step input 70

60

αmax [deg]

50

40 CLEARED

30

20

10

0

0

1

2

3

4 5 Flight condition

6

7

8

9

Fig. 15.11. Step input, 1000mm/sec rate limit, largest exceedance of AoA, squares (nominal), circles (uncertainties applied)

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Nzmax [g]

10

CLEARED 5

0

0

1

2

3

4 5 Flight condition

6

7

8

9

Fig. 15.12. Step input, 1000mm/sec rate limit, largest exceedance of nz , squares (nominal), circles (uncertainties applied)

AoA limit exceedance criterion − Ramp input 70

60

αmax [deg]

50

40 CLEARED

30

20

10

0

0

1

2

3

4 5 Flight condition

6

7

8

9

Fig. 15.13. Ramp input, 40mm/sec rate limit, largest exceedance of AoA, squares (nominal), circles (uncertainties applied)

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Nz limit exceedance criterion − Ramp input

15

Nzmax [g]

10

CLEARED 5

0

0

1

2

3

4 5 Flight condition

6

7

8

9

Fig. 15.14. Ramp input, 40mm/sec rate limit, largest exceedance of Nz, squares (nominal), circles (uncertainties applied)

FC3, 40mm/sec ramp input response

AoA[°]

60 40 20 0

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

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0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5 Time [sec]

6

7

8

9

10

beta[°]

10 0 −10

nz[−]

3 2 1 0

M[−]

0.6 0.4 0.2 0

Fig. 15.15. Nominal FC3 time domain response of a 40mm/sec ramp input

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15.8 Conclusions The objective of a clearance is to assess the safety of a closed-loop aircraft system. When necessary, a clearance can lead to flight restrictions for particular parts of the flight envelope or it indicates areas where the controller needs to be improved. In the HIRM+ analysis, several non-compliances have been determined for the different clearance criteria that have been evaluated. Linear criteria. The linear criteria have shown acceptable analysis results although some angle-of-attack restrictions have been found. The results of the linear analysis are summarised in Table 15.1. The table shows the minimum and maximum angle-of-attack values for which the aircraft is cleared. In the column on the right the overall cleared angle-of-attack is given, based on analysis of all linear criteria. Table 15.1. Cleared angle-of-attack ranges for linear analysis criteria

FC 1 2 3 4 5 6 7 8

Stability margin Tsym Tdiff Rdr -9/29 -9/33 -9/29 -15/29 -15/33 -15/31 -15/29 -15/31 -15/31 -15/31 -15/29 -15/27 -15/29 -15/31 -15/29 -9/29 -9/29 -9/27 -1/11 -1/11 -1/11 -15/29 -15/31 -15/29

Eigenvalues long lat -5/33 -9/33 -15/33 -15/33 -15/33 -15/33 -15/27 -15/27 -15/33 -15/31 -9/23 -9/25 -1/11 -1/11 -15/33 -15/31

PIO APR AA Cleared AoA -9/21 -9/35 -5/21 -15/35 -15/35 -15/29 -15/35 -15/35 -15/29 -15/35 -15/35 -15/27 -15/35 -15/35 -15/29 -9/29 -9/29 -9/23 -1/11 -1/11 -1/11 -15/35 -15/35 -15/29

For the linear criteria, the Nichols plot criterion is responsible for most of the flight clearance restrictions. For higher angle-of-attack, the stability degree decreases rapidly for all 3 control surfaces. Therefore, these limitations can be regarded as mandatory and must not be violated in flight. The minor violation of the average phase rate criterion for an angle-of-attack greater than 21◦ for FC1 can probably be cleared using extended manned simulation and pilot ratings. This might allow the limit to be expanded up to 29 ◦ angleof-attack. The configuration with the highest dynamic pressure (FC7) has a very limited operational angle-of-attack region due to the limits on load factor and within these limits, no restrictions are required, based on the evaluation of the clearance criteria. The clearance result of Table 15.1 is also illustrated in Fig. 15.16.

15 Baseline Solution Final clearance results

4

5

261

x 10

4.5 4

FC3

FC8

α ∈ [ −15, 29]

α ∈ [ −15, 29]

3.5 FC5

Altitude (ft)

3

α ∈ [ −15, 29] FC2

2.5

α ∈ [ −15, 29] FC6

2

α ∈ [ −9, 23]

FC4

1.5

α ∈ [ −15, 27]

1 0.5 0

0

0.1

0.2

FC1

FC7

α ∈ [ −5, 21]

α ∈ [ −1, 11]

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mach Number

Fig. 15.16. Final clearance results for the linear criteria (compare with Table 15.1)

Nonlinear criteria. The nonlinear criteria have shown major deficiencies of the controller, and in particular of the angle-of-attack limiting capability. The analysis result of the nonlinear criteria is given in Table 15.2. Flight conditions 3, 4, 5 and 7 cannot be cleared due to deficiencies in the angle-of-attack limiting functionality, which allows a rapid decay of the airspeed and subsequently a loss of control. Unlike for the linear analysis, no flight restrictions are given in terms of angle-of-attack. This result implies that improvements are required for the design of the HIRM+ angle-of-attack limiting capability in order to prevent stall or departures from controlled flight. Table 15.2. Nonlinear criteria clearance results: y (cleared), n (not-cleared)

FC 1 2 3 4 5 6 7 8

AoA/nz -limit excedance criteria Step Ramp y y y y n n n y y n y y n n y y

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Overall result. In Fig. 15.17 the overall clearance result of both the linear and nonlinear criteria is illustrated. The nonlinear analysis results highlight the importance of nonlinear time simulations. The approach used in this baseline clearance assessment has proven to be generally applicable. The method certainly finds a case which is worse than other parameter combinations (local minimum), but whether it is the worst case (global minimum), cannot be stated.

Final clearance results

4

5

x 10

4.5 4

FC3

FC8

not cleared

α ∈ [ −15, 29]

3.5 FC5

Altitude (ft)

3

not cleared FC2

2.5

α ∈ [ −15, 29] FC6

2

α ∈ [ −9, 23]

FC4

1.5

not cleared

1 0.5 0

0

0.1

0.2

FC1

FC7

α ∈ [ −5, 21]

not cleared

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mach Number

Fig. 15.17. Final clearance results with nonlinear criteria included (compare with Table 15.1)

Final remarks. In current industrial practice, the clearance task of modern fly-by-wire aircraft comprises both linear criteria and analysis of nonlinear aspects by time simulation. One of the main issues in finding problem areas in modern fly-by-wire aircraft is related to nonlinear analysis. Here, above all, research into so-called ’mission/role-oriented handling qualities’ is of current research interest [5] i.e. to find manoeuvre tasks representing worst-case scenarios for the FCS, under realistic operational circumstances. In this analysis, PIO phenomena due to nonlinear effects, such as rate saturation, have not been considered. These types of PIO are often not detected by using the common PIO criteria. A possible analysis approach for future evaluations is given by the OLOP criterion. The Open-Loop Onset Point (OLOP) is defined as the frequency response value of the openloop aircraft or aircraft-pilot system at the closed-loop onset frequency. The

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OLOP-parameter can be used to predict stability problems of rate saturated closed-loop systems. The results from the application of criteria are often difficult to understand, especially if the handling qualities and nonlinear manoeuvre criteria evaluations are based on qualitative comments from pilots about certain characteristics of the aircraft. To understand their applicability to different control systems is therefore not always straightforward, and the background of each handling criterion has to be carefully considered.

References 1. Koehler R., Wilmes T. Revision of control law design and clearance requirements DLR-IB-111-2000/10, February 2000. 2. GARTEUR FM(AG11) Clearance of HIRM+RIDE nonlinear augmented aircraft model. Baseline solution. GARTEUR/TP-119-09, Oct 2000. 3. Duus G., Duda H. HAREM: Ein Programmpaket zur Flugeigenschaftsanalyse unter MATLAB DLR-IB-111-1997/22, Februar 1998 4. H¨ ohne G., Koehler R. Aufbereitung der Flugeigenschaftskriterien f¨ ur die Optimierung. DLR-IB-111-1997/30, Juni 1997 5. Wilmes V.T. Survey of role related manoeuvres for fighter aircraft. DLR-ARNO111-123-01, May 2001

16 µ-Analysis of Linear Stability Criteria Declan G. Bates, Ridwan Kureemun, Martin J. Hayes, and Ian Postlethwaite Control and Instrumentation Research Group, Department of Engineering, University of Leicester, University Road, Leicester, LE1 7RH, UK. [email protected], [email protected], [email protected], [email protected]

Summary. In this chapter we present results of the clearance of the HIRM+ RIDE flight control law using µ-analysis techniques. The analysis covers both the longitudinal and lateral flight dynamics. Of the five clearance criteria specified for investigation in Chapter 10, we address the Stability Margin and Unstable Eigenvalue criteria in detail. Comments on the applicability of µ-analysis to the other criteria are included at the end of the chapter.

16.1 Introduction In the following analysis, three flight cases, namely, straight and level flight (unaccelerated γ-trim) and the two pull-up manoeuvers (pullup-pushoverα-trim and pullup-pushover-nz -trim), are taken into account. For some of the flight cases we do not restrict ourselves to the discrete set of points in the flight envelope specified for analysis in Chapter 10, but instead seek to clear the control law over continuous regions of the flight envelope, without resorting to traditional gridding approaches. An approach based on generating uncertainty bounds is adopted for representing the uncertain aircraft parameters in the form of structured LFT-based uncertainty models required for µ-analysis. This approach, which is described in detail in Chapter 13, is simple and fast, but conservative, i.e. our uncertainty model may contain extra uncertainties which could not be generated in practice by the actual set of uncertain parameters. Therefore, in this analysis we can only propose positive results, e.g., we can prove stability at a particular point or over a particular region of the flight envelope. If we cannot do this it means that there is some ∆ in our uncertainty model which causes instability - however it will not be possible to say for sure if this ∆ would in fact be generated by a particular combination of the actual uncertain aircraft parameters. Bounds on µ are computed using new methods, introduced in Chapter 3, which allow the computation of tight bounds on high order real µ problems. µ tests are formulated in terms of avoidance of elliptical Nichols plot exclusion regions to allow fast and easy evaluation of the control law’s robust stability properties. These regions are not exactly the same as those currently used by industry but they do allow the specification of appropriate levels of Gain and C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 265-284, 2002. Springer-Verlag Berlin Heidelberg 2002

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Phase Margins. Techniques for presenting µ-analysis results as multivariable root locus plots are developed and applied at various flight conditions for both the longitudinal and lateral analyses. These techniques are shown to be a sensible means of illustrating the relative movement of the various closed loop poles for variations in the level of parameter uncertainty.

16.2 Analysis of the Longitudinal HIRM+ Dynamics In this section, we detail results from the analysis of the longitudinal dynamics of the HIRM+ RIDE control law, using the µ tools and methods described in Chapter 3. All three flight cases (straight level flight and pull-up manoeuvres) for the longitudinal dynamics are considered. 16.2.1 Unaccelerated γ-Trim In the following we present results for the γ-trimmed at 0, unaccelerated flight case over the flight envelope proposed in Chapter 10. Tests for the various criteria are performed for longitudinal dynamics only, with all Category 1 and 2 uncertainties applied simultaneously. Stability over the flight envelope: Before attempting to identify over which regions of the flight envelope the RIDE control law satisfies the Stability Margin criterion, we first of all identify the regions over which the aircraft remains stable. The region that satisfies the Stability Margin criterion will then be a subset of this region. For three points in the flight envelope, FC1, FC2 and FC3, see Fig. 16.1, our µ tests fail to guarantee stability, while for two regions of the flight envelope (shown in Fig. 16.1 in dashed lines) covering points FC4 to FC8, stability is guaranteed by the µ-analysis. By refining these two regions, the portion of the flight envelope over which stability is guaranteed was found to be as shown in Fig. 16.2. Stability margin criterion: In the following we present clearance results for the Stability Margin Criterion, with respect to violation of elliptical Nichols plot exclusion regions as defined in Chapter 3. We focus on the two regions of the flight envelope, covering flight conditions FC4 to FC8, for which our initial µ tests guaranteed stability. For these regions we test for avoidance of elliptical exclusion regions providing Gain and Phase Margins of ±4.5dB and ±28.44◦ respectively. The open-loop Nichols plots of the frequency response are obtained by breaking the loop of the symmetrical taileron actuator, leaving all the other loops closed. The resulting µ bounds and corresponding worst case Nichols plots are shown in Figs. 16.3, 16.4 and 16.5 for the region enclosing flight conditions FC5 to FC8. Note that the test for avoidance of the inner ellipse is passed, while the test for avoidance of the outer ellipse is failed. This result is verified by the worst case Nichols plot

16 µ-Analysis of Linear Stability Criteria Flight Envelope

4

4.5

267

x 10

FC3

4

FC8

3.5 FC5

Altitude (ft)

3 FC2

2.5

FC6 2 FC4

1.5

1

0

FC7

FC1

0.5

0

0.1

0.2

0.3

0.4

Mach

0.5

0.6

0.7

0.8

0.9

Fig. 16.1. Regions and points in flight envelope for analysis Flight Envelope

4

4.5

x 10

FC3

P1: (0.23,0) P2: (0.3,15000) P3: (0.37,25000) P4: (0.53,40000)

4

P4

FC8

3.5 FC5

Altitude (ft)

3 FC2

2.5

P3

FC6 2 FC4

P2

1.5

1

0

FC7

FC1

0.5

P1

0

0.1

0.2

0.3

0.4

Mach

0.5

0.6

0.7

0.8

0.9

Fig. 16.2. Cleared portion of the flight envelope - stability

shown in Fig. 16.5. The variable quality of the lower bound for µ shown in the Figs. 16.3 and 16.4 is due to the fact that the real parametric uncertainty is dominating over the complex uncertainty associated with the Nichols region, and this is causing convergence problems for the mixed µ lower bound algorithm of [1]. The worst case Nichols plots correspond to the peak of the computed µ lower bounds, and so to be sure of having identified the true

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worst case we need the lower bound to be close to the upper bound at the peak value of µ. From the figures we can see that this is indeed the case. A comparison of the different regions of the flight envelope cleared for the basic stability criterion (the region enclosed by L1 to L4), and the stability margin criterion (the region enclosed by M1 to M9) is shown in Fig. 16.6. Fig. 16.7 summarises the results for the stability margin tests (symmetrical taileron loop cut) at each of the specified points in the flight envelope. The figure shows that flight conditions FC1, FC2 and FC3 fail this criterion, but that all other points pass. 0.9

0.8

0.7

0.6

Mu

0.5

0.4

0.3

0.2

0.1

0 −3 10

−2

10

−1

10

0

10

1

w (rad/s)

10

2

10

3

10

4

10

Fig. 16.3. Mixed µ bounds, Nichols exclusion region (inner ellipse) test, FE region FC5 to FC8 1.4

1.2

1

Mu

0.8

0.6

0.4

0.2

0 −3 10

−2

10

−1

10

0

10

1

w (rad/s)

10

2

10

3

10

4

10

Fig. 16.4. Mixed µ bounds, Nichols exclusion region (outer elipse) test, FE region FC5 to FC8

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Effect of symmetrical taileron loop cut

50

40

Nominal Worst case

30

20

10

0

−10

−20

−30

−40

−50 −240

−230

−220

−210

−200

−190

−180

−170

−160

−150

−140

−130

−120

−110

−100

−90

Fig. 16.5. Nominal and worst case Nichols plot, FE region FC5 to FC8 Flight Envelope

4

4.5

x 10

L1: (0.23,0) L2: (0.3,15000) L3: (0.37,25000) L4: (0.53,40000)

4

L4 M9

3.5

M8 M1: (0.25,0) M2: (0.26,5000) M3: (0.28,10000) M4: (0.31,15000) M5: (0.34,20000) M6: (0.38,25000) M7: (0.43,30000) M8: (0.49,35000) M9: (0.54,40000)

Altitude (ft)

3

2.5

M7 L3

2

M5 L2

1.5

1

M4

M3 M2

0.5

0

M6

L1 0

0.1

0.2

M1 0.3

0.4

Mach

0.5

0.6

0.7

0.8

0.9

Fig. 16.6. Comparison of portions of FE cleared for stability and stability margin criteria

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4

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x 10

4

FC8 1.37

FC3 0.98

3.5

Altitude (ft)

3

FC5 1.43

2.5

FC2 0.93

2

FC6 1.46

1.5

FC4 1.42

1

0

FC7 1.57

FC1 0.91

0.5

0

0.1

0.2

0.3

0.4

0.5 Mach

0.6

0.7

0.8

0.9

1

Fig. 16.7. Summary of results for the worst case stability margins, unaccelerated γ-trim

Worst case eigenvalue criterion: Real µ tests of the kind described in Chapter 3 are applied to the HIRM+ model to address the Unstable Eigenvalue Criterion. In this study we concentrated on eigenvalues associated with the the aircraft’s dynamic modes. The other eigenvalues which are due to the actuator dynamics, engine and controller states could be considered in future work. The eigenvalues corresponding to the phugoid and short period modes were obtained after the system matrix was stripped down to only four states: VT , α, θ and q. These four states give rise to two complex-conjugate pairs of eigenvalues, which correspond to the two oscillatory modes. Note that in the small disturbance case, normal velocity w can be used interchangeably with α, [2]. In the lateral axis analysis,, the lateral states were reduced to β, φ, p and r to give two real eigenvalues and a complex-conjugate pair, the latter corresponding to the dutch roll mode. µ-analysis tests were performed for the worst case (in terms of the magnitude of the real part) unstable eigenvalues for each of the three flight conditions FC1, FC2 and FC3, and the two FE regions enclosing flight conditions FC4 to FC8 shown in Fig. 16.1. As an illustrative example, the worst case eigenvalues at the flight condition FC3 are shown in Fig. 16.8, with a superimposed plot of the boundaries for the unstable eigenvalue requirement proposed in Chapter 2. As an example of the multivariable root-locus method, the movement of the two eigenvalues of the short-period mode at flight con-

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dition FC3 for different percentages (10%, 20%,..., 100%) of the worst case uncertainty is shown in Fig. 16.9. Fig. 16.10 summarises the results for the worst case eigenvalue tests at the specified points in the flight envelope. The unstable eigenvalue criterion fails for flight conditions FC1 and FC2 as the real parts of these two eigenvalues lie outside the boundaries specified in Chapter 10. Worst−Case Delta Pole−Zero 0.3

0.2

0.1

0

−0.1

−0.2

−0.3 −0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Fig. 16.8. Worst case unstable eigenvalue for flight condition FC3

16.2.2 Pullup-Pushover-α-Trim In this trim condition, α is considered as an uncertain parameter which varies between -15 and +35 degrees and thus enters the LFT-based uncertainty model as an additional uncertainty. In linearising the model, it must be ensured that the load factor limits (-3g ≤ nz ≤ 7g) are not violated. However, there exist certain values of α which cause the load factor to go beyond these limits for some flight conditions. Table 16.1 below summarises the allowable range of α for each of the eight flight conditions. Due to the variation in the allowable range of α at different flight conditions we restrict the analysis to considering the eight discrete points of the flight envelope specified in Chapter 2, rather than considering regions of the flight envelope as done previously. However, the same approach can readily be applied to cover a portion of the envelope for a range of α values. Finally, to avoid introducing

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10%

0%

1.3

20%

30% 40%

Nominal

50%

1

60% 70% 80%

0.7

90%

0.4 100% Worst case

Im

0.1

Worst case

−0.2

100%

−0.5 90%

−0.8

80% 70% 60%

−1.1

50%

Nominal

−1.4

0%

−1.7

10%

−1.2

−1

20%

−0.8

30%

Re

40%

−0.6

−0.4

−0.2

0

Fig. 16.9. Migration of both eigenvalues (short-period mode) for flight condition FC3 HIRM+ 1.2 Ride 1.0, µ−analysis 12−Dec−2000, c3, Long Cat 1+2

4

4.5

x 10

4

FC8 1.81e−3

FC3 0.008

3.5

Altitude (ft)

3

FC5 1.43e−3

2.5

FC2 0.460

2

FC6 1.21e−3

1.5

FC4 1.20e−3

1 FC1 0.435

0.5

0

0

0.1

0.2

FC7 1.92e−3

0.3

0.4

0.5 Mach

0.6

0.7

0.8

0.9

1

Fig. 16.10. Summary of results for the unstable eigenvalues, unaccelerated γ-trim

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273

excessive conservatism in the analysis, the value of α was allowed to vary in intervals of ∆α = 2o over the allowable range. Table 16.1. Allowable ranges of α Flight condition 1 2 3 4 5 6 7 8

α (o ) -15 to +35 -15 to +35 -15 to +35 -12 to +28 -15 to +35 -6 to +21 -1 to +9 -13 to +35

Stability over the flight envelope: FC1 and FC2 were found to be unstable (µ > 1) for some value of α, while FC3 to FC8 preserved stability over the allowed range of α values, as shown in Table 16.2. The range of α which maximises the value of µ is also shown. Table 16.2. µ values (Trim Option 3) Flight condition 1 2 3 4 5 6 7 8

µ Range of α (o ) that maximises µ 1.63 [33,35] 1.65 [33,35] 0.72 [33,35] 0.33 [26,28] 0.32 [33,35] 0.25 [17,19] 0.24 [7,9] 0.22 [33,35]

Stability margin criterion: Fig. 16.11 summarises the results for the stability margin (symmetrical taileron loop cut) analysis. The figure shows the ranges of α at each point in the flight envelope where the criterion is not cleared. For instance, the stability margin criterion fails for FC5 if α is in the range 27o ≤ α ≤ 35o . Unstable eigenvalue criterion: Fig. 16.12 summarises the results for the unstable eigenvalue criterion. Again, the figure shows the ranges of α at each point in the flight envelope where the criterion is not cleared.

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4

4.5

x 10

4

FC8 [29,35] [Ua4,Ua4]

FC3 [27,35] [Ua4,Ua4]

3.5

Altitude (ft)

3

FC5 [27,35] [Ua4,Ua4]

2.5

FC2 [−15,−12][27,35] [Ua4,Ua4]

2

FC6 [−15,−4][24,35] [Ua4,Ua4][Ua4,Ua4]

1.5

FC4 [28,35] [Ua4,Ua4]

1 FC1 [−15,−8][28,35] [Ua4,Ua4][Ua4,Ua4]

0.5

0

0

0.1

0.2

0.3

0.4

FC7 [−15,−3][9,35] [Ua4,Ua4][Ua4,Ua4] 0.5 Mach

0.6

0.7

0.8

0.9

1

Fig. 16.11. Summary of results for the stability margin, pullup-pushover-α-trim HIRM+ 1.2 Ride 1.0, µ−analysis 12−Dec−2000, c3, Long Cat 1+2

4

4.5

x 10

4

FC8 [−15,−13] [33,35] [Ua4,Ua4] [Ua4,Ua4]

FC3 [30,35] [Ua4,Ua4]

3.5

Altitude (ft)

3

FC5 [29,35] [Ua4,Ua4]

2.5

FC2 [32,35] [Ua4,Ua4]

2

FC6 [−15,−5][22,35] [Ua4,Ua4][Ua4,Ua4]

1.5

FC4 [−15,−11][23,35] [Ua4,Ua4][Ua4,Ua4]

1

0.5

0

FC1 [−15,−7][35] [Ua4,Ua4][Ua4] 0

0.1

0.2

0.3

0.4

FC7 [−15,−2][10,35] [Ua4,Ua4][Ua4,Ua4] 0.5 Mach

0.6

0.7

0.8

0.9

1

Fig. 16.12. Summary of results for the unstable eigenvalues, pullup-pushover-αtrim

16.2.3 Pullup-Pushover-nz-Trim In this trim condition, nz is considered as an uncertain parameter which varies between -3g and 7g and thus enters the LFT-based uncertainty model as an additional uncertainty . Whilst linearising the model, it must be ensured that the limits of α (−15o ≤ α ≤ 35o ) are not exceeded. However,

16 µ-Analysis of Linear Stability Criteria

275

there exist certain values of the load factor which cause this condition to be violated. As a result, the load factor range must be reduced for these particular flight conditions. Table 16.3 lists the allowable range of the load factor which satisfies the above condition for α. Figs. 16.13 and 16.14 summarise the Table 16.3. Allowable range of load factor for longitudinal trim option 4 Flight condition 1 2 3 4 5 6 7 8

nz (g) -3 to 2 -3 to 2 -3 to 2 -3 to 3 -3 to 3 -3 to 6 -3 to 7 -3 to 3

results for the stability margins (symmetrical taileron loop cut) and unstable eigenvalues respectively. Only flight condition FC7 was cleared for the whole range of load factor specified. HIRM+ 1.2, RIDE 1.0, µ−analysis 12−Dec−2000, c1_symm, Long Cat 1+2

4

4.5

x 10

4

FC8 [3,7] [Ua4,Ua4]

FC3 [3,7] [Ua4,Ua4]

3.5

Altitude (ft)

3

FC5 [3,7] [Ua4,Ua4]

2.5

FC2 [2,7] [Ua4,Ua4]

2

FC6 [7] [Ua4]

1.5

FC4 [3,7] [Ua4,Ua4]

1 FC1 [2,7] [Ua4,Ua4]

0.5

0

0

0.1

0.2

0.3

FC7 [] 0.4

0.5 Mach

0.6

0.7

0.8

0.9

1

Fig. 16.13. Summary of results for the stability margins, pullup-pushover-nz -trim

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4.5

x 10

4

FC8 [3,7] [Ua4,Ua4]

FC3 [3,7] [Ua4]

3.5

Altitude (ft)

3

FC5 [3,7] [Ua4,Ua4]

2.5

FC2 [2,7] [Ua4,Ua4]

2

FC6 [7] [Ua4]

1.5

FC4 [3,7] [Ua4,Ua4]

1

0.5

0

FC7 []

FC1 [2,7] [Ua4,Ua4] 0

0.1

0.2

0.3

0.4

0.5 Mach

0.6

0.7

0.8

0.9

1

Fig. 16.14. Summary of results for the unstable eigenvalues, pullup-pushover-nz trim

16.3 Analysis of the Lateral HIRM+ Dynamics This section presents the results obtained from the analysis of the lateral dynamics of the HIRM+ RIDE control laws. Six Category 1 uncertain parameters (YCG , Ix , Iz , Clβ , Cnβ and Cnr ) are taken into account, and a reduction factor of 0.46 is applied since three aerodynamic uncertainties are used. For Trim option 1, the flight envelope was partitioned as shown in Fig. 16.1, ie regions as well as points in the flight envelope were considered. Due to space limitations, the analysis results presented here have been restricted to only the eight discrete flight conditions defined in Chapter 2. The Stability Margin criterion is evaluated for both differential tailplane and rudder loop cuts. 16.3.1 Unaccelerated γ-Trim Figs. 16.15, 16.16 and 16.17 summarise the results for the stability margins (differential taileron and rudder loop cuts) and the unstable eigenvalues. 16.3.2 Pullup-Pushover-α-Trim Figs. 16.18, 16.19 and 16.20 summarise the results for the stability margins (differential taileron and rudder loop cuts) and the unstable eigenvalues.

16 µ-Analysis of Linear Stability Criteria HIRM+ 1.2, RIDE 1.0, µ−analysis 12−Dec−2000, c1_diff, Lat Cat 1

4

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277

x 10

4

FC8 1.68

FC3 1.59

3.5

Altitude (ft)

3

FC5 1.71

2.5

FC2 1.50

2

FC6 1.85

1.5

FC4 1.78

1

0

FC7 1.61

FC1 1.48

0.5

0

0.1

0.2

0.3

0.4

0.5 Mach

0.6

0.7

0.8

0.9

1

Fig. 16.15. Summary of results for the stability margins due to differential taileron loop cut, unaccelerated γ-trim HIRM+ 1.2, RIDE 1.0, µ−analysis 12−Dec−2000, c1_rudd, Lat Cat 1

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4.5

x 10

4

FC8 1.50

FC3 1.42

3.5

Altitude (ft)

3

FC5 1.48

2.5

FC2 1.40

2

FC6 1.60

1.5

FC4 1.53

1

0

FC7 1.70

FC1 1.39

0.5

0

0.1

0.2

0.3

0.4

0.5 Mach

0.6

0.7

0.8

0.9

1

Fig. 16.16. Summary of results for the stability margins due to rudder loop cut, unaccelerated γ-trim

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x 10

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FC8 1.10e−12

FC3 6.66e−14

3.5

Altitude (ft)

3

FC5 1.05e−12

2.5

FC2 1.00e−14

2

FC6 1.60e−12

1.5

FC4 3.19e−12

1 FC1 9.57e−14

0.5

0

0

0.1

0.2

0.3

FC7 1.80e−13

0.4

0.5 Mach

0.6

0.7

0.8

0.9

1

Fig. 16.17. Summary of results for the unstable eigenvalues, unaccelerated γ-trim HIRM+ 1.2, RIDE 1.0, µ−analysis 12−Dec−2000, c1_diff, Lat Cat 1

4

4.5

x 10

4

FC8 [27,35] [Ua3,Ua3]

FC3 [26,35] [Ua3,Ua3]

3.5

Altitude (ft)

3

FC5 [28,35] [Ua3,Ua3]

2.5

FC2 [−15,−13][32,35] [Ua3,Ua3]

2

FC6 [−15,−8][28,35] [Ua3,Ua3][Ua3,Ua3]

1.5

FC4 [27,35] [Ua3,Ua3]

1 FC1 [−15,−8][33,35] [Ua3,Ua3][Ua3,Ua3]

0.5

0

0

0.1

0.2

0.3

0.4

FC7 [−15,−2][12,35] [Ua3,Ua3][Ua3,Ua3] 0.5 Mach

0.6

0.7

0.8

0.9

1

Fig. 16.18. Summary of results for the stability margins due to differential taileron loop cut, pullup-pushover-α-trim

16 µ-Analysis of Linear Stability Criteria HIRM+ 1.2, RIDE 1.0, µ−analysis 12−Dec−2000, c1_rudd, Lat Cat 1

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x 10

4

FC8 [28,35] [Ua3,Ua3]

FC3 [26,35] [Ua3,Ua3]

3.5

Altitude (ft)

3

FC5 [27,35] [Ua3,Ua3]

2.5

FC2 [−15,−13][29,35] [Ua3,Ua3][Ua3,Ua3]

2

FC6 [−15,−8][28,35] [Ua3,Ua3][Ua3,Ua3]

1.5

FC4 [28,35] [Ua3,Ua3]

1

FC1 [−15,−9][30,35] [Ua3,Ua3][Ua3,Ua3]

0.5

0

0

0.1

0.2

0.3

0.4

FC7 [−15,−2][13,35] [Ua3,Ua3][Ua3,Ua3] 0.5 Mach

0.6

0.7

0.8

0.9

1

Fig. 16.19. Summary of results for the stability margins due to rudder loop cut, pullup-pushover-α-trim HIRM+ 1.2 Ride 1.0, µ−analysis 12−Dec−2000, c3, Lat Cat 1

4

4.5

x 10

4

FC8 [−15][33,35] [Ua3][Ua3,Ua3]

FC3 [35] [Ua3]

3.5

Altitude (ft)

3

FC5 [33,35] [Ua4,Ua4]

2.5

FC2 [35]

2

FC6 [−15,−8][24,35] [Ua3,Ua3][Ua3,Ua3]

1.5

FC4 [−15,−14][26,35] [Ua3,Ua3][Ua3,Ua3]

1

0.5

0

FC1 [−15,−8][33,35] [Ua3,Ua3][Ua3,Ua3] 0

0.1

0.2

0.3

0.4

FC7 [−15,−1][13,35] [Ua3,Ua3][Ua3,Ua3] 0.5 Mach

0.6

0.7

0.8

0.9

1

Fig. 16.20. Summary of results for the unstable eigenvalues, pullup-pushover-αtrim

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16.3.3 Pullup-Pushover-nz-Trim Figs. 16.21, 16.22 and 16.23 summarise the results for the stability margins (differential taileron and rudder loop cuts) and the unstable eigenvalues. HIRM+ 1.2, RIDE 1.0, µ−analysis 12−Dec−2000, c1_diff, Lat Cat 1

4

4.5

x 10

4

FC8 [3,7] [Ua3,Ua3]

FC3 [3,7] [Ua3,Ua3]

3.5

Altitude (ft)

3

FC5 [4,7] [Ua3,Ua3]

2.5

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Fig. 16.21. Summary of results for the stability margins due to differential taileron loop cut, pullup-pushover-nz -trim

16.4 A Note on LFT Modelling As stated in the introduction to this chapter, the method for LFT uncertainty modelling used in our analysis is fast and easy, but conservative. As a result, the corresponding µ tests provide only sufficient conditions for the clearance criterion in question, and the corresponding worst case uncertainty is given in terms of a particular state-space system, not in terms of the original uncertain parameters. In order to provide necessary and sufficient conditions, and in order to be able to identify the corresponding worst case set of uncertain parameters, a more complex LFT modelling approach (e.g., using symbolic linearisation, or the trends and bands method - see Chapters 12 and 13) is required. The question of to whether this extra modelling effort is justified can be more clearly answered by first applying the fast but conservative approach and analysing the results. If for example, we can positively clear a control law

16 µ-Analysis of Linear Stability Criteria HIRM+ 1.2, RIDE 1.0, µ−analysis 12−Dec−2000, c1_rudd, Lat Cat 1

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Fig. 16.22. Summary of results for the stability margins due to rudder loop cut, pullup-pushover-nz -trim HIRM+ 1.2 Ride 1.0, µ−analysis 12−Dec−2000, c3, Lat Cat 1

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Fig. 16.23. Summary of results for the unstable eigenvalues, pullup-pushover-nz trim

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over the whole envelope with this approach, little will be gained by deriving a more exact LFT description. Also, if only a few flight cases cannot be cleared using the conservative approach, it may be sufficient to examine these cases in detail using traditional approaches and/or time domain simulations. If, on the other hand, large numbers of flight conditions throughout the flight envelope cannot be cleared, there is no option but to resort to the more complex approach, in order to determine whether the problem lies with the conservatism introduced in the LFT modelling or with the control law itself.

16.5 Comparison with the Baseline Analysis Some comparisons can be made between the results obtained in our analysis and those obtained in the baseline analysis (Chapter 15) and the analysis via worst case optimisation (Chapter 21), as well as with those produced by the other team using µ-analysis methods (Chapter 17). In the following we will focus on the major agreement and/or discrepancies between the various analysis results for the pullup-pushover-α-trim case. A general observation is that most of the results are in agreement. Analysis of the longitudinal axis, in which the effect of the symmetrical taileron loop was investigated, produced results shown in Fig. 16.11 for the worst-case stability margin criterion. Note that the full set of uncertain parameters was considered. Since the baseline analysis and the µ-analysis results of Chapter 17 were restricted to Category 1 uncertain parameters, exact comparison can only be made with the optimisation-based clearance approach of Chapter 21. It can be observed that most of the results are consistent. The discrepancies can be explained by the fact that the optimisation-based clearance used the standard diamond-shaped Nichols exclusion while we used the more stringent elliptical Nichols exclusion regions. These differences, together with the conservatism introduced by the LFT-modelling method used with the µ-analysis, are clearly reflected in the clearance results where the ranges of α for which the control law is cleared are smaller than those cleared using the optimisation based approach. For instance, flight condition FC3 from Fig. 16.11 was cleared for −15o ≤ α ≤ 26o while the result using worst-case optimisation for the same flight condition gave a bigger clearance range of −15o ≤ α ≤ 29o . Similar comments apply for the other flight conditions. The analysis of the lateral axis was performed using the Category 1 set of uncertain parameters for both the differential taileron and rudder loops. Most results again agree largely with the baseline and worst case optimisation approaches, although conservatism due to the elliptical Nichols exclusion regions has reduced the cleared range of α for all the flight conditions. A comparison of the results produced by the other µ-analysis study shows very good agreement, with our results being slightly more conservative due to larger intervals of α being used at some flight conditions.

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Since the full clearance process covers a wide range of tasks (initial acquaintance with the HIRM+ RIDE model, formulation of LFTs and the application of the different analysis techniques, ...etc), accurate timing of the overall process is difficult. An approximate guide to the timing results is provided as shown in Table 16.4. Note that the timings for the full parameter set analysis in the case of the baseline approach are not readily available and hence only the reduced set analysis timings will be taken into consideration. (N/A = not available) Table 16.4. Timing results in hours Analysis problem Baseline µ-analysis Worst-case optimisation Symmetrical taileron loop N/A 10.50 11.85 Differential taileron loop 6.96 6.00 6.40 Rudder loop 7.52 6.00 6.22 Longitudinal worst-case eigenvalue N/A 11.00 9.83 Lateral worst-case eigenvalue 6.24 5.00 2.86

16.6 Conclusions The approach adopted in our analysis seems to have been quite successful, in the sense that we can clear the RIDE control law at most of the required flight conditions for several criteria with modest modelling and computational effort. It was also shown, through illustrative examples for some flight cases, that whole portions of the flight envelope can be cleared by including Mach No. and altitude as uncertain parameters, thus removing the need to resort to gridding the flight envelope. The capability to clear the FCS for continuous intervals of α and nz was also demonstrated successfully. The approach to LFT generation adopted in this analysis is fast and easy, but conservative, and does not allow the computation of the worst case in terms of particular values of the uncertain parameters. Thus, as indicated in the proposed analysis cycle, we feel that this approach is best suited for an initial analysis, where the main objective is to identify those regions of the flight envelope where the control law passes the clearance criteria. These regions can then be eliminated from subsequent analyses, leading to a significant saving in time and effort. µ-analysis is a frequency domain technique, and is based on H∞ robust control and linear systems theory. Therefore it is not surprising that the criteria which are most easily addressed with this tool are robust stability, avoidance of Nichols plane exclusion regions, identification of worst case eigenvalues etc. Within these general types of criteria, it seems likely that

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new µ-tools can be developed to address the Average Phase Rate and Average Amplitude criteria. The other type of clearance criterion defined in this study is that based on time domain performance robustness with respect to uncertainties and non-linear actuator effects (e.g., the Clonk Criterion). Some preliminary work on the development of µ-tools to address time-domain criteria (Clearance Criteria 5) is described in [3]. The proposed approach seems to have the potential to indicate worst case uncertainty with respect to a related frequency-domain performance measure, at least in the context of a linear robustness analysis. Incorporation of non-linear effects (such as actuator rate saturation) into a µ framework is also possible (via describing function analysis for example), and is the subject of on-going research.

References 1. Balas, G. J., Doyle, J. C., Glover, K., Packard, A. and Smith, R., µ-Analysis and Synthesis Toolbox User’s Guide, The Mathworks, 1995. 2. Etkin, B. and Reid, L. D., Dynamics of Flight - Stability and Control, John Wiley and Sons Inc., New York, 1996. 3. Bates, D. G., Kureemum, R., Hayes, M. J. and Postlethwaite, I., Clearance of the HIRM+ RIDE flight control law: A µ-analysis approach, GARTEUR Technical-Publication TP-119-11, 2001.

17 µ-Analysis of Stability Margin Criteria Thomas Mannchen and Klaus H. Well University of Stuttgart Institute of Flight Mechanics and Control Pfaffenwaldring 7 a 70550 Stuttgart Germany [email protected] [email protected]

Summary. In this chapter µ-analysis is applied to HIRM+RIDE using Trends and Bands type linear fractional transformation (LFT) based uncertainty models. An LFT representation for the stability margin criterion using Pad´e approximations is developed. Different general analyses are performed, such as tests for parametric interdependencies, decreased size of intervals for AoA and µ assessments treating Mach number and altitude as uncertain parameters. An optimisation based skew µ calculation is developed which avoids the risk of missing steep µ peaks, which has proven to be essential for problems with pure real value parameters. Both singleloop and multi-loop stability margin criteria are assessed for lateral and longitudinal dynamics.

17.1 Introduction In this chapter results are presented which have been obtained using the Trends and Bands (Chap. 13) LFT creation techniques and applying µanalysis methods to the HIRM+ (Chap. 8) and the RIDE controller (Chap. 9). Both stability margin and unstable eigenvalue criteria have been addressed, but due to space limitations only the stability margin criterion results are presented in this chapter. For details and results of the unstable eigenvalue criterion see [1]. The proposed analysis cycle is an efficient way to find the worst case uncertain parameter combination for each flight case with one LFT test rather than many system evaluations. In order to be used in a µ-analysis, the Nichols stability margin criterion is reformulated using a Pad´e approximation. This also allows to consider several loops simultaneously, referred to as multi-loop analysis, in contrast to single-loop analysis, where only one loop at a time is considered. The chapter is organized as follows: Section 17.2 outlines some relevant aspects of the model. The generation of LFT models is described in Sect. 17.3. A Pad´e approximation based stability margin criterion formulation is defined in Sect. 17.4. Section 17.5 presents the results of miscellaneous general analyses. C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 285-311, 2002. Springer-Verlag Berlin Heidelberg 2002

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The results of the stability margin clearance are contained in Sect. 17.6. A discussion and comparison with results obtained by others is given in Sect. 17.7. Finally, Subject. 17.9 contains conclusions, the summarized analysis cycle and suggestions for further work.

17.2 Model Definition and Flight Conditions The model was split up into lateral and longitudinal dynamics. According to the HIRM+ description [2], [3], a cross-influence is given for AoA larger than 20◦ only. Therefore no problems are expected below this limit, whereas the results may be inaccurate for AoA between 20 ◦ and 35◦ . The range for AoA lies between −15◦ and +35◦ , whereas the load factor should not exceed -3 g or +7 g. The trimming was performed with trim option three for AoA trim. AoA was varied between the limits with equally spaced steps of 2 ◦ . Based on the range for AoA, the HIRM+ has been trimmed at each flight case (FC) and the calculated load factor has been checked. Results are given in Table 17.1. Intervals where the load factor limits are exceeded are not further investigated. The table shows that the load factor limit leads to a decreased AoA range in FC6 and FC7. In order to receive coinciding 2 ◦ AoA subintervals, all angles have odd values. Therefore a marginal violation of the load factor limits was accepted. Conditions which could not be trimmed and where violations of the control surface deflection limits (|δTS + δTD | ≤ 10◦ ) occurred in FC1 are also excluded from further analysis. During further analysis (with uncertainties applied to the model), no additional checks were performed to ensure that control surface deflections are within allowed limits. Table 17.1. Flight conditions and corresponding Ma numbers, altitudes and AoA ranges, load factor limits FC

1

2

3

4

5

6

7

8

AoA [-61 ,35] [-15,35] [-15,35] [-15,35] [-15,35] [-11,31] [-3,13] [-15,35] 1

Limited due to deflection constraints

To split the dynamics into longitudinal and lateral dynamics, the influences of the different states of the linearised sub-dynamics were examined. Values needed for controller scheduling, which are not part of the actual sub-model, for instance the longitudinal velocity for the lateral motion, were considered to equal the initial trim values. The complete analysis cycle was separately conducted for the lateral and the longitudinal dynamics.

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17.3 Generation of LFT Model 17.3.1 Lateral HIRM+ Dynamics The lateral dynamics of the HIRM+ including controller, actuator and sensor dynamics have seven uncertain parameters to be considered: – – – – – – –

Angle of attack (AoA) Centre of gravity offset in y-direction (Ycg ) Uncertainty of moment of inertia in x-direction (Ix ) Uncertainty of moment of inertia in z-direction (Iz ) Uncertainty in clβ derivative Uncertainty in cnβ derivative Uncertainty of yawing moment derivative cnr due to normalised yaw rate

The examination of the structure showed that actuators and sensors dynamics did not depend on any uncertain parameter at all. The lateral flight dynamics are influenced by nearly all considered uncertain parameters whereas the controller only depends on AoA, Ix and Iz . All three aerodynamic derivatives showed linear influences. The model was linearised at each flight case within the AoA range as given in Tab. 17.1. All other uncertain parameters were varied over the full range. To assess nonlinear dependencies of the state-space matrix elements on the uncertain parameters, five points for each uncertain parameter were evaluated, whereas AoA was considered with a gridding of 2 ◦ yielding 26 points. Linear dependencies were computed at two points only. Hence, the overall number of trim points was npoints = 26 · 53 · 23 = 26000 for the flight dynamics (without Ycg , see below for explanation), the controller dynamics was evaluated at npoints = 26 · 52 = 650 trim points. Since three aerodynamic parameters were considered at the same time, a reduction factor of 0.46 was applied as proposed in [6]. Based on the linearised models, LFT parametric uncertainty models were generated using the Trends and Bands technique (Chap. 13). The subblocks were connected as shown in Fig. 17.1. The ∆ sets of the subblocks differed in the number of parameters, in the block size for each parameter and the type of uncertainty representation. The structure of the uncertainty model given in Table 17.2 was obtained for FC7. As shown in the table for FC7, the influence of Ycg was filtered out in all flight cases, although it was declared as a class one parameter. The number of uncertainties, especially those of the compensation parameters, vary for different flight cases. Combined, the ∆ block of the global LFT is set to: ∆ = diag(I9 δ1 , I6 δ3 , I6 δ4 , δ5 , δ6 , δ7 , δcp,1 , . . . , δcp,15 ) .

(17.1)

The parameter variations resulted in LFT representations with more than 50 internal inputs and outputs, but were still suitable for the MATLAB µ tools [4].

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M fd

ysim

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+ +

out

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∆ con pilot demands

M con

Fig. 17.1. LFT representation of the HIRM+ lateral dynamics Table 17.2. Subsets of perturbations for the lateral dynamics, FC7 Uncertainty

flight dynamics

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compensation parameters

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17.3.2 Longitudinal HIRM+ Dynamics The longitudinal dynamics of the HIRM+ including controller, engine, actuator and sensor dynamics have six uncertain parameters to be taken into account: – – – – –

AoA Centre of gravity offset in x-direction (Xcg ) Uncertainty of moment of inertia in y-direction (Iy ) Uncertainty in cmα derivative Uncertainty of pitching moment derivative cmδT S due to symmetrical tailplane deflection

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– Uncertainty of pitching moment derivative cmq due to normalised pitch rate Preliminary examinations of parameter dependencies showed that engines, actuators and sensors are independent on all considered parameters. For the analysis of the longitudinal flight dynamics and the controller, all considered parameters were taken into account. Again, the aerodynamic uncertainties showed linear influences. The number of overall trim points was 5200 for both flight dynamics and controller. Again, three aerodynamic parameters were considered at the same time, reduced by a factor of 0.46 as proposed in [6]. As an example, the structure of the uncertainty model for FC7 is given in Table 17.3. Table 17.3. Subsets of perturbations for the longitudinal dynamics, FC7 Uncertainty

flight dynamics

controller

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Iy variability

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As shown in the table, the influence of cmδT S was very small and was therefore filtered out and taken care of by compensation parameters, although it was declared as a class one parameter. Note that this does not neglect the influence of the parameter but accounts for it by compensation parameters instead, due to numerical efficiency reasons. A slightly different parameter setting of the Trends and Bands routines could be used to produce LFTs with cmδT S included. Combined, the ∆ block of the global LFT is set to: ∆ = diag(I12 δ1 , I8 δ2 , I7 δ3 , I2 δ4 , δ6 , δcp,1 , · · · , δcp,43 ) .

(17.2)

17.4 Approximations of Nichols Plot Exclusion Zones To analyse simultaneous gain and phase margins, the stability margin criterion (Chap. 10) is used. To check this criterion in a µ-analysis context, the

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criterion has to be transformed into a suitable form. The Nichols exclusion zone can be approximated by an ellipse, as done in Chap. 16. The main advantage is that the elliptical approximation leads to one single complex value δ only. On the other hand, this approximation introduces some conservatism, because the uncertainty region is larger than the required exclusion zone, see Fig. 17.2. It is therefore desirable to find an exact representation or at least a closer approximation for the exclusion zones. In the following, a new method to model the Nichols exclusion zone will be presented using a Pad´e approximation.

Nichols exclusion zone Elliptical approximation

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-2

-3 0

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15 20 Phase offset [˚]

25

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Fig. 17.2. Nichols exclusion zone, comparison with elliptical approximation

To relate the requirements for gain and phase margin, the following equations are introduced. The phase offset φ is represented as ¶ ¶ µ µ φmax + φmin φmax − φmin δ2 + , (17.3) φ= 2 2 | | {z } {z } γ1

γ2

where δ2 is a real value δ. The gain offset a depends on the actual phase margin, therefore it must be modelled using two (real value) δ’s: a [dB] = δ1 (t − mδ2 ) .

(17.4)

t and m characterize the top limit line of the exclusion zone. For instance, the multi-loop exclusion zone in Fig. 17.2 requires t and m to equal two and one, respectively. For use in an LFT representation this description must be

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transformed to the form ae−jφ , with the negative sign for phase lag. This yields ae−jφ = ecδ1 (t−mδ2 )−j(γ1 δ2 +γ2 ) = e−jγ2 ecδ1 (t−mδ2 )−jγ1 δ2

(17.5)

with c = (ln 10)/20. This form is still not suited for the use in an LFT representation, but a first order Pad´e approximation can be used e−τ s ≈ 1 −

τs , 1 + τ2s

(17.6)

where −τ s is defined as −τ s = cδ1 (t − mδ2 ) − jγ1 δ2 .

(17.7)

This first order approximation is adequate up to phase margins of approximately 90◦ [5]. An LFT representation of this first order Pad´e approximation is given in Fig. 17.3. It is based on two real value δ’s, a single one and a twice repeated one, which leads to the ∆ subblock ¸ · δ1 0 . (17.8) ∆margins = 0 δ2 I2 This LFT represents a gain- and phase offset and adds the required uncertainty to the signal and can be inserted in the paths to be analysed with respect to the stability margin criterion. Note that this method matches the diamond-shape of the Nichols plot exclusion zones very well as can be seen in Fig. 17.4 for the single-loop exclusion zone. A disadvantage of this method is that it leads to a pure real µ problem in contrast to the elliptical approach, which introduces a complex δ. Furthermore, one of the δ’s of the Pad´e approximation is repeated (twice). Both effects can cause problems to compute bounds on µ.

17.5 Robust Stability Analysis In this section some fundamental analysis results are presented. The topics addressed are: The significance of trends, a reduction of conservatism with smaller AoA-intervals, the parametric interdependency and finally Mach number and altitude are treated as variabilities. Some of the results consider a smaller AoA interval (−10◦ to 30◦ ) and lateral dynamics, only. Nevertheless, the results offer some interesting insights into the capabilities of µ-analysis for the clearance task.

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+

δ1

c

-

δ2

m

γ

δ2

j

1

-

+

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+

-

+

e

-j γ 2

Fig. 17.3. LFT representation of Nichols plot exclusion zones using first order Pad´e approximation Nichols Plot

5 4 3

Open loop gain [dB]

2 1 0 -1 -2 -3 -4 -5 185

180

175

170

165 160 Open loop phase [˚]

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Fig. 17.4. Nichols plot exclusion zone using a Pad´e approximation

17.5.1 Trends versus No Trends At the beginning of the analysis work only robust stability was examined. The results showed that a stability proof is possible for nearly all flight conditions with the µ-analysis and the Trends and Bands LFT generation method, even partly for large variations of AoA. Moreover, the Trends and Bands method can reduce the involved conservatism significantly, i.e. it could be strongly reduced compared with the element-wise “min-max” approach. For the test a frequency gridding was introduced between the frequencies of 10−1 rad/s and 103 rad/s. The grid consists of 150 points, logarithmically spaced. The plot in Fig. 17.5 (solid line) shows the value of µ versus the

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frequency for FC1. The µ upper bound for the Trends and Bands based LFT has a peak at the frequency of about three rad/s, the maximum value of µ is 0.957. This proves that the complete system is stable over the whole parameter range considered. 1.5

Compensation parameters "Trends and Bands" Linear approximation

Coverage of nonlinear dependencies

Reduced µ peak value due to less conservatism

1

Increased µ peak value due to coverage of nonlinear dependencies

µ

Covering nonlinear dependencies using "Trends and Bands"

0.5

Linear approximation

0 -1 10

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10 Frequency [rad/sec]

2

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Fig. 17.5. µ versus frequency for robust stability at FC1, lateral dynamics, AoA between −10◦ and +30◦ , all other class one parameters applied using reduction factors

For comparison, the µ-analysis was also performed for an LFT with all trends neglected. Only compensation parameters have been introduced to cover the nonlinear dependencies. This is basically the “min-max” approach. The result is also plotted in Fig. 17.5 (dashed line). The µ peak value reaches a value of 1.41. This is nearly one and a half times the value obtained by using the Trends and Bands method. In other words, conservatism was reduced by one third using the Trends and Bands technique. Finally, the same computation was conducted using only the linear approximation (trends) without compensation parameters. This is represented by the dash-dotted line in Fig. 17.5. The µ peak value is decreased due to the fact that the nonlinearities in the dependency of the state-space elements on the parameters are not covered any more. This bears the risk of missing unstable regions. So the Trends and Bands technique represents a good trade-off between minimum conservatism and coverage of nonlinear dependencies. 17.5.2 Validation of LFTs The final goal of the clearance process is to analyse the properties of the original non-linear system. Clearing an LFT model representing the original

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system is an intermediate step, i.e. an aid to complete the task. To safely conclude from the results of the LFT model to the stability and performance of the original system, it must be assured that the LFT is adequately describing the original system’s properties. A critical point in the Trends and Bands type LFT creation procedure is the selection of points per physical uncertain parameter, as a too small number of points bears the risk to miss system’s behaviour in between points and a too large number severely increases the time required to trim and linearise. The method described in Chap. 13, i.e. continuously increasing the number of points for a parameter and subsequently performing µ-analyses with respect to the criterion of interest, was used for the HIRM+. Resulting µ curves look like Fig. 17.5, with the Trends and Band µ curve starting at the lower limit of the linear approximation type LFT µ curve for two points and increasing toward a limit value of the curve for an increasing number of points. The analysis showed that a number of five points is sufficient to cover the system’s dependency on a physical parameter and thus to yield the limit Trends and Bands curve, see Fig. 17.5. For the linear parameters, a number of two parameters was found to be sufficient, as was expected from the fact that these parameters enter the equations linearly. 17.5.3 Robust Stability with Reduced AoA-Intervals In general, there are two possible reasons for a µ peak value to exceed the limit of unity: either it may be caused by an unstable system or the conservatism is too large. Since AoA is treated as an uncertain parameter, conservatism strongly depends on the range of variation. Different ranges have been investigated. Instead of using only one interval covering AoA from −10◦ to +30◦ , it is also possible to split the interval into two sub-intervals. The new maximum value of µ corresponding to the full range is then the greatest µ peak value of the two reduced intervals. Due to the fact that conservatism is reduced by splitting up the range of values for AoA, the new maximum value of µ is decreased. This procedure can be continued by a further reduction of the interval sizes, for instance by using two, four or ten sub-intervals. For example, Fig. 17.6 shows the robust stability µ peak values corresponding to different numbers of intervals for FC5. The plot shows that the maximum µ peak value is decreased for an increased number of subintervals. In order to reduce conservatism, this method will be used for all flight cases in further investigations. In Fig. 17.6 all maximum values of µ, each with the different numbers of intervals, are shown. It can be seen that µ values below unity can already be reached by quartering AoA interval. Since in this case, robust stability is proven for each of the four intervals, robust stability is verified for the whole

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range of AoA, covering −10◦ to +30◦ . The necessary number of subintervals depends on the considered flight case. Note that the smallest value of µ would be obtained using discrete values for AoA. But then robustness can only be guaranteed for discrete values of AoA, not for a range of AoA. This has to be kept in mind when comparing results based on 2◦ subintervals with results of discrete AoA tests. 1.5

µ peak value(s)

1

0.5

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0

1

2

3

4

5

6

7

8

9

10 11 12 13 Number of intervals

14

15

16

17

18

19

20

21

22

Fig. 17.6. µ peak value versus number of intervals for AoA

17.5.4 Investigation of Parametric Interdependency Trimming and linearisation is very time consuming if there are numerous uncertain parameters to be taken into account. Due to this fact it may be very useful to check the interdependencies of the parameters, i.e. to find out, whether the influence of state-space elements on one parameter is affected by other parameters. For that purpose a set of already linearised matrices was used as a reference to compare with, resulting in (21 · 5 · 5 · 5 · 2 · 2 · 2) possible combinations. Because of the smaller AoA range there are 21 instead of 26 values for AoA considered. The next step is to create a new set of matrices (by trimming and linearising), where the investigated parameter is constant (set to the nominal value). Thereafter, the procedure is repeated, but now only the considered parameter is varied. This yields a number of [(21 · 5 · 5 · 5 · 2 · 2) + 2] possible permutations when examining an aerodynamic parameter.

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After creating an LFT model (using the Trends and Bands technique) it has to be checked whether the structure of the new ∆-matrices has changed. This may be a first indication for parametric interdependency. Generally, it can be stated that an unchanged configuration of the ∆ set is necessary. In that case the submatrices representing the trends for the individual parameter are added to the new representation and a µ-analysis is performed. In order to demonstrate this method, results from FC4 are shown below. The generated ∆ set for the flight dynamics (all class one parameters applied) was   −7 0 AoA  −4 0  Ix    −3 0  Iz    c −1 0 . (17.9) ∆= lβ    −1 0  c n β    −1 0  cnr −27 0 comp. parameters The notation for ∆ sets is defined in [4]. As an exemple, [−7 0] stands for a repeated real value delta with 7 copies. Again, the variability Ycg was filtered out. The same procedure has produced the following ∆-matrices for one aerodynamic parameter constant and just considering one uncertain parameter clβ , cnβ or cnr :   −7 0  −4 0     −3 0    (17.10) ∆new =    −1 0   −1 0  −27 0 £ ¤ ∆clβ = ∆cnβ = ∆cnr = −1 0 .

(17.11)

As claimed before, the structure of the new ∆ (at each case without one aerodynamic parameter) has not changed, i.e. the influence of these aerodynamic tolerances to the influence of other uncertain parameters was not too strong. For comparison, the results of a µ-analysis both with the original system and the new additionally supplemented uncertainty model is shown in Fig. 17.7. Note that there is no significant difference between the original µ curve and the new ones. Whereas the µ plot for the added clβ uncertainty is nearly the same, the two others only result in an increased µ peak value of four to five percent. A great application of this examination is the reduction of time needed for trimming and linearisation. By taking out one aerodynamic parameter the time needed is reduced by the factor of two.

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297

Other uncertain parameters, such as Ix and Iz , have also been examined, but the influences on the remaining uncertainties were too large. This was indicated by a change of ∆, the “measurement” of parametric interdependencies. 1.5 µ, all parameters applied simultaneously µ, Clβ added supplementary µ, Cnr added supplementary µ, Cnβ added supplementary

µ

1

0.5

0 -2 10

10

-1

0

10

1

10 Frequency [rad/sec]

2

10

3

10

4

10

Fig. 17.7. µ-analysis, standard method compared to new method with added uncertainties, FC4, full AoA range

17.5.5 Mach Number and Altitude as Variabilities In view of the aim to finally cover the entire flight envelope with µ-analysis, it is of high interest to treat Mach number and altitude as uncertain parameters. Therefore FC4 is chosen and the model has been linearised within a Mach number range between 0.45 and 0.55 and an altitude range of 12500 ft up to 17500 ft. A graphical representation of this area of the flight envelope is given as a hatched box in Fig. 17.8. With the addition of two parameters, both with five points to be evaluated due to nonlinear dependency assessments, the overall number of trim points will be increased by the factor of 25. Therefore, the results of the last subsection have been used here to reduce computing time. The uncertainty in the clβ stability derivative has been taken out of the trimming and linearisation process and was added afterwards instead, as described in the last subsection. Thus a bisection of the computing time was gained.

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Fig. 17.8. Flight envelope, hatched area analysed as continuous interval

Treating Mach number and altitude as variabilities yields a peak value of µ = 2.0. It is remarkable that this value is not much bigger than the value of µ = 1.4 for fixed Mach number and altitude, although not even a splitting of the AoA interval has been used. This allows an optimistic view for future analyses. Here a reduction of conservatism could be attained by splitting up the Mach-altitude-zone into four equally sized sub-intervals and simultaneously dividing the range of AoA. However, this study was not stipulated for the assessment of HIRM+ and therefore all further investigations will be based on a specific flight condition with constant values for Mach number and altitude. 17.5.6 Optimisation Based Skew µ To demonstrate the importance of frequency grid selection when dealing with systems with a pure real parameter structure, three µ-analyses of the same system have been performed using three different grids. Two analyses were carried out as usual, the first with a grid of 50, the second with a grid of 55 logarithmically spaced frequency points. While the peak of the first plot stays well below unity (thus indicating robust stability), the one with only five additional points produces a peak larger than unity, proving the system to be not robustly stable. To reduce the risk of missing the critical frequency point due to an inappropriate grid selection, a new method was developed to calculate µ. The µ calculation is formulated as an optimisation problem,

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max µ (ω) ,

299

(17.12)

ω∈<

where the cost function is a classical µ calculation using standard µ algorithms for a single frequency. Maximizing the cost function with respect to frequency is then expected to converge to the frequency of the maximum µ value. The third µ-analysis uses a rough initial grid of only ten logarithmically spaces points (as initial guess for the optimisation), but adds an optimisation based skew µ refinement afterwards. After 40 additional iterations in the optimisation procedure (thus consuming the same computing time as the 50 point grid), a peak value of µ of over 1.7 is reached. Figure 17.9 shows these three upper bound plots around the interesting frequency.

no skew, 50 point gridding no skew, 55 point gridding skew, 50 steps

1.6 1.4

µ

1.2 1 0.8 0.6 0.4 1

2

3

4

5

Frequency [rad/s]

Fig. 17.9. Different values for µ peaks obtained using two different grids and one optimisation based skew µ calculation, lateral category 1 parameters, FC8, AoA=32◦

The results show the superior performance of the optimisation based skew µ calculation compared to a simple grid refinement approach, not only in terms of the result but also in terms of the computational effort. The narrow peaks encountered in systems with all real parameters could only be safely identified with acceptable effort when performing an additional frequency optimisation as explained above. Note that no additional conservatism is added since the original µ problem is unchanged, in contrast to approaches, where additional uncertainties are added to include frequency dependency in the LFT model itself.

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17.6 Analysis Results for Stability Margin Criterion The results from the robust performance assessments are given in the following. Here, the performance criterion is the stability margin criterion. Although the Nichols exclusion zone is related to stability (and not to timedomain performance), it is nevertheless referred to as a performance criterion. This is following the definition that a µ test which is not only for pure stability, i.e. a µ test for a system augmented with any kind of performance ∆-block, is called a robust performance test. In [7], an LFT model of the X-38 crew return vehicle is used to analyze gain/phase margins of the closed-loop system. The stability margin (Chap. 10) is calculated iteratively. To scale the Nichols exclusion zone linearly as required by the criterion, a scaling factor is incorporated into the LFT representation. This factor is changed until the transfer function of the system is just touching the scaled exclusion zone in the Nichols plot. Transferred to µ-analysis this corresponds to a µ peak value of unity. The current scaling factor is then defined as the stability margin ρ. In order to ensure the comparability of the different flight cases, all results are computed for a 2◦ interval of AoA. However, computing ρ is not possible for all flight cases due to the fact that a reduction of performance requirements can not reduce the µ peak value to unity, if robust stability was not guaranteed beforehand. Those cases are defined to have a stability margin of naught. A stability margin larger than unity can be achieved whenever robust performance was fulfilled. As the scaling procedure performed to obtain µ = 1 is an iterative process that sometimes proves to be quite difficult, the scaling was declared successful when µ reached a peak value of 1 ±0.005. Particular µ plots are not presented here since the stability margin is more meaningful in the flight clearance context. When comparing the following results with results obtained with other methods, it should be kept in mind that not only robust performance for one discrete value of AoA but an AoA range of 2 ◦ is investigated. This can lead to a smaller stability margin compared to looking at one discrete value of AoA only. For the single-loop and multi-loop assessment the corresponding Nichols plot exclusion zones according to [6] have been used. The total computing time is summarized in Table 17.4. Due to space limitations only results of the stability margin are given here. For the results of the worst case parameter combinations see [1]. 17.6.1 Differential Tailplanes The results from the robust performance assessment are given in Fig. 17.10. Figure 17.11 shows the stability margin versus AoA subintervals for all eight flight cases. As shown in Fig. 17.11, the curves representing FC1, FC6 and

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301

Table 17.4. Computational Effort Computing time

1

[h]

7

Task Trimming and linearising the model

0.1

Trends and Bands LFT generation

15

Checking stability margin criteria

95

Scaling of the exclusion zone (yielding the stability margin)

70

Calculating lower bounds on µ

(yielding a “criteria fulfilled yes/no” answer)

(yielding the worst case uncertain parameter combinations) 1

Pentium III 0.5 GHz PC

FC7 do not cover the full AoA range according to Table 17.1. At the right hand side of the figures, i.e. corresponding to high AoA, the curves drop rapidly down to naught. A value of naught for the stability margin ρ indicates the failed robust stability assessment. 17.6.2 Rudder The Nichols plot exclusion zones single-loop evaluation of the rudder is realized in the same way as presented for the differential tailplanes before. Again, only ranges allowed by Table 17.1 were analysed and listed. Figure 17.13 shows the stability margin versus AoA subintervals for the eight flight conditions. The rapid drop down of the curves on the right hand margin indicates the failed robust stability assessment as mentioned before. Again, the values are set to naught to indicate this. 17.6.3 Multi-loop: Differential Tailplanes and Rudder The multi-loop performance assessment was performed using the same principle as for the single-loop tests. Two uncertainty blocks were implemented in the LFT representation which allow independent variation of phase and gain for the tailplane and for the rudder channel. The maximum gain and phase margins for the multi-loop were decreased from ±4.5 dB to ±3 dB and from 35◦ to 30◦ as stated in [6]. Again, only ranges allowed by Table 17.1 were analysed and listed. 17.6.4 Symmetrical Tailplanes The symmetrical tailplane single-loop criterion is inserted into the longitudinal flight dynamics of the HIRM+. Nevertheless, the analysis procedure is still the same.

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5

x 10

4.5 FC

FC8

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4

α ∈ [ -15, 27]

α ∈ [ -15, 25]

3.5 FC5

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3

α ∈ [ -15, 31] FC2

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α ∈ [ -15, 31] FC

α ∈ [ -11, 25]

6

2 FC4

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α ∈ [ -15, 29]

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FC

α ∈ [ -6,33]

α ∈ [ -3, 13]

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mach Number

Fig. 17.10. Flight envelope, stability margin criterion, differential tailplane, lateral category 1 parameters

2 1.8 1.6

Stability Margin ρ

1.4 1.2 1 0.8 0.6 0.4 0.2 0 -15

FC1 FC2 FC3 FC4 FC5 FC6 FC7 FC8 -10

-5

0

5

10 AoA [˚]

15

20

25

30

35

Fig. 17.11. Stability margin criterion, differential tailplane, lateral category 1 parameters

17 µ-Analysis of Stability Margin Criteria

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4

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x 10

4.5 FC

FC8

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4

α ∈ [ -15, 29]

α ∈ [ -15, 27]

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2.5

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α ∈ [ -11, 27]

6

2 FC4

1.5

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0

FC

FC

α ∈ [ -6,29]

α ∈ [ -3, 13]

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0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mach Number

Fig. 17.12. Flight envelope, stability margin criterion, rudder, lateral category 1 parameters

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1.4

Stability Margin ρ

1.2

1

0.8

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0 -15

FC1 FC2 FC3 FC4 FC5 FC6 FC7 FC8 -10

-5

0

5

10 AoA [˚]

15

20

25

30

35

Fig. 17.13. Stability margin criterion, rudder, lateral category 1 parameters

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x 10

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FC8

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α ∈ [ -15, 27]

3.5 FC5

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2.5

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α ∈ [ -11, 27]

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2 FC4

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α ∈ [ -6,29]

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0.2

0.3

0.4

0.5

0.6

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Fig. 17.14. Flight envelope, stability margin criterion, multi-loop differential tailplane and rudder, lateral category 1 parameters

1.8

1.6

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-5

0

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10 AoA [˚]

15

20

25

30

35

Fig. 17.15. Stability margin criterion, multi-loop differential tailplane and rudder, lateral category 1 parameters

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4

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x 10

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3

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α ∈ [ -15, 17] [ 19,33]

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Fig. 17.16. Flight envelope, stability margin criterion, symmetrical tailplanes, longitudinal category 1 parameters

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10 AoA [˚]

15

20

25

30

35

Fig. 17.17. Stability margin criterion, symmetrical tailplanes, longitudinal category 1 parameters

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The stability margin versus AoA subintervals for the eight flight conditions is shown in Fig. 17.17. Again, only ranges allowed by Table 17.1 were analysed and listed. Contrary to the former analyses, where a drop down was encountered at about 26◦ AoA, here an additional drop down at 16 ◦ and 18◦ can be observed. For larger AoA the margin increases again. This might indicate a scheduling problem in the controller. The peak of the plots for FC5 and FC8 at 30 ◦ AoA is discussed in Sect. 17.7.3. 17.6.5 Comparison of Single-loop and Multi-loop Analysis

2

2

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0 -15 -10 -5 2

0

ρ

ρ

In the lateral motion two actuators, differential tailerons and rudder, were analysed. This opens up the possibility to either analyse one actuator at a time, single-loop, or to look at the Nichols gain and phase margins in both loops simultaneously. Here, a gain and phase offset of ±4.5 dB and 35◦ was used for the single-loop analysis, and ±3 dB and 30◦ for the multi-loop analysis, where the gain and phase offsets in the two loops were independent.

5

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ρ

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10 15 20 25 30 35 AoA [˚]

FC8 dt FC8 r FC8 dt+r

0 -15 -10 -5 0

Fig. 17.18. Comparison of single-loop and multi-loop analysis for differential tailerons (dt) and rudder (r)

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307

The multi-loop results tend to give similar margins as the more critical of the two single-loop analysis margins (here: rudder was more critical) in the high angle of attack regime. In the low angle of attack regime the multi-loop stability margins are mostly larger than the single-loop margins, possibly due to a smaller cross-coupling and the smaller gain and phase offsets in the multi-loop Nichols exclusion zone.

17.7 Discussion and Comparison of Results To compare the analysis results using µ-analysis with the classical gridding approach, the stability margin criterion for the symmetrical tailplanes in the longitudinal motion is looked at in more detail as an illustrative example. Table 17.5 shows the uncleared ranges of AoA. The results from µ-analysis using Trends and Band type LFT are compared with µ-analysis and gridding results from other analysis teams. Figure 17.19 shows the comparison of µanalysis results with the baseline solution in more detail by plotting stability margin over AoA for all eight flight cases. In general the agreement of the results is very good. Some discrepancies and possible explanations are discussed in the following. 17.7.1 Influence of Lateral/Longitudinal Dynamics Only For the µ-analysis of the stability margin criteria, the dynamics were split up into longitudinal respectively lateral dynamics only. Analysing the symmetrical taileron loop, the longitudinal uncertainties were applied to the longitudinal dynamics and for the differential taileron and rudder loops, the lateral uncertainties were applied to the lateral dynamics. As opposed to that, the gridding approach always used the full dynamics: Analysing the symmetrical taileron loop, the longitudinal uncertainties were applied to the full dynamics and for the differential taileron and rudder loops, the lateral uncertainties were applied to the full dynamics. For AoA values larger than 20◦ , a cross-influence of lateral and longitudinal dynamics is given [2]. This might lead to different results for large AoA as could be the case in FC2, AoA=[29,35], where the µ-analysis results indicate that the criterion is fulfilled (ρ = 1.01) and the baseline solution indicates a violation (ρ = 0.94), see Fig. 17.19. 17.7.2 Continuous Coverage of AoA Intervals By definition, in the gridding approach only discrete points in the parameter space are looked at, whereas µ-analysis allows to cover continuous parameter ranges. Here, both the width of the intervals and the step size for the grid is 2◦ AoA. Figure 17.20 shows the stability margin for FC3, AoA=[15,21],

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Table 17.5. Comparison of results, AoA ranges violating the stability margin criterion for symmetrical tailerons

FC1 FC2

µ Trends and Bands LFTa

µ “min-max” LFTb

[−15, −6]ef , [7, 17], [19, 35]

[]g

Gridding 2◦c

Gridding 1◦d

[−15, −8]ef , [28, 35]

[−15, −9]e , [29, 35]

[−15, −6]ef , [29, 35]

[−15, −12], [27, 35]

[29, 35]

[29, 35]

FC3

[17, 19]h , [33, 35]

[27, 35]

[29, 35]

[29, 35]

FC4

[31, 35]

[28, 35]

[31, 35]

[31, 35]

FC5

[29, 35]

[27, 35]

[29, 35]

[30, 35]

FC7

[−15, −3]i , [13, 35]i

[−15, −4], [24, 35] [−15, −3]i , [9, 35]i

[−15, −9]i , [29, 35]i [−15, −1]i , [11, 35]i

[−15, −2]i , [12, 35]i

FC8

[15, 19], [29, 35]

[29, 35]

[29, 35]

[30, 35]

FC6

a

b

c d e f g h i

[−15, −11]i , [17, 19], [31, 35]i

[−15, −9]i , [29, 35]i

Continuous AoA intervals of 2 ◦ , longitudinal dynamics only, Pad´e-based exact Nichols exclusion zones, class one longitudinal uncertainty parameters Continuous AoA intervals of 2 ◦ , longitudinal dynamics only, elliptical Nichols exclusion zones, all longitudinal uncertainty parameters, results taken from [9] Discrete AoA values, spacing 2 ◦ , odd AoA values, class one longitudinal uncertainty parameters, results taken from [8] Discrete AoA values, spacing 1 ◦ , all longitudinal uncertainty parameters, results taken from [10] not trimmable |δTS + δTD | < 10◦ violated See discussion of possible influence of lateral/longitudinal dynamics only See discussion of difference of discrete points and continuous intervals Load factor limit violated (small discrepancies due to different rounding)

for the symmetrical taileron stability margin criterion, calculated both with µ-analysis and gridding. Based on the 2 ◦ step size grid [8] values only, the continuous interval values seem to be conservative, but the calculation of one additional point [10] between two grid point reveals the capability of the continuous interval strategy to reduce the risk to miss critical combinations in between grid points. Here, the 2 ◦ -analysis result for the stability margin (ρ = 0.997) leads to the uncleared interval [17,19] in Table 17.5. 17.7.3 Unstable Real Eigenvalues The apparent “discontinuity” at 30 ◦ AoA in FC5 and FC8 in the µ-analysis results is due to an unstable complex conjugate low frequency pole pair. The corresponding µ peak occurs at a frequency < 0.15 rad/s, whereas the typical frequency for the Nichols exclusion zone µ peak ranges approximately between 3 rad/s and 15 rad/s, depending on whether the Nichols plot touches the upper or lower side of the exclusion zone. Thus in this particular case, the stability margin of naught is indicating a violation of the unstable eigenvalue criterion and not of the stability margin criterion, distinguishable by the µ peak frequency. These violations are confirmed by the tests of the unstable

17 µ-Analysis of Stability Margin Criteria

1

1 ρ

1.5

ρ

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10 15 20 25 30 35 AoA [˚]

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0 -15 -10 -5 0

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10 15 20 25 30 35 AoA [˚]

Fig. 17.19. Comparison of µ-analysis results with baseline results (gridding approach) for stability margin criterion for symmetrical tailerons

eigenvalue criterion [1]. To avoid this, the minimum frequency for the µ test could be increased.

17.8 Summary of Analysis Cycle The proposed analysis cycle can be summarized as follows: 1. Define the uncertain parameters and decide on the number of trim points (iteratively, experience based) 2. Trim and linearise both model and controller (fully automated) 3. Generate LFT model (fully automated, with user settings to be selected) 4. Check criteria (fully automated) 5. Calculate margins if desired (fully automated) 6. Calculate worst case uncertain parameter combination if desired (fully automated)

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Grid with ∆=2˚ Grid with extra point at AoA=18˚ Continuous intervals of ∆=2˚

1.2

Stability Margin ρ

1.15

1.1

1.05

1

0.95 15

16

17

18 AoA [˚]

19

20

21

Fig. 17.20. Comparison of µ-analysis with continuous intervals and grid with discrete points for stability margin criterion, symmetrical tailerons, FC3, lateral category 1 parameters

17.9 Conclusions µ-analysis techniques were applied to HIRM+ using Trends and Bands type LFT uncertainty models. The analysis guarantees robustness for a continuous range of AoA and all possible parameter combinations if a robustness test is passed, in contrast to the gridding approach, which merely evaluates the criteria on a finite number of data points. The stability margin criterion was analysed using a new Pad´e based formulation, which allows a closer approximation of the exclusion zone. The approach is flexible since the criteria can be exchanged without modification of the method itself. Both single- and multiloop analyses were performed. The multi-loop results (with smaller gain and phase offsets) yielded similar stability margins as the most critical of the corresponding single-loop analysis margins (with larger gain and phase offsets). An optimisation based skew µ calculation was developed which avoids the risk of missing steep µ peaks, which has proven to be essential for problems with pure real value δ’s. In general it can be stated that the comparison of the results both with other µ-analysis calculations and with the classical gridding solution showed good agreement.

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References 1. Mannchen, T., Klett, Y., Pertermann, C., Weinert, B. and Z¨obelein, T., Flight Control Law Clearance of the HIRM+ Fighter Aircraft Model Using µ-analysis, GARTEUR technical report TP-119-12-v2, 2001. 2. Muir, E., et al., Robust Flight Control Design Challenge Problem Formulation and Manual: the High Incidence Research Model (HIRM), GARTEUR technical report TP-088-4, 1997. 3. Moormann, D. and Bennett, D., The HIRMplus Aircraft Model and Control Laws Development, GARTEUR technical report TP-119-2, 1999. 4. Balas, G. J., Doyle, J. C., Glover, K., Packard, A. and Smith, R., µ-Analysis and Synthesis Toolbox, The MathWorks, 1998. 5. Miotto, P. and Paduano, J. D., Application of Real Structured Singular Values to Flight Control Law Validation Issues, AIAA-95-3190-CP, 1995. 6. Karlsson, F., Korte, U. and Scala, S., Selected Criteria for Clearance of the HIRMplus Flight Control Laws, GARTEUR technical report TP-119-2-A1-v1, 1999. 7. Shin, J. Y., Balas, G. J. and Packard, A. K., Worst-case analysis of the X38 crew return vehicle flight control system, AIAA Journal of Guidance and Control, 24(2), pp. 261–269, 2001. 8. Wilmes, T., Clearance of HIRM+RIDE non-linear augmented aircraft model, GARTEUR technical report TP-119-09-v2, 2000. 9. Bates, D. G., Kureemun, R., Hayes, M. J. and Postlethwaite, I., Clearance of the HIRMplus RIDE Flight Control Law: A µ-analysis approach, GARTEUR technical report TP-119-11, 2001. 10. Varga, A., Robust stability and performance analysis of flight control laws using optimisation-based worst-case search: Linear stability and handling criteria, GARTEUR technical report TP-119-15-v2, 2000.

18 ν-Gap Analysis of Stability Margin Criteria John Steele and Glenn Vinnicombe University of Cambridge, Department of Engineering Trumpington Street, Cambridge CB2 1PZ,UK [email protected], [email protected]

Summary. In this chapter, we present an approach to the HIRM+ stability margin clearance task which makes use of the tools introduced in Chapter 4; namely, the ν-gap metric and the generalised stability margin. For the single-loop criteria our results are largely in agreement with the baseline method. Indeed, the single-loop analysis may be regarded as an exercise in calibrating the generalised stability margin against the standard Nichols exclusion regions. However, the generalised stability margin also applies directly to a multi-loop analysis (and is just as easily calculated in that case). For this (non-standard) multi-loop criterion we find many extra violations occurring well inside the flight envelope. These violations are not an artifact of any conservatism in the analysis since, at these flight points, we can demonstrate relatively small gain/phase offsets which, when applied simultaneously at each input and output, lead to instability.

18.1 Introduction We present a variation of the usual µ based robustness analysis, and apply it to the HIRM+ stability margin criteria clearance task. Key features of the method are: 1) Use of the ν-gap for assessing LFT approximation errors and for ranking the parameters in terms of their potential effect on the clearance criteria. 2) Use of the ²scaled -margin, a generalised stability margin, as a multivariable gain/phase margin. 3) Frequency domain LFT’s are generated using a small number of trim plus linearisation steps. 4) Calculation of µ is replaced by a simple polynomial time approximation which exploits bounds derived from the ν-gap of individual perturbations. Of these, 1) and 2) are presented as fundamental contributions. 3) and 4) are presented as potentially useful alternatives to established techniques; these alternatives have been found to work well for the HIRM+ analysis but need further testing on more realistic aircraft models. For the single-loop criteria our results are largely in agreement with the baseline method, with the aircraft being cleared except at a few points near the extremes of the envelope. Indeed, the single-loop analysis may be regarded as an exercise in calibrating the ²scaled -margin against the standard C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 313-332, 2002. Springer-Verlag Berlin Heidelberg 2002

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Nichols exclusion regions. The advantage of the ²scaled -margin is that it applies directly to a multi-loop analysis (and is just as easily calculated in that case). For this (non-standard) multi-loop criterion we find many extra violations occurring well inside the flight envelope. These violations are not an artifact of any conservatism in the analysis since, at these flight points, we can demonstrate relatively small gain/phase offsets which, when applied simultaneously at each input and output, lead to instability. . Although we have just considered the stability margin criteria, since these are the criteria for which the approach is most well suited, we believe that aspects of this method (particularly 1) ) should be useful for all criteria. In addition, individual aspects of the method could be used in conjunction with other techniques. For example, the ²scaled -margin could be used in conjunction with LFTs generated by other methods and a conventional µ test; or the νgap could be used to help choose grids for a conventional analysis.

18.2 Flight Clearance Objectives The HIRM+RIDE benchmark problem proposes three linear and two nonlinear measures of performance. In this study, we will investigate the stability margin, one of the linear criteria, which is related to the measure ²scaled margin introduced in Definition 4 of Chapter 4 . This clearance criterion defines both single- and multi-loop types of stability analysis. 18.2.1 Single-loop Stability Margin The usual single-loop stability margin considers a generalisation of traditional gain and phase margin analysis. A single-input, single-output system obtained by breaking the closed-loop system at one of the plant inputs is used to assess the robustness of the loop to complex multiplicative perturbations of this input signal. The system should be stable if any complex gain/phase offset from within the dashed region of the Nichols plot in Fig. 18.1 is multiplicatively inserted into the loop at the broken input. This analysis is equivalent to saying that the Nichols frequency response of this single-input, single-output system avoids the +1 point by a region the size of that in Fig. 18.1 (centred at +1). The Nichols stability margin is defined as the largest linear scaling factor which can be applied to the Nichols exclusion region such that the Nichols plot of the single-loop system avoids the scaled exclusion region. A stability margin greater than 1.0 indicates the system used to generate that Nichols frequency response meets the clearance criteria. This single-loop specification is similar to the ²scaled -margin single-loop result shown in Fig. 4.4 of Chapter 4 . For this study, a ²scaled -margin threshold of 0.27 is used to approximate the original specification. Figure 18.1 indicates the allowable gain/phase offset region for this ²scaled -margin specification.

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Notice this ellipsoidal region is a good approximation of the usual polygonal region. 8

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Fig. 18.1. The single-loop stability margin clearance criterion is that the closedloop system should remain stable when a complex multiplicative gain (referred to as a gain/phase offset) from within the polygonal area is inserted into any one loop. When the ²scaled -margin is greater than 0.27, the closed loop is guaranteed to remain stable when an offset from within the ellipsoidal region is inserted into any one loop. We use this result to approximate the usual clearance criteria.

The HIRM+ benchmark specifies that single-loop analyses be performed for the symmetric tailplane longitudinal response, the differential tailplane lateral response, and the rudder lateral response. For each of these cases, the lateral or longitudinal closed-loop system is formed, and then the loop to be studied is broken at the appropriate input to the HIRM+ model. The plant is defined as the closed-loop system with input and output at the broken loop, and the controller is considered be +1 (in positive feedback). 18.2.2 Multi-loop Stability Margin The usual multi-loop stability margin criteria is a generalisation of the singleloop criteria. The closed-loop system should remain stable when any independent gain/phase offsets from within the dashed region of the Nichols plot in Fig. 18.2 are simultaneously applied to each of the plant inputs. No efficient method for computation of this multi-loop stability margin is known. However, as shown in Section 4.3.2 of Chapter 4 , the (easily computed) multi-loop ²scaled -margin guarantees an ellipsoidal region in the Nichols plot from which independent and simultaneous input and output gain/phase offsets can be

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applied while still retaining closed-loop stability. Due to the nature of H∞ loop shaping as a method of considering the closed-loop effect of input and output disturbances, the ²scaled -margin result cannot easily be restricted to consider input gain/phase offsets only (as in the usual multi-loop specification). However, in light of experience with H∞ loop-shaping design discussed in Section 4.3.2 of Chapter 4 , consideration of both input and output offsets is proposed as a more logical and powerful measure of robustness. For this study, a ²scaled -margin threshold of 0.27 is used to approximate the original specification. Figure 18.2 indicates the allowable gain/phase offset region for this level. Notice that the resulting region is smaller than the usual polygonal region, but bear in mind that stability is guaranteed when offsets from the elliptical region are applied simultaneously at both the input and the outputs. 8

max. permissible multi−loop input ONLY offsets for GARTEUR max. permissible multi−loop input AND output offsets for ε(P,C) ≥ 0.27

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Fig. 18.2. The multi-loop stability margin clearance criterion is that the closedloop system should remain stable when independent complex multiplicative gains (referred to as gain/phase offsets) from within the polygonal area are inserted onto each of the plant inputs. When the ²scaled -margin is greater than 0.27, the closed loop is guaranteed to remain stable when independent offsets from within the ellipsoidal region are applied to each of the plant inputs and outputs. We use this result to approximate the stability margin clearance criteria.

18.3 Limitations of the HIRM+RIDE Model A number of trim points specified by the HIRM+ benchmark were excluded from analysis due to aircraft limitations or failures of the model. Those points

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where the allowed load-factor (−3g to 7g) was exceeded or actuators were saturated (for either the nominal of perturbed aircraft) are marked as points not cleared in the summary figures. In addition, some trim points were excluded from the analysis on the basis that the frequency response of the linearised plant was discontinuous with respect to certain parameter values. The discontinuities, which were encountered with respect to parameters XCG, clβ , and Iz , can be verified with a singular value plot of the frequency response of a sequence of plants as the parameter is varied over its allowable range.

18.4 Analysis Cycle An outline of the analysis cycle applied to the HIRM+RIDE model is provided here to clarify how the theoretical tools of the previous chapter are used to assess performance of the HIRM+RIDE closed-loop system with respect to the measures described in Section 18.2. Results of this analysis method will be demonstrated for a representative trim point in Section 18.5. We assume that a priori engineering judgement is used to specify a grid of trim points and a grid of frequencies at which to assess the stability margin. For the HIRM+RIDE analysis, the trim points discussed above are used, and the frequency grid is 100 logarithmically spaced points between 0.06 Hz and 15.9 Hz. A frequency-wise analysis is performed since the HIRM+RIDE closed-loop system is allowed to have unstable poles below a frequency of 0.06 Hz. The pilot is expected to stabilise such slow modes. For each specified trim point, we first generate the nominal model and determine input and output weights for frequency-wise ²scaled -margin analysis. Algorithm 1 (Initial calculations for nominal plant) 1. Trim the HIRM+ and RIDE models at the specified trim point for constant α-trim pull-up/push-over trajectory. 2. In the case of single-loop analysis, construct the closed-loop system with the desired single broken loop. P0 is then the closed-loop system with the single broken loop and and C = 1. (In the multi-loop case do nothing.) 3. For each frequency in consideration, generate frequency-wise optimal ²scaled -margin input and output weights, W0i and W0o , for the nominal plant and controller according to the optimisation described in Definition 3 of Chapter 4 . After making these calculations for the nominal plant, we then attempt to verify that the margins of the linearised models at each trim point will remain above the ²scaled -margin clearance threshold for all parameter values.

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Algorithm 2 (Worst-case stability margin analysis) 1. Compute the frequency-wise nominal stability margin ¢ ¡ ¢ ¡ −1 −1 (jω). CW0o ρscaled P0 (jω), C(jω) = ρ W0o P0 W0i , W0i

2. 3.

4.

5.

If this falls below 0.27 at any point, then the nominal system fails to meet clearance specifications. STOP. Select the parameters to consider (e.g. longitudinal category 1). By following Steps 1 to 2 of Algorithm 1, create linearised perturbed plants, Pδi , for each individual parameter at a value of δi = 0.01δ i where δ i is the maximum allowable value of that parameter (excluding reduction factors). Compute Xi = 1/0.01 ∗ XW0o Pδi W0i using (4.9) of Chapter 4 for each parameter using the optimal nominal weights from Step 3 of Algorithm 1. This calculation of Xi is a numerical approximation of the derivative from (4.12) of Chapter 4 Note that Xi approximates the weighted ν-gap effect of applying the ith parameter at a value of δi = δ i . Compute the frequency-wise approximation of the weighted ν-gap between the nominal plant and each maximally perturbed plant: ¢ ¡ σ(Xi (jω)) . κ W0o Pδi W0i , W0o P0 W0i (jω) = p 1 − σ 2 (Xi (jω))

6. Compute the lower bound on the approximate frequency-wise nominal stability margin using (4.15) of Chapter 4 : ¡ ¢ ρscaled Papp (jω), C(jω) n X ¢ ¡ −1 −1 (jω) − CW0o σ(Xi (jω)). & ρ W0o P0 W0i , W0i i=1

If the lower bound is above the clearance threshold of 0.27 at all frequencies, then the margins should be good for all combinations of parameters. STOP. 7. Now we must perform frequency-wise estimation of worst-case parameter combinations. To avoid unnecessary computation, the frequencies to be considered should be limited to those where the lower bound for the ²scaled margin is below the clearance threshold.1 At each frequency to consider for worst-case stability margin: 1

In fact, the frequencies to be considered can usually be restricted to those around the minimum of the stability margin lower bound curve as these frequencies are usually those near crossover where ²scaled -margin is most sensitive. However, in the results presented in this report, the approximation analysis is performed over the same frequency grid as that used for the nominal stability margin assessment in Step 1 in order to present more instructive ²scaled -margin and ν-gap plots.

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a) Sort the parameters according to decreasing values of their frequencywise ν-gap at the current frequency as computed in Step 5. b) Discard parameters which have insignificant ν-gap effect at this frequency¡ ¢ (e.g. κ W0o Pδi W0i , W0o P0 W0i (jω) < 0.001). c) Use the following procedure to perform a sequential-application heuristic method of estimating worst-case parameter values. i. Since the aerodynamic reduction factor makes the search over the parameter space combinatorial in the number of aerodynamic uncertainties, the worst-case set of parameters for the linear plant approximation Papp could contain one aerodynamic uncertainty parameter applied at full scale, or it could contain a combination of three such parameters applied at the maximum value allowed by the appropriate reduction factor. Therefore, we will perform several analyses each with a different number of active aerodynamic uncertainties. For each different analysis, we start with the Xi from Step 4 and the significant parameters from Step 7b. The following steps, A through I, should be repeated for values of N ranging from 1 to the number of significant aerodynamic parameters: A. Select the most significant N aerodynamic parameters from those deemed significant in Step 7b and discard the other aerodynamic parameters. B. Scale the Xi objects associated with significant aerodynamic parameters by the reduction factor for N aerodynamic parameters. C. Set Xapp = 0. D. Of those remaining under consideration, select the most significant uncertainty parameter (in terms of individual parameter ν-gap size at this frequency as calculated in Step 5). This parameter will be referred to as δi . + − = Xapp + Xi and Xapp E. Compute Xapp = Xapp − Xi . Which correspond to approximating the effect of this parameter on Xapp when δi = +δ i and when δi = −δ i respectively. + + − F. Using (4.9) of Chapter 4 , compute Papp from Xapp and Papp − from Xapp . ¢ ¡ −1 −1 + (jω), the weighted staW0i , W0i CW0o G. Compute ρ W0o Papp + bility margin of Papp at this frequency. Also compute the − weighted margin for Papp . Determine if δi = +δ i or δi = −δ i gives worse margins and set δi to this worst-case direction. + − H. Set Xapp = Xapp or Xapp = Xapp to reflect the worst-case direction determined in the previous step. I. Remove the parameter δi from consideration. If other parameters remain to be considered, return to Step 7(c)iD and con-

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tinue the process of sequentially applying the parameters in order of their closed-loop significance. ii. When the loop analyses for all relevant values of N have been performed, the user can determine what number of aerodynamic parameters yielded the worst-case stability margin estimate. d) Following Steps 1 to 2 of Algorithm 1, generate Ptrue , the linearised HIRM+ model with the worst-case parameter values from the sequential-application algorithm of Step 7c for this trim point and frequency. ¡ ¢ −1 −1 (jω), the actual stabile) Compare ρ W0o Ptrue W0i , W0i CW0o ity margin, with the margin of the ν-gap-approximate model, ¡ ¢ −1 −1 ρ W0o Papp W0i , W0i (jω). If the value using Papp differs CW0o largely from that using Ptrue , then the approximation has failed.¢2 ¡ Note that these values only approximate ρscaled Ptrue (jω), C(jω) since the true ²scaled -margin requires that the analysis weights be re-optimised. ¡ ¢ f ) Compute ρscaled Ptrue (jω), C(jω) , the actual stability margin achieved with Ptrue using Definition 3. If the effect of the parameters is not too drastic, ¢ then this value should be close to ¡ −1 −1 (jω). The later value is what is approxCW0o ρ W0o Ptrue W0i , W0i imated by both the µ and the sequential-application analyses.

18.5 Detailed Results for a Sample Trim Point The concepts behind the analysis algorithms of Section 18.4 are illustrated in this section by highlighting the key points of an analysis of a specific trim condition of the HIRM+RIDE system. Flight condition 5 at 27 ◦ angle of attack was selected as a representative trim point. Unless otherwise stated, references to “Steps” in this section refer to the analysis of Algorithm 2. 18.5.1 Lateral Analysis We first use Algorithm 1 to compute the nominal trimmed lateral plant P0 along with input and output weights, W0i and W0o , which provide the nominal ²scaled -margin. nominal lateral multi-loop ²scaled ¡ The frequency-wise ¢ margin curve, ρscaled P0 (jω), C(jω) , from Step 1 (of Algorithm 2) is show as the solid line in Fig. 18.3. (The other lines in this figure will be discussed later.) The nominal model satisfies our clearance threshold of 0.27, so we cannot reject the model based upon its¡nominal stability margins. ¢ The frequency-wise ν-gap values, κ W0o Pδi W0i , W0o P0 W0i (jω), for the effect of each individual lateral uncertainty parameter are calculated as in 2

This type of observation was used to identify many conditions where the HIRM+ model is discontinuous in parameters as summarised in Section 18.3.

18 ν-Gap Analysis of Stability Margin Criteria 1 0.9

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Fig. 18.3. The frequency-wise ²scaled -margin shows that stability is most critical near 0.41 Hz for the lateral multi-loop analysis of flight condition 5 at 27 ◦ angle of attack. The nominal margin is just above the minimum acceptable level, and the approximate lower bound falls below 0.1 indicating potentially rather poor behaviour. The dash-dotted line shows the frequency-wise approximate worst-case stability margin identified with the sequential-application search method (Step 7e of Algorithm 2). The worst-case parameters yield an approximate margin of 0.191.

Step 5. The results for the category 1 parameters are shown in Fig. 18.4, and those for category 2 parameters are shown in Fig. 18.5. The magnitude of the category 2 parameters is quite small (< 0.004), so the suggestion that these parameters are not important for the analysis is supported by this analysis of their individual effects. We will consider all of the category 1 parameters for our ν-gap approximation analysis. In Step 6, we obtain an approximate lower bound on the approximate worst-case stability margins from the ν-gap-approximate plant model by subtracting (frequency-wise) the sum of the ν-gap values for the category 1 parameters from the nominal ²scaled -margin. This lower bound is shown as a dashed line in Fig. 18.3. Since the lower bound is below our clearance threshold of 0.27 between 0.06 and 3 Hz, we can restrict our frequency-wise analysis to this region. The sequential-parameter-application method for estimating worst-case parameters of Step 7c identified parameters at each frequency which give the worst-case curve shown as a dash-dotted line in Fig. 18.3. In general, the worst-case set of parameters varies with frequency, but the worst-case parameters for closed-loop stability margins are those which produce the smallest number across all frequencies. This minimum occurs at 0.41 Hz. At

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Fig. 18.4. Parameters which may be significant to multi-loop lateral closed-loop stability margins are revealed in a plot of the frequency-wise ν-gap approximation for the effect of each independent full-scale category 1 parameter on the lateral dynamics of flight condition 5 at 27 ◦ angle of attack. 0.004 clδR 0.0035

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Fig. 18.5. All the lateral mode category 2 parameters have small multi-loop lateral closed-loop effect as revealed in a plot of the small magnitudes of the frequency-wise ν-gap approximations for the effects of each independent full-scale parameter on the lateral dynamics of flight condition 5 at 27 ◦ angle of attack. Notice that the scale of this plot is approximately 100 times smaller than that of Fig. 18.4.

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this frequency, the ν-gap effect of the parameter YCG was less than 0.001 so this parameter was excluded from consideration. As discussed in Step 7(c)i, the HIRM+RIDE model parameter uncertainty ranges vary according to the number of active aerodynamic uncertainties. If more aerodynamic parameters are applied, a smaller range of uncertainty in each parameter is permitted (as defined by the reduction factors described elsewhere), so the search for the worst-case set of parameters must consider all possible quantities of these active parameters. Using the frequency-wise ν-gap plots in Fig. 18.4, we can rank the aerodynamic parameters in terms of their significance to closed-loop behaviour, and then use the sequentialapplication method to estimate the worst-case margins obtained from applying the top 1, 2, or 3 most significant aerodynamic parameters. This process is a polynomial-time algorithm which in this case estimates that the worstcase stability margin occurs at 0.41 Hz with only one active aerodynamic parameter. The parameter values and stability margins for each iteration of this analysis for different numbers of aerodynamic uncertainty parameters are shown in Table 18.1. Allowing for an error of 0.01 due to the exclusion of parameter YCG from the analysis, these worst-case approximate ²scaled margin values in the last column of that table are in fact the worst-case approximate margins over all the vertices of the associated active parameter space (as verified by exhaustive search). Table 18.1. HIRM+RIDE FC5, 27◦ lateral multi-loop analysis of variable number of aerodynamic uncertainties Number of aero. param. considered 1 2 3

YCG [m] 0 0 0

uncertainty parameters I x Iz cl β cnβ cnr [-] [-] [1/rad] [1/rad] [1/rad] 0.20 0.08 0.04 0 0 0.20 0.08 0.0248 -0.0248 0 0.20 0.08 0.0184 -0.0184 0.0230

approx ²scaled (Papp , C) [-] 0.183 0.201 0.216

Analyses using µ methods may miss these type of worst-case parameter combinations where considering fewer aerodynamic parameters with a larger range may be worse than considering all such parameters over a reduced range. When considering that three aerodynamic uncertainty parameters are deemed significant by looking at Fig. 18.4 at 0.41 Hz, a worst-case margin of 0.211 at that frequency was estimated using µ-analysis techniques. Of course, the user could combine these methods by using a sequential-application approximation to determine which parameters to use in a subsequent muanalysis. Such an approach is not performed here in order to investigate the potential of the sequential-application method alone. After selecting the worst-case parameters for this analysis (those from Table 18.1 which give an approximate ²scaled -margin of 0.183),

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we can apply these parameters to the non-linear model and re-trim as in Step 7d to obtain Ptrue . Using the frequency-wise weights, W0i and W0o , which are optimal for the¡ nominal plant¢ in Step 7e, we compute an approximation of ρscaled Ptrue (jω), C(jω) at 0.41 Hz as ¢ ¡ −1 −1 (j0.41(2π)) = 0.185. This calculation shows that ρ W0o Ptrue W0i , W0i CW0o the ν-gap approximation has worked well (error of¢ 0.002). Finally, we should ¡ compute the true level of ρscaled Ptrue (jω), C(jω) obtained by re-optimising the input and output analysis weights as described in Step 7f. The resulting stability margin is 0.191 which shows that the ν-gap approximation technique also works well for estimating the perturbed ²scaled -margin at this trim point. An illustration of the meaning of an ²scaled -margin of 0.191 is shown in Fig. 18.6 where the multi-loop gain/phase offset regions for the usual multiloop stability margin criteria and for ²scaled (Ptrue , C) > 0.27 are shown. The smaller elliptical offset region corresponding to a ²scaled -margin of 0.191 is also shown along with a set of input and output offsets within this region (computed from the µ lower bound) which destabilise the closed-loop linear system. Note that these offsets are quite small in relation to the usual region, but the closed-loop system is still destabilised by these offsets. 8

GARTEUR input offsets limit offset limit for ε(P,C) ≥ 0.27 offset limit for ε(P,C) ≥ 0.19 δ − diff. tailplane input gain offset TD δ − rudder input gain offset

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Fig. 18.6. The worst-case estimated lateral multi-loop ²scaled -margin for flight condition 5 at 27◦ angle of attack is 0.191. The controller fails clearance at this flight condition since multi-loop input and output offsets can be identified within the region of allowable offsets which destabilise the closed-loop system.

Applying the same analysis of Algorithm 2 to the differential tailplane and rudder single-loop lateral models does not identify any parameters which violate the 0.27 ²scaled -margin clearance threshold. The Nichols frequency responses for the worst-case parameters for each of these two models are

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shown in Figs. 18.7 and 18.8. Notice that the Nichols frequency responses for both loops lies outside of both the ²scaled -margin and the standard clearance exclusion regions. 8

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Fig. 18.7. The Nichols frequency response of the lateral differential tailplane loop for flight condition 5 at 27◦ angle of attack remains outside the minimum acceptable standard and ²scaled -margin offset regions (centred on the +1 point). The exclusion regions for the achieved standard stability margin and ²scaled -margin are plotted to illustrate how those margins guarantee stability of the loop with respect to larger offsets than those permitted under the clearance specifications.

18.5.2 Longitudinal Analysis The frequency-wise ²scaled -margin and ν-gap curves for the multi-loop longitudinal analysis at flight condition 5 with 27 ◦ angle of attack are shown in Figs. 18.9, 18.10, and 18.11. Notice that the longitudinal category 2 parameters are of negligible magnitude (< 0.001). The approximate ²scaled -margin lower bound from Step 6 does not violate the clearance threshold of 0.27, so the analysis can be terminated at Step 6 without needing to perform the worst-case parameter analysis. Such an analysis was performed for completeness, and the resulting worst-case stability margin curve is also shown in Fig. 18.9.

18.6 Results for all Trim Points Results from all single- and multi-loop analyses at all trim points are presented in the following sections. The estimated worst-case parameter values

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Fig. 18.8. The Nichols frequency response of the lateral rudder loop for flight condition 5 at 27◦ angle of attack remains outside the minimum acceptable standard and ²scaled -margin offset regions (centred on the +1 point). The exclusion regions for the achieved standard stability margin and ²scaled -margin are plotted to illustrate how those margins guarantee stability of the loop with respect to larger offsets than those permitted under the clearance specifications. 1

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Fig. 18.9. The frequency-wise ²scaled -margin lower bound for the longitudinal multiloop analysis of flight condition 5 at 27 ◦ angle of attack is above the clearance threshold of 0.27, so no further analysis was required.

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Fig. 18.11. All the longitudinal mode category 2 parameters have small multi-loop longitudinal closed-loop effect as revealed in a plot of the small magnitudes of the frequency-wise ν-gap approximations for the effects of each independent full-scale parameter on the longitudinal dynamics of flight condition 5 at 27 ◦ angle of attack. Notice that the scale of this plot is approximately 400 times smaller than that of Fig. 18.10.

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from the sequential-application heuristic algorithm are presented here due to the ability of that method to cope with the aerodynamic parameter reduction factors in a simple fashion (without running multiple µ-analysis tests as discussed in Section 18.5). 18.6.1 Relating the ²scaled -Margin and Standard Nichols Exclusion Regions For the single-loop cases, after using Algorithm 2 to estimate the worstcase parameters (and the associated worst-case plant Ptrue ), we can compute ²scaled (Ptrue , C) along with the usual single-loop stability margin. Figure 18.12 compares these measurements on a scatter plot in order to justify the use of ²scaled -margin as an measure of stability which provides the same sort of results as the standard single-loop measure. Notice that the scatter plot is approximately linear and that roughly the same plants which violate the standard specification (< 1) also violate a ²scaled -margin clearance threshold of 0.27. 0.7

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Fig. 18.12. We can compare the ²scaled -margin and the usual stability margin measures for this study by making a scatter plot of those two measures of stability for the worst-case parameter values of the HIRM+RIDE model in each of the three single-loop analyses. A minimum acceptable ²scaled -margin of 0.27 correlates well with the usual Nichols stability margin minimum clearance level of 1.0.

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18.6.2 Clearance Results A summary of all single-loop clearance violations is provided in Fig. 18.13. A comprehensive summary of all clearance violations (single- and multi-loop) is provided in Fig. 18.14. Clearly, there are a large number of lateral trim conditions where the HIRM+RIDE system passes both the single-loop clearance tests but fails the multi-loop clearance test. No such situations occurred in the longitudinal analyses. The existence of these results for the lateral case demonstrates that traditional methods of applying loop-at-a-time analysis to multi-loop systems can miss loop-coupling effects which may destabilise the system. The proper use of ²-margin as an analysis tool can allow such multi-loop stability issues to be identified. 50000 45000 FC3 [31, 35] discnt: -5, -3, 9, [17, 25]

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18.6.3 Validating the ν-Gap Approximation In this section, we consider the error in using the ν-gap-approximate models of (4.11) of Chapter 4 with nominal weights W0i and W0o to estimate the

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¢ ¡ −1 −1 (jω) as in Step 7e of Algorithm 2. actual value of ρ W0o Ptrue W0i , W0i CW0o A histogram of this error for each of the worst-case parameters at each trim point in the lateral and the longitudinal multi-loop analyses is shown in Figs. 18.15 and 18.16. The errors are tightly clustered around zero, indicating the ν-gap approximation is accurate for this model. These results do not consider the error in approximating ²scaled (Ptrue , C) (which requires a re-optimisation of the analysis weights) as described in Steps 7e and 7f. Such errors are discussed in Section 18.6.4. 18.6.4 Effect of Weights Finally we consider the error in using the ²scaled -margin optimal input/output weights for the nominal plant, W0i and W0o , to estimate ²scaled (Ptrue , C) for the perturbed plant as discussed in Steps 7e and 7f. The computation of ²scaled (Ptrue , C) requires a re-optimisation of the analysis weights which cannot be approximated in a linear fashion. The error is essentially zero for longitudinal analyses, but the lateral analyses contain a number of errors up to a value of 0.025 (and a few outliers beyond that level). However, this error is always conservative since the re-optimisation of analysis weights can only improve the weighted ²-margin.

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18.7 Conclusions We have presented the ν-gap tangent plane approximation method as a computationally simple method for obtaining an approximate model which is close in the frequency-wise ν-gap metric to a system with parametric uncertainty. The approximate model is linear fractional in the parameters, so tools for analysing linear fractional uncertainty can be applied. The approximation should work for systems whose dependence upon the uncertainty parameters is sufficiently smooth, and has been demonstrated to work well

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for the HIRM+ model in all regions except those where the HIRM+ model exhibits discontinuities due to the aerodynamic lookup tables used. The advantage of this method is that the number of trim+linearisations required is equal to the number of parameters variations being considered, rather than being exponential in the number of parameters as in a vertex enumeration method. Single- and multi-loop clearance criteria defined in terms of ²scaled -margin are proposed as alternatives to the standard stability margin clearance criteria. The ²scaled -margin multi-loop criterion defines allowable combinations of simultaneous input and output gain/phase offsets which can be applied while still retaining closed-loop stability, but yet is no more difficult to calculate than the single-loop criteria. The use of this multi-loop criterion identified many trim points which have poor multi-loop ²scaled -margin while satisfying the single-loop stability margin criteria. Such trim points may exhibit poor closed-loop behaviour in practice considering the small input and output gain/phase offsets which will destabilise the aircraft. As part of both the usual stability margin clearance procedure and the proposed modification, the ν-gap has been shown to be useful for classifying parameters in terms of their potential effect on the stability margin, on a frequency by frequency and flight condition by flight condition basis. This has facilitated the use of a simple algorithm instead of the usual µ lower bound to find worst case parameter combinations.

19 Polynomial-Based Clearance of Eigenvalue Criteria Leopoldo Verde and Federico Corraro Centro Italiano Ricerche Aerospaziali Flight System Department Via Maiorise, 81043, Capua (CE) Italy [email protected],[email protected]

Summary. This chapter deals with application of a polynomial based analysis method to the HIRM+RIDE benchmark clearance problem. We only present the results of the eigenvalue clearance criterion over the selected flight conditions and uncertain parameter sets. Some additional analyses have been performed in order to exploit the full potential of this method. These results mainly illustrate that this method allows a criterion in an entire region of the uncertain parameter space to be checked, thus giving the possibility to determine the estimated cleared and not cleared regions (not simple points) in the uncertain parameter space.

19.1 Description of the Analysis Cycle In this chapter we address only the eigenvalue clearance criterion described in Chapter 10. Instead of finding a worst case point, with this method regions within the flight envelope will be identified in terms of Mach Number, altitude and AoA where the eigenvalues of the HIRM+RIDE linearized model are at the right side of the domain in the complex plane shown in Fig. 19.1 (i.e. the not cleared regions). This is the domain of the complex plane to be considered for checking the eigenvalue clearance criterion Chapter 10. As also required in Chapter 10, the reported final results have been obtained by simultaneously varying all five category 1 uncertain parameters for longitudinal analysis and all six category 1 uncertain parameters for lateral analysis. In order to exploit all the features of this method, two preliminary analyses have been performed and relevant results are also illustrated. The first one concerns the analysis of the complete HIRM+RIDE flight envelope Chapter 8, with all the uncertain parameters set to their nominal values. The results of this analysis well exploit the capability of this method to identify the not cleared regions, not just simple points. Moreover, a side result of this analysis is also determination of the ‘true’ physical flight envelope of HIRM+RIDE. The second preliminary analysis addresses continuous variation of AoA and one uncertain parameter at a time, at a predefined point in the flight envelope and with all other uncertain parameters set at their nominal values. The obtained results can be considered a follow-up of the first preliminary analyC. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 333-353, 2002. Springer-Verlag Berlin Heidelberg 2002

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sis and can be used to understand which parameters are most important for clearance and which ranges of AoA are of most interest so as to reduce effort when performing a clearance analysis with many simultaneously varying uncertainties. In summary, we actually performed three analysis. A preliminary one by using the nominal system, a second preliminary analysis with single uncertainty parameter and AoA, and a final analysis with all uncertain parameters. 1

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All three above mentioned analyses can be basically performed using the same analysis cycle based on the mathematical results mainly described in [1] and [2] and implemented in an algorithm suitable for clearance analysis as reported in Chapter 5. As this method is based on analysis of characteristic polynomial coefficients of a linear uncertain system, a function must be written which computes the value of these coefficients in a given point π of the uncertain parameter space. A software tool has been developed under MATLAB/SIMULINK [2], which basically implements the above stated algorithm. This tool allows a user to easily specify algorithm parameters and defines the structure of the above described function for evaluating characteristic polynomial coefficients in a given point of the uncertain parameter space. Some further utilities are available to aid a user to tune algorithm parameters and to visualize computation results. All computation has been performed by using the above described tool under MATLAB Rel.11 on a Pentium III 500MHz machine with 256MB RAM, running Windows NT4. The main steps of the analysis cycle based on the use of this tool are below described.

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19.1.1 Trimming and Linearisation Routine Development The first step in applying this technique concerns the development of an appropriate routine which outputs characteristic polynomial coefficients in a given point π of the uncertain parameter space. This routine basically performs trimming and linearization of the aircraft model in a predefined point and outputs also a flag which states the validity of such computations. This flag is used to deal with non-continuous functions that map the uncertainties into the characteristic polynomial coefficients, non-rectangular uncertain domains, trimming and linearization errors and trimming limitations. Because the algorithm needs to repeatedly evaluate the characteristic polynomial coefficients of the linearized system in given points of the uncertain parameter space, a fast and reliable trim and linearization is needed. In all the performed analyses, a trim routine has been used which basically trims and linearises aircraft dynamics separately from the RIDE control laws Chapter 8. The aircraft trim routine, in particular, uses an inverse dynamic model of the HIRM+ resulting in a very accurate and fast calculation. Some modifications to the trimming and linearization routine described in Chapter 8 have been performed in order to allow execution within a batch calculation with minimum execution time. These modifications only affected marginal aspects of the trimming and linearization algorithms, whose key calculations were not changed. Moreover, in order to take into account all the limits of the ‘true’ flight envelope, as specified in Chapter 8, the following limitations on the load factor and on the taileron and rudder actuator deflections were used Chapter 8: −3g ≤ nz ≤ 7g −40◦ ≤ δT S + δT D ≤ 10◦ −40◦ ≤ δT S − δT D ≤ 10◦ −30◦ ≤ δR ≤ 30◦

(19.1)

Violation of conditions (19.1) and an excessive trim error set the above mentioned validity flag so that the adaptive grid generation algorithm can tag the relative uncertain point as not-valid (which is automatically a notcleared point). While examining each box, if a not-valid point is found the box is further divided up to the resolution of the adaptive grid generation procedure (see Chapter 5, for details). In this way the algorithm can also determine the ’true’ physical flight envelope of the HIRM+RIDE. The linearized model produced by this routine is then used to compute the characteristic polynomial coefficients. Because no model reduction has been performed in all the analysis reported below, the complete closed loop HIRM+RIDE model has been used which has 61 states. Finally, in order to demonstrate that this techniques allows a ‘black box’ approach to be used (i.e. no information on the aircraft dynamics is actually needed), no preliminary analysis on the aircraft dynamics has been performed.

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19.1.2 Stability Domain and Uncertainty Range Specification The proposed polynomial based analysis method requires the domain D to be specified in the complex plane where eigenvalues shall belong (i.e. the Dstability domain defined in Chapter 5). To this end, minimum and maximum damping factors and natural frequencies (in case of close domains) or an array of domain contour points (in case of open domains) shall be specified. In this latter case, the algorithm always assumes that the criterion is fulfilled when eigenvalues are located on the left side of the specified contour. In this analysis, equations (10.1) of Chapter 10 have been used to generate the contour points. Also, because the contour is specified as a finite dimensional array, a maximum imaginary part of 30 has been used. This is equivalent to assuming that no pure imaginary eigenvalue will have a modulus bigger than 30, which is almost always true with aircraft systems. Finally, the range of uncertainties shall be specified in terms of the interval centre point and semi-dimension. The algorithm only allows rectangular uncertainty regions to be specified. Should a different region shape have to be analyzed, the validity flag of the routine described in section 19.1.2 can be used to account for the constraints following from the uncertainty parameters that result in the desired shape for the initially specified rectangular region. This method has been used for the nominal flight envelope analysis in order to exclude the upper-left corner (high altitudes, low Mach numbers) from the computation. 19.1.3 Algorithm Parameter Setting As described in Chapter 5, the proposed analysis method actually consists of two procedures which are executed in cascade: an adaptive grid generation procedure (AGP) where a grid is generated for the map between uncertainties and polynomial coefficients, say a(.), and a D-stability region shape computation procedure (RSC) where the clearance criterion is actually checked. Both procedures need to specify some configuration parameters that mainly affect the accuracy in determining D-stability region shape, the number of trimming and linearization points and the computation time. The AGP (where aircraft trimming and linearization is performed several times) needs two parameters: the maximum estimated error and the minimum grid size. The first parameter is the maximum allowable value of an error’s measure between a(.) and its local affine approximation (see Chapter 5). The second parameter is the ultimate grid size that shall be considered. By using these two parameters, the algorithm generates large grid sizes in the regions of the uncertain parameter space where the estimated error is low (i.e. under its maximum value) and smaller grids, up to the minimum grid size, in the regions where the estimated error is high. A sample output result is shown in Fig. 19.2, where it can be noted how the algorithm adapts the grid size using a smaller size where the function a(.) is mostly not linear.

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These two parameters mainly affect the error when using a local affine approximation of a(.) and the number of trimming and linearization points which in turn is the most time consuming task. For this reason, the AGP parameters need to be interactively tuned for obtaining maximum accuracy with a minimum number of trim and linearization points. Even if some utilities have been developed to aid a user to perform this tuning, some means to automate this task can also be investigated, but this is out of the scope of this work. In Fig. 19.3 a comparison is shown between eigenvalue locations of the affine a(.) approximation and the actual ones (the crosses are the exact eigenvalue locations and the squares the approximated ones). It can be noted how the slow eigenvalues are better fitted than the fast ones, which is exactly the result we need. Finally, as concerns the RSC procedure, a user has to specify the resolution for the D-stability region shape determination. As this procedure generally involves less than 20% of the total computational effort, the setting of this parameter basically depends on user wishes. A high resolution will result in a better determination of the cleared and not cleared region shapes at the expense of a limited increase in computational effort. For the analysis results below reported, this phase took a total of about two weeks work for one person. 19.1.4 Procedure Running and Result Visualization Once all the above steps are completed, the analysis can run without any further user intervention. Moreover, the AGP and RSC procedures can be executed sequentially and inserted in a batch computation in order to, for example, change flight conditions. Outputs of this technique are the lists of not valid, cleared, not cleared and undetermined boxes (i.e. hyper-rectangles with the dimension equal to the number of uncertainties) whose union gives the shape of the respective regions. Undetermined boxes are included because for these boxes the maximum resolution of the RSC procedure has been reached and the technique cannot determine if they are cleared or not cleared. Most often these boxes delimit the cleared areas from the not-cleared ones, while sometimes they denote some kind of discontinuities in the function a(.). These output lists can easily be visualized when analyzing two or three uncertainties. In this case a 2 or 3 dimensional region can be plotted. In the case of more than 3 simultaneous uncertainties, special algorithms should be used to visualize the analysis results, but their complete overview is out of the scope of this paper. In the following, an algorithm which plots the 2-D projections (on two uncertainty axes) of the biggest not-cleared multidimensional areas has been used (so called worst areas ).

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19.2 Analysis Results for Nominal HIRM+RIDE Flight Envelope One of the main capabilities of this method is that it allows an analysis to be performed with a ’black box’ approach. This means that basically no preliminary analysis in the flight envelope of aircraft dynamics has to be performed, because this method allows regions or points which seem more relevant for the analysis to be selected. To demonstrate this capability, an analysis of the eigenvalue criterion for the nominal HIRM+RIDE model (i.e. with all the uncertainties set to their nominal values) has been set-up. The steps described in section 19.1 have been performed by considering Mach number and Altitude as uncertain parameters, ranging respectively in [0.2, 0.8] and [1000ft, 40000ft]. An additional constraint on allowable Mach numbers and altitudes has been added to the trimming and linearization routine in order to avoid the points in the upper-left side of the envelope. The AGP and RSC procedures have been inserted in a batch computation in order to analyse the flight envelope in the full AoA range (from –15˚ to 35˚) with 2˚ steps. The minimum grid size set in the AGP is 0.0313 while the region shape resolution in the RSC procedure is 0.0156. Because both these parameters refer to a unity uncertainty interval, in this case the minimum grid size for Mach Number is 0.0187 and for altitude is 507ft (as we set the altitude range in the interval [1000ft, 40000ft]). These values are the maximum errors in determining the not ’true’ flight envelope regions (which is a side result of the AGP). As the resolution in the RSC procedure has been set to half the minimum grid size, maximum errors in determining the boundary between a cleared and not-cleared region are 0.0094 for Mach and 253.5ft for altitude. Selected results of this analysis have been shown in Figs. 19.4, 19.5, 19.6, and 19.7 where the cleared and not cleared regions for four analyses with AoA respectively of 23˚, 27˚, 31˚ and 33˚ have been included. Just for reference, in each plot the contour of the HIRM+ envelope and the eight flight conditions defined in Chapter 10 have also been shown. Each plot in these figures can be read as follows: • the white region (labelled not part of the ’true’ flight envelope) inside the flight envelope is a region where the aircraft cannot be trimmed because of some limitation, as specified in section 19.1; • the grey region (labelled cleared region) is the cleared region (i.e. where all aircraft eigenvalues are at the left of the boundary shown in Fig. 19.1); • the dark grey region (labelled not cleared region) is the not cleared region (i.e. where, at least one eigenvalue is at the right of the boundary shown in Fig. 19.1); • the light grey region (not labelled) between the cleared and not cleared regions is the region where nothing can be said because the algorithm reaches the maximum resolution in determining region shape.

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From these figures it can be noted how the boundary between the cleared and not-cleared region moves, changing its shape and increasing the notcleared region as the AoA increases. By looking at all the plots for each AoA value, the results can be summarised as follows: • the nominal HIRM+RIDE model has no problem for AoA ranging in [– 1˚;9˚] (i.e. in this range and for all points in the whole flight envelope the aircraft can be trimmed and linearized and the eigenvalues are all at the left of the boundary shown in Fig. 19.1); • not cleared regions can be found only for AoA in the range [21˚;35˚]; • not trimmable regions can be found for AoA in the ranges [-15˚;-3˚] and [11˚;35˚]; • most problematic regions (in terms of trimmability) appear to be the rightlower part of the flight envelope, while a small not trimmable region appears in the lower-left corner for AoA ranging in [-15˚;-11˚]; • a small grey region appears for AoA=35˚ in the central-left part of the flight envelope (higher resolution would be needed to better determine the shape), which actually includes FC2 (i.e. FC2 is the only flight condition which is cleared for AoA=35˚). Many other considerations can be made by using just the results from this nominal flight envelope analysis of the HIRM+RIDE model.

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In particular, the user gets pretty good information to choose the points or regions within the flight envelope that are more relevant for performing further analysis for the non-nominal HIRM+RIDE model. In this way the effort required for performing analyses in non-nominal cases can be dramatically reduced. Other than the type and quality of results, computational effort is another important aspect for evaluation of this method compared to conventional gridding methods. As already said, the most time consuming task is trimming and linearisation of the aircraft model in a given point of the uncertain parameter space (this took typically more than 80% of total computational effort). In our case, about 1 second is needed for trimming and linearization of the HIRM+RIDE model in a given point. The completed analysis took about 4.5 hours computation time with a total of about 13000 evaluated points. A conventional gridding method would obtain results of the same accuracy (i.e. with an equivalent resolution of 0.0094 for Mach number and 253.5ft for altitude) by evaluating 105625 total points and spending about 29.3hours computational time (accounting only for the trim and linearization computations). These timing figures clearly demonstrate the capability of the proposed clearance method to reduce computational effort up to 10 times less than conventional gridding ones, provided that the same resolution is used in determining region boundaries. This is obviously obtained at the expense of a

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much more complicated algorithm and of some degree of approximation in determining the cleared region shape (for details, see discussion in Chapter 5).

19.3 Analysis Results for Single Uncertain Parameter and AOA Once an analysis in the whole flight envelope for the nominal aircraft has been performed and some points or regions have been selected where further investigations are needed, a user can consider increasing the number of uncertain parameters varied simultaneously. This can again lead to an analysis in regions of the flight envelope or to other analyses with a different choice of uncertain parameters. In the following we present selected results of an analysis performed by varying simultaneously the AoA and one uncertain parameter chosen among the category 1 parameters for longitudinal and lateral analysis defined in Chapter 10. The analysis has been performed in all the flight conditions selected for the HIRM+RIDE benchmark problem Chapter 10. In this case, the AoA can be considered to vary continuously in its whole range of [-15˚, 35˚], because this technique can guarantee fulfilment of the eigenvalue criterion in a whole interval for the parameters defined in the algorithm as uncertainties. Furthermore, instead of using a pre-fixed grid size for AoA (for example, the suggested 2˚ steps), it is the algorithm that computes the values of the AoA where the eigenvalue criterion is not fulfilled trying to minimise the number of trimming and linearization points. These are key differences with the analysis described in the following section (with respect to AoA variation) and with the conventional techniques, where a pre-defined fixed gridding is used (leading to a fixed number of evaluated trim points, even if many of these calculations could turn out to be unnecessary) and only local information (in the points of the grid) can be rigorously guaranteed. This calculation has been performed by setting the RSC resolution (referred to a unity range) to 0.0156 which, in this analysis, results in an error of less than 0.8˚ for the AoA determination. In the Figs. 19.8, 19.9, 19.10 and 19.11 only the analysis results for flight condition 6 (FC6) are shown; the second uncertain parameter (shown on the horizontal axis) in each case is: uncertainty on centre of gravity x-axis position xCGunc , the uncertainty level of a component of the inertia matrix along the aircraft longitudinal axis Iyunc , the uncertainty on the variation (due to AoA) of the longitudinal aerodynamic moment coefficient Cmα unc and the uncertainty on centre of gravity y-axis position yCGunc respectively. For each of these parameters, the ranges defined in Chapter 10 have been used without any reduction factor. As in the previous section, each plot shows the cleared (grey), not cleared (dark grey), not trimmable (white) and not defined (light grey) regions. For example, looking at the analysis for (xCGunc , AoA) in Fig. 19.8, it can be seen

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that the aircraft is allowed to fly (considering only the eigenvalue criterion) for AoA in the interval of [-5.6˚; 24.2˚], for any uncertainty of xCG in the range of [-0.15m, 0.15m]. 35 Not part of the "true" flight envelope

30 Not cleared region

25 20

α [deg]

15 Cleared region

10 5 0 −5 −10 −15

Not part of the "true" flight envelope

−0.1

−0.05

x

0

CGunc

[m]

0.05

0.1

0.15

Fig. 19.8. Clearance results for (xCGunc , AoA) analysis in FC6

Also, by looking at the vertical line of each figure passing through the nominal value of the uncertain parameter, the results of the nominal HIRM+RIDE flight envelope analysis can be checked for FC6. Furthermore, it can be noted that, at least for the analysis of (Iyunc , AoA) in the FC6, the not cleared range of AoA has to be found by looking at middle values of the uncertain parameter Iy . This demonstrates that it is not always true that worst cases have to be found at the extremes of an interval (which is mostly the assumption undertaken in the industrial clearance method described in Chapter 15). A side result of this analysis is the determination of not trimmable regions also in non-nominal conditions (see for example Figs. 19.8 and 19.11, where the worst AoA values have to be found at the extremes of the uncertainty interval). The computational effort required for performing these analyses is similar to the nominal flight envelope analyses. Also in this case, a single analysis with two uncertain parameters in one flight condition requires evaluation of less than 500 trim and linearization points and about 10 minutes computation on a PIII 500MHz machine. Again it should be noted that a pre-fixed grid

19 Polynomial-Based Clearance of Eigenvalue Criteria 35 Not part of the "true" flight envelope

30 Not cleared region

25 20

α [deg]

15 10 Cleared region

5 0 −5 −10 −15 −0.05

Not part of the "true" flight envelope

−0.04

−0.03

−0.02

−0.01

0 IYunc

0.01

0.02

0.03

0.04

0.05

Fig. 19.9. Clearance results for (IY unc , AoA) analysis in FC6 35 Not part of the "true" flight envelope

30 Not cleared region

25 20

α [deg]

15 10

Cleared region

5 0 −5 −10 Not part of the "true" flight envelope

−15 −0.1

−0.08

−0.06

−0.04

−0.02

0 Cm unc

0.02

0.04

0.06

0.08

0.1

α

Fig. 19.10. Clearance results for (Cmα unc ,AoA) analysis in FC6

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Not part of the "true" flight envelope

30 Not cleared region

25 20

α [deg]

15 Cleared region

10 5 0 −5 −10 −15 −0.1

Not part of the "true" flight envelope

−0.08

−0.06

−0.04

−0.02

0 yCGunc [m]

0.02

0.04

0.06

0.08

0.1

Fig. 19.11. Clearance results for (yCGunc , AoA) analysis in FC6

size chosen to get similar result accuracy (even if they are guaranteed only in the grid points) would require evaluation of about 4225 trim and linearization points with a computation time of more than one hour (accounting only for trimming and linearization computations).

19.4 Analysis Results for the Complete Set of Parameters Starting from the results for AoA and a single uncertain parameter variation, it is possible to gradually increase the number of simultaneously varied uncertain parameters while reducing the AoA range under investigation. For example, taking into account the results of Figs. 19.9, 19.10 and 19.11, it is not necessary to check clearance of more than one simultaneous longitudinal parameters for AoA ranging in [25˚, 35˚] and [-5˚, -15˚], because these intervals are already not cleared. In this way it is possible to minimise the AoA range where an analysis is still required with a high number of simultaneous parameters. As it is clear from Chapter 5 that this method has a complexity of 2k (k being the number of uncertain parameters), reducing the AoA range is actually important to dramatically reduce computation time. In the following the suggestion outlined above was not followed, in order to produce timing results which can be compared with the other proposed meth-

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ods. Thus, the completed analysis was performed by simultaneously varying both 5 longitudinal and 6 lateral uncertain parameters, for all eight conditions and in the whole AoA range (with a grid of 2˚ step). 35 Not part of the "true" flight envelope

30 Not cleared region

25 20

α [deg]

15 10

Cleared region

5 0 −5 −10 Not part of the "true" flight envelope

−15

−0.1

−0.05

x

0

CGunc

[m]

0.05

0.1

0.15

Fig. 19.12. Worst areas projection on (xCGunc , AoA) for complete longitudinal analysis in FC6

Both longitudinal and lateral unstable eigenvalue criterion analyses have been performed by using only the category 1 uncertain parameters in the range specified in Chapter 10 and applying reduction factors to the aerodynamic coefficients as per rule specified in Chapter 10. It is noted here that the terms longitudinal and lateral analyses stand for analyses performed by varying uncertain parameters which are relevant for the longitudinal and the lateral axis of an aircraft, while in both analyses the full model (61 states) has been used. Furthermore, in both analyses the minimum resolution (equal to 0.25, referred to a unity uncertainty interval) which allowed the reliability of results and acceptable computational effort has been used. Once the algorithm configuration parameters (see section 19.1.3) have been defined for each set of uncertain parameters, these two analyses (longitudinal and lateral) can both be performed in a batch computation without any user intervention. It is highlighted here that, by using the above mentioned suggestion, a reduction in the number of evaluations (i.e. trim and

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linearization points) can be obtained which has been estimated to be more than 40% of the total evaluations reported below for the complete analysis. 35 Not part of the "true" flight envelope

30 Not cleared region

25 20

α [deg]

15 10

Cleared region

5 0 −5 −10 Not part of the "true" flight envelope

−15 −0.05

−0.04

−0.03

−0.02

−0.01

I

0

0.01

0.02

0.03

0.04

0.05

Yunc

Fig. 19.13. Worst areas projection on (IY unc , AoA) for complete longitudinal analysis in FC6

To allow a comparison, selected analysis results are shown in Figs. 19.12, 19.13, 19.14 and 19.15 for the same flight condition (FC6), longitudinal and lateral uncertain parameters whose analysis results (obtained by leaving the other parameters to the nominal value) are reported in the previous section. As said, in this case, the regions computed by this algorithm have more than 3 dimension (5 for longitudinal and 6 for lateral), thus an algorithm shall be used to visualise the results on a 2-D or 3-D space. In the reported figures, even if a fixed grid of 2˚ step is used for the AoA (unlike the previous section, where it was configured in the algorithm as uncertainty), this parameter has been considered as an additional uncertainty. To this end, just for the scope of visualisation, the results obtained in the chosen AoA grid points have been considered valid also in an interval of +1˚ and –1˚ around the evaluated AoA values. This allows to show, in these figures, the bigger 2-D projections (so called worst areas ) of the non-trimmable areas (white regions) and notcleared or undefined areas (dark and light grey regions, respectively) on the plane formed by each uncertain parameter and AoA. The cleared regions (grey) are obtained as the remaining areas.

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35 Not part of the "true" flight envelope

30 Not cleared region

25 20

α [deg]

15 10

Cleared region

5 0 −5 −10 −15

Not part of the "true" flight envelope

−0.04

−0.03

−0.02

−0.01

0 Cm unc

0.01

0.02

0.03

0.04

α

Fig. 19.14. Worst areas projection on (Cmα unc , AoA) for longitudinal analysis in FC6

With this kind of visualisation, if one chooses, for example, a dark grey, white or light grey point (respectively not cleared, not trimmable and undefined) on a figure, say x∗CGunc and AoA∗ for a fixed flight condition, it can be guaranteed that at least one combination of the other uncertain parameters exists that does not fulfil the eigenvalue criterion (not cleared point) or the aircraft is not trimmable (not trimmable point) or the algorithm has not examined that point because the maximum resolution has been reached (undefined point). Instead, if one chooses a grey point (cleared point) on a figure, say x0CGunc and AoA’ for a fixed flight condition, it can be guaranteed that the eigenvalue criterion is fulfilled for each possible combination of the other uncertain parameters, within their own uncertain intervals. By comparing the Figs. 19.12, 19.13, 19.14 and 19.15 with homologous figures of the previous section, it can be noted that the simultaneous variation of the uncertain parameters does not generate not cleared regions that are much wider from the ones generated in the analysis of section 19.3 (except for the different resolutions used). Thus, at least for these parameter sets and for the examined eigenvalue criterion, simultaneous variation of more than one parameter does not imply a noticeable increase in not cleared regions. Also, due to the use of this specific visualisation algorithm (with reference to the AoA treatment, see discussion above), of a fixed grid for AoA and

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30 Not cleared region

25 20

α [deg]

15 10

Cleared region

5 0 −5 −10 Not part of the "true" flight envelope

−15 −0.1

−0.08

−0.06

−0.04

−0.02

0 yCGunc

0.02

0.04

0.06

0.08

0.1

Fig. 19.15. Worst areas projection on (yCGunc , AoA) for complete longitudinal analysis in FC6

of the different resolutions employed, some discrepancies could be identified between homologous figures. By using these kind of plots it is possible to summarise the overall clearance results for eigenvalue criterion obtained with the proposed method. These results are shown in Fig. 19.16 for longitudinal analysis and in Fig. 19.17 for lateral analysis. Shown in these figures are the AoA ranges where the eigenvalue criterion is not satisfied (not cleared points). It can be noted that these results are substantially in agreement with the baseline results Chapter 15. In particular, no points are missing, while some additional not cleared points have been revealed which are mainly not trimmable. Also in this case it is important to consider the analysis computational effort and to compare it with conventional gridding methods and the min/max approach used for the baseline results Chapter 15. The full longitudinal axis analysis (5 uncertain parameters varied simultaneously over the whole AoA range and 8 flight conditions) requested evaluation of about 70000 trimming and linearization points for a total of about 24 hours continuous computation. The full lateral axis analysis (the same as before but with 6 uncertain parameters) required about 120000 trimming and linearizations for a total of about 54hours computation time. These timing figures state the high computational burden required by this method in order to perform analysis with

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4

5

x 10

4.5 4

FC3

FC8

α ∈ [ -15, 33]

α ∈ [ -15, 33]

3.5 FC5

Altitude (ft)

3

α ∈ [ -15, 33] FC2

2.5

α ∈ [ -13,33] FC6

2

α ∈ [ -7,23] FC4

1.5

α ∈ [ -13,27]

1

0

FC7

FC1

0.5

α ∈ [ -5, 33] 0

0.1

0.2

0.3

α ∈ [ -1,11] 0.4

0.5

0.6

0.7

0.8

0.9

1

Mach Number

Fig. 19.16. Longitudinal parameter clearance results for the eigenvalue criterion 4

5

x 10

4.5 4

FC3

FC8

α ∈ [ -15, 33]

α ∈ [ -15, 31]

3.5 FC5

Altitude (ft)

3

α ∈ [ -15, 31] FC2

2.5

α ∈ [ -11, 33] FC6

2

α ∈ [ -7, 23]

FC4

1.5

α ∈ [ -13, 27]

1 0.5 0

0

0.1

0.2

FC1

FC7

α ∈ [ -5, 33]

α ∈ [ -1, 11]

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mach Number

Fig. 19.17. Lateral parameter clearance results for the eigenvalue criterion

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more than three parameters varying simultaneously. Specifically, it takes more than nine times that required by the baseline solution Chapter 15, where only all combinations of min/max uncertainty values are evaluated. Indeed, if we compare these timing results with a conventional gridding method used to obtain the same resolution, this last method shows even worst results. By choosing a fixed grid size to obtain the same resolution, the total evaluated trim points are 625000 for longitudinal and 3125000 for lateral. This leads to a computational time (with a PIII 500Mhz machine) of more than seven days for longitudinal analysis and more than 36 days computation time (accounting only for trim and linearization computations).

19.5 Conclusions In this paper we illustrated the clearance results for the eigenvalue criterion obtained with the polynomial based approach proposed in Chapter 5. This method basically determines the region(s) in the uncertainty space where all eigenvalues of a (linear) uncertain dynamic system belong to a pre-defined domain D in the complex plane. This technique can be used for analysing all clearance criteria which can be mapped in the locations of a system’s eigenvalue. Also, any kind of dependence between the characteristic polynomial coefficient and the uncertain parameters can be taken into account. A key advantage with respect to classical methods, where analysis is conducted on simple discrete points, is that this technique allows the shape of cleared, not-cleared and trimmable regions in the uncertain parameter space to be determined. Because no assumption has been undertaken when choosing the kind of uncertain parameter to be investigated, this technique also allows the flight envelope to be investigated continuously to determine the cleared regions in order to discover hidden weaknesses in this envelope and/or to gather information about further analysis to be conducted. On the other hand, even if theoretical results guarantee that a minimum number of evaluations (i.e. computation of the characteristic polynomial coefficients) will be performed, this is still high if compared to the number of evaluations needed in the clearance methods based on a min/max approach (see Chapter 15 for details). However, the results demonstrate that the proposed method dramatically reduces (up to ten times) the required computational burden, if a fixed grid size is used to get the same accuracy. Some further developments could also improve the results and applicability of this technique. For example, the adaptive grid generation algorithm can be improved to identify the parameters that are mostly not linear (this will give the possibility to reduce complexity to ∼ 2l where l is less than the number of uncertain parameters). Presently, the algorithm decreases the grid size where the function is not linear by decreasing the grid for all uncertain parameters, whether or not they are the actual cause for the non linearity. More sophisticated algorithms could decrease the grid size only for some of

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the uncertain parameters thus reducing dramatically the number of points (i.e. trimming and linearization) to be evaluated. Also, some techniques can be investigated for extending applicability to other linear clearance criteria (like the stability margin criterion) which do not map directly in a condition on the location of the eigenvalues. For example, it is possible to design an appropriate parametric system that in cascade with the open loop aircraft model can artificially cause a closed loop instability. In this way, if the parameters of this additional system can be linked to the gain and phase margins, it is possible to map the stability margin criterion to a problem of robust stability which can be analysed with the proposed method. Extending the applicability of this method to other criteria gives, in principle, the opportunity to perform the analysis by using the same trim and linearization points evaluated by the adaptive grid generation algorithm (i.e. the adaptive grid is generated only once), thus leading to a computational effort only slightly higher than required for analysis of only one criterion. Finally, the use of the proposed polynomial based method is rather straightforward and no particular skills are required for performing an analysis. Furthermore the results can be very easily interpreted: one can directly look at the regions in the uncertain parameter space where a clearance criteria is fulfilled. However, this image can be very difficult to plot and interpret when the number of uncertain parameters under consideration is more than three.

References 1. Cavallo A. De Maria G. Verde L., Robust Flight Control Systems: a Parameter Space Design. In Journal of Guidance, Control and Dynamics, Vol.15, n.5, pp. 1207-1215, 1992. 2. Amato F. Canzolino P. Verde L., A Software Tool for Robustness Analysis in Plant Parameter Space (ROBAN). In Proc. IEEE CACSD Conference, Alaska, USA, 2000.

20 Bifurcation-Based Clearance of Linear Stability Criteria Mark Lowenberg and Thomas Richardson Department of Aerospace Engineering University of Bristol, Queens Building University Walk, Bristol, BS8 1TR, UK. [email protected]

Summary. The background to the bifurcation and continuation method was given in Chapter 6, together with the analysis cycle adopted in the GARTEUR FM(AG11) project. In this chapter, the application of the methodology to the HIRM+RIDE clearance task is described. Sample output produced during the implementation is shown and the overall clearance results for the stability margin and unstable eigenvalues criteria are presented.

20.1 Introduction The clearance analysis of the HIRM+RIDE model was performed using the bifurcation and continuation technique, following the methodology outlined in Chapter 6. This approach includes the generation of a path of trim points across the incidence range at each flight condition (FC). Thus it was deemed to offer most benefit when applied to linear criteria (nonlinear criteria are initiated only at the 1-g trim point). The full set of results for all eight FCs, using category 1 uncertainties and the single loop Nichols stability margin and unstable eigenvalues criteria, is given in [2]. This chapter illustrates the analysis cycle as applied to HIRM+RIDE, the objective being to demonstrate the advantages offered by the approach. This is achieved by discussing in detail the results for one FC. Thereafter, the cleared angle of attack (α) ranges are given for all the FCs and these may be compared with results produced by other analysis techniques. An indication of set-up and computation times is given, after which the main features of the method are summarised and recommendations made for further studies.

20.2 HIRM+RIDE Model Implementation The model was supplied to the Action Group in MATLAB/Simulink form, and is described in Chapters 8 and 9. It consists of the 52-state HIRM+ openloop model (flight dynamics, actuators, engine dynamics and sensors) and 9state RIDE controller – making a total of 61 states. - For implementation in the C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 355-383, 2002. Springer-Verlag Berlin Heidelberg 2002

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continuation method using the Fortran-coded PCS program, an interface to the model was written, allowing the right-hand side of the system equations (6.1) to be passed between the two. Thus the method uses the nonlinear equations as provided and not re-coded or altered in any way. 20.2.1 Quasi-Trim Solutions The linear clearance criteria are evaluated at trim points over the specified angle of attack range. Since only one pitch axis control effector is used (symmetric tailplane), there is only one true trim point – i.e. pitch rate q = 0 – for each flight condition. The so-called trim points at which the model is linearised over the α range are in fact “quasi-trims” corresponding to the bottom of a symmetric pull-up (or top of a symmetric push-over for negative q). All these points, other than where q = 0, describe accelerated flight (normal acceleration nz 6= 1 g). The HIRM+RIDE software provided to the Action Group included a trimming routine called trimhirmplus. The option trimopt=3 provides the symmetric pull-up/push-over trim case with throttle fixed at 1 (47 kN sea level thrust per engine) and hence a non-zero rate of change of total velocity, V˙ T 6= 0. Also, by definition, at the bottom of a symmetric pull-up or top of symmetric push-over, θ˙ 6= 0; θ = α, θ˙ = q and γ = 0 (where θ is pitch angle and γ is flight path angle). For implementation in the bifurcation/continuation code, it was necessary to choose the order and setup of the model used for finding quasi-trim points such that it corresponded to these conditions. The θ˙ and V˙ T equations were omitted, VT was fixed according to the flight condition Mach number and altitude, and θ was set equal to α. This reduced the order of the model by two. A further eight equations – associated with variables that play no role in the system dynamics – could also be omitted: – the x and y earth axes positions of the aircraft and the heading angle, ψ, have no influence on the rest of the model; this also allows the heading angle sensor states to be omitted (ψsensor and ψ˙ sensor ); – to perform runs that correspond to the given flight conditions in terms of defined altitude and Mach number, the z state (vertical earth axis position) was fixed at the appropriate value for each FC; – the nose suction is not utilised, so this state could be set to zero; and – there was one unused RIDE state (zero gain in the model). The continuation method implementation used to generate results for quasi-trim solutions therefore used a model reduced in order by 10 (51 st order)1 . 1

The order could actually have been further reduced by 15 when solving for equilibria: as with the nose suction, the canards are not used and their four actuator

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˙ For true 1-g trims, the order of the equations would be higher, as the θ, V˙ T and z˙ equations would be retained. The GARTEUR specifications for quasi-trim points required the RIDE controller command path to be ignored. In essence, this refers to the αlimiter, which was therefore omitted in the continuation method quasi-trim solutions. 20.2.2 Linearisation Having effectively found a trim condition as described above for each solution point in the continuation process, the linear clearance criteria must be evaluated with respect to the full unconstrained model: i.e. including the command path and without applying pull-up/push-over conditions. The continuation method program was therefore implemented in a dual approach: using the constrained model to solve for quasi-trim points, and the 61st -order unconstrained model with command path for linearisation. The PCS software accommodates this dual-model form of operation 2 . When setting up the bifurcation/continuation analysis care must be taken to ensure that linearisations match those conducted in the baseline clearance. In the latter, the HIRM+ model is linearised first, followed by the RIDE controller; then the two state-space forms are combined into a closed-loop linearised form. The operating point about which the linearisation is taken comes from the inverse trimming method, which sets the “reference actuator demand” inputs to the values required for trim, while the “controller actuator demand” inputs for HIRM+ are zero (HIRM+ control surface actuator inputs are the sum of the reference values and the controller demand values). This is a logical approach, given that each trim point is calculated independently in the baseline clearance. In the continuation method, however, it is possible to trim and linearise the entire aircraft-plus-controller model at each solution point, such that the controller generates non-zero actuator demands. This can result in different linearisations. Another issue is that when RIDE is linearised by trimhirmplus, the “measured” states are available to it but the demands are all zero (except for the speed demand). Thus it receives the value of q for the quasi-trim point but always has a zero value for qdem . In the bifurcation analysis application, it is possible to linearise with q = qdem – but this elicits different results compared to the baseline clearance.

2

states could have been omitted. Also, with constant thrust, the four thrust ac˙ nx tuator/engine states and seven sensor states (for VT , Mach number, h, θ, θ, and n˙ x ) could have been left out of the system. So equilibria could have been sought using a 36th -order system. In practice, where one model is simply a subset of the other, the full model is called each time but only the relevant part is used for quasi-trim solutions. Conversely, more than two models can be used, e.g. if this is useful when checking several criteria during a run.

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Therefore, in order to allow comparison of results with those of the baseline, the same trimming and linearising procedure has been followed.

20.3 Clearance Criteria Analysed The criteria evaluated for each of the eight flight conditions were: – single loop Nichols stability margin criterion for the three actuator loops: symmetric tailplane (δT S ) loop broken, differential tailplane (δT D ) loop broken, and rudder (δR ) loop broken; – unstable eigenvalues criterion. The multiple loop Nichols stability criterion and the average phase rate and absolute amplitude criteria could have been implemented in essentially the same manner as the above: they are also linear criteria and the bifurcation/continuation approach applies the clearance tests in the standard way. In principle, nonlinear criteria can also be incorporated into the methodology but, as pointed out in Chapter 6, it offers fewer benefits than in the case of linear criteria. The step/ramp largest exceedence criteria are time history-based, starting always at the q = 0 trim flight condition. The advantage of obtaining a steady-state path through an incidence range is therefore not applicable here. However, the bifurcation-based method could be used to identify worst case uncertainties, via ita “nonlinear sensitivities” step. In order to develop and assess the analysis technique within the time available, only category 1 uncertainty parameters were applied. Thus, following the rules stipulated for the FM(AG11) HIRM+RIDE clearance task [1], the following five cases were considered when determining worst-case uncertainty parameter combinations: (i) (ii) (iii) (iv) (v)

stability margin, δT S loop: stability margin, δT D loop: stability margin, δR loop: unstable eigenvalues: unstable eigenvalues:

longitudinal uncertainties (Long Cat 1) lateral uncertainties (Lat Cat 1) lateral uncertainties (Lat Cat 1) longitudinal uncertainties (Long Cat 1) lateral uncertainties (Lat Cat 1)

where “Long Cat 1” are the category 1 longitudinal uncertainty parameters and “Lat Cat 1” are the category 1 lateral uncertainty parameters. In the case of (iv) and (v) the worst-case unstable eigenvalue can be real or complex. Where one of each fails the criterion, only the one that fails at the lower α is investigated. It was found in all HIRM+RIDE unstable eigenvalues criterion cases, that violations arose as the critical complex eigenvalue moved approximately “horizontally” (roughly constant imaginary part) to the right, across the

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criterion boundary on the imaginary axis. It was therefore possible to use the maximum real part of the critical eigenvalue when analysing a violation in respect of complex eigenvalues. A more general measure could be applied if necessary.

20.4 Presentation of Results Where bifurcation diagrams are presented below, the x-axis is always the continuation parameter and the y-axis in each plot is a state variable. Conventional bifurcation diagrams use line-type to indicate local stability along the solution paths; the boundary between stable and unstable eigenvalues is the imaginary axis. For the HIRM+RIDE clearance, however, it is convenient to show the worst case unstable eigenvalues criterion. This defines a boundary of acceptable eigenvalues that permits some incursion into the right-half plane (based on acceptable time to double amplitude) — see Chapter 10. The modified line-type protocol adopted is: – solid line: all eigenvalues acceptable in terms of the criterion (to the “left” of the boundary); – dashed line: one or more real eigenvalues violates the criterion; – dotted line: one or more complex eigenvalues violates the criterion. In this section, the analysis cycle is illustrated using sample results from the clearance of FC6 (the full results can be found in [2]). The plots presented are: Nominal case: Bifurcation diagrams for the following six state variables, each plotted against the continuation parameter, qdem : α, β, p, q, nz and δR (angles of attack and sideslip, roll and pitch rates, normal load factor and rudder deflection). These have been selected from the 51 states in the quasi-trim system to reveal a reasonable quantity of information about the physics of the solutions. Limits of α, qdem and nz are indicated on the diagrams, as is the δT S range (to allow tailplane surface saturation to be observed, in cases where it occurs); if the rudder saturates, this can be seen on the δR diagram. Maximum real eigenvalue versus α (corresponding to the bifurcation diagram solution path). Limits such as control surface saturation can be indicated on these diagrams, and any crossing of the critical value – signifying non-compliance by the nominal system of the criterion – is shown. Complex eigenvalues. It was decided that the clearest way of showing the variation of complex pairs relative to the criterion – which vary according to imaginary part value – is on a root locus plot (corresponding to the bifur-

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cation diagram solution path)3 . Two plots are given, the lower one being a magnification of the upper one near the criterion boundary. All eigenvalues are shown (within the chosen axes limits), not just the worst-case complex pair. The boundary is indicated on the plots, as well as the α value at or just after violation. Nichols stability margin versus α (corresponding to the bifurcation diagram solution path). Each figure contains three plots, one for each of the broken feedback loops. The critical stability limit is indicated. At points where this is transgressed, the following information is noted: upper (U) or lower (L) Nichols plot region (upper meaning gain > 0 dB; lower means gain < 0 dB), frequency in Hz; and highest achievable α prior to violation. Uncertainty cases: “Nonlinear sensitivity” plots are shown for all critical points for the stability margin and unstable eigenvalues criteria. A critical point here means a point at which the nominal case violates or nearly violates a particular criterion; the sensitivity is with respect to that criterion, at a constant α corresponding to the critical point. Local bifurcation diagrams are presented for each of the above cases, to indicate how the worst-case uncertainty combination reduces the cleared envelope for the criterion applicable to each critical point. At the end of the chapter, a summary of clearance results for each criterion for all eight FCs is shown in a series of flight envelope diagrams.

20.5 FC6 Clearance Results Recall that in all the results: – the bifurcation diagram solution paths are from the 51 st -order quasi-trim model; – bifurcation diagram line types represent the worst-case eigenvalue criterion; – linearisations and clearance criteria are for the full 61 st -order model. 20.5.1 Nominal Results Fig. 20.1 contains bifurcation diagrams of selected states for the nominal case. Fig. 20.2 shows the worst-case real eigenvalue over the α range. Fig. 20.3 is a root locus (the lower plot is a magnification of the upper one) to reveal worst-case complex eigenvalues. 3

Root locus refers here to loci of roots as the continuation parameter varies (rather than a gain varying).

20 Bifurcation-Based Clearance of Linear Stability Criteria

361

Table 20.1. FC6 cleared α ranges – nominal case. Envelope limits4 nz ÿ

−9◦ , 30◦

þ

Clearance criteria

δT

δR

δT S loop

δT D loop

–

–

ÿ þ −15◦ , 31.9◦

–

U 1.31 Hz

δR loop

unstable e’val.

ÿ þ ÿ þ −15◦ , 26.0◦ −15◦ , 25.0◦

U 0.44 Hz

(real) þ ÿ −15◦ , 31.4◦ (complex)

Fig. 20.4 shows the Nichols exclusion region stability index over the α range for each of the three broken loops. We observe from the bifurcation diagram (Fig. 20.1) that only one solution branch is shown, namely that of conventional flight. It is likely that, even with the controller active, there are other branches – possibly spin modes, probably indicative of one or more aspects of the closed-loop system saturating. Such additional branches have not been sought; if they were, they could be helpful in understanding results from nonlinear simulations where the α-limiter fails and the system appears to depart. Fig. 20.1 does reveal that at the higher incidences the controller fails to maintain fully symmetric flight: β and p become non-zero. The δR plot shows the deflections commanded by the RIDE controller to try to maintain symmetry. This feature of the bifurcation diagram is advantageous, showing what might be regarded as an inadequacy in the controller. However, the degree of asymmetry is fairly small. (Strictly, the implementation of the quasitrim solutions in the continuation method should have been for exactly β = p = 0, so as to match the baseline clearance.) From the nominal results we observe that the following criteria are violated: δT S loop stability margin, δR loop stability margin and unstable eigenvalues. Also, nz limits are reached within the α range. The δT D loop stability margin criterion does not fail but reaches its worst-case value (closest to critical) at α = 35◦ . The clearance details are summarised in Table 20.1. The values indicated within square brackets in Table 20.1 are the cleared ranges of α for each envelope limitation or clearance criterion. If no such values are shown, then there is no violation of the criterion within the specified 4

qdem also has limits but these are functions of the command path gearing and are not considered to be operating boundaries. The qdem limits were not encountered for FC6; in cases where they were, these were noted but not used to constrain the α range at which clearance analysis was performed.

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M. Lowenberg and T. Richardson

FC6 − Nominal 30

FC6 − Nominal

5

α limit

10

Fails (real eigenvalue)

β (deg)

α (deg)

20 0

0 −10 −10

0

10

−5

20

5

−10

0

10

20

−10

0

10

20

−10

0 q

10 (deg/s)

20

20 15 10 Q (deg/s)

P (deg/s)

qdem range 0

5 0 −5 −10 −15

−5

−10

0

10

−20

20

10

30 20 max

ANZ (g)

Rudder (deg)

5

0 min −5

−10

10 0 −10 −20

0 q

dem

10 (deg/s)

20

−30

dem

Fig. 20.1. FC6 bifurcation diagram – nominal case.

20 Bifurcation-Based Clearance of Linear Stability Criteria FC6 − Nominal 0.14

0.12

Maximum real eigenvalue

0.1

0.08 25.0 degrees α 0.06

0.04

0.02

0

−0.02 −15

−10

−5

0

5

10 α (deg)

15

20

25

30

35

Fig. 20.2. FC6 worst-case real eigenvalue – nominal case. FC6 Eigenvalues

Imaginary axis

5

31.4 degrees α 0

−5 −2

−1.8

−1.6

−1.4

−1.2

−1 −0.8 Real axis

−0.6

−0.4

−0.2

0

0.2

Imaginary axis

0.4 0.2 0 25.0 degrees α

−0.2 −0.4 −0.05

0

0.05 Real axis

0.1

Fig. 20.3. FC6 eigenvalues – nominal case.

0.15

363

364

M. Lowenberg and T. Richardson FC6 Nominal − Symmetric Tailplane (c1−symm) Stability Margin

1.5 1

Cleared Not Cleared

0 −15

Stability Margin

U 1.31Hz α cleared = 31.9 deg

0.5

−10

−5

0

−5

0

5 10 15 Differential Tailplane (c1−diff)

20

25

30

35

20

25

30

35

1.5 1

Cleared Not Cleared

0.5 0 −15

−10

5

10 15 Rudder (c1−rudd)

Stability Margin

1.5 1

Cleared Not Cleared

U 0.44Hz α cleared = 26.0 deg

0.5 0 −15

−10

−5

0

5

10 α (deg)

15

20

25

30

35

Fig. 20.4. FC6 single loop Nichols stability margin criterion – nominal case.

operating envelope of −15◦ ≤ α ≤ 35◦ . A boxed value of α represents an envelope limit (i.e. load factor bound or control surface saturation, in which case the system remains uncleared beyond this value) or a violation of a clearance criterion – which then requires further investigation in terms of worst-case uncertainty parameters. The nz limit is an operating constraint so that any violation that occurs outside the −9◦ ≤ α ≤ 30◦ range does not need to be considered further (in the context of worst-case uncertainties). This should apply to δT S loop stability margin in the case of FC6 but, since its violation point is not far from the nz limit, this criterion is investigated (just in case the application of worst-case uncertainties induces a transgression at an α that is lower than the nz limit). The δT D case is also studied, for the same reason. The stability margin criteria include information concerning the point on the frequency response at which non-compliance occurs. For example, the notation: i h −15◦ , 31.9◦ U 1.31 Hz

20 Bifurcation-Based Clearance of Linear Stability Criteria

365

used in Table 20.1 means that the violation at α = 31.9◦ arose due to the frequency response curve crossing the Nichols exclusion zone in the upper half of the Nichols diagram (gain>0dB) at a frequency of 1.31 Hz. An “L” instead of a “U” would represent a violation in the lower half of the Nichols plot (gain<0dB). In the case of the unstable eigenvalue criterion, an indication is given as to whether the allowable eigenvalue boundary is crossed by a real or a complex eigenvalue. From Table 20.1 the critical points for evaluation of criteria with uncertainties included are: stability margin, δT S loop: stability margin, δT D loop: stability margin, δR loop: unstable eigenvalues:

α = 31.9◦ α = 35.0◦ α = 26.0◦ α = 25.0◦

(no nominal violation)

20.5.2 Results with Worst-Case Uncertainties Stability margin criterion, δT S loop The nonlinear sensitivities to variations in normalised longitudinal uncertainties are shown in Fig. 20.5 (normalised uncertainties are: 0 for nominal case, −1 for minimum value and +1 for maximum value). The trends for the Iy , Cmα , CmδT S and Cmq uncertainties are smooth and near-linear. Also, the impact of Cmα and Cmq are significantly smaller than for the other two. Using the same scale for all the five sensitivity plots would make this clear; the reason this is not done is to make it easier to pick off numerical values for the aerodynamic uncertainties, to calculate the effects of applying reduction factors. It is evident that the Nichols margin criterion is most sensitive to the Xcg uncertainty, and that there is a “jump” in its influence at a normalised value of about 0.25. This indicates that application of this uncertainty alone will shift the criterion violation point. From Fig. 20.5 the worst-case normalised value for each uncertainty is deduced. These are shown in Table 20.2, where the boxed entries denote those chosen as the worst-case combination (accounting for aerodynamic uncertainty reduction factors). Note that the worst-case normalised value for Xcg U nc is given as +1, although in the sensitivity diagram the worst case appears at around +0.3. It was decided that, in order to make the clearance analysis results comparable with the baseline and other methods, worst-case uncertainties of either −1 or +1 would be used; the difference here is small. An advantage of the bifurcation/continuation method is that such intermediate worst-case values are identified and can be used – but the GARTEUR specifications required only the extreme values to be considered.

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M. Lowenberg and T. Richardson

0.04

0.1

0.03

0

0.02

∆ Nichols Margin

∆ Nichols Margin

FC6 (c1symm) − XCG

−0.1 −0.2 −0.3 −0.4 −0.5

0.01 0 −0.01 −0.02 −0.03

−1

−0.5 −3

4

FC6 (c1symm) − Iy

x 10

0

0.5

−0.04 −1

1

FC6 (c1symm) − Cmα

0.05

−0.5

0

0.5

1

FC6 (c1symm) − CmδTS

2

∆ Nichols Margin

∆ Nichols Margin

3

1 0 −1 −2

0

−3 −4 −1

−0.5 −3

1.5

x 10

0

0.5

1

−0.05 −1

−0.5 0 0.5 Normalised uncertainty (−1/+1)

1

FC6 (c1symm) − C

mq

∆ Nichols Margin

1 0.5 0 −0.5 −1 −1.5 −1

−0.5 0 0.5 Normalised uncertainty (−1/+1)

1

Fig. 20.5. FC6 nonlinear sensitivity computed at nominal critical point: symmetric tailplane loop stability margin criterion (α = 31.9◦ ).

20 Bifurcation-Based Clearance of Linear Stability Criteria

367

Table 20.2. FC6 worst-case uncertainty combinations – stability margin, δT S loop. Uncertainties (Long. Cat. 1) Xcg U nc

Iy U nc

Cmα U nc

CmδT S U nc

Cmq U nc

+1

–1

–1

–1

+1

The result of the local bifurcation run used to locate the critical point in the presence of this worst-case uncertainty parameter combination is shown in Fig. 20.6 (upper plot). This shows that the violation now occurs at a slightly lower value of α and gives the following cleared region: FC6, stability margin criterion, δT S loop: [ −15◦ , 31.8◦ ] Ua1 U 1.49 where the “Ua1” means that the worst-case uncertainty combination includes one aerodynamic uncertainty parameter (“Ua2” means two aerodynamic uncertainties, etc.). Note that the point at which the criterion is violated lies outside the flight envelope defined by nz limits. Stability margin criterion, δT D loop The nonlinear sensitivities to lateral uncertainties are shown in Fig. 20.7. Note that the “steps” in the Cnr uncertainty effect are numerical in nature, its influence being two orders of magnitude less than that of the next smallest uncertainty sensitivity. The worst-case normalised value for each uncertainty is deduced from Fig. 20.7 and shown in Table 20.3. Boxed entries are those chosen as the worst-case combination (accounting for aerodynamic uncertainty reduction factors). Table 20.3. FC6 worst-case uncertainty combinations – stability margin, δT D loop. Uncertainties (Lat. Cat. 1) Ycg U nc

Ix U nc

Iz U nc

Clβ U nc

Cnβ U nc

Cnr U nc

–1

–1

–1

–1

–1

+1

368

M. Lowenberg and T. Richardson

FC6 Worst case − Symmetric Tailplane (c1−symm)

Stability Margin

1.5

1

0.5

0 30

30.5

31

31.5

32.5

33

33.5

34

34.5

35

Differential Tailplane (c1−diff)

1.5

Stability Margin

32

1

0.5

0 28

29

30

31

32

33

34

35

Rudder (c1−rudd) 1.6

Stability Margin

1.4 1.2 1 0.8 0.6 0.4 0.2 0 20

21

22

23

24

25 α (deg)

26

27

28

29

30

Fig. 20.6. FC6 local bifurcation runs, stability margin criteria: solid line: nominal case; dot-dashed line: worst case.

20 Bifurcation-Based Clearance of Linear Stability Criteria

−4

6

x 10

FC6 (c1diff) − YCG

∆ Nichols Margin

∆ Nichols Margin

0.1

2 0 −2 −4

−8 −1

−0.5 0 0.5 Normalised uncertainty (−1/+1) −3

x 10

1

lβ

0.03 0.02

2 0 −2 −4

0.01 0 −0.01 −0.02

−6 −8 −1

−0.5 0 0.5 Normalised uncertainty (−1/+1) −3

x 10

1

−0.03 −1

FC6 (c1diff) − Cnβ

x 10

1

FC6 (c1diff) − C

nr

∆ Nichols Margin

0

1 0 −1

−5 −10 −15

−2 −3 −1

−0.5 0 0.5 Normalised uncertainty (−1/+1) −5

5

2 ∆ Nichols Margin

−0.5 0 0.5 Normalised uncertainty (−1/+1) FC6 (c1diff) − C

FC6 (c1diff) − Iz

∆ Nichols Margin

∆ Nichols Margin

−0.1

−0.3 −1

1

4

3

0

−0.2

−6

6

FC6 (c1diff) − Ix

0.2

4

369

−0.5 0 0.5 Normalised uncertainty (−1/+1)

1

−20 −1

−0.5 0 0.5 Normalised uncertainty (−1/+1)

1

Fig. 20.7. FC6 nonlinear sensitivity computed at nominal worst-case point: differential tailplane loop stability margin criterion (α = 35◦ ).

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M. Lowenberg and T. Richardson

The result of the local bifurcation run used to locate the critical point in the presence of this worst-case uncertainty parameter combination is shown in Fig. 20.6 (middle plot). We note that, whereas the nominal system did not violate this criterion within the specified α range, the worst-case situation does. The cleared region is thus: FC6, stability margin criterion, δT D loop: [ −15◦ , 33.3◦ ] Ua1 L 2.80 Note that the point at which this criterion is violated also lies outside the flight envelope defined by nz limits. Stability margin criterion, δR loop The nonlinear sensitivities to lateral uncertainties are shown in Fig. 20.8, from which the worst-case normalised value for each uncertainty is deduced (the same comment applies to the Cnr uncertainty as in Fig. 20.7). These are shown in Table 20.4, where the boxed entries denote those chosen as the worst-case combination (which incorporates two aerodynamic uncertainties in this case – Clβ U nc and Cnβ U nc – and therefore a reduction factor of 0.62). Note that the worst-case normalised value for Cnβ U nc is given as −1, although in the sensitivity diagram the worst case appears at approx. −0.8. As in the case of the δT S loop, it was decided to allocate a value of either −1 or +1 to the uncertainties (for comparison with other methods); again, the difference is not large. Table 20.4. FC6 worst-case uncertainty combinations – stability margin, δR loop. Uncertainties (Lat. Cat. 1) Ycg U nc

Ix U nc

Iz U nc

Clβ U nc

Cnβ U nc

Cnr U nc

+1

+1

+1

+1

–1

–1

The result of the local bifurcation run used to locate the critical point in the presence of the worst-case uncertainty parameters is shown in Fig. 20.6 (lower plot). This gives the following cleared region: FC6, stability margin criterion, δR loop: [ −15◦ , 25.0◦ ] Ua2 U 0.43

20 Bifurcation-Based Clearance of Linear Stability Criteria

−3

3

x 10

FC6 (c1rudd) − YCG

0.1

371

FC6 (c1rudd) − Ix

0.05

1

∆ Nichols Margin

∆ Nichols Margin

2

0 −1 −2

0

−0.05

−3 −4 −1

0.1

−0.5 0 0.5 Normalised uncertainty (−1/+1)

−0.1 −1

1

FC6 (c1rudd) − C

FC6 (c1rudd) − Iz

0.2

∆ Nichols Margin

∆ Nichols Margin

−0.1 −0.2 −0.3

−0.5 0 0.5 Normalised uncertainty (−1/+1) nβ

−0.4 −0.6 −0.8

0.01

−0.5 0 0.5 Normalised uncertainty (−1/+1)

1

FC6 (c1rudd) − Cnr

0.005 ∆ Nichols Margin

∆ Nichols Margin

−0.2

−1.2 −1

1

0

−0.5

−1

−1.5 −1

lβ

−1

FC6 (c1rudd) − C 0.5

1

0

0

−0.4 −1

−0.5 0 0.5 Normalised uncertainty (−1/+1)

0

−0.005

−0.5 0 0.5 Normalised uncertainty (−1/+1)

1

−0.01 −1

−0.5 0 0.5 Normalised uncertainty (−1/+1)

1

Fig. 20.8. FC6 nonlinear sensitivity computed at nominal critical point: rudder loop stability margin criterion (α = 26.0◦ ).

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M. Lowenberg and T. Richardson

Unstable Eigenvalues Criterion The nonlinear sensitivities to longitudinal and lateral uncertainties are shown in Fig. 20.9 and Fig. 20.10, from which the worst-case normalised value for each uncertainty is deduced. These are shown in Table 20.5, where the boxed entries denote those chosen as the worst-case combination. Note that the longitudinal worst-case combination accounts for the reduction factor of 0.62 for the two aerodynamic uncertainties (Cmα U nc and CmδT S U nc ). Table 20.5. FC6 worst-case uncertainty combinations – unstable eigenvalues criterion. Long. Cat. 1

Lat. Cat. 1

Xcg U nc

Iy U nc

Cmα U nc

CmδT S U nc

Cmq U nc

–1

+1

+1

–1

+1

Ycg U nc

Ix U nc

Iz U nc

Clβ U nc

Cnβ U nc

Cnr U nc

–1

+1

–1

+1

–1

–1

The result of the local bifurcation run used to locate the critical point in the presence of the worst-case uncertainty parameters is shown in Fig. 20.11. This gives the following cleared region: FC6, unstable eigenvalues, longitudinal uncertainties: [ −15◦ , 24.2◦ ] Ua2 FC6, unstable eigenvalues, lateral uncertainties: [ −15◦ , 24.7◦ ] Ua1 Note from Fig. 20.11 that the real eigenvalue is influenced more strongly by the longitudinal uncertainties than the lateral uncertainties. The lateral case in Fig. 20.11 (lower graph) also shows a comparison between the “actual” worst-case combination (accounting for aerodynamic uncertainty reduction factors, used in the continuation/bifurcation approach) and the “worst-case of five” adopted in the baseline clearance study. It is clear that in this case the use of the one aerodynamic uncertainty parameter is worse than using all the parameters. The above results agree with those of the baseline clearance, and this was essentially the case for all the FCs (bearing in mind that the baseline clearance considered 2◦ intervals in α). To some extent, this is expected because

20 Bifurcation-Based Clearance of Linear Stability Criteria

FC6 − XCG

−3

x 10

∆ Maximum Eigenvalue

∆ Maximum Eigenvalue

6 4 2 0 −2 −4

−0.5

0.5 0 −0.5 −1

0.5

1

−1.5 −1

FC6 − Cmα

−8

4

0

x 10

∆ Maximum Eigenvalue

∆ Maximum Eigenvalue

2 1 0 −1 −2 −3 −0.5 −9

∆ Maximum Eigenvalue

x 10

0

−0.5 −8

3

3

−4 −1

x 10

1

−6 −1

FC6 − Iy

−5

1.5

0.5

1

373

x 10

0

0.5

1

FC6 − CmδTS

2 1 0 −1 −2 −1

−0.5 0 0.5 Normalised uncertainty (−1/+1)

1

FC6 − C

mq

1

0

−1 −1

−0.5 0 0.5 Normalised uncertainty (−1/+1)

1

Fig. 20.9. FC6 nonlinear sensitivity computed at nominal critical point: unstable eigenvalues criterion, longitudinal uncertainties (α = 25.0◦ ).

the application of clearance criteria mirrors that of the baseline clearance. However, the selection of worst case uncertainty combinations is certainly performed differently, utilising the nonlinear sensitivity process described in Chapter 6. As implemented, it does not guarantee the true worst cases – but it did succeed in finding them for the HIRM+RIDE analyses (in some

374

M. Lowenberg and T. Richardson

FC6 − YCG

−5

3

x 10

∆ Maximum Eigenvalue

∆ Maximum Eigenvalue

x 10

1.5

2 1 0 −1 −2 −3

1 0.5 0 −0.5 −1 −1.5

−4 −1

−0.5

0

0.5

−2 −1

1

FC6 − IZ

−5

2

FC6 − IX

−6

2

x 10

−0.5

0.5

1

0.5

1

FC6 − Clβ

−3

2

0

x 10

∆ Maximum Eigenvalue

∆ Maximum Eigenvalue

1.5 1

0

−1

1 0.5 0 −0.5 −1 −1.5

−2 −1

−0.5

∆ Maximum Eigenvalue

x 10

0.5

FC6 − Cnβ

−0.5 0 0.5 Normalised uncertainty (−1/+1)

−0.5 −7

5

0

−1 −1

−2 −1

1

∆ Maximum Eigenvalue

−5

1

0

1

x 10

0 FC6 − C

nr

0

−5 −1

−0.5 0 0.5 Normalised uncertainty (−1/+1)

1

Fig. 20.10. FC6 nonlinear sensitivity computed at nominal critical point: unstable eigenvalues criterion, lateral uncertainties (α = 25.0◦ ).

instances the selection of a worst-case uncertainty value involves a discrepancy – e.g. −1 instead of +1 – but only for uncertainties that have negligible impact on the criterion). An extension to the process aimed at improving the confidence with which the true worst cases are generated is explained in Chapter 6.

20 Bifurcation-Based Clearance of Linear Stability Criteria

375

FC6 worst−case eigenvalue criterion (c3): max. real eigenvalue − longitudinal uncertainties

0.102

0.101

Maximum real eigenvalue

worst−case Long Cat 1 uncertainties 0.1

0.099

not cleared cleared

0.098 nominal 0.097

0.096 23.5

24

24.5 α (deg)

25

25.5

FC6 worst−case eigenvalue criterion (c3): max. real eigenvalue − lateral uncertainties

0.105 0.104 0.103

Maximum real eigenvalue

0.102 0.101

worst−case Lat Cat 1 uncertainties

"worst−case of 5" Lat Cat 1 uncertainties

0.1 0.099

not cleared cleared

0.098 nominal 0.097 0.096 0.095 23.5

24

24.5

α (deg)

25

25.5

26

Fig. 20.11. FC6 local bifurcation runs, unstable eigenvalues criterion. Upper graph: longitudinal uncertainties; lower graph: lateral uncertainties.

376

M. Lowenberg and T. Richardson

The clearance task as specified for the GARTEUR group accepts worst cases made up of combinations of the actual number of uncertainties (e.g. six for lateral category 1). In other words, it ignores the fact that reduction factors will often – possibly usually – mean that a subset of the total number of uncertainties will yield a worse case than taking all of them simultaneously. This is clearly demonstrated in Fig. 20.11, where the selection of four uncertainties is a worse case than using all six (although only to a small degree). Therefore, it can be construed as slightly incongruous to demand a guarantee that the true worst case combination of the k uncertainties is found, when in fact it is likely that the real true worst case will be with fewer than k uncertainties. In a sense, therefore, the bifurcation and continuation method shows an improvement over the baseline approach, while at the same time suffering the disadvantage of not in itself providing a guarantee for the worst-case combination.

20.6 Results Summary for All Flight Conditions The analysis cycle illustrated above for FC6 was carried out for all the flight conditions. A summary of the results is presented in the form of flight envelope diagrams. One diagram is used for each of the five criteria (and includes envelope limits due to nz , control surface saturation, etc.): Fig. Fig. Fig. Fig. Fig.

20.12: 20.13: 20.14: 20.15: 20.16:

stability margin, δT S loop stability margin, δT D loop stability margin, δR loop unstable eigenvalues unstable eigenvalues

longitudinal category 1 uncertainties lateral category 1 uncertainties lateral category 1 uncertainties longitudinal category 1 uncertainties lateral category 1 uncertainties

The final overall clearance, combining the limits from the above criteria, is shown in Fig. 20.17. These HIRM+RIDE flight envelope diagrams indicate the range of α for which each FC is cleared. Details of the uncertainty combinations (i.e. number of aerodynamic uncertainties used), frequencies at violation of stability margin criteria and the operational limits (nz and control surface saturation) can be included on these diagrams but are omitted here for consistency with presentation in other chapters.

20.7 Man-Hour and Computation Time Estimates The following is an estimate of the time required to apply the bifurcation and continuation method to clearance of the HIRM+RIDE (assuming a user with reasonable experience of the software and the clearance procedure).

20 Bifurcation-Based Clearance of Linear Stability Criteria Clearance results for symmetric tailplane loop

4

5

377

x 10

4.5 4

FC3

FC

α ∈ [ −15, 29.8]

α ∈ [ −15, 30.3]

8

3.5 FC5

Altitude (ft)

3

α ∈ [ −15, 30.6] FC2

2.5

α ∈ [ −15, 29.6] FC6

2

α ∈ [ −9, 30]

FC

4

1.5

α ∈ [ −15, 31.3]

1 0.5 0

0

0.1

FC1

FC

α ∈ [ −9, 30]

α ∈ [ −3, 12]

0.2

7

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mach Number

Fig. 20.12. Cleared α ranges – Nichols stability margin, symmetric tailplane loop. Clearance results for differential tailplane loop

4

5

x 10

4.5 4

FC3

FC

α ∈ [ −15, 30.4]

α ∈ [ −15, 31]

8

3.5 FC

5

Altitude (ft)

3

α ∈ [ −15, 32.5] FC

2

2.5

α ∈ [ −15, 33.1] FC

α ∈ [ −9, 30]

6

2 FC

4

1.5

α ∈ [ −15, 34.1]

1 0.5 0

0

0.1

0.2

FC1

FC7

α ∈ [ −9, 35]

α ∈ [ −3, 12]

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mach Number

Fig. 20.13. Cleared α ranges – Nichols stability margin, differential tailplane loop.

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M. Lowenberg and T. Richardson Clearance results for rudder loop

4

5

x 10

4.5 4

FC3

FC

α ∈ [ −15, 29.6]

α ∈ [ −15, 26.5]

8

3.5 FC5

Altitude (ft)

3

α ∈ [ −15, 26.5] FC2

2.5

α ∈ [ −15, 30.7] FC6

2

α ∈ [ −9, 25]

FC

4

1.5

α ∈ [ −15, 26]

1 0.5 0

0

0.1

FC1

FC

α ∈ [ −9, 30]

α ∈ [ −3, 12]

0.2

7

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mach Number

Fig. 20.14. Cleared α ranges – Nichols stability margin, rudder loop. Clearance results for maximum eigenvalue − longitudinal uncertainties

4

5

x 10

4.5 4

FC3

FC

α ∈ [ −15, 34.7]

α ∈ [ −15, 33.7]

8

3.5 FC

5

Altitude (ft)

3

α ∈ [ −15, 33.7] FC

2

2.5

α ∈ [ −15, 34.7] FC

α ∈ [ −9, 24.2]

6

2 FC

4

1.5

α ∈ [ −15, 27.1]

1 0.5 0

0

0.1

0.2

FC1

FC7

α ∈ [ −9, 34.8]

α ∈ [ −3, 12]

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mach Number

Fig. 20.15. Cleared α ranges – unstable eigenvalues, longitudinal uncertainties.

20 Bifurcation-Based Clearance of Linear Stability Criteria 4

5

x 10

379

Clearance results for maximum eigenvalue − lateral uncertainties

4.5 4

FC3

FC

α ∈ [ −15, 34.5]

α ∈ [ −15, 31.6]

8

3.5 FC5

Altitude (ft)

3

α ∈ [ −15, 31.7] FC2

2.5

α ∈ [ −15, 34.7] FC6

2

α ∈ [ −9, 24.7]

FC

4

1.5

α ∈ [ −15, 27.8]

1 0.5 0

0

0.1

FC1

FC

α ∈ [ −9, 34.8]

α ∈ [ −3, 12]

0.2

7

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mach Number

Fig. 20.16. Cleared α ranges – unstable eigenvalues, lateral uncertainties. 4

5

x 10

Final clearance results (stability margins+unstable eigenvalues)

4.5 4

FC3

FC

α ∈ [ −15, 29.6]

α ∈ [ −15, 26.5]

8

3.5 FC

5

Altitude (ft)

3

α ∈ [ −15, 26.5] FC

2

2.5

α ∈ [ −15, 29.6] FC

α ∈ [ −9, 24.2]

6

2 FC

4

1.5

α ∈ [ −15, 26]

1 0.5 0

0

0.1

0.2

FC1

FC7

α ∈ [ −9, 30]

α ∈ [ −3, 12]

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mach Number

Fig. 20.17. Final clearance – all stability margin and unstable eigenvalues criteria.

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M. Lowenberg and T. Richardson

Setting up software with given model: This includes implementation of the clearance criteria and will usually require the “dual-model” form of the code: the trim (or quasi-trim) flight conditions must be accounted for when generating the solution paths and the correct system used for clearance evaluation. The code structure should facilitate a range of continuation parameter choices, including the uncertainty parameters. Estimated time: 30 man-hours. Performing runs: Once set up, the software can simultaneously evaluate all criteria during runs. The user, however, needs to note all critical and worst-case points for the nominal runs, and then set up each nonlinear sensitivity run accordingly. This would typically take 15 minutes per critical or worst-case point. 34 such points were explored for HIRM+RIDE, amounting to a total of approximately 9 hours. In the baseline method, 2k combinations of uncertainty parameter are evaluated for each criterion at each incidence angle at each FC. There are 26 values of α over the specified range and 8 FCs, giving 208 × 2k , which is 6656 combinations for longitudinal category 1 and 13312 for lateral category 1 uncertainties. In the bifurcation analysis method, the 2 k combinations are considered only in the critical point regions, thereby saving computation time relative to the gridding approach. Furthermore, the continuation method admits the possibility that, due to reduction factors on aerodynamic uncertainties, the worst case is often a combination of fewer than the k uncertainties. In the baseline method, the 2k computations at each point accounts only for the use of all k uncertainties simultaneously. A search for the true worst case, including effects of reduction factors, would actually require the following number of combinations to be considered at each point: k−1 X i=0

2k−i k! i!(k − i)!

This gives 242 possible longitudinal category 1 combinations for each and every critical point, making a full baseline clearance extremely timeconsuming. The advantage of the continuation method is, therefore, increased when the clearance task demands a worst case combination of any number of the k uncertainties (although the bifurcation method does not guarantee the result). It also identifies if intermediate values of the uncertainties are the worst case (not just the vertices). Once the nonlinear sensitivity plots have been obtained, the user is required to select from them the worst-case combination for each case. This only takes a couple of minutes per case. A local bifurcation diagram is then

20 Bifurcation-Based Clearance of Linear Stability Criteria

381

produced from each nominal critical point, typically taking about 5 minutes of run time. This yields the worst case criterion violation point for each run, the total of which required about four hours for the 34 HIRM+RIDE clearance points. Thus the total man-hour count for performing runs to generate the HIRM+RIDE results was approximately 13 hours. Computation times: Computation times are very variable, as is often the case with nonlinear systems. The run times on a Pentium III 300 MHz PC were: – Nominal continuation plot with qdem as continuation parameter, implementing unstable eigenvalues criterion and three single loop stability margins at each solution point: ≤ 60 minutes (30 minutes without stability margin criteria) for each FC. – Nonlinear sensitivity plot, in terms of unstable eigenvalues criterion: 5 minutes per uncertainty parameter (per FC). – Nonlinear sensitivity plot, in terms of stability margin criteria: 10 minutes per uncertainty parameter (per FC). – Local continuation with qdem using worst-case uncertainties, for either unstable eigenvalues criterion or Nichols stability margin criterion: 5-10 minutes. Thus, once model and software have been set up properly, the full analysis cycle reported here can take approximately 15 hours of computation time. These estimated times are summarised in Table 20.6. Although man-hour and computation times would not usually be lumped together when assessing resource requirements, this is done here in order to give an indication of the “total time” involved in producing the results described in this chapter. Table 20.6. Estimated man-hour and computation times for HIRM+RIDE clearance, stability margin and unstable eigenvalues criteria. Man-hours — setting up: Man-hours — performing runs: Computation time: TOTAL:

30 hours 13 hours 15 hours 58 hours

20.8 Conclusion and Recommendations In certain respects, the HIRM+RIDE clearance examples do not extract the maximum benefit obtainable from the bifurcation/continuation approach. This is because, for the most part, the bifurcation diagram branch solved

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M. Lowenberg and T. Richardson

for in each FC does not contain strong nonlinearities (e.g. due to control surface saturation). In [2] an example is shown of a bifurcation run in which the aircraft velocity and thrust are permitted to vary (still a bottom-of-pull-up/top-of-pushover situation but, unlike in the quasi-trim cases, not at fixed velocity). The diagrams reveal a fold in the solution branches caused by thrust saturation at about Mach 0.5, α ≈ 20◦ and as a result, virtually no positive q solutions exist. Whilst this clearly does not alter the clearance study as defined, it does illustrate the type of phenomena which bifurcation/continuation can capture and explain. The sample results shown for the FC6 stability margin and unstable eigenvalues criteria do, however, reveal the main strengths of the methodology in the context of the HIRM+RIDE clearance: – it is systematic and transparent, using the model as defined; – the results are non-conservative, as the clearance criteria are implemented directly (as in baseline clearance); – under the assumptions for the system model as outlined in Chapter 6 (which actually applies to any analysis method), the technique substantially reduces computation by firstly identifying where clearance boundaries are violated or approached on the nominal system; – the user is consistently in touch with the physics of the problem; – the worst-case combination of uncertainty parameters can, under the assumptions of Chapter 6 (i.e. not guaranteeing a worst case), be selected to consider not just the worst of all the parameters, but the actual worst case incorporating reduction factors on aerodynamic uncertainties — and an example was given in this chapter where this reduced the cleared region relative to the baseline solution; – the nonlinear sensitivity approach for choosing worst-case uncertainty parameter combinations allows intermediate values of parameters to be selected where relevant (i.e. normalised value between −1 and +1). There is scope for improvement and extension of the method. It is recommended that future work investigates: (i) An improvement in the confidence with which worst-case uncertainty combinations are defined. The issue is essentially that of the implied lack of coupling between the uncertainties in terms of selecting whether to use normalised values of −1 or +1 (or intermediate values) for each. Increased confidence would be achieved if the selection were checked at the critical point for the normalised values found under the present approach; if a discrepency were to arise, a further iteration would commence. (ii) Implementation of additional clearance criteria would further reveal the versatility of the approach.

20 Bifurcation-Based Clearance of Linear Stability Criteria

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(iii) It is likely that a particularly fruitful development of the continuation/ bifurcation methodology would be to combine it with nonlinear optimisation. The latter could supplement the nonlinear sensitivity method so as to provide a “guarantee” for worst-case parameter combinations at critical points. (iv) The nature of the bifurcation/continuation method makes it suitable for extending the fixed-point clearance procedure to one of generating worstcase loci through the flight envelope.

References 1. F. Karlsson, U. Korte and S. Scala. Selected Criteria for Clearance of the HIRM+ Flight Control Laws, GARTEUR/TP-119-2-A1, 1999 (addendum to GARTEUR- TP-119-2 “The HIRM+ Aircraft Model and Control Laws Development”). 2. M. H. Lowenberg and T. S. Richardson. HIRMPRIDE Clearance using Bifurcation Analysis, GARTEUR/TP-119-13v3, 2001.

21 Optimisation-Based Clearance: The Linear Analysis Andras Varga German Aerospace Center DLR - Oberpfaffenhofen Institute of Robotics and Mechatronics D-82234 Wessling, Germany. [email protected] Summary. We present the clearance results of the HIRM+RIDE control configuration for the linear stability and handling criteria mostly used in the current industrial practice. The performed analysis is based on an optimisation-driven worst-case search. Two classes of linear stability related criteria are considered: the Nichols exclusion region based stability margin criterion and the unstable eigenvalues criterion. The considered handling criteria are the average phase rate and the absolute amplitude criteria. The analysis results clearly illustrate the high potential of the optimisation-based approach in reliably solving clearance problems with many simultaneous uncertain parameters.

21.1 Optimisation-Based Clearance of Linear Criteria Our analysis addresses the linear stability and handling criteria defined in Chapter 10, namely (a) the stability margin criterion, (b) the unstable eigenvalues criterion, (c) the average phase rate criterion, (d) the absolute amplitude criterion. Because of space limitations, we are forced to restrict our presentation to selected results which best illustrate the different aspects of an optimisationbased clearance approach. In this introductory section, we discuss some aspects which are common to all clearance tasks formulated above. Specific aspects and detailed analysis results are presented in separate sections dedicated to particular classes of criteria. The complete results, including results for the classical gridding-based approach, are presented in a GARTEUR AG11 Report [1]. The definition of suitable distance functions is a crucial step in an optimisation-based clearance approach (see Chapter 7). The main requirement for a satisfactory distance function is to enforce at minimum, worst cases where the clearance conditions are potentially most violated. Additionally, a satisfactory distance function to be used for an optimisation-driven C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 385-413, 2002. Springer-Verlag Berlin Heidelberg 2002

386

A. Varga

worst-case search must not introduce additional local minima, must be continuous and have continuous derivatives, and finally, must be easy to compute. Two categories of clearance problems can be identified, which lead to different types of distance functions. In the first category, for the clearance problems represented by (b) and (d), the clearance criterion is a mathematically expressible function c(p, F C) depending smoothly on the parameters grouped in a vector p and flight condition grouped in a vector F C. Such a criterion usually serves for controller design, and therefore, let us assume that the design results are better for lower values of c(p, F C). If co is the limiting acceptable performance level, then as a ”distance” function, to be used by a function minimiser, we can define d(p, F C) = −c(p, F C) + c0 . By minimising d(p, F C) (i.e., maximising c(p, F C)), worst-case parameter/flight conditions are determined. In the second category of clearance problems, represented by (a) and (c), exclusion regions are defined which, for satisfactory performance, must have empty intersection with certain sets of points used for graphical evaluations (e.g., frequency response Nichols-plots or performance plots). The boundary of the exclusion region can be associated with a fictitious limiting performance level c0 for an appropriate criterion c(p, F C) (to be chosen). If we define the signed ”distance” function as d(p, F C) = c(p, F C)−c0 then, by minimising d(p, F C), we can determine worst-case parameter/flight condition combinations. Note that the term ”worst-case” is always to be considered in connection with the chosen distance function. For both categories of clearance problems, if for a fixed F C, d(p, F C) is negative for some parameter values of p, then the clearance requirement is not fulfilled in F C and the point F C is not cleared. Otherwise, we define F C as cleared. A fast and reliable criteria evaluation is another aspect of paramount importance for the effectiveness of an optimisation-based worst case search. Typically, when evaluating criteria on the basis of linearised augmented aircraft models, all criteria evaluations can be done using a unique parameterised nonlinear model which describes the HIRM+ dynamics (see Chapter 8) in a feedback connection with the RIDE controller (see Chapter 9). A particular feature of HIRM+ is that for large values of AoA, a longitudinal/lateral coupling in the aircraft dynamics takes place. Therefore, contrary to the approach used in current industrial practice, we did not use separate linearised models for longitudinal and lateral axis dynamics. In our presentation, the terms ”longitudinal axis analysis” and ”lateral axis analysis” merely indicate that the analysis has been performed for the uncertain parameter set relevant to either the longitudinal axis or the lateral axis, respectively. As a consequence of using a unique nonlinear model to evaluate the defined clearance criteria (a)-(d), each function evaluation involves trimming and linearisation for a given flight condition and given parameter values. To speed up these computations, the HIRM+ and RIDE are trimmed and linearised separately and the final closed-loop linearised model is constructed by appropriate feedback coupling. The trimming of HIRM+ is done on the basis of special inverse

21 Optimisation-Based Clearance: The Linear Analysis

387

models, which ensure very fast and accurate trim computations (for details see Chapter 8). For criteria involving frequency-responses, model reduction techniques [2] have been used for further speeding up of computations. This technique is very effective, since the full HIRM+RIDE closed-loop model has order 61, while the reduced order models used to evaluate particular criteria have orders usually bellow 20. The numerically reliable evaluation of criteria means avoiding discontinuities originating from numerical computations (e.g., 360◦ phase jumps when computing the phase for the frequency response), improper tolerance settings (e.g., large truncation errors), or even failures of function evaluations (e.g., for points outside of the flight envelope). A key aspect of the optimisation-based approach is the choice of adequate optimisation software. Here adequate means to employ the best suited software for each clearance task, taking into account the possible existence of multiple local minima, level of noise in function evaluation, possible discontinuities of functions or derivatives etc. Because of their expected higher performance, the gradient-based methods like the sequential quadratic programming (SQP) or projected quasi-Newton (e.g., L-BFGS-B) always represent the first choice. Still, these techniques are not always able to produce the best results, especially when discontinuities in function/derivatives and/or noise in function values are present. Alternatively, the derivative-free linear approximation based trust-region method COBYLA, or the slower but often more robust pattern search (PS) method can be employed. If the presence of multiple local minima is to be expected, global search methods like the genetic algorithm (GA) or simulated annealing (SA) can be employed, either to locate initialisation points for local search based methods or, although expensive, to perform a global search for selected flight conditions. The analysis results presented in this paper have been obtained for the eight representative flight conditions F Ci , i = 1, . . . , 8 specified in Chapter 10, for values of the angle of attack α, ranging from −15◦ to 35◦ with a step size of ∆α = 1◦ . To define the ”true” physical flight envelope, a preliminary analysis of the open-loop HIRM+ has been performed for all flight conditions for the nominal values of parameters to check if the conditions −3 g ≤ nz ≤ 7 g

(21.1)

are violated or not, and to check if the HIRM+ is trimmable within the allowed limits of the deflections of taileron and rudder actuators given in Chapter 8 −40◦ ≤ δT S + δT D ≤ 10◦ −40◦ ≤ δT S − δT D ≤ 10◦ (21.2) −30◦ ≤ δR ≤ 30◦ The analysis revealed (see [1] and Fig. 21.27) that, because of violation of conditions (21.1) and (21.2), F C1 is defined only for α ∈ [−9◦ , 35◦ ], F C6 is defined only for α ∈ [−9◦ , 29◦ ] and F C7 is defined only for α ∈ [−2◦ , 12◦ ]. Violations of conditions (21.1) and (21.2) have been incidentally detected

388

A. Varga

also for the computed worst-case parameter combinations. In both cases, the corresponding points are automatically not cleared, because they do not belong to the admissible flight envelope. To save computational times, the analysis automatically detects points where the conditions (21.1) and (21.2) are violated for the nominal values of parameters. For such points, no further analysis is performed. Another saving in computational time is possible by skipping further analysis in points where the clearance conditions for the nominal values of parameters are not satisfied. However, this was not done, just to show that even worse results can be computed by an optimisationbased search. It is worth mentioning that, because of the presence of an AoA-limiter at 29◦ in the RIDE controller, the ”true” flight envelope must be probably further restricted to values of AoA α ≤ 29◦ , otherwise some analysis results for linear criteria are questionable for values of AoA α > 29◦ . The analysis of clearance criteria (a)-(d) has been performed for the both the small and the full parameter sets defined in Chapter 10 for both longitudinal and lateral axis analyses. For the longitudinal axis analysis there are 5 parameters in the small (most relevant) parameter set and 9 parameters in the full parameter set, while for the lateral axis analysis there are respectively 6 and 14 parameters in these sets. Complete results for the analysis of all criteria are presented in [1], where results for both optimisation as well as for gridding-based search are given. Due to space restrictions, we present in this contribution only a selection of the most relevant results. All computations have been performed on a Pentium II 400 MHz machine running Matlab 5.3 and Simulink 3.0 under Windows NT 4.0. The basis for the performed analysis was the analysis cycle described in Fig. 7.1. For each analysed criterion a corresponding procedure has been implemented as a Matlab script which performs the whole analysis for all flight conditions, saves intermediary and final results, and evaluates and documents the results through appropriate plots. All scripts allow easy switching between different solvers available in a dedicated optimisation environment. The employed optimisation software includes the updated SQP and PS software from the RASP library [3], L-BFGS-B using the implementation of [4], the COBYLA software implemented by Powell [5], and a binary coding based GA software adapted from David Carroll’s code [6]. The open software architecture in Fig. 21.1 underlies our optimisation-based clearance. It allows us to easily add new solvers and to use the available solvers interchangeably for analysis. A unique Matlab interface mex -function nlpmex offers a neutral interface to a generic solver for general nonlinear programming problems (NLPs). The solvers are launched as independent child processes (tasks) which communicate with the parent process (i.e., nlpmex) via a problem dependent process communication dialog. For example, each function and/or gradient evaluation involves transferring the current values of optimisation parameters from the solver (i.e., child process) to the mex -function (i.e., parent process) which calls the m-function typically used to implement the clear-

21 Optimisation-Based Clearance: The Linear Analysis

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Fig. 21.1. An open software architecture for nonlinear programming

ance criteria. This function usually calls appropriate trimming and linearisation routines (partly based on Simulink). The computed function and/or gradient values are then sent back to the solver. This reverse-communication based software architecture completely separates the function evaluations from the optimisation software. This allows to easily integrate a heterogeneous collection of optimisation tools in a single flexible optimisation environment. Thus, the underlying tools can be written in different programming languages, can have different options and parameter lists, can use different function and gradient evaluation schemes, etc. This open architecture allows us to easily add new solvers as a need arises.

21.2 Results for the Stability Margin Criterion The goal of the analysis is to identify all flight conditions in terms of Mach number M , altitude h, and AoA α, and all combinations of uncertain parameters where the Nichols plot stability margin boundaries are most violated. In Chapter 10 requirements for both single-loop as well as multi-loop analysis are defined. In our analysis, we consider only the single-loop analysis. The analysis requirement is to check if the open-loop Nichols plot of the frequency response obtained by breaking the loop at the input of the actuators for the symmetric taileron, differential taileron or rudder, avoids the exclusion region delimited by the polygonal boundary B0 in Fig. 21.2. The analysis for the symmetric taileron loop has been performed only for the longitudinal parameter sets, while for the differential taileron and rudder loops only the lateral parameter sets have been considered.

390

A. Varga Nichols plane exclusion region

6

4.5

B

4

1

B0

2

Gain [dB]

1.5 −180

0

−145

−1.5 −2

−4 −4.5

−6 −200

−190

−180

−170

−160 Phase [°]

−150

−140

−130

−120

Fig. 21.2. Exact and approximate Nichols plot exclusion regions for gain and phase

To define an appropriate distance function, we approximate the boundary B0 of the exclusion region in Fig. 21.2 by an arc of an ellipse B1 defined by ³ g ´2 µ ϕ + 180 ¶2 + =1 4.5 35 where g and ϕ are the gain and phase, respectively. Instead of using B0 , we can interpret B1 as the given boundary for the limiting acceptable performance and we can use it to define a smooth distance function for an optimisation-driven worst-case search. Note that defining the distance function with respect to B0 would lead to a non-differentiable distance function. For a given frequency dependent gain-phase pair (g(ω), ϕ(ω)), a normalised signed ”distance” to B1 can be computed by using the formula ˜ d(ω) =

µ

g(ω) 4.5

¶2

µ +

ϕ(ω) + 180 35

¶2 − 1.

˜ Note that d(ω) > 0 if the point (g(ω), ϕ(ω)) lies outside of the ellipse and ˜ ˜ d(ω) ≤ 0 otherwise. The minimum of d(ω) over a given frequency range [ωmin , ωmax ] defines the least distance d(p, F C) to B1 for a given uncertain parameter vector p and flight condition F C. Note that p and F C are the values used to determine the linearised model which serves for evaluating the frequency response.

21 Optimisation-Based Clearance: The Linear Analysis

391

The minimum value d(F C) of the distance function with respect to the parameters in p can be used to define the following stability margin for flight condition F C p (21.3) ρs (F C) = 1 + d(F C). With this stability margin, a flight condition F C could be categorised as not cleared if ρs (F C) ≤ 1 and cleared otherwise. Since ρs is defined on the basis of a distance function with respect to the approximate boundary B1 , it could happen that values of ρs marginally less than 1 can be still cleared, because, in reality, no intersection with the Nichols exclusion region occur. Conversely, values of ρs marginally greater than 1, can be categorised as not cleared, because the Nichols plot intersects the exclusion region. This is why, we used the stability margin defined in (21.3) only for the visualisation of our computational results. Still, all reported clearance results rely on strict checks of intersection/no intersection of worst-case Nichols plots with the exclusion region defined by the boundary B0 in Fig. 21.2. For the success of an optimisation-based search, the function evaluations must be as fast as possible. There are two critical, time demanding computations to evaluate d(p, F C), for given p and F C. The first critical computation is the trimming of the HIRM+. A fast trimming for HIRM+ is done using the inverse model approach, as described in Chapter 8 and we will not further discuss this aspect here. The second expensive computation in evaluating the distance function is the computation of frequency responses for the 57-th order (4 state components omitted) closed-loop SISO models formed from the linearised HIRM+ and RIDE. Taking into account the need to compute the minimum distance accurately, the number of frequency points to be used must be sufficiently high. For the optimisation-driven search, we employed 100 frequency values ranging logarithmically in the interval [10 −2 , 10]. Some timing results obtained by evaluating the distance function for the differential taileron loop on a 400 MHz Pentium II machine are interesting to keep in mind. The time necessary to trim, linearise and build the closedloop SISO evaluation model is about 1.11 seconds, from which 0.06 seconds account for trimming HIRM+ using the inverse model, 0.13 seconds for the linearisation of HIRM+, and 0.9 seconds for the linearisation of the RIDE controller. Note that although the order of the controller is 9, about 90% of the time is necessary to linearise it using a SIMULINK model. The total times to evaluate d(p, F C) for 100 and 1000 frequency values are 1.45 seconds and 3.9 seconds, respectively, of which, the times necessary to compute only the frequency responses are 0.36 seconds and 2.85 seconds, respectively. The times for frequency response computations can be significantly reduced, by observing that the resulting SISO closed-loop systems used were always nonminimal. To speed up the computations, a preliminary order reduction is performed by using the recently developed high quality model reduction tools [2]. The resulting minimal order models for the symmetric taileron, differential taileron and rudder loops have dimensions 14, 24, and 22, respectively. For

392

A. Varga

the differential taileron loop, the model reduction requires only 0.07 seconds, but has the effect of reducing the time to evaluate the frequency responses for 100 and 1000 points to 0.06 seconds and 0.61 seconds, respectively. Thus, the total times to evaluate d(p, F C) for 100 and 1000 frequency points become 1.15 seconds and 1.7 seconds, respectively, with performance gains of 20% and 50% in these two cases. Note that the time saving is more substantial for a larger number of frequency values. One main goal of our analysis was to compare the necessary computational efforts for the gridding-based approach and for optimisation-driven worst-case search methods. The number of maximum function evaluations required by the gridding-based approach for the 8 × 51 flight conditions can be easily computed for the small parameter sets for longitudinal and lateral axis analyses. The longitudinal axis analysis requires at most 2 5 × 8 × 51 = 13056 function evaluations while the lateral axis analysis requires twice as many 26 × 8 × 51 = 26112. About 10%–20% of function evaluations can be saved, by restricting the search only to those points which belong to the ”true” flight envelope. In Table 21.1 we present some timing results for the performed analysis of the three loops on the small parameter sets: the symmetric taileron loop (STL) for the longitudinal axis parameters, and the differential taileron loop (DTL) and rudder loop (RL) for the lateral axis parameters. The solvers implementing the local search methods SQP, PS, COBYLA and L-BFGS-B (see Section 21.1) have been used for analysis, with the accuracy tolerance set to 10−5 . Table 21.1 also includes results for the classical gridding-based approach. Table 21.1. Timing results for gridding and optimisation-based analysis Method

Gridding

SQP

PS

COBYLA

L-BFGS-B

Times for STL (sec)

12943

19796

38188

20840

19426

Times for DTL (sec)

25074

23064

45333

38973

24024

Times for RL (sec)

27087

22374

41808

61657

20977

The time for the gridding-based solution for 5 parameters is always less than the times required by the optimisation-based search. However, this is not the case for 6 parameters, where gradient based methods like SQP or L-BFGS-B are more efficient than the gridding-based approach. The computational effort increases exponentially with the number of parameters and therefore a gridding-based solution is not affordable for the full sets of 9 longitudinal and 15 lateral axis parameters due to very high computational costs (24 = 16 and 29 ≈ 500 larger times, respectively). By using the optimisationbased approach it was possible to obtain complete clearance results for the full parameter sets. Table 21.2 summarises the number of function evaluations (NFE) and the computational times required by using the SQP method.

21 Optimisation-Based Clearance: The Linear Analysis

393

Table 21.2. Performance results for stability margin analysis Gridding

SQP (Small sets)

SQP (Full sets)

NFE

Time (sec)

NFE

Time (sec)

NFE

Time (sec)

STL

11382

12943

17519

19796

38649

42662

DTL

22710

25074

19220

23064

49596

55081

RL

23088

27087

18645

22374

43693

50903

Detailed analysis results for both gridding and optimisation-based search, for all three loops, are presented in [1]. The results for the differential taileron loop are summarised in Fig. 21.3, where we give, in each flight condition, the domains of α-values which are considered as ”cleared” for three cases: (1) gridding-based analysis; (2) optimisation-based analysis for the small parameter set; and (3) optimisation-based analysis for the full parameter set. While for the small parameter set both gridding and optimisation-based search produce almost the same results in term of not cleared flight conditions, the optimisation-based search performed on the full parameter set revealed many additional not-cleared points. This clearly illustrates the power of the optimisation-based approach to simultaneously address many uncertain parameters. Clearance results for differential taileron loop

4

5

x 10

(1) Gridding−based search (small parameter set) (2) Optimisation−based search (small parameter set) (3) Optimisation−based search (full parameter set) FC3

4.5 4 3.5

Altitude (ft)

8

(1) α ∈ [ −15, 31]

(2) α ∈ [ −15, 31]

(2) α ∈ [ −15, 31]

(3) α ∈ [ −15, 24]

(3) α ∈ [ −15, 23]

FC

5

3

(1) α ∈ [ −15, 32] (2) α ∈ [ −15, 32]

FC2

2.5

(3) α ∈ [ −15, 24] ∪ [26,28]

(1) α ∈ [ −12, 33] (2) α ∈ [ −14, 32]

2

FC6

(3) α ∈ [ −14, 29]

(1) α ∈ [ −9, 29] (2) α ∈ [ −9, 29] (3) α ∈ [ −9, 24] ∪

FC

4

1.5

[26,29]

(1) α ∈ [ −15, 30] ∪ {35} (2) α ∈ [ −15, 31] ∪ {35}

1

(3) α ∈ [ −15, 30] FC1

0.5 0

FC

(1) α ∈ [ −15, 31]

(1) α ∈ [ −6, 33]

FC7

(1) α ∈ [ −2, 12]

(2) α ∈ [ −9, 33] (3) α ∈ [ −9, 34] 0

0.1

0.2

0.3

0.4

(2) α ∈ [ −2, 12] (3) α ∈ [ −2, 12] 0.5

0.6

0.7

0.8

0.9

1

Mach Number

Fig. 21.3. Clearance results for the differential taileron loop

Fig. 21.4 shows the values of the stability degree ρs versus α for the small parameter set. These values are visibly greater than the corresponding

394

A. Varga

values of ρs for the full parameter set in Fig. 21.5. Thus, as expected, the HIRM+RIDE configuration has less stability margin in most of the points when more uncertain parameters are allowed to simultaneously vary. Worst−case stability margins − differential taileron

2.5

Stability degree ρs

2

1.5

1

0.5

FC 1 FC2 FC 3 FC 4 FC5 FC6 FC 7 FC 8

0 −15

−10

−5

0

5

10 AoA [deg]

15

20

25

30

35

Fig. 21.4. Worst-case stability margins for DTL (small parameter set)

Worst−case stability margins − differential taileron

2.5

Stability degree ρs

2

1.5

1

0.5

0 −15

FC1 FC2 FC 3 FC4 FC5 FC6 FC7 FC8 −10

−5

0

5

10 AoA [deg]

15

20

25

30

35

Fig. 21.5. Worst-case stability margins for DTL (full parameter set)

21 Optimisation-Based Clearance: The Linear Analysis

395

In each performed analysis we determined an ”overall” worst-case parameter combination over all flight conditions. In Figs. 21.6 and 21.7, we present the worst-case parameter combination for the differential taileron loop and the corresponding frequency response, respectively. Note that in this case, the optimisation-based search was able to determine a global minimum which led to crossing of the exclusion region through its centre (0, −180◦ ). Worst−case stability margin − differential taileron

4

x 10

5

Pull−up/push−over alpha−trim: Mach = 0.5 Altitude = 15000 ft γ = 0° α = 32° φ = 0° β = 0° Throttle = 1 nz = 4.7244 δ = −12.393° TS δ = −0.81754° TD δ = 0° CS δ = 0° CD δ = 13.3582° R

4.5 4

Altitude (ft)

3.5 3 2.5 2

Lateral parameters: Y = −0.047603m CG I = −0.064491 x I = 0.061721 z

−1

C = 0.004503 rad lβ C = −0.0085859 rad−1 nβ Cnr = 0.00087234

ClδTD = −0.00098157 rad−1 C = −0.0051115 rad−1 lδCD −1 ClδR = −0.00021683 rad C = −1.8462e−005 rad−1 nδTD −1 CnδCD = −0.0023855 rad −1 C = 0.0051227 rad nδR C = 0.010717 lp C = −0.0054737 lr C = 0.0042731 np

ρs = 0.00058226

1.5

f = 0.36767Hz

1 0.5 0

0

0.1

0.2

0.3

0.4

0.5 0.6 0.7 Mach Number

0.8

0.9

1

1.1

1.2

Fig. 21.6. Worst-case parameters for DTL (full parameter set) Worst−case frequency response − differential taileron

30

Nominal Perturbed

20

10

Gain [dB]

ρ = 0.00058226 s

f = 0.36767Hz 0

ρ = 0.89893 s

−10

f = 0.36767Hz

−20

−30 −260

−240

−220

−200

−180 Phase [°]

−160

−140

−120

−100

Fig. 21.7. Worst-case frequency response for DTL (full parameter set)

396

A. Varga

In Figs. 21.8 and 21.9 we present the clearance results for the symmetric taileron and rudder loops. Clearance results for symmetric taileron loop

4

5

x 10

(1) Gridding−based search (small parameter set) (2) Optimisation−based search (small parameter set) (3) Optimisation−based search (full parameter set) FC

4.5

3.5

Altitude (ft)

8

(1) α ∈ [ −15, 29]

(1) α ∈ [ −15, 30]

(2) α ∈ [ −15, 29]

(2) α ∈ [ −15, 30]

(3) α ∈ [ −15, 29]

(3) α ∈ [ −15, 30]

FC5

3

(1) α ∈ [ −15, 32] FC

(2) α ∈ [ −15, 29]

(1) α ∈ [ −15, 29]

(3) α ∈ [ −15, 30]

2

2.5

FC6

(2) α ∈ [ −13, 29]

2

(3) α ∈ [ −15, 29]

(1) α ∈ [ −9, 29] (2) α ∈ [ −9, 29] (3) α ∈ [ −9, 29]

FC

4

1.5

(1) α ∈ [ −15, 31] (2) α ∈ [ −15, 31]

1

(3) α ∈ [ −15, 30] FC1

0.5 0

FC

3

4

FC7

(1) α ∈ [ −9, 29]

(1) α ∈ [ −2, 12]

(2) α ∈ {−9} ∪ [ −6, 29] (3) α ∈ [ −8, 29] 0

0.1

0.2

0.3

0.4

(2) α ∈ [ −2, 12] (3) α ∈ [ −2, 12]

0.5

0.6

0.7

0.8

0.9

1

Mach Number

Fig. 21.8. Clearance results for the symmetric taileron loop Clearance results for rudder loop

4

5

x 10

(1) Gridding−based search (small parameter set) (2) Optimisation−based search (small parameter set) (3) Optimisation−based search (full parameter set) FC3

4.5 4 3.5

(1) α ∈ [ −15, 29]

(2) α ∈ [ −15, 30]

(2) α ∈ [ −15, 29]

(3) α ∈ [ −15, 30]

(3) α ∈ [ −15, 28]

FC

5

Altitude (ft)

3

(1) α ∈ [ −15, 32] FC

(2) α ∈ [ −15, 29]

(1) α ∈ [ −12, 31]

(3) α ∈ [ −15, 28]

2

2.5

FC6

(2) α ∈ {−15} ∪ [ −13, 31]

2

(3) α ∈ [ −15, 30]

(1) α ∈ [ −9, 28] (2) α ∈ [ −9, 29] (3) α ∈ [ −9, 27]

FC

4

1.5

(1) α ∈ [ −15, 28] ∪ {35} (2) α ∈ [ −15, 28] ∪ {35}

1

(3) α ∈ [ −15, 28] FC1

0.5 0

FC8

(1) α ∈ [ −15, 30]

(1) α ∈ [ −6, 30]

FC7

(1) α ∈ [ −2, 12]

(2) α ∈ [ −8, 30] (3) α ∈ [ −8, 29] 0

0.1

0.2

0.3

0.4

(2) α ∈ [ −2, 12] (3) α ∈ [ −2, 12] 0.5

0.6

0.7

0.8

0.9

1

Mach Number

Fig. 21.9. Clearance results for the rudder loop

From the clearance results presented in Figs. 21.3, 21.8 and 21.9 we can see that in some cases, points which are not cleared by the classical

21 Optimisation-Based Clearance: The Linear Analysis

397

gridding-based search approach, appear to be cleared by the more powerful optimisation-based search approach for the small or even for the full parameter sets. The explanation of this apparent paradox is that the worst-case parameter combinations computed by different methods occasionally lead to violations of the condition −3 ≤ nz ≤ 7 or of the trimming constraints (21.2). Note that such points, found only incidentally, automatically restrict the effective flight envelope of HIRM+ (see Fig. 21.27) and can be skipped in further analysis.

21.3 Results for the Unstable Eigenvalues Criterion The goal of the analysis is to identify all flight conditions in terms of Mach number M , altitude h, and angle of attack α, and all combinations of uncertain parameters where unstable eigenvalues are found. The clearance task formulated in Chapter 10 is to check if the closed-loop eigenvalues lie to the left of the boundary defined in Fig. 21.10. Boundary for a closed−loop eigenvalue λ = λ ± i λ r

0.9

i

Imaginary part λi

0.45

0.15 λ ≤ ln 2 / 20 r

0

λ ≤ ln 2 / 7 , λ = 0 r

i

−0.15 ≤ λ ≤ 0.15 i

−0.15

−0.45

−0.9 −0.05

0

0.05 Real part λ

0.1

0.15

r

Fig. 21.10. Boundary for eigenvalues.

Mathematically, if λ = λr + i λi is an eigenvalue of the state matrix Acl of the linearised closed-loop model, then the real part λr must satisfy the following conditions  |λi | ≥ 0.15  0, λr ≤ ln 2/20, 0 < |λi | < 0.15  ln 2/7, λi = 0

398

A. Varga

For a given flight condition F C and a parameter vector p, a straightforward ”distance” function can be defined as d(p, F C) = − max{Re λ|λ ∈ Λ(Acl )}

(21.4)

By minimising d(p, F C), we determine the worst-case parameter combination leading to a maximal real part of the closed-loop eigenvalues. This real part corresponds either to a purely real eigenvalue or to a pair of complex conjugate eigenvalues. Each evaluation of d(p, F C) involves the construction of the closed-loop state matrix Acl using the linearised models of the HIRM+ and RIDE controller and the computation of eigenvalues of a matrix of order 61. To evaluate d(p, F C) there is no need to separate the longitudinal and lateral dynamics to perform the analysis for the two categories of uncertain parameters. For both cases, the unique nonlinear parameter uncertain model of the HIRM+RIDE closed-loop system can be used to evaluate the distance function. In general, the distance function (21.4) does not cover all possible cases of the analysis because, by minimising the maximum real part, the real axis segment in Fig. 21.10 is primarily favoured due to the presence of at least one unstable real pole in the linearised model. To address more specifically the regions delimited in Fig. 21.10, distance functions can be defined for each region according to the values of the imaginary parts. For example, to restrict the analysis strictly to purely real eigenvalues, the ”distance” function d1 (p, F C) = ln 2/7 − max{λr |λr + i λi ∈ Λ(Acl ), λi = 0}

(21.5)

can be used. To restrict the analysis to the strips defined by imaginary values satisfying 0 < |ω| ≤ 0.15 the ”distance” function d2 (p, F C) = ln 2/20 − max{λr |λr + i λi ∈ Λ(Acl ), 0 < λi ≤ 0.15}

(21.6)

can be used. Finally, to restrict the analysis to the regions for 0.15 < |ω|, the ”distance” function d3 (p, F C) = − max{λr |λr + i λi ∈ Λ(Acl ), 0.15 < λi }

(21.7)

is appropriate. It is possible to perform a simultaneous analysis for all three regions by considering as distance function d(p, F C) = min{d1 (p, F C), d2 (p, F C), d3 (p, F C)}

(21.8)

This function is not continuous. Due to the shape of the eigenvalue boundary in Fig. 21.10, the migration of eigenvalues from one region to another leads to abrupt changes in the values of d(p, F C). Still, our experiments have shown that for the HIRM+RIDE configuration, the distance function d2 was practically never active for the given flight conditions, thus the discontinuity in function values manifests practically never. Complete analysis results obtained with both distance functions (21.4) and (21.8) have shown that there

21 Optimisation-Based Clearance: The Linear Analysis

399

are no qualitative differences for the clearance when using the simpler, but continuous distance function (21.4) instead of the discontinuous one (21.8). This strongly supports our approach in [1] to use (21.4) as a distance function. In what follows, we present complete clearance results obtained using the SQP approach for the small and full parameter sets for both longitudinal and lateral axis analyses using the distance function (21.8). Table 21.3 shows the number of function evaluations (NFE) and the times for the gridding and the optimisation-based search methods. Table 21.3. Performance results for maximum real part analysis Gridding NFE

SQP (Small sets)

SQP (Full sets)

Time (sec)

NFE

Time (sec)

NFE

Time (sec)

Longitudinal 11382

12053

15060

15975

34496

35421

Lateral

22482

9628

10300

25159

25832

22710

In Fig. 21.11, we present the clearance results for the longitudinal axis analysis. As it can be observed, the analysis for the full parameter set revealed many not cleared points, which were previously cleared on the basis of the analysis performed using the gridding-based approach or optimisation-based search on the small parameter set. Clearance results for maximum real part − longitudinal axis

4

5

x 10

(1) Gridding−based search (small parameter set) (2) Optimisation−based search (small parameter set) (3) Optimisation−based search (full parameter set) FC

4.5

3.5

(1) α ∈ [ −15, 34]

(1) α ∈ [ −15, 34]

(2) α ∈ [ −15, 34]

(2) α ∈ [ −15, 33]

(3) α ∈ [ −15, 33]

(3) α ∈ [ −13, 32]

FC5

Altitude (ft)

3

(1) α ∈ [ −15, 34] FC

(2) α ∈ [ −15, 33]

(1) α ∈ [ −13, 34]

(3) α ∈ [ −15, 30]

2

2.5

FC

(2) α ∈ [ −15, 34]

2

6

(3) α ∈ [ −15, 34]

(1) α ∈ [ −8, 24] (2) α ∈ [ −8, 24] (3) α ∈ [ −7, 21]

FC

4

1.5

(1) α ∈ [ −14, 27] (2) α ∈ [ −14, 27]

1

(3) α ∈ [ −11, 23] FC1

0.5 0

FC8

3

4

FC

7

(1) α ∈ [ −6, 34]

(1) α ∈ [ −2, 11]

(2) α ∈ [ −9, 34] (3) α ∈ [ −9, 31] 0

0.1

0.2

0.3

0.4

(2) α ∈ [ −2, 11] (3) α ∈ [ −2, 9] 0.5

Mach Number

0.6

0.7

0.8

0.9

1

Fig. 21.11. Clearance results for the maximum real part - longitudinal axis

400

A. Varga

This can also be easily observed by comparing Figs. 21.12 and 21.13, which present the worst-case maximum real parts and corresponding imaginary parts versus AoA for the small parameter set, with the similar Figs. 21.14 and 21.15 for the full parameter sets. These results clearly illustrate the capability of optimisation-based worst-case search to solve clearance problems for many simultaneous parameters. Worst−case largest real parts − longitudinal axis 0.25

0.2

FC1 FC2 FC3 FC4 FC 5 FC 6 FC 7 FC

Largest real parts

8

0.15

0.1

ln2 / 7

0.05 ln2 / 20

0 −15

−10

−5

0

5

10

15

20

25

30

35

AoA [deg]

Fig. 21.12. Worst-case maximum real parts (longitudinal, small parameter set) Worst−case imaginary parts − longitudinal axis 3

2.5

Imaginary parts

2

FC1 FC 2 FC3 FC4 FC 5 FC 6 FC 7 FC8

1.5

1

0.5

0.15 0 −15

−10

−5

0

5

10

15

20

25

30

35

AoA [deg]

Fig. 21.13. Worst-case imaginary parts (longitudinal, small parameter set)

21 Optimisation-Based Clearance: The Linear Analysis 0.25

Largest real parts

0.2

401

Worst−case largest real parts − longitudinal axis FC 1 FC 2 FC 3 FC 4 FC 5 FC6 FC7 FC8

0.15

0.1

ln2 / 7

0.05 ln2 / 20

0 −15

−10

−5

0

5

10

15

20

25

30

35

AoA [deg]

Fig. 21.14. Worst-case maximum real parts (longitudinal, full parameter set) Worst−case imaginary parts − longitudinal axis 3

2.5

Imaginary parts

2

FC 1 FC 2 FC 3 FC4 FC 5 FC 6 FC 7 FC8

1.5

1

0.5

0.15 0 −15

−10

−5

0

5

10

15

20

25

30

35

AoA [deg]

Fig. 21.15. Worst-case imaginary parts (longitudinal, full parameter set)

From Figs. 21.13 and 21.15 it is apparent that in most flight conditions d1 was active, in very few cases d3 was active and d2 was never active. This confirms that the analysis for HIRM+RIDE configuration can be carried out reliably by using (21.4) as an alternative distance function.

402

A. Varga

In Fig. 21.16 we present the clearance results for the lateral axis analysis. As it can be observed, the results of gridding-based search and of the optimisation-based search are almost identical. Clearance results for maximum real part − lateral axis

4

5

x 10

(1) Gridding−based search (small parameter set) (2) Optimisation−based search (small parameter set) (3) Optimisation−based search (full parameter set) FC3

4.5 4 3.5

Altitude (ft)

8

(1) α ∈ [ −15, 32]

(2) α ∈ [ −15, 34]

(2) α ∈ [ −15, 32]

(3) α ∈ [ −15, 34]

(3) α ∈ [ −15, 32]

FC5

3

(1) α ∈ [ −15, 33] FC2

2.5

(2) α ∈ [ −15, 32] (3) α ∈ [ −15, 32]

(1) α ∈ [ −12, 34]

FC6

(2) α ∈ [ −15, 35]

2

(3) α ∈ [ −15, 35]

(1) α ∈ [ −9, 24] (2) α ∈ [ −9, 24] (3) α ∈ [ −9, 24]

FC

4

1.5

(1) α ∈ [ −15, 27] (2) α ∈ [ −15, 27]

1

(3) α ∈ [ −15, 27] FC1

0.5 0

FC

(1) α ∈ [ −15, 34]

FC

7

(1) α ∈ [ −6, 33]

(1) α ∈ [ −2, 12]

(2) α ∈ [ −9, 34] (3) α ∈ [ −9, 34] 0

0.1

0.2

0.3

0.4

(2) α ∈ [ −2, 12] (3) α ∈ [ −2, 12] 0.5

0.6

0.7

0.8

0.9

1

Mach Number

Fig. 21.16. Clearance results for the maximum real part - lateral axis

21.4 Results for the Average Phase Rate and Absolute Amplitude Criteria The average phase rate (APR) and absolute amplitude (AA) criteria are intended to identify pilot induced oscillation (PIO) tendencies in the pitch and roll axis control loops. The precise mathematical definition of these criteria can be done on the basis of the Nichols plot in Fig. 21.17 for the transferfunction g(s) of the transmission between the longitudinal or lateral stick force and the corresponding pitch or bank angle, respectively. For a given frequency f , let g(j2πf ) be the corresponding gain-phase representation of the frequency response g(j2πf ) = Af ejΦf where Af and Φf are the gain (in dB) and the phase angle (in degrees) at frequency f , respectively. The APR is defined as AP R =

Φfc − Φ2fc −180◦ − Φ2fc = fc fc

21 Optimisation-Based Clearance: The Linear Analysis

403

where fc is the (crossover) frequency (in Hertz) where the phase angle Φfc = −180◦ . The absolute amplitude is defined as the gain (in dB) corresponding to fc AA = Afc 10

Average phase rate and absolute amplitude definition

0

−10

Gain [dB]

−20

−30

APR = (−180 − Φ

2f

)/fc = 63.6366°/Hz

A = −32.2718 f

c

c

f = 1.2941Hz

−40

c

Φ2f = −262.3517° c

−50

2f = 2.5882Hz c

−60

−70 −300

−260

−220

−180 Phase [°]

−140

−100

Fig. 21.17. Definition of average phase rate and absolute amplitude criteria

The goal of our analysis is to identify all flight conditions in terms of Mach number M , altitude h, and angle of attack α, and all combinations of uncertain parameters where the Level 1 boundary defined in Fig. 21.18 is violated or where the absolute amplitude exceeds -29dB (i.e., crosses the bold line in Fig. 21.17). Note that for the HIRM+RIDE control configuration the Level 2 specifications can be easily fulfilled, and therefore our analysis goal is more stringent than that formulated in Chapter 10. For a given flight condition F C and a parameter vector p, a straightforward ”distance” function for the analysis of the APR criterion is d(p, F C) = 1/AP R =

fc −180 − Φ2fc

(21.9)

Minimising this function is equivalent to maximising the APR (by minimising fc ) and thus the worst-case criterion values tend to exceed the boundary of the Level 1 region. For the analysis of the absolute amplitude criterion, the natural candidate for a ”distance” function is d(p, F C) = −Afc

(21.10)

404

A. Varga Average phase rate analysis performance levels

200

Average Phase Rate [degrees/Hz]

Level 3

150

Level 2

100

Level 1

50

0

0

0.2

0.4

0.6 0.8 1 1.2 1.4 Frequency, fc, at −180 degrees phase [Hz]

1.6

1.8

2

Fig. 21.18. Average phase rate criterion level boundaries

The evaluation of d(p, F C) involves in each case, the construction of the closed-loop linearised model for the transfer function from the longitudinal or lateral stick force to the corresponding pitch or bank angle, respectively, the evaluation of the corresponding frequency response, and the determination of the crossover frequency fc using, for instance, linear interpolation between two consecutive points to the left and to the right of the −180◦ axis. To speed up the computation of frequency responses, model reduction techniques can be employed to get minimal order state-space representations for the respective transfer functions. An important aspect for an optimisation-based search is the reliable numerical evaluation of criteria. For the APR criterion we encountered several difficulties which led to the need to experiment with several gradient-free methods. For example, because of random phase jumps of ±360◦ in the initial phase values, we occasionally obtained completely erroneous values of the estimated crossover frequency fc . To prevent such jumps, the phase matching approach usually employed when drawing Nichols plots has been extended to include an initial phase of about −100◦ at the frequency ωmin = .1. Another difficulty which we encountered was the occurrence of multiple crossover frequencies. In some cases, for values of α > 29◦ , we encountered points where multiple frequency values satisfy the condition Φfc = −180◦ . To handle such cases, we defined fc as the largest value of the frequency where a crossing occurs. This allowed us, in most cases, to compute an APR which was within the expected range of values. The cause of this difficulty probably lies in the presence of the α-limiter in the RIDE controller at α = 29◦ . This is why the cleared points for values of α > 29◦ can only be cautiously accepted. Finally,

21 Optimisation-Based Clearance: The Linear Analysis

405

discontinuities in the gradient can be expected due to inaccurate localisation of the crossover frequency fc . This could lead to noisy function evaluations and therefore, to difficulties when using gradient-based methods. The above aspects can partly explain the somewhat poorer results obtained with gradient-based methods like SQP for the APR criterion, than those resulted from a gridding-based approach. To overcome the difficulties caused by noisy function evaluations, we ran the gradient based search with SQP in conjunction with the GA, where the GA has been used to compute initial points for the gradient-based search. Furthermore, we used the more accurate central difference approximation for the gradient instead of forward difference approximation, with the immediate consequence of higher computational costs. The best results for the APR criterion for the longitudinal axis parameters have been obtained by using the PS method, but the computational times were about 2.6 times larger than for the SQP method and 3.5 times larger than for COBYLA. Complete clearance results have been obtained using the SQP approach for the small and full parameter sets for the APR criterion (both longitudinal and lateral axes) and the AA criterion (only longitudinal axis). Table 21.4 shows the number of function evaluations (NFE) and the times for the gridding and the optimisation-based search. Note the larger times resulted partly because of more expensive gradient computations. Table 21.4. Performance results for APR and AA criteria analysis Gridding

SQP (Small sets)

SQP (Full sets)

NFE

Time (sec)

NFE

Time (sec)

NFE

Time (sec)

APR (long)

11382

13716

24539

32095

48711

61694

APR (lat)

22710

29935

24354

32256

68810

87008

AA

10421

16202

11377

17696

29592

38022

(long)

In Fig. 21.19 we present the clearance results for the longitudinal axis analysis of APR criterion. As can be observed, there are practically no differences in the results for the gridding-based and optimisation-based approaches. Practically, all points in the flyable flight envelope exhibit Level 1 handling characteristics, as can be seen from the computed worst-case APR values shown in Fig. 21.20. In Fig. 21.21 we present the clearance results for the lateral axis analysis of the APR criterion. This time, the analysis for the full parameter set revealed many points which do not fulfill Level 1 performance specifications, but which where considered as satisfying Level 1 handling characteristics on the basis of the analysis performed using a gridding or optimisation-based search on the small parameter set. These findings are also clearly visible from Figs. 21.22

406

A. Varga Level 1 clearance results for the APR − longitudinal axis

4

5

x 10

(1) Gridding−based search (small parameter set) (2) Optimisation−based search (small parameter set) (3) Optimisation−based search (full parameter set) FC

4.5

3.5

Altitude (ft)

8

(1) α ∈ [ −15, 29]

(1) α ∈ [ −15, 30]

(2) α ∈ [ −15, 29] (3) α ∈ [ −15, 29]

(2) α ∈ [ −15, 30] (3) α ∈ [ −15, 30]

FC5

3

(1) α ∈ [ −15, 30] FC

(2) α ∈ [ −15, 30]

(1) α ∈ [ −13, 29]

(3) α ∈ [ −15, 30]

2

2.5

FC

(2) α ∈ [ −15, 29]

2

6

(3) α ∈ [ −15, 29]

(1) α ∈ [ −8, 29] (2) α ∈ [ −9, 29] (3) α ∈ [ −9, 29]

FC

4

1.5

(1) α ∈ [ −14, 31] (2) α ∈ [ −14,31]

1

(3) α ∈ [ −14, 31] FC

1

0.5 0

FC

3

4

(1) α ∈ [ −6, 30]

FC7

(1) α ∈ [ −2, 12]

(2) α ∈ [ −9, 30] (3) α ∈ [ −9, 30] 0

0.1

0.2

0.3

0.4

(2) α ∈ [ −2, 12] (3) α ∈ [ −2, 12] 0.5

0.6

0.7

0.8

0.9

1

Mach Number

Fig. 21.19. Clearance results for the APR criterion - longitudinal axis Average phase rate analysis − longitudinal

200

Average Phase Rate [degrees/Hz]

Level 3

150

Level 2

100

Level 1

50

0

0

0.2

0.4

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Fig. 21.20. Worst-case APR criterion analysis (longitudinal, full parameter set)

and 21.23, which present the worst-case APRs versus AoA for the small and full parameter sets, respectively. Once again, our results illustrate the power of optimisation-based worst-case search to solve clearance problems with many simultaneous parameters.

21 Optimisation-Based Clearance: The Linear Analysis Level 1 clearance results for the APR criterion − lateral axis

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The analysis for the full parameter set revealed that most of the points in the flyable flight envelope manifest for the worst-case parameter combinations, only Level 2 handling characteristics. This can be clearly seen from the computed worst-case APR values shown in Fig. 21.24.

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In Fig. 21.25 we present the clearance results for the longitudinal axis analysis of the AA criterion. Here, there are no notable differences in the results of the gridding and optimisation-based approaches. The only differences arise because of incidental detection of points which do not belong to

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the flyable envelope for the computed worst-case parameter combinations. As can be seen in Fig. 21.26, practically, all points in the flyable flight envelope exhibit satisfactory handling characteristics in terms of the computed worst-case AA values. Clearance results for the absolute amplitude criterion

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The analysis of the APR and AA criteria revealed several points which are not cleared because the controller gain resulting from the linearisation is zero. This is the case for values α > 29◦ in F C2 and F C3 , α > 30◦ for F C5 and F C8 , and α > 31◦ for F C4 . The probable explanation of these cases is the presence of the AoA-limiter at α = 29◦ in the RIDE controller.

21.5 Evaluation of Results and Conclusions In this section we summarise the main results achieved by our analysis, discuss the main advantages of the optimisation-based approach to clearance, and some requirements to be fulfilled to use this approach. We also briefly indicate some directions for further investigations. A first result which emerged from our analysis is an updated flight envelope of HIRM+ which covers the allowed parametric variations of the model. Both the gridding-based search as well as the optimisation-based search incidentally revealed points for which the worst-case parameter combinations led to violation of conditions (21.1) and (21.2). In both cases, the corresponding points are automatically not cleared. Figure 21.27 collects all results reported in [1] and shows comparatively the flight envelopes which can be flown by HIRM+ for nominal and worst-case values of parameters. HIRM+ true flight envelopes

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Fig. 21.27. Updated flight envelope for HIRM+.

The stability related clearance results for full parameter sets revealed many points where the HIRM+RIDE configuration is not cleared, but which

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were cleared by using gridding or optimisation-based search on the small parameter sets. Note that this is a qualitatively new aspect of the optimisationbased clearance when compared with the classical gridding-based approach. Because of its exponential computational complexity, the gridding-based approach is strictly limited to handle only a small number of simultaneous parameters, and is not able to produce results comparable with those obtained by an optimisation-based search, with reasonable costs. The clearance results for handling criteria revealed that the HIRM+RIDE configuration practically does not fulfill Level 1 performance specifications. Because the handling qualities provides merely indications of possible poor behaviour of the augmented aircraft 1 , these criteria are not strict from the point of view of aircraft clearance. Note that for all additional not cleared points detected in handling criteria analysis, the achieved performance is still Level 2 (see Fig. 21.24) which can be acceptable in certain conditions. The analysis of the APR and AA criteria revealed several points where the controller gain resulting from the linearisation, for values of angle of attack above 29◦ is zero. Since such a behaviour can be explained only by the presence of the α-limiter, all results relying on linearisation of the non-linear HIRM+RIDE configuration for values of angle of attack exceeding 29 ◦ must be cautiously treated. The optimisation-based approach to clearance has two main advantages over the classical gridding-based approach, both clearly illustrated by our analysis. While the classical approach is limited to analysis with at most 8 − 9 simultaneous parameters, the optimisation based approach has no such limitations. Interestingly, in many cases the analysis with the full parameter sets determined parameter combinations where the control configuration is not cleared, although for the small sets, the system is cleared. Note that such cases can not be found by the classical approach. The second important advantage of the optimisation-based clearance is the increased reliability in locating worst-case parameter combinations. While the classical approach evaluates the criteria only in the min/max vertex points of the parameter space, the optimisation-based continuous search found many worst-case parameter combinations lying in the interior of the uncertainty region. The main technical challenge for the applicability of the optimisationbased approach is an efficient and reliable identification of worst-case parameter perturbation combinations/flight condition for the given clearance criteria. Several requirements must be fulfilled for successful usage of this approach. First, adequate parametric aircraft models must be available which allow fast trimming, reliable linearisation and accurate simulation of the closed-loop system. Note that many of the existing models used for nonlinear simulations by the industry, already satisfy these requirements or can be easily adapted for this purpose. Fast trimming and reliable linearisation are of paramount importance for the success of this approach, because these 1

U. Korte, private communication

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computations are necessary at each evaluation of the clearance criteria. A major requirement is the availability of adequate, numerically robust and computationally efficient optimisation software. Since the optimisation problems often involve noisy functions having multiple local minima, alternative methods to the usual gradient search techniques, as for example, derivativefree or global search method must be available. The optimisation-based approach can be seen as a straightforward enhancement of the current industrial practice by replacing the traditional gridding-based search with an optimisation-driven search to locate worst-case parameter combinations. Since the application of this new clearance approach does not require special additional skills from the users, its acceptance by the industry must be a serious option to be considered to improve the clearance process for the next generation of aircraft. There are many aspects of the optimisation-based clearance which need further investigations. One such aspect is the use of more powerful optimisation algorithms which are best suited to the class of NLPs which typically appear in clearance problems. A promising direction is the use of gradientfree methods with fast convergence rates, able to address the minimisation of noisy and expensive functions. For example, trust-region methods working mostly with surrogate function models to perform optimisation, are very efficient in terms of the required number of function evaluations. Therefore, using the recently developed trust-region methods as underlying optimisation tools can drastically improve the costs and reliability of the optimisation-based clearance. Another direction is the use of optimisation algorithms for mixed integercontinuous problems, where some of the variables have discrete variation and others have continuous variation. By using mixed integer/continuous optimisation, it is possible to combine the discrete grid-based search (e.g., for those parameters with known monotonic effects) with a continuous exploration for the rest of parameters, thus increasing the overall efficiency of the optimisation-based search. Using global optimisation techniques to solve clearance problems is worth investigating in depth. For example, for problems with possible multiple local minima, global search can be used in conjunction with local search algorithms to locate good initialisation points. New and very promising developments occur in parallel methods for global optimisation, where a large number of function evaluations can be done in parallel (e.g., in a genetic algorithm to evaluate a new population). With the advent of cheap parallel architecture machines, the high computational costs associated with global search methods can be significantly reduced, thus making their standard usage as clearance tools affordable.

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References 1. A. Varga. Robust stability and performance analysis of flight control laws using optimization-based worst-case search: Linear stability and handling criteria. Technical report, GARTEUR FM(AG11)/TP-119-15, September 2001. 2. A. Varga. Model reduction software in the SLICOT library. In B. N. Datta, editor, Applied and Computational Control, Signals and Circuits, volume 629 of The Kluwer International Series in Engineering and Computer Science, pages 239–282. Kluwer Academic Publishers, Boston, 2001. 3. G. Gru ¨bel and H.-D. Joos. RASP and RSYST - two complementary program libraries for concurrent control engineering. In Prepr. 5th IFAC/IMACS Symp. CADCS’91, Swansea, UK, pages 101–106. Pergamon Press, Oxford, 1991. 4. C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal. Algorithm 778. L-BFGS-B: Fortran subroutines for Large-Scale bound constrained optimization. ACM Transactions on Mathematical Software, 23:550–560, 1997. 5. M.J.D. Powell. A direct search optimization method that models the objective and constraint functions by linear interpolation. In S. Gomez and J.P. Hennart, editor, Advances in optimization and numerical analysis, pages 51–677. Kluwer Academic Publishers, 1994. 6. D. L. Carroll. FORTRAN genetic algorithm (GA) driver. World Wide Web, http://www.cuaerospace.com/carroll/ga.html, 1999.

22 Optimisation-Based Clearance: The Nonlinear Analysis Lars Forssell and Andreas Sandblom Swedish Defence Research Agency, (FOI) System Technology SE-172 90 Stockholm, Sweden [email protected]

Summary. In this chapter the nonlinear clearance analysis of the HIRM+ is presented by considering the nonlinear AoA/nz -limit exceedance criterion. The main idea behind the presented analysis is to reformulate a robust stability or performance problem as an optimisation problem. Nonlinear analysis of the optimisationbased clearance using both local and global search techniques has been performed. Results from using a Sequential Quadratic Program (SQP), a Genetic Algorithm (GA) and the Multilevel Co-ordinate Search (MCS) optimisation algorithm are presented. The results show that the proposed optimisation-based clearance method has a high potential to reduce the computing time and cost associated with the flight clearance process. The results also indicate that the optimisation-based clearance method results in a higher reliability than a traditional grid-based search and has wide applicability.

22.1 Introduction This chapter presents some of the results of the analysis carried out at The Swedish Defence Research Agency (FOI) within the project. The analysis task, according to Chapter 10 is to find, for a given set of flight conditions, the combination of uncertainties that results in the worst violation (if any) of the criteria. If there are violations for a criterion, then the analysis should be able to reveal all parameter combinations of the model uncertainties that give violations for that specific criterion. The presented results summarise the work presented in the GARTEUR report TP-119-19, Optimisation-Based Nonlinear Analysis of the HIRM+, [2]. Our approach closely follows the optimisation-based clearance framework presented in Chapter 7. The analysis has been done using a parametric model developed in MATLAB/Simulink for the nonlinear simulation of the closed-loop HIRM+RIDE control. From the criteria listed in Chapter 10 we address the nonlinear simulation based AoA/nz -limit exceedance criterion. This criterion has been designed to find, for a given speed and altitude, the combinations of uncertainties that might stall the aircraft or exceed its structural stress limit. The criterion and a detailed description of how it should be implemented C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 415-430, 2002. Springer-Verlag Berlin Heidelberg 2002

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are available in Chapter 10. The HIRM and the RIDE-controller used in this analysis are described in Chapters 8 and 9.

22.2 Optimisation-Based Clearance of the AoA/nÿ -Limit Exceedance Criteria The principal idea behind the optimisation-based clearance method is to take a robust stability or performance problem and reformulate it as an optimisation problem. It is based upon the assumption that an optimisation algorithm will find the combination of parameters that causes the largest violation for a given stability/performance criterion, faster and with more accuracy, than a traditional gridding-based search. The analysis relies on the assumption that the analysed criterion has a limiting value defining the boundary between acceptable (i.e. cleared) and nonacceptable (i.e. not cleared) values. This value can be used to define a suitable distance function d(F C, p), see Chapter 7, which serves as an optimisation criterion to find worst-case parameter combinations in an optimisation-based clearance approach. The choice of distance function will have a profound impact on how well posed the resulting optimisation problem will be. The clearance task for the AoA/nz - limit exceedance criterion can be formulated as the question: which combination of uncertainty parameters, given a specific flight condition, will result in the highest α and nz of the HIRM+ when subjected to a prescribed step/ramp input signal? This question can then be regarded as an anti-optimisation problem if α and nz are considered as distance functions that are allowed to vary over a parameter space. Then the optimisation algorithm, that tries to maximise α and nz , will search over the allowed parameter space and find the combination of parameters that results in the highest α and nz , which is then the worst case of the AoA/nz - limit exceedance criterion. The proposed optimisation-based analysis method and how it is applied in the flight clearance process is described in more detail in the tutorial Chapter 7. For the AoA/nz -limit exceedance criterion, two different distance functions have been defined: αmax (F Ci , p) = max α(F Ci , p, t)

(22.1)

nzmax (F Ci , p) = max nz (F Ci , p, t)

(22.2)

t∈T

and t∈T

where the functions α and nz are the output signals derived from the numerical integration of the HIRM+ over the time interval 0 6 t 6 10, for the

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flight condition F Ci ∈ F C, with the model uncertainty parameter vector p defined as: ¤ £ p = XcgUnc IyUnc Cmα Unc CmδT S Unc Cmq Unc .

(22.3)

The given model uncertainties and the given set of flight conditions are described in Chapter 8. In this analysis, only longitudinal uncertainties are considered. The parametric uncertainty space, defined by P , with an aerodynamic reduction factor are defined in table 22.1. Table 22.1. The parametric uncertainty space P with reduction factor (ω = 0.46). pi XcgUnc IyUnc CmαUnc CmδT SUnc CmqUnc [min; max] [−0.15; 0.15] [−0.05; 0.05] [−0.046; 0.046] [−0.018; 0.0180] [−0.046; 0.046]

Fig. 22.1 exemplifies the defined distance functions αmax and nzmax and their corresponding signals α and nz for FC3 with no parametric uncertainties applied, i.e. the nominal case.

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For a given flight condition (F Ci ∈ F C) an optimisation problem can then be formulated: p∗1 = max αmax (p) (22.4) p∈P and p∗2 = max nzmax (p) p∈P

(22.5)

where the solutions p∗1 and p∗2 are the combinations of uncertainty parameters that will result in the highest value of αmax and nzmax . If then for a given solution, αmax (p∗1 ) > αlimit

(22.6)

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the aircraft is stated as not cleared with respect to the AoA/nz -limit exceedance criterion for that given flight condition. A preliminary analysis of the distance functions αmax and nzmax , which are output signals calculated by numerical integration of the ODEs of the HIRM+ in the time domain, were conducted by varying each uncertainty parameter independently. The results for the quick response can be seen in Fig. 22.2, where the effects of the uncertainties upon nzmax are shown. It can be observed that XcgU nc is likely to have the most profound effect on the value of nzmax . It can also be noted that the maximum value of nzmax when XcgU nc is varied, is located, not at the extreme values (-1 and 1) but at 0.8 of the normalised uncertainty parameter. This indicates that the worst-case will be found in the interior of the admissible uncertainty space. Based upon that analysis, it was concluded that the resulting optimisation problems, see 22.4 and 22.5, are nonlinear and continuous. The results also suggest that it might be a concave function, which implies that the stated optimisation problem is convex. This is an important quality of the stated problem, since if the optimisation problem is convex, the solution obtained with a local optimisation algorithm will then be a global optimum. Note that in an optimisation problem, the distance function plus the constraints, i.e. the parameter space where the distance function is defined, together determine whether the stated problem is convex or not. Here, however, it turns out not to be the case (e.g. several local maxima are found by the local search method for each flight condition). In order to solve the transformed flight clearance problem, three different optimisation algorithms where chosen: a Sequential Quadratic Program (SQP), a Genetic Algorithm (GA) and the Multilevel Coordinate Search (MCS) routine.

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The SQP is a constrained local optimisation routine (fmincon) provided in MATLAB’s Optimisation Toolbox. SQP tries to find a local minimum of a scalar function (e.g. αmax ) of several variables using gradient-based search in the admissible parameter space. The routine is described in more detail in [6]. In order to capture all possible local maxima for the stated problems (22.4 and 22.5), each flight-case was solved 32 times, with the initial guess for the SQP-routine chosen as the same corner points in the admissible parameter space, as the ones that were used in the Baseline Solution (BS). The BS, which is a traditional grid-based search, was conducted by applying the criteria to the closed-loop nonlinear simulation of the HIRM+RIDE control configuration, with the model uncertainty parameters p in combination, set to the extreme values of the parameter space P . This resulted in the criterion being applied 32 times with different combinations of uncertainty parameter vectors for each flight condition. This is here referred to as the corner points in the in the hyper-cube spanned by the admissible parameter space P . The worst-case for each flight condition was then found by selecting the highest value for each FC. The full results from BS are described in Chapter 15.

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The general idea behind GA optimisation methods is based upon findings in evolution biology, and is an attempt to mimic, what is called the survival of the fittest, i.e. the principle of natural selection hypothesised by Darwin. The optimisation is initiated with an initial population of parameter vectors pi ∈ P . Here, the initial population was randomly distributed over the admissible parameter space. Genetic operators (GO) that are applied on the population perform the basic search algorithm. The GO that are used to generate new generations are crossover and mutation. Crossover takes two individuals and creates two new ones, while mutation acts on a single individual. The choice of the individuals that will form the next generation, i.e. the individuals that the GO acts on, is based upon probabilistic selection. A probabilistic selection is performed, based upon the fitness of the individuals, so that the better ones have a higher probability of being selected. The specific algorithm used here and its implementation in MATLAB is described in [7]. The main principle behind the MCS algorithm is that the parameter space is divided into smaller subspaces recursively, by branching. The splitting of the parameter space is done in a heuristic non-uniform way, favouring areas where optimal function values are expected. Based upon the variation of the cost function with respect to a certain parameter space component, it is possible to determine both the component to be split and the position of the split. In contradiction with other Branch and Bound methods, the MCS splits only one parameter, each time it iterates. For each split, a base-point is assigned to every new subspace. Each new base point differs from the old one in at least one coordinate. In order to increase the rate of convergence, the MCS algorithm then uses a sequential quadratic program to perform local searches in the final subspaces. The local search starts in the final base-point of each sub-space. According to [8], the MCS algorithm is guaranteed to converge to a global optimal solution, if the objective function is continuous in the vicinity of the global solution. The algorithm is described in more detail in [8]. Fig. 22.3 shows how the optimisation algorithms have managed to find more extreme solutions to the quick ramp AoA/nz clearance task, than the BS. The resulting combinations of uncertainties can be found in Table 22.6, where it can be seen that the optimisation algorithms found solution points within the interior of the parameter space. These solutions are missed by the very sparse search grid (only 32 points) applied by the BS. In order to find solutions with the same accuracy as the optimisation algorithms, with a traditional grid-based search, like BS, it has to be conducted in a grid containing 1010 points. This search will take approximately 5.3125 · 1011 seconds, based upon the computing times presented in Table 22.7. The results presented clearly demonstrate the advantages with the proposed optimisation-based analysis, which searches continuously over the entire uncertainty parameter space, compared to the grid-based search conducted for discrete points (i.e. the corner points) which is done for the BS.

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22.3 Analysis Results 22.3.1 Description of the Criterion The definitions of the clearance criteria are covered in Chapter 10. Here, only the nonlinear AoA/nz -limit exceedance criteria are considered. In the AoA/nz -limit exceedance criteria, all flight conditions are to be identified that, in the pull-up manoeuvre defined below, exceed the AoA/nz limits. The AoA and nz limits are 35◦ and 7 g respectively. The combination of uncertainty parameters which yields the largest violation is to be identified. There are two different aircraft responses which are to be assessed, a full stick rapid pull and a pull in 3 seconds, here referred to as the quick and slow ramp. The input signals can be seen in Fig. 22.4. Both commands must be applied from a trimmed condition of straight and level flight, and the simulation should be run for 10 seconds. The criterion is satisfied, at the given flight condition, if αmax is less than 35◦ and nzmax is less than 7 g. Note that it is also of interest to establish, for each flight condition, every combination of uncertainties that will result

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in a violation, and not just the worst-case. The AoA/nz -limit exceedance criterion is described in more detail in Chapter 10. 22.3.2 Analysis Results for Slow Ramp In Figs. 22.5 and 22.6 the analysis results achieved by the optimisation-based worst-case search with the three optimisation algorithms and the baseline solution are presented. The obtained results for maximum angle of attack, αmax , essentially agree with those obtained with the BS. There is one exception though - for FC3 the SQP found the same worst-case as the BS, at 58.8 ◦ AoA but the MCS found a lower value than the BS, of 54.6◦ . The result obtained with GA is exceptional, with a maximum angle of attack of 69.6 ◦ . The corresponding sets of uncertainties for the maximum angles of attack obtained for FC3 are displayed in Table 22.2. The GA and MCS has found solution points in the interior of the parameter space, while the SQP, which starts its searches in the 32 corner points, has ended up in one of the corner points. For all sets of uncertainties found by the optimisation-based analysis for all flight cases, see [2]. For the maximum normal load factor, the same tendency could be noticed. The results obtained with the optimisation-based search essentially agree

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with those obtained with the BS, except for some maximum normal load factors obtained with the GA. In FC4 and 6 - 8, the GA has found maximum normal load factors that are 0.1 to 0.6 g higher than those found with the other algorithms. The corresponding uncertainty parameters can be found in Tables 22.3 and 22.4 Clearance results slow step response criterion − αmax [deg]

4

5

x 10

α

4.5

SQP

α α

MCS

4

α

BS

= 58.8

= 69.6

GA

FC7

= 54.6

= 58.8

α

SQP

α

FC

GA

3

α

MCS

3.5

α α

SQP

Altitude [ft]

3

FC5 FC2

2.5

GA

α

BS

SQP

α

GA

2

α

BS

= 35.6

= 35.5

= 29.8

α

SQP

= 29.8

MCS

α

= 35.5

α

FC

= 29.8

GA

6

MCS

α

BS

FC

= 29.3

= 29.3

α

= 29.8

1.5

= 27.7

= 27.7

= 35.6

MCS

α α

= 29.3

= 29.3

4

α

1

α

SQP

α

GA

0.5

α

FC

α

0

0.1

0.2

SQP

= 30.0

α

GA

= 30.0

MCS

1

BS

0

α

BS

= 27.7

= 27.7

α

MCS

= 30.0

α

BS

= 30.0

0.3

= 29.8

= 29.8

α

SQP

= 29.8

α

GA

FC8

= 29.8

α

MCS

α

BS

0.4

0.5 0.6 Mach number []

0.7

0.8

= 30.0

= 30.0 = 30.0

= 30.0

0.9

1

Fig. 22.5. Clearance results for the slow ramp and AoA/nz criterion - αmax [deg]

Table 22.2. Clearance results for the slow ramp and AoA/nz exceedance criterion, FC3. Algorithm αmax [deg] SQP 58.8 GA 69.6 MCS 54.6 BS 58.8

XcgUnc [m] -0.1500 -0.1303 0.0256 -0.1500

IyUnc [-] -0.0500 -0.0493 0.0500 -0.0500

CmαUnc CmδT SUnc [1/rad] [1/rad] -0.0460 0.0180 0.0392 -0.0110 0.0460 0.0180 -0.0460 0.0180

CmqUnc Cleared [-] 0.0460 No 0.0460 No 0.0460 No 0.0460 No

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L. Forssell and A. Sandblom Clearance results slow step response criterion − nzmax [deg]

4

5

x 10

n

4.5

SQP

n

GA

n

MCS

4

n

BS

= 1.6

= 1.6

FC7

= 1.6

= 1.6

3

= 10.5

n

MCS

n n

SQP

3 Altitude [ft]

= 9.9

n

GA

3.5 n

FC

5

FC2

2.5

GA

n

BS

GA

2

n

BS

= 3.5

n

SQP

n

FC6

= 1.2

GA

MCS

n

BS

FC4 n

1

n

SQP

n

GA

0.5

n

FC1

BS

0

0.1

0.2

n

GA

= 1.1

MCS

n

SQP

= 1.1

n

= 1.1

n

BS

= 1.1

n

SQP

= 4.3

n

GA

FC8

= 4.3

n

0.4

0.5 0.6 mach Number []

0.7

0.8

= 3.9

= 4.0

MCS

n

BS

0.3

= 6.4

= 6.4

= 4.3

= 4.8

MCS

= 6.4

= 6.7

n

= 1.2

1.5

= 9.9

= 9.9

= 3.5

= 1.2

MCS

n

= 3.5

= 1.2

SQP

n

BS

= 3.5

MCS

n n

0

n

SQP

FC

= 3.9

= 3.9

0.9

1

Fig. 22.6. Clearance results for the slow ramp and AoA/nz criterion - nzmax [g] Table 22.3. Clearance results for the slow ramp and AoA/nz exceedance criterion, FC6. Algorithm nzmax [g] SQP 6.43 GA 6.70 MCS 6.43 BS 6.43

XcgUnc [m] -0.1500 -0.1499 -0.1500 -0.1500

IyUnc [-] 0.0500 0.0500 0.0500 0.0500

CmαUnc CmδT SUnc [1/rad] [1/rad] 0.0460 0.0180 0.0460 0.0180 0.0460 0.0180 0.0460 0.0180

CmqUnc Cleared [-] 0.0460 Yes 0.0460 Yes 0.0460 Yes 0.0460 Yes

22.3.3 Analysis Results for Quick Ramp In Fig. 22.7 the clearance results for the quick ramp AoA/nz -limit exceedance criterion with respect to the maximum angle of attack are presented. For six of the flight cases the results obtained with different optimisation methods match each other, while the results for FC3 and FC4 differ to some extent. One interesting remark is that for FC3, the MCS and the SQP algorithms have found higher values of αmax than the GA, although the solution found by GA is still higher than the BS. The parameter combinations found by the

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Table 22.4. Clearance results for the slow ramp and AoA/nz exceedance criterion, FC7. Algorithm nzmax [g] SQP 9.94 GA 10.51 MCS 9.94 BS 9.94

XcgUnc [m] -0.1500 0.0474 -0.1500 -0.1500

IyUnc [-] 0.0500 0.0500 0.0500 0.0500

CmαUnc CmδT SUnc [1/rad] [1/rad] 0.0460 0.0180 0.0460 0.0180 0.0460 0.0180 0.0460 0.0180

CmqUnc Cleared [-] 0.0460 No 0.0460 No 0.0460 No 0.0460 No

different optimisation algorithms in FC3 and FC4 can be found in Tables 22.5 and 22.6. The analysis results for the quick AoA/nz -limit exceedance criterion with respect to the maximum normal load factor are presented in Fig. 22.8. All the algorithms have resulted in the same values of nzmax . Clearance results quick step response criterion − α

4

5

[deg]

max

x 10

α

4.5

SQP

α

GA

α

MCS

4

α

BS

= 47.3

= 47.0

FC7

= 47.5

= 37.9

α

SQP

α

FC

GA

3

α

MCS

3.5

α α

SQP

Altitude [ft]

3

5

FC2

2.5

GA

α

BS

SQP

α

GA

2

α

BS

= 29.6

= 29.5

= 29.8

α

SQP

= 29.8

MCS

α

= 29.6

α

FC

= 29.8

GA

6

MCS

α

BS

FC4 α

1

α

SQP

α

GA

0.5

α

FC

α

BS

0

0.1

0.2

α

GA

= 30.0

MCS

1

SQP

= 30.0

α

= 30.0

α

BS

= 30.0

0.3

α

SQP

= 90.0

α

GA

FC8

= 77.5

α

0.5 0.6 Mach number []

0.7

0.8

= 29.7

= 29.7

MCS

α

BS

0.4

= 29.2

= 29.2

= 89.3

= 90.0

MCS

= 29.2

= 29.2

α

= 29.8

1.5

= 27.8

= 27.8

= 29.6

MCS

α α

0

α

FC

BS

= 27.8

= 27.8

= 29.7

= 29.7

0.9

1

Fig. 22.7. Clearance results for the quick ramp and AoA/nz criterion - αmax [deg]

426

L. Forssell and A. Sandblom Clearance results quick step response criterion − nzmax [deg]

4

5

x 10

n

4.5

SQP

n

GA

n

MCS

4

n

BS

= 1.6

= 1.6

FC7

= 1.6

= 1.6

n

SQP

n

FC

GA

3

n

MCS

3.5

n n

SQP

Altitude [ft]

3

FC

5

FC2

2.5

GA

n

BS

SQP

n

GA

2

n

BS

= 3.5

= 3.5

= 1.2

n

SQP

= 1.2

MCS

n

= 3.5

n

FC6

= 1.2

GA

MCS

n

BS

FC4 n

1

n

SQP

n

GA

0.5

n

FC1

BS

0

0.1

0.2

n

GA

= 1.1

MCS

n

SQP

= 1.1

n

= 1.1

n

BS

= 1.1

n

SQP

= 4.8

n

GA

FC8

= 4.8

n

0.4

0.5 0.6 mach Number []

0.7

0.8

= 4.0

= 4.0

MCS

n

BS

0.3

= 6.7

= 6.7

= 4.8

= 4.8

MCS

= 6.7

= 6.7

n

= 1.2

1.5

= 10.5

= 10.5

= 3.5

MCS

n n

0

n

BS

= 10.5

= 10.5

= 4.0

= 4.0

0.9

1

Fig. 22.8. Clearance results for the quick ramp and AoA/nz criterion - nzmax [g] Table 22.5. Clearance results for the quick ramp and AoA/nz exceedance criterion, FC3. Algorithm αmax [deg] SQP 47.3 GA 47.0 MCS 47.5 BS 37.9

XcgUnc [m] -0.1500 -0.1463 -0.1500 -0.1500

IyUnc [-] 0.0154 0.0310 0.0500 0.0500

CmαUnc CmδT SUnc [1/rad] [1/rad] 0.0460 0.0180 0.0460 0.0141 0.0460 0.0063 -0.0460 0.0180

CmqUnc Cleared [-] 0.0460 No 0.0458 No 0.0460 No -0.0460 No

22.3.4 Comparison In Fig. 22.3 the quick ramp responses for the worst-cases found by the four algorithms are compared with the nominal case. In this particular flight case (FC3), the three optimisation-based search algorithms have found combinations of uncertainties that result in severe violations of the angle of attack limitation. A grid-based baseline solution of the type used here, only indicates a slight violation of the AoA/nz -limit exceedance criterion. The results show the importance of investigating the handling criteria of the aircraft, not only

22 Optimisation-Based Clearance: The Nonlinear Analysis

427

Table 22.6. Clearance results for the quick ramp and AoA/nz exceedance criterion, FC4. Algorithm αmax [deg] SQP 89.3 GA 90.0 MCS 90.0 BS 77.5

XcgUnc [m] -0.1442 -0.1499 -0.1500 -0.1500

IyUnc [-] 0.0500 0.0424 0.0500 0.0500

CmαUnc CmδT SUnc [1/rad] [1/rad] 0.0460 0.0180 0.0458 0.0180 0.0460 0.0097 0.0460 0.0180

CmqUnc Cleared [-] 0.0460 No -0.0460 No 0.0460 No -0.0460 No

for the grid points of the baseline solution, but over the entire uncertainty parameter space. This implies that the use of a search method, like the optimisation methods used here, is necessary for the type of clearance problems which may have maxima not lying at the extreme values of the uncertainty parameter space. In Table 22.7, the computing times and the required number of function evaluations (NFE) for the different optimisation algorithms are shown. It is clearly seen that although the SQP algorithm was executed 32 times, it is the most efficient algorithm in respect of computing effort, for solving this type of flight clearance problem. A drawback with SQP is that the algorithm can only provide local solutions. Regarding the global algorithms, the GA algorithm generally takes almost four time longer and the MCS algorithm more than eight times longer, to find a solution to the clearance task for one flight condition, compared to the SQP method. On the other hand, the use of the global algorithms generally results in more reliable solutions. Table 22.7. Performance results for FC3 (quick ramp) with respect to AoA. BS SQP MCS GA (32 points) NFE 32 472 986 837 Time (sec) 1700 10500 89600 38900

22.4 Extended Analysis The given analysis task is to find the combination of uncertainties that, for a given flight condition, results in the worst violation of the specified criterion, and if possible, find all combinations of uncertainties that result in violations of the criterion. The proposed method is not just applicable to the analysis of a given FC, it can also be used to give the worst flight condition within the entire envelope. If the optimisation algorithm is allowed to search, not

428

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just over the parameter space, but all over the flight envelope, i.e. Mach number and altitude, this will result in the worst-case flight condition with corresponding uncertainties. A drawback is that when the optimisation-based analysis is performed it will only find a maximum (or minimum) value of the distance function. Either this value is below the limit specified in the criterion, with the result that the aircraft could be cleared with respect to the analysed criterion, or the value is above the specified limit, which indicates that the aircraft is not cleared with respect to the criterion. If the aircraft is concluded to be not cleared for a given flight condition, the analysis will not reveal all combinations of uncertainties that violates the criterion. Step 1

1

Altitude [10000m]

0.8

Altitude [10000m]

0.8 0.6

0.6

0.4

0.4

0.2 0

0.2

0

0.2

0.4 0.6 Mach [M]

0.8

0

1

Step 3

1

0

0.2

0.4 0.6 Mach [M]

0.8

1

0.8

1

Step 4

1

Altitude [10000m]

0.8

Altitude [10000m]

0.8 0.6

0.6

0.4

0.4

0.2 0

Step 2

1

0.2

0

0.2

0.4 0.6 Mach [M]

0.8

1

0

0

0.2

0.4 0.6 Mach [M]

Fig. 22.9. Description of the splitting of the flight envelope (grey areas = cleared).

If information about all flight conditions with corresponding sets of uncertainties that violate the criterion is requested, a single optimisation-based search of the complete flight envelope is not enough. In order to cope with this situation, the idea is to first use the time-saving optimisation-based search for the entire envelope to decide whether the aircraft violates the criterion at any flight condition. If this is the case, the envelope should be split into smaller

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sub-areas (see Fig. 22.9) and new local optimisation-based searches should then be performed in these sub-areas. If there are sub-areas without violations (grey areas in Fig. 22.9) these should be declared as cleared, while the remaining non-cleared sub-areas should be split once again and subjected to further optimisation-based searches. This process of searching for maximum violations, declaring sub-areas as cleared and splitting sub-areas into smaller ones, should continue until the non-cleared areas are small enough for limited local baseline solutions. This approach, based upon ADMIRE and its related results, is described in [9].

22.5 Conclusions We have presented an optimisation-based approach for parametric robustness assessment of flight control systems with respect to nonlinear stability and performance criteria. The main idea behind our approach is to reformulate the stability and performance robustness assessment problem as an optimisation problem to find worst-case parameter combinations. The main aspect of the optimisation-based analysis method (and also of the traditional approach based on simple grid-based search) is that the method is primarily oriented toward finding a worst-case parameter combination that results in the largest violation of a given criterion. Thus, the method is especially well suited to locate points where design weaknesses are present. However, the performed optimisation-based approach cannot address the detection of all parameter combinations which violate a given criterion. Three optimisation methods have been applied to solve the HIRM+ nonlinear clearance problem: the gradient-based local search algorithm SQP, the global search based GA and the hybrid approach MCS, combining local (SQP) and global search methods. The results of our analysis illustrate the following facts: 1. For all flight conditions, the optimisation-based approach produced worstcase parameter combinations which led to performance degradations at least as severe as those obtained with the traditional gridding-based search used in the BS. 2. In some flight conditions, all three methods where able to find worstcase parameter combinations which corresponded to larger degradations in performance than those achieved for the BS. Since these solutions were between the grid points, they could not be localised by a simple gridding-based approach. 3. The cost to be paid for this increased reliability in locating worst-case parameter combinations is a higher computational effort. Note that typically, the required computational effort of using global search approaches such as GA or MCS, is substantially higher than the effort required by local search based methods like the SQP.

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Our analysis has shown that even a local search method like SQP, can be successfully employed to perform the robustness analysis of a class of nonlinear simulation-based stability and performance criteria. An additional gain in reliability, though involving higher computational costs, can be achieved by employing alternative methods using global search strategies, like GA. A hybrid algorithm like MCS, combining local and global search methods, has the potential capability to achieve a good compromise between the higher rate of convergence of local search methods (typical for gradient-based search) and the increased reliability in avoiding local minima of global search based methods. The proposed optimisation-based analysis method has a high potential to solve flight clearance problems, with much higher reliability than a traditional grid-based search, but at the price of a higher cost in terms of computational effort. Still, since our analysis was limited to only a few cases with a relatively small number of uncertain parameters, we cannot draw definitive and general conclusions about the capability of the different algorithms for handling largescale analysis problems.

References 1. GARTEUR FM(EG17). Terms of Reference for GARTEUR Flight Mechanics Action Group FM(AG11) on: New Analysis Techniques for Clearance of Flight Control Laws FM(EG17)M1-v1, 19 February 1999. 2. L. Forssell and A. Sandblom. Optimisation-Based Non-linear Analysis of the HIRM model. GARTEUR/TP-119-19, 2002 3. A. Varga Robust stability and performance analysis of flight control laws using optimization-based worst-case search: Linear stability criteria. GARTEUR/TP119-15 4. F. Karlsson, U. Korte and S. Scala. Selected Criteria for Clearance of the HIRMplus Flight Control Laws,GARTEUR/TP-119-2-A1(addendum to GARTEUR/TP-119-2). GARTEUR FM(AG11), Nov. 1999. 5. D. Moormann and D. Bennett. The HIRMplus Aircraft Model and Control Laws Development. GARTEUR/TP-119-2, 1999. 6. The MathWorks Inc. Optimization Toolbox Users Guide December, 1996. 7. R. Houck,J. Jones and M. Kay. A Genetic Algorithm for Function Optimization: A Matlab Implementation. North Carolina State University. 8. W. Huyer and A. Neumaier. Global Optimization by Multilevel Co-ordinate Search. Kluver Academic Publishers, 1998. 9. L. Forssell and A. Hyden Optimisation Based Worst Case Search of ADMIRE GARTEUR/TP-119-26, 2002 10. A. Varga et. al. Post-Design Stability Robustness Assessment of the RCAM Controller Design Entries GARTEUR/TP-088-35, 1997

23 Industrial Evaluation Fredrik Karlsson1 and Chris Fielding2 1

2

Saab AB Flying/Handling Qualities 581 88 Link¨ oping, Sweden [email protected] BAE SYSTEMS Aerodynamics (W427D) Warton, Preston PR4 1AX, UK [email protected]

Summary. The analysis methods described in Chapters 3 to 7 of this book have been applied to the HIRM+ to determine how the methods would perform in relation to the aircraft flight clearance task. The results, which are summarised in Chapters 15 to 22, were reported in detailed technical reports for evaluation by a team of industrial flight control specialists. Their findings are described with recommendations for future development work.

23.1 Introduction Modern fighter aircraft are designed to be naturally unstable for performance reasons. Therefore, they can only be flown by means of control laws that provide artificial stability. As the safety and performance of the aircraft are dependent on the control system, it must be proven to the authorities that the system is functioning correctly throughout the flight envelope and that the aircraft fulfils its stability and handling requirements, as defined in specific clearance documents. It has to be demonstrated that the aircraft is fit for flight, for normal operation, and for cases where hardware failures might have occurred. The calculations needed to support the flight clearance need to consider the nominal, expected dynamics and those resulting from a range of uncertainties, such as those defined in Chapter 8, which occur due to inaccuracy of predicted aerodynamic data, or measurement errors within the air data system. Such uncertainties are relatively large for a new aircraft that has not flown, but are then reduced with aircraft maturity, as flight-matched information is established. Producing the aircraft clearance is a time consuming and expensive process, which involves making some critical decisions about whether the aircraft is suitable for flight. For a complex problem, a detailed analysis will be required and the large amount of information that is generated will need to be presented in a format that is readily understood by all who are involved in the

C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 433-445, 2002. Springer-Verlag Berlin Heidelberg 2002

434

F. Karlsson and C. Fielding

clearance process. Clearly, suitable criteria are needed to help the designers and customers reach such important decisions.

23.2 Clearance Criteria and Analysis Methods For the purposes of the work carried out by GARTEUR FM(AG11), a set of criteria for clearance of flight control laws was agreed by the group, as defined in Chapter 10. The defined clearance criteria are summarised as: – Worst-case stability margin – to address linear stability, – Worst-case unstable real eigenvalues – also to address linear stability, – Average phase rate and absolute amplitude criterion – to address aircraft handling qualities, including PIO susceptibility, – Largest exceedance of angle of attack and normal load factor limits for a pull-up manoeuvre – to address nonlinear response. The criteria were evaluated at discrete points of the flight envelope as defined by the variations in Mach number, altitude and angle of attack that are relevant to the HIRM+ aircraft simulation. The criteria were considered for possible application with six different analysis techniques, which have been described in Chapters 3 to 7 and 11: – – – – – –

Optimisation Based Worst-case Search, Bifurcation and Continuation Method, µ-Analysis, Polynomial Based Method, ν-gap Analysis, Baseline Solution.

The Baseline Solution addressed all four clearance criteria, thereby providing visibility of the closed-loop characteristics of the HIRM+ and more importantly, a basis against which the other methods could be compared. Each new analysis technique has been applied to selected criteria, depending on method compatibility and the tool developments achievable within the project timescales. The evaluation was carried out by the industrial partners of the FM(AG11) group, with support from the research teams, according to the procedure described below. 23.2.1 Evaluation Procedure The evaluation procedure was formed by using the experience gained from the previous work carried out by the GARTEUR group FM(AG08) and presented in [2]. The evaluations of the different analysis techniques were entirely based on the analysis reports produced within GARTEUR FM(AG11). A number

23 Industrial Evaluation

435

of questions were asked, where the evaluator could rate different aspects of each method by using a scale for the following opinions: unacceptable, unsatisfactory, acceptable, good, very good. The questions on tutorial value, generality, reliability, conservatism, and effort to understand, learn and set up the method, all requested such a rating. The questions on effort are more subjective and mainly rely on the evaluator’s experience. The evaluations were performed by a number of different people with different backgrounds, and from different industrial partners. Each report was evaluated by at least two industrial partners. The evaluators were asked to self-assess their professional experience in flight control development, to allow assessment of any effects on the evaluation. In total, 15 individual evaluations form the basis for the total evaluation, with all the completed questionnaires being used as an input to this chapter. 23.2.2 Industrial Evaluation Guidelines and Questionnaire Each analysis team provided answers to a second questionnaire, aimed at giving a quick overview of each method’s expected capabilities. This was also helpful in the industrial evaluation, by providing focused information to support the detailed technical reports. The questionnaire also gave the analysis teams an indication of the needs of industry. The industrial partners of the FM(AG11) group that performed the evaluation were Airbus Deutschland GmbH, BAE SYSTEMS, Dassault Aviation, EADS Milit¨arflugzeuge, Saab AB, and The MathWorks (UK). The objective of the evaluation was to determine whether new analysis methods make it possible to complete the flight clearance process with less effort and/or more reliability, compared to current methods. The new methods were assessed with respect to the flight clearance criteria that they can readily address, their reliability, generality and conservatism. The amount of effort and degree of competence needed to use and understand the methods were also evaluated. The evaluators were instructed to focus on the positive aspects of the results and to ensure that any criticisms that they made, were constructive and justified, with positive recommendations, where possible. It was essential that the reports were evaluated by persons whose primary experience was in flight clearance and aircraft certification, but persons with other backgrounds also evaluated the reports. Within each industrial organisation, the analysis reports were evaluated by persons with different backgrounds in flight control, covering the following areas: – – – – –

Flight clearance / certification, Flight mechanics / aircraft stability, Flight control law design, Piloted simulation / handling qualities, Flight control computer implementation and testing.

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One of the questions in the evaluation questionnaire asked for the evaluators to provide their background and the information gathered for the ten evaluators is presented in Table 23.1. The fact that different evaluators have different experience allowed this factor to be considered in the interpretation of the evaluators’ comments about the methods. The evaluators’ self-assessment ratings follow the scale: 1. 2. 3. 4. 5.

Very little knowledge and experience, Basic knowledge and some experience, Good knowledge and experience, High level of knowledge and experience, Expert.

Table 23.1. Evaluators’ self-assessed level of expertise. Evaluator: Flight clearance/ certification Flight mechanics / Aircraft stability Flight control law design Piloted Simulations / Handling qualities FCC implementation and testing

1 3

2 2

3 2

4 5

5 4

6 4

7 3

8 3

9 3

10 4

4

4

3

5

5

3

4

4

2

4

5

4

4

5

5

2

4

3

2

5

4

2

2

5

5

4

3

3

4

3

2

2

3

3

4

2

2

1

4

3

23.3 Evaluated Analysis Techniques 23.3.1 Comparison of Analysis Results An overall comparison of analysis results for flight conditions 1-8 is given in Fig. 23.1, which shows the non-cleared angle of attack regions, as determined by the different methods. The criterion used in this comparison is the Nichols criterion for symmetric tailplane stability margins. One can see that the results agree for many points, but there are also some significant differences that require explanation. The results from the optimisation-based method agree with the baseline solution but the method introduces a greater restriction on the negative angle of attack for flight conditions 1 and 2, as it found some worse cases that the baseline solution did not find. All other differences can be explained by the different resolutions that have been used – the optimisation-based method

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used an angle of attack interval of 1◦ in the analysis, whereas, 2◦ was used for the baseline solution. The results from the ν-gap analysis follow the baseline results fairly well, but some trim points were excluded from analysis. The excluded trim points are marked as lighter shaded areas in Fig. 23.1. The reason for the excluded points is that the method found that the frequency of the linearised model was discontinuous with respect to certain parameter values in several cases. That is, those points were neither cleared nor failed in the ν-gap analysis. The method also encountered such points in the middle of the angle of attack range. More details are given in Chapter 18. The µ-analysis, as presented by UST, gave conservative results for several of the flight conditions and particularly for flight condition 1. The method has identified some worst cases in the middle of the angle of attack range for flight conditions 3, 6 and 8. These cases were not detected by the other methods because, they were judged to be marginally acceptable, as opposed to being marginally unacceptable – in practice these could be the same. Therefore, Fig. 23.1 might be regarded as misleading for the UST results because such borderline cases might be clearable, based on more detailed assessments, possibly including nonlinear simulations. This is endorsed by Fig. 19 in Chapter 17, which shows that the UST results are much closer to the baseline than is suggested by Fig. 23.1. For flight condition 2, UST has cleared the region above 30◦ angle of attack, but others have not – again, the earlier figure shows that this is another borderline case and that the UST results are again, close to the baseline. The results of the µ-analysis as presented by ULES are close to the baseline but give slightly conservative results for some flight conditions. The results from the bifurcation method agree very well with the baseline solution. This is to be expected, as the bifurcation method implements the criterion in the same way as the baseline solution. There are some differences caused by the fact that the baseline solution used the 2 ◦ angle of attack intervals and the bifurcation method, in principle, found the exact point at which violation of criteria occurred. It is clear from Fig. 23.1 that although the trends are the same there is not total agreement. If it were a real aircraft clearance, then in some cases, unnecessary flight limitations might be imposed – and in others, flight clearances might be given that might put the aircraft at risk. To ensure reliability, the next step would be to prove or disprove any ambigous clearance results by using more detailed or different analyses. In industry, any such discrepancies would need to be completely understood before flight clearance could be given. 23.3.2 Baseline Solution All defined criteria were addressed in the baseline solution, which was performed in order to provide a benchmark for comparison with the evaluated new methods. The baseline solution was not subjected to an evaluation, but

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Fig. 23.1. Comparison of non-cleared angle of attack regions. The lighter shaded areas represent regions that have been excluded from analysis.

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it was noted that the report did not contain sufficient information on the effort involved, in terms of man-time and computing time. The evaluations in the following pages on this aspect are therefore, essentially judged based on each evaluator’s experience with current methods used in industry. 23.3.3 ν-Gap Technique Generality: the method’s generality was considered to be low by the evaluators, since only the stability margin criterion had been addressed directly. As with any other method relying on linearisation, it cannot address nonlinear response criteria directly. Reliability and conservatism: the formulation of the Nichols stability margin into the ε-margin can affect the reliability but it was considered that any errors introduced would be acceptable. The method introduces some conservatism in the same way as µ-analysis. Understanding and effort : in terms of effort, the method requires considerable knowledge of linear algebra and mathematics in general. Significant expertise is needed to implement the method and to verify the results but from a user’s perspective, if reliable results are obtained with good visualisation and physical interpretation, then the level of understanding needed will be low. Computing time should be low compared to current industrial methods, taking into account that all combinations of class 1 and 2 uncertainties were investigated. If the method could be fully automated and integrated with nonlinear aircraft models within a suitable computing environment, with guaranteed reliability of the results, then it is expected that the method would be more efficient than current methods. Strength: the method has good visualisation of the effects of the parameter variations on robustness, as a function of frequency, taking into account the effects of tolerance size and the design clearance boundary. This potentially helps to simplify the process by eliminating second-order effects. The idea of considering each parameter separately and then estimating the effects of multiple parameter variations is a welcomed simplification. Weakness: the main weaknesses of the method are that the theory is hard to understand and that nonlinear criteria cannot be addressed. 23.3.4 µ-Analysis Techniques Generality: µ-analysis can take uncertainties and variabilities directly into account via a suitable LFT model, but it is restricted to linear models. It may be used to check stability margin and eigenvalues, but time domain criteria are not readily addressed. Reliability: the approach of µ-analysis by using a continuous variation of uncertainty parameters is potentially more reliable than the conventional

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gridding approach where parameters are only checked for discrete combinations. The reliability of the method is very dependent on the quality of the LFT model. Conservatism: the distance between upper and lower bounds will need to be tightened to reduce the conservatism of the method. Conservatism may also be introduced when approximating the trapezoidal-shaped Nichols exclusion zone with an elliptical-shaped zone. Understanding and effort : considerable knowledge of robust flight control theory is needed to implement this method on a new project and good knowledge of the aircraft dynamics and the flight control system is needed to get a good LFT model. Once the method is implemented and operational, the level of understanding to use the method is probably significantly less if it can be used like a ‘black box’ tool. The effort in terms of man power and computing time is comparable with current methods when the method has been set up. The big effort will be to set up the method and create LFT models. Strength: the method’s strength is that it can quickly identify the worst cases, which might be hard to find, or even missed. Another interesting aspect is that whole regions of the flight envelope can be cleared in one test. The robust stability in these regions is also guaranteed for the assumed tolerance set, as a consequence of the method’s conservatism. Weakness: the weaknesses of the method are the effort for learning and implementation, and that time domain criteria cannot be checked directly. Another concern is that with a conservative LFT model, the result could be an unnecessarily constrained cleared envelope. The µ-analysis is best suited as a last step of the design, or at an early stage of the clearance process but reduced conservatism is essential, if this method is to be used in the final clearance of a flight control system. 23.3.5 Bifurcation and Continuation Method Generality and reliability: the stability margin and eigenvalue criteria were both addressed. The bifurcation method was used for the identification of critical regimes only and was supplemented by appropriate classical or advanced methods that computed the analysis criteria. Based on the reported results, estimation of the worst-case uncertainty combinations uses information generated by varying one parameter at a time. Hence, although actual worst-cases were successfully found in the HIRM+ analysis, it is not known whether a true worst case had been identified, reducing the method’s reliability. However, the authors state that confidence in the choice of worst-case uncertainty combination can be substantially improved by iterating their selection process. Conservatism: the bifurcation method does not add any conservatism, compared to current industrial practice. It could potentially find worst cases of single and possibly multiple parameter uncertainty that current methods might not find.

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Understanding and effort : significant effort and expertise are needed to set up the method correctly, requiring additional knowledge of bifurcation and continuation theory, beyond the flight mechanics and classical control theory needed for current methods. The user will need less knowledge of the theory, but some learning will be needed for interpretation of the bifurcation diagrams. Strength: the method has its strength in its good links with the physics of flight, an invaluable attribute for familiarisation with an aircraft’s dynamic characteristics. It is also of benefit that, despite not being demonstrated by the analysis team, nonlinear criteria can be addressed. Worst-cases that lie between the grid points can also be detected by the systematic identification of worst-case manoeuvres and uncertainty values. The method can identify if a subset of the uncertainties is worse than combining all the uncertainties, due to the reduction factors applied to aerodynamic uncertainty parameters. Weakness: there is a weakness in that the method does not guarantee the worst combinations of parameters, but it does identify the critical areas for closer investigation. To enhance current methods and gain additional confidence, the bifurcation approach could be used as a complement to the current gridding approach, to find regions where closer investigation is needed, possibly by increasing the density of gridding in certain areas. A promising role for the method would be to link it to the optimisation approach, with the latter iterating the uncertainty combination suggested by the bifurcation method in order to guarantee the worst-case. 23.3.6 Polynomial Based Method Generality: as presented in this book, the generality is regarded as low since the method only addresses the requirements based on eigenvalue locations, but it is claimed by the authors that the method could potentially address any linear criteria. As presented, the method is well suited for the eigenvalue criterion application. Reliability: the method is considered to be reliable and it should be noted that this method is potentially more reliable than current industrial practice, particularly when conventional analysis assumes that worst-case parameter uncertainties always occur when parameters take on their extreme values. This method is also found to be more reliable than conventional methods, due to the possibility of clearing entire regions of the uncertain parameter space. Conservatism: the method itself is not conservative and might show up weaknesses not previously considered. However, the parameter settings of the adaptive grid generation might introduce some conservatism. Understanding: basic understanding is required for correct and effective application of the method, and it is not considered to be difficult to learn the method within a reasonable time since with appropriate tools, there is no need

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for the user to gain a detailed understanding of the underlying mathematics. However, to be able to successfully tune the parameter setting of the adaptive grid generation, a detailed knowledge of the principles is required. Effort: the development of the trimming and linearisation routines may require some effort to suit this specific method, and the algorithm parameter setting may require some effort to obtain good results. The method can be applied as a batch routine and the necessary man-time should therefore be low. The main effort is with respect to computing power when an increased number of uncertainties is applied. It is questionable whether an extremely fine grid is required, since in a real industrial assessment, computing of eigenvalues with conventional methods has not been a problem. Strength: the method has its strength in its potential for making a continuous investigation of whole regions of uncertain parameters in the entire flight envelope. The types of plot produced, although not tied to this specific method, are very illuminating. Weakness: the weaknesses are that the method is presently restricted to the eigenvalue computation and the required computational power may be high. 23.3.7 Optimisation-Based Worst-Case Search Generality: the method is very general, as it can be used for both frequency-domain and time-domain analyses, for linear and nonlinear criteria. Application of criteria for stability margin, eigenvalues and average phase rate are demonstrated in this book. Reliability: the method is potentially more reliable than those used in current industrial practice, as points in between the vertices of the parameter box are checked. However, the user has to be careful in the choice of optimisation technique. There are examples where some points are cleared with the Sequential Quadratic Programming method, but not if the computationally more expensive Pattern Search method is used. This reduces the reliability. Conservatism: the method itself does not add any conservatism to the results from the conventional methods upon which it relies. The only exception is the use of the elliptical-shaped Nichols exclusion zone instead of the trapezoidal-shaped exclusion zone. Understanding and effort : knowledge of different optimisation methods is required to apply the method on a new project. The interpretation of the optimisation results and the tuning of any parameters unique to the method will require significant knowledge of the applied optimisation method. Using the method for simple optimisation tasks should not require much learning for any engineer with a reasonable mathematical background. For cases of multiple maxima or minima or any steep gradients, the required manual intervention may be considerable and time consuming. The computing time is about the same as for current practice but for the case of a higher number of

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uncertainties, the computing time will be considerably less with the optimisation method. Overall, the effort in terms of computing time and man-time, should be marginally higher than for current methods, but that is the price for increased reliability. Strength: the primary strength of the method is its flexibility in that it can be used to check all nonlinear and linear criteria. The optimisation method can also detect violations that current methods may miss, for example, if they occur between grid points. If combinations of more than five parameters need to be investigated, then it is estimated that computing time becomes significantly lower than with current methods. Weakness: there is a weakness in that the most economic optimisation method is not sufficient to find the worst cases and hence loses some reliability. There are no global optimisation algorithms that guarantee not to miss worst cases. Hence, this method would be used in conjunction with conventional methods and gridding still might need to be performed.

23.4 Conclusions The research activity described in this book is believed to be unique in the sense that the focus of the work is on aircraft flight clearance, from a flight control perspective. As the work progressed, it was evident that there were challenges beyond those that were in the original plans – primarily associated with capturing and communicating what is meant by the ‘flight clearance task’. This was successfully achieved by using the HIRM model from an earlier project [2], by adding uncertainty representations and providing flight clearance criteria to the analysis teams, who worked hard to provide answers to the problem that they were posed. To evaluate the results, a strong industrial team was established, that included experts from some of Europe’s major aerospace companies. Throughout the project, there has been a positive exchange of views and information, and it is this aspect that has made this book possible. One of the main lessons learned from the project is that for any method, good knowledge of aircraft stability, control and flight mechanics is essential to achieve a reliable implementation of the aircraft model and to provide a reliable interface with any analysis method. The effort involved in establishing a sufficiently realistic model should not be underestimated – this includes the ability to reliably trim and linearise the complete model, and to establish any method-specific model interface that is required. All the methods presented have shown benefits, but they also have their drawbacks. The new methods generally give more information and a better coverage of the uncertainty parameter space. However, this can require increased computational effort, and can also introduce some conservatism into the analysis. Many of the methods can be considered as complements to current methods and might initially be used to underpin the current approach,

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with the possibility of replacing some current tools, if sufficient maturity and confidence can be established. It might also be worth looking into combinations of the presented methods to establish any synergy that might be possible through such an integrated approach. The bifurcation analysis has its main benefits in its connection to the physics of flight and is very useful for familiarisation with a new aircraft model. The ν-gap analysis has presented a good way to find the relative effects of the different parameters, as a function of frequency. It can also be used to find critical areas for further analysis, by conventional gridding or optimisation techniques. The more conservative µ-analysis should be considered for initial use in the clearance cycle to detect critical areas of the flight envelope for more detailed analysis. The polynomial-based method, with its continuous investigation of parameters, is appropriate for closer investigation of areas where the system is close to violating limits. The overall view from the industrial evaluation was that the most promising of the methods is the optimisation-based worst-case search, due to its flexibility. It increases the test coverage with reasonable computational effort, and it can be used for all the defined criteria. However, it has a drawback in that its reliability is dependent on the optimisation method used and high levels of confidence require correspondingly high levels of computation. Most of the methods investigated are linear, in the sense that they rely on small perturbations about a trimmed aircraft state and can, therefore, only be used reliably about the steady state. Such methods need to be supported with nonlinear simulations for large perturbations – just as with the current industrial approach. Most of the presented new methods are focused on linear frequency domain criteria, whereas within the current industrial clearance process, most effort is usually required on assessments against nonlinear or time domain criteria. This is partly due to the limitations of linear control theory for covering complex nonlinear systems. Perhaps, more research is needed to determine what might be extracted from the nonlinear model directly, without the use of linearisation. This might be used as a crude ‘first pass’ to determine the areas of design weakness, prior to a more detailed linear investigation (it is noted that the principles involved should be general and could be applied to any control system). Further research is also needed to bridge the gaps between the method researchers’ domain and that of the flight dynamicist, for example, it is not easy to generate an LFT model from uncertainties in a nonlinear aerodynamic dataset and a nonlinear control system – a high degree of automation will be needed. Work is also needed to determine the best means of visualising complex models, prior to the analysis and afterwards, to enable an effective understanding of what the particular flight control problem involves and to demonstrate the quality of any design solution. All the methods have a learning curve that industry will need to tackle in order to be able to successfully apply them and it will take several man-

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months of effort to get a new method running with an existing model. Finally, it should be noted that the software implementation of the method must be written to the appropriate standards, if it is to be used to provide information for aircraft clearances. It must be remembered that the aircraft’s safety is dependent on the analysis and, therefore, the supporting information must be reliable. Acknowledgement. This chapter would not have been possible without the support of the Evaluation Team which included the following personnel: Chris Fielding (BAE-S), Paul Flux (BAE-S), Laurent Goerig (DAv), Jonathan Irving (BAE-S), Rick Hyde (TMW), Fredrik Karlsson (SAAB), Udo Korte (EADS-M), Robert Luckner (Airbus), Andreas Persson (SAAB) and Karin St˚ ahl-Gunnarsson (SAAB).

References 1. L. Rundqwist, K. St˚ ahl-Gunnarsson and J. Enhagen. Rate limiters with phase compensation in JAS 39 Gripen. European Control Conference 1997, Paper 998, Brussels, Belgium, 1-4 July 1997. 2. C. Fielding and R. Luckner. The Industrial View. In J.F. Magni, S. Bennani and J. Terlouw (eds.), Robust Flight Control – A Design Challenge, Springer Verlag London Limited 1997.

24 Considerations for Clearance of Civil Transport Aircraft Robert Luckner Airbus Deutschland GmbH Stability and Control / Flight Dynamics P.O.Box 95 01 09, D–21111 Hamburg, Germany. [email protected]

Summary. This chapter reflects on the previous chapters to present a civil aircraft perspective to clearance. It briefly describes the common areas and differences between the clearance of flight control laws (FCLs) of civil transport aircraft and fighter aircraft. It evaluates and discusses the applicability of the presented FCL clearance techniques for civil aircraft and gives some recommendations for future research.

24.1 Introduction Although this book has been based on a military aircraft flight clearance investigation, the results from the HIRM+ benchmark problem can be interpreted in terms of their applicability to civil aircraft, as the design, implementation and flight control law (FCL) clearance processes have many common characteristics – it is mainly the details that are different. Those common areas and differences will be discussed in the following sections. The industrial FCL clearance process has to be qualified 1 like any other aircraft design and production process. So, each civil and military aircraft manufacturer has established and qualified its own proprietary FCL clearance process, and is constantly improving its processes, striving for better quality and greater efficiency, see [1]. The two key elements for process improvements are: firstly, the capability to master the uncertainties in design models and model data, which are inherent in aircraft development; secondly, the capability to achieve a higher level of automation without compromising insight into the physics and understanding of each design step. These key elements are of the same importance in clearing the flight envelope of a fighter and that of a civil transport aircraft 2 . Therefore, it is not surprising that the main tasks and needs for the FCL clearance process, as described in Chapter 1

2

The term ‘qualified’ refers to the procedural requirements that an aircraft manufacturer has to fulfil for achieving the approval as a design organisation; for example according to JAR 21, see [2]. Here civil transport aircraft or civil aircraft comprise commercial aircraft over 25 tonnes that are certified on basis of FAR/JAR 25.

C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 447-456, 2002. Springer-Verlag Berlin Heidelberg 2002

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2 for fighter aircraft, are essentially the same for civil transport aircraft. In this general aspect, there exists no difference between a civil and a military clearance process. However, if we go into the details, civil and military clearance processes do differ and have their specific needs. It is quite natural that the variations in the process, which are like different dialects in a language, result mainly from the differences between the civil and the military world. Company-specific evolution of the process during previous development programmes and the influence of different engineering schools may contribute to some variations as well. The latter two factors are not dependent on affiliation to the civil or military business and contribute also to process differences between different civil companies. To better understand the differences between civil and military needs, it is necessary to gain a deeper insight into the details. The different needs arise from different certification regulations, differences in manufacturer / customer relations, and different aircraft design objectives that result from aircraft mission requirements, the required flight envelope and the required aircraft configurations. This chapter briefly describes the common areas and differences between the clearance of FCLs of civil transport aircraft and fighter aircraft. It evaluates and discusses the applicability of the presented FCL clearance techniques for civil aircraft and gives some recommendations for future research. To better understand the common areas and the differences between the clearance process for civil transport aircraft and fighter aircraft, the similarities and dissimilarities in customer and certification requirements and in mission requirements are explained in the next two sections.

24.2 Customer and Certification Requirements The general objective in designing a civil transport aircraft can be formulated as follows: ‘the transport of a given number of passengers and/or load from A to B: safely, for minimum cost, taking care of ecological constraints and taking operational needs into account’ (all weather operation, field length required for take-off and landing, highly reliable systems, etc.). The aircraft manufacturers perform market surveys, analyse the requirements of potential airline customers and derive detailed specification to achieve a competitive product. The customers want an aircraft that fits their operational needs and that they can safely operate at minimum costs. Therefore, FCL requirements are not an airlines’ primary concern. They simply expect, when buying an aircraft, that it has adequate flight control laws. This approach is clearly different to the one that has been used for military fighter or transport aircraft in the past: the military customer defined detailed FCL requirements, which were specified in MIL-F8785C [5]. As this approach proved to be expensive and inflexible towards new functions and technologies, it evolved via the standard MIL-STD-1797A [6] to the handbook MIL-HDBK-1797 [7]. Both

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have identical text but by redesignating the standard as a handbook, it is used as a guideline and no longer as a requirement today. Presently, the civil and military approaches are very similar. The civil FCL requirements stem from the airworthiness regulations, which are issued by the airworthiness certification authorities. The aircraft manufacturer has to demonstrate for each newly designed aircraft that it complies with the airworthiness requirements, which are specified by the European and US regulations JAR Part 25 [3] and FAR Part 25 [4]. The certification requirements relating to stability, control and handling qualities, and implicitly to the flight control laws, are not very specific. Therefore, each manufacturer develops its own proprietary design requirements and criteria, often making use of the more specific military design specifications and guidelines, for example [5]. The demonstration of compliance requires the same systematic, understandable and well-documented design process, extensive testing and the proof of correct functioning under all possible operational conditions (including all kinds of failures) in simulation and flight tests, as for a military aircraft.

24.3 Mission Requirements The mission requirements for a civil transport aircraft comprise terminal and non-terminal flight phases with gradual manoeuvres and accurate flight-path control. Those missions are less demanding – in terms of dynamic response – compared to high-manoeuvrability fighter aircraft that have to fly missions, which require precise tracking and rapid manoeuvres. Examples are: in-flight refuelling, air-to-air combat with highly dynamic manoeuvres, short or vertical take-off and landing, take-off and landing on a carrier, low-level flight, and terrain following. This leads to additional assessment criteria for fighter aircraft.

24.4 Uncertainties in Aircraft Dynamics The parameter variations, uncertainties and tolerances that are described in Section 2.3.5 are typical for the civil FCL clearance process as well. As for military aircraft, they comprise: aerodynamic uncertainties, uncertainties of system components such as actuators, and sensors (air data and inertial reference systems, radio altimeters, accelerometers). The numerous uncertainties result in a large number of combinations that have to be investigated. This number is comparable to the military clearance task. Uncertainties in civil aircraft dynamic models are related to prediction capabilities, such as how accurately aerodynamic derivatives can be determined by CFD computations or wind tunnel measurements. They are further related to model fidelity, especially in relation to unsteady aerodynamic

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effects, structural dynamics and system components, such as actuators and sensors. As all design work is done concurrently, uncertainties are typically higher in earlier development phases, but they still exist, when FCL clearance has to be performed. Another class of uncertainties is related to the in-flight measurement accuracy of important parameters, such as the c.g. position, which affect controller gains, and of inertial or air data signals, which are fed back into the FCLs. For civil and military aircraft, the modelling of structural dynamics is another important aspect, but it was not considered in this research activity due to the limited available resources. The fidelity of the structural dynamics model that is used in FCL design and clearance is of particular importance for aircraft with large slender wings and fuselage such as Airbus A340-600 and Boeing 757. Dynamic pressure and mass distribution, which vary with load and fuel consumption, are the main parameters. While from a structural point of view, the slenderness of the structure defines the main difficulties for civil aircraft, the challenge for military aircraft is the high number of store configurations in combination with high load factors. Furthermore, each store configuration has different aerodynamic characteristics.

24.5 Clearance Criteria and Clearance Tasks 24.5.1 Flight Envelope and Flight Conditions The typical flight envelope of a civil transport aircraft for airline operation includes fewer critical areas than the flight envelope of a fighter aircraft in military operations. The margin to minimum speed is higher, the maximum speed lies in the transonic region, whereas it is supersonic for most fighters, and load factors are much lower. As for military aircraft, it must be proven to the certification authorities that the controller is functioning correctly throughout the flight envelope and that the aircraft fulfils the stability and handling requirements, as defined in the specific certification documents [3], [4]. This is achieved by using gridding-based search techniques. Parameters that define a flight condition and have to be investigated include: aircraft mass, moments of inertia, c.g. position, different aircraft configurations (slat/flap settings), air speed or Mach number and altitude. The grid mesh has to be denser for high angles of attack and for high Mach numbers (typically above M = 0.6), as nonlinearities strongly affect dynamic characteristics in those regions. A variation of AoA at a given flight condition (h, M) and weight, which means trimming with steady load factors not equal to 1g (HIRM+: between -3g and + 7g) is not performed for a civil transport aircraft. Steady non-one-g flight conditions, such as turns, in which longitudinal and lateral dynamics are coupled, are investigated in the time domain. Therefore, the number of test conditions (in non-failure cases) for a civil clearance is significantly lower compared to a military clearance.

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Another important difference is the number of failure cases which have to be investigated and cleared. This number is much higher for civil aircraft. It is related to a higher redundancy of control surfaces (e.g. an Airbus A340 has 14 control surfaces for roll control, 5 roll spoilers and 2 ailerons on each wing) and different safety objectives. The civil regulations (JAR/FAR 25.1309, [3], [4]) require, for a flight critical system such as a flight control system, that a catastrophic technical failure is less probable than 10 −9 per flight hour. The equivalent number for a military system is 10 −6 per flight hour. As a single electronic and hydraulic component has a typical failure probability in the order of 10−4 per flight hour, a system has to be built with multiple redundant lanes. The higher redundancy for a civil aircraft implies more possible failure combinations3 . Each combination has to be investigated in the FCL clearance process, leading to a high number of failure cases – typically about 2000 cases for a modern civil flight control system 4 . 24.5.2 Stability Requirements Current civil transport aircraft are designed for high aerodynamic performance, which means minimum drag or minimum fuel consumption. This requirement is contradicting the requirement to have a naturally stable aircraft throughout the whole flight envelope. For example, minimum drag is achieved with rear c.g. positions, which reduces longitudinal stability, and with small vertical tail surfaces, which reduces directional stability and hence, dutch-roll damping. Due to performance reasons, there is a clear trend in current and future designs towards less stable and even unstable configurations. Therefore, as for military aircraft, damping systems of high integrity are required in today’s transport aircraft and they will become increasingly important. However, civil transports do not have manoeuvring requirements that can lead to a reduction of stability margins. Stability requirements (eigenvalues and stability margins) are first checked by linear analysis and subsequently by nonlinear analysis. A method that would directly compute the criteria values from the more accurate nonlinear models would improve the clearance process significantly. 3

4

For example, civil transport aircraft with full-authority electronic flight control systems and no mechanical back-up system have three or four hydraulic systems and two to four engines, which drive electric generators and hydraulic pumps. Fighter aircraft usually have two hydraulic systems and one or two engines. It has to be noted that the requirements are less severe for those failure cases that have a probability of less than 10 −5 per flight hour. This makes it easier to design the control laws, but requires discrimination of different cases including criteria in the design and in the clearance process.

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24.5.3 Handling Qualities Requirements The best references for handling qualities requirements are the military specifications [5], [6], [7] and the background information [8]. While the requirements specified there have been compulsory in the military clearance process in the past, they are used nowadays as guidelines in both civil and military processes. For a civil aircraft, it is less demanding to fulfil the handling qualities requirements, as its dynamic characteristics are normally more benign than those of a fighter aircraft. 24.5.4 Controllability and Manoeuvrability Requirements Controllability and manoeuvrability requirements for military aircraft are explained in Section 2.4. Those requirements are dependent on the flight control philosophy, especially on the flight parameters that the pilot commands by his inceptors. In principle, they are similar to the ones that are applied for civil clearance. Requirements that are specific to civil aircraft are specified in [3], [4] and address, for example, take-off and landing in crosswind conditions, roll rates with all engines operating and with engine failure(s). All controllability and manoeuvrability criteria are checked in off-line (non-realtime) simulations. So, this task is very similar for the civil and the military clearance processes. 24.5.5 Summary of Common Areas and Differences It can be concluded that the civil and the military FCL clearance processes have much in common. The tasks that have to be performed are similar although there are differences in the number and type of flight conditions and failure cases, which have to be checked. The flight envelope of a civil transport aircraft covers less critical regions and the dynamic characteristics are less complex. A notable difference to the military FCL clearance is the preponderance of the nonlinear criteria with respect to linear criteria in civil clearance tasks.

24.6 Applicability of the Presented FCL Clearance Techniques for Civil Aircraft Seven teams have demonstrated five new analysis methods to analyse criteria that they have selected to best demonstrate the advantages of their method. The presented solutions had to be developed in a limited time frame. Therefore, the proposed solutions are not necessarily the best possible ones, and certainly can be further improved. The choice of methods was unrestricted, resulting in a concentration around linear methods. More emphasis on nonlinear criteria and methods would have been favourable from the civil point

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of view. Only the bifurcation and the optimisation-based approaches allow addressing of nonlinear criteria, although it has not been demonstrated for the bifurcation method. The industrial evaluation in Chapter 23 discusses all methods in detail. This discussion is valid for the applicability of these methods in the civil clearance process as well. Some additional remarks regarding the methods´ strengths and weaknesses follow, taking into account two specific aspects of the civil clearance process, namely, the preponderance of nonlinear criteria and the benign dynamic characteristics of the aircraft. As consequence, methods able to address only linear analysis criteria are less suited to improve the current civil clearance process. Nevertheless, since all presented linear methods offer interesting features in addressing robustness aspects, they could play an important role in the design of FCLs. Optimisation-Based Worst-Case Search: The strength of this method is that it can address linear and nonlinear criteria. The use of classical methods to compute criteria values makes a smooth implementation of this method into the current industrial process possible. Nevertheless, there is no restriction to integrate more sophisticated methods. The option to investigate combinations of more than five uncertain parameters simultaneously, without excessive computing time requirements, allows designers to check more combinations than today and is relevant as it has the potential to improve quality and may reduce risk, as critical cases can be detected earlier. Worst-cases or violation of criteria can be found between grid points that cannot be detected with current methods. It makes use of the computing performance that today’s PCs and workstations offer and is certainly a very promising approach. It is applicable to the civil process and addresses its needs very well, as it allows analysis to include nonlinear criteria. Bifurcation Analysis: The strength of Bifurcation Analysis is its ability to determine the critical flight conditions, in which clearance criteria are likely to be violated, from the nonlinear equations of motion. Worst-cases that are in between grid points can be detected. For civil subsonic transport aircraft that typically have benign dynamic characteristics and which have to be cleared for low dynamic manoeuvres only – compared to military combat or aerobatic aircraft – the Bifurcation Analysis cannot demonstrate its potential. The bifurcation diagram may consist of one unbifurcated branch only, which would not give any useful information. µ-Analysis and LFT generation: The strength of µ-Analysis is its ability to analyse whole regions of the flight envelope or to investigate various uncertain parameters simultaneously in one test. This requires the generation of very accurate LFT models. Research on LFT model generation is ongoing. Different possibilities exist, which are classified into analytical and griddingbased methods, as described in Chapter 11. The analytical approaches include

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linearisation, approximation of data tables (aerodynamic derivatives and engine data are typically in 2 or 3 dimensional tables) and model reduction, as the complex aircraft dynamics would otherwise result in systems of orders that are impossible to handle. Each step contains approximations and simplifications, which makes a validation of LFT models necessary. Typically, validation requires comparing a set of linear simulation results with nonlinear simulations. All these steps make LFT generation expensive. Gridding-based LFT generation techniques, which are less expensive, might be better for application to civil aircraft than for fighter aircraft, as civil aircraft have benign dynamic characteristics and the danger of missing worst-case conditions between grid points is lower. If µ-Analysis is used in the clearance process, the validity of the LFT models must be proven. Currently, no automated method exists for validation and the required effort may cause prohibitive costs in an industrial project. An option would be to use non-validated LFT models to find worst-cases and check the results with a validated nonlinear model. Another important requirement is the possibility to trace back worst cases to the physical parameters. LFT generation methods that do not allow this are not suited for clearance. As µ-Analysis is restricted to the analysis of linear criteria, it can only be applied to a part of civil clearance criteria. A trade-off between this restriction, the potential high costs for LFT generation and validation and the potential benefits might be unfavourable for the application of the method in a civil clearance process. But µ-Synthesis is an excellent method for the design of robust FCLs. If it is used during design, it will certainly have a positive effect on the clearance process. Polynomial-Based Method: The strength of the method is its ability to investigate eigenvalue criteria and clear a region in the flight envelope or in the space of uncertain parameters. Computing effort is high and as it is limited to eigenvalue criteria and cannot handle nonlinear criteria, it would not improve the civil clearance process. ν-Gap Analysis: The strength of the method is its ability to find worst cases quickly, with significantly lower computing time requirements compared to the current gridding method, as it has been demonstrated for all class 1 and 2 uncertainty parameter combinations. Nonlinear criteria cannot be addressed and the method is restricted to linear analysis with single-loop and multi-loop Nichols criteria. Therefore, the method is not suited for the civil clearance process. However, the ²-metric and the ν-gap approach offer a very compact way to analyse linear systems with multiple uncertainties. This excellent capability should be considered for use in the design of robust FCLs.

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24.7 Recommendations for Future Research Competition and economic pressure are forcing industry to reduce costs and time to market. Industry is continually restructuring and automating its processes to increase efficiency. Engineers are asked to design faster, ever more complex systems. This will remain as a running demand. Even the analysis methods for FCL clearance are affected by such practical demands. Therefore, methods and techniques that support an efficient automated FCL clearance process – as the optimisation-based worst-case search method – are needed and should attract researchers’ attention. To investigate and demonstrate the benefits of new analysis methods for the civil clearance process, an equivalent civil benchmark problem should be formulated. This benchmark should consist of a civil aircraft model plus controller and a baseline solution. Special emphasis should be placed on the analysis of nonlinear criteria. A combination of these techniques with optimisation–based worst–case search should be considered. Finally, research centres and universities are encouraged to exploit the benefits of other advanced analysis techniques than demonstrated here, e.g. artificial intelligence methods like neural networks and genetic algorithms that have emerged more recently. The strength of these novel methods must not necessarily lie in flight control design or clearance. Other applications, such as in–flight PIO detection [9] or failure detection are imaginable. The potential of such new techniques should be systematically investigated in integrated teams with members from industry, research and universities, in order to reduce the time from method development to its application in industry. Acknowledgements. The author would like to acknowledge the support of Chris Fielding (BAE Systems) and Philippe Menard (Airbus France).

References 1. C. Fielding, R. Luckner. Industrial Considerations for Flight Control. In: Flight Control Systems. Co–published by IEE Control Engineering Series, London, UK, 2000 and American Institute of Aeronautics and Astronautics (AIAA), USA, 2000. 2. Joint Aviation Requirements (JAR) – 21: ‘Certification Procedures for Aircraft and Related Products and Parts’. Joint Aviation Authorities, Hoofddorp, The Netherlands 3. Joint Aviation Requirements (JAR) – 25: ‘Large Aeroplanes’. Joint Aviation Authorities, Hoofddorp, The Netherlands. 4. Federal Aviation Regulations (FAR) Part 25: ‘Airworthiness Standards Transport Category Airplanes’. Federal Aviation Administration (FAA), USA. 5. USAF MIL–8785C: ‘Military Specification, Flying Qualities of Piloted Airplanes’, 1980. 6. USAF MIL–STD–1797A: ‘Military Standard, Flying Qualities of Piloted Vehicles’, 1990.

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7. USAF MIL–HDBK–1797: ‘Department of Defense Handbook – Flying Qualities of Piloted Vehicles’, 1997. 8. C.R. Chalk et al. Background Information and User Guide for MIL–F–8785B (ASG) – Military Specification, Flying Qualities of Piloted Airplanes. AFFDL– TR–74–9, 1969. 9. N. Raimbault, P. Fabre. Probabilistic Neural Detector of Pilot–Induced Oscillations. AIAA Guidance, Navigation, and Control Conference, paper AIAA– 2001–4353, 6–9.8.2001, Montreal, Canada.

25 Concluding Remarks Michiel Selier1 , Rick Hyde2 , and Chris Fielding3 1

2 3

National Aerospace Laboratory NLR Flight Mechanics Department, Anthony Fokkerweg 2, 1059 CM Amsterdam, The Netherlands. [email protected] The MathWorks Ltd., Coombe Lodge, Blagdon, Bristol, BS40 7RG, UK. [email protected] BAE SYSTEMS, Aerodynamics (W427D), Warton, Preston PR4 1AX, UK [email protected]

25.1 Summary of Achievements The main goal of the project was to explore the potential benefits from applying alternative mathematical analysis techniques to the industrial flight clearance process used for military aircraft, in terms of their flight control laws. In the tutorial section of this book, the selected techniques were described in relation to how they could be applied to the clearance process, prior to their application and demonstration on the basis of the HIRM+ clearance problem. This model captured some of the typical features of an aircraft’s flight clearance and included a representative set of analysis criteria, which were to be evaluated for a limited number of points in the flight envelope. As a reference, a baseline solution was generated, using the current industrial methods for flight clearance. Each technique was then applied to show its potential for improving the process, with interesting concepts and innovations being demonstrated. Finally, in Chapters 23 and 24 , the strengths and weaknesses of the techniques have been qualitatively assessed by industry. The research activity has been beneficial for all participants, with a healthy interchange of ideas and information. By undertaking this research, universities and research establishments have been able to familiarise themselves with an industrial task, and experience the practical problems which can arise. In return, European aircraft manufacturers have gained insight into the potential of the state-of-the-art analysis techniques that are available and are undergoing development within the scientific community. By carrying out the research, undergraduate and post-graduate students have had the opportunity to perform part of their academic work on a subject with a high level of industrial relevance. In this respect, it is considered that the objective of this research activity has been achieved and it has been shown that GARTEUR research provides an excellent opportunity to improve the communication and exchange of knowledge between industry and the scientific community. C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 457-459, 2002. Springer-Verlag Berlin Heidelberg 2002

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During the project, three aircraft models were developed to a mature state, for use on this project and on future projects. In particular, the ADMIRE and HWEM are of great potential value for future research activities, since these are very realistic nonlinear aircraft models, which are not usually available to universities. The ADMIRE model is publicly available on the FOI Web site (http://www.foi.se/admire). It is important to note that the analysis techniques that have been applied are still undergoing development and therefore, improvements are to be expected. The knowledge and experience gained from this project will help the scientific world to tailor their analysis techniques, so that they are more specifically aimed at complex nonlinear problems such as the flight clearance task. Until now, only limited research in this area could be carried out, since the relevant information was not available in the public domain. The research effort described in this book should be viewed as a sound exploration of analysis techniques of different levels of maturity, instead of a final comparison between fully mature techniques.

25.2 Future Research In this research activity, relatively new analysis techniques have been applied, to assess the stability and performance of flight control laws by using classical analysis criteria and measures, such as stability margins and eigenvalue criteria. The newer methods often have their own measures and an important point of attention is the interpretation of such new measures. For designers and clearance authorities, it is important that the analysis results can be translated back to the physical world, to enable decisions to be made in relation to the aircraft’s design and its safe operation. If the aircraft does not fulfil its requirements and the controller needs adjustment, it is important that the engineers can deduce what is necessary to solve the problem. The new measures for stability and performance might be regarded as extensions to existing measures, for example the µ-value and ν-gap measures - which build on the concept of gain margin. Furthermore, completely radical analysis methods might be developed, such as those based on artificial intelligence techniques, which could be especially beneficial for nonlinear analysis. This book has concentrated on alternative methods for adding confidence to the flight clearance process, in terms of the thoroughness and efficiency of an aircraft’s closed-loop stability and handling assessment. Another direction, which can bring benefits to the aircraft manufacturers and researchers, is that of enhanced visualisation tools. This aspect was explored as part of the project, leading to a specification for an analysis and visualisation tool to calculate specific clearance criteria over defined flight envelopes. This facility would interrogate and present results already stored in a database, and to enable reliable trimming and linearisation of models defined by graphical

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tools. The development of tool-supported processes for flight control law assessment deserves more attention by the software vendors, and it is hoped that the research performed by this group will accelerate progress in this area. It is considered that all of the methods need further work to provide tight integration with the aircraft’s nonlinear flight dynamics environment in terms of its aerodynamic dataset, equations of motion and flight control system model. Fundamentally, the methods need to be as user-friendly as they can be, to enable their use by a wide number of people with different skills. There is also a requirement for the development of verification tools, to prove the reliability and accuracy of the methods. Most of the methods applied during this project are linear, but the behaviour of the aircraft and its controller is always nonlinear. This is especially true for the high angle-of-attack region, where vortices and local flow separations cause rapid changes in the forces and moments acting on the aircraft. In current industrial practice, many nonlinear simulations are performed in parallel with the linear analysis, to analyse the nonlinear behaviour. A major efficiency improvement for the flight clearance process would be the ability to extract information from the nonlinear model directly, without the need for gridding and linearisation. Potential weaknesses in the controller design that are found in this way, could give a first indication of areas which have to be further analysed by man-in-the-loop simulations or by more detailed linear analyses. As mentioned in Chapter 24, nonlinear analysis and nonlinear simulations are also very important for the clearance of flight control laws for civil transport aircraft. Although this book has focused on military applications, the civil industry is also very interested in approaches that will increase the efficiency of the aircraft’s flight clearance process. For the future, it is recommended that a similar study is performed on a civil benchmark problem, with typical civil aircraft flight clearance criteria. More information on the project can be obtained at the public project website: http://www.nlr.nl/public/hosted-sites/garteur/rfc.html. At this web site, the project’s Technical Publications will be made available as much as possible. The availability depends on the permission of the authors and their organisations to publish the documents into the public domain. For all the Technical Publications, the essential elements have been included in this book.

A Nomenclature and Acronyms

A.1 List of Acronyms Acronym AA ADMIRE AG Airbus AoA APC APR BAE-S BU CAP CIRA c.g. CP DAv DERA DLR DSTL DUT EADS EADS-M FBW FC FCL FCS FE FM FM(AG08) FM(AG11) FOI GA GARTEUR

Definition Absolute Amplitude Aero-Data Model In Research Environment Action Group Airbus Deutschland GmbH Angle of Attack Aircraft-Pilot Coupling Average Phase Rate BAE SYSTEMS The University of Bristol Control Anticipation Parameter Centro Italiano Ricerche Aerospaziali (Italian Aerospace Research Centre) Centre of Gravity Continuation Parameter Dassault Aviation Defence Evaluation and Research Agency Deutsches Zentrum f¨ ur Luft- und Raumfahrt (German Aerospace Centre) Defence Science & Technology Laboratories Delft University of Technology European Aeronautic Defence and Space Company EADS Deutschland GmbH, Military Aircraft Fly-By-Wire Flight Condition Flight Control Laws Flight Control System Flight Envelope Flight Mechanics Flight Mechanics (Action Group 8) Flight Mechanics (Action Group 11) Totalf¨ orsvarets Forskningsinstitut (Swedish Defence Research Agency) Genetic Algorithms Group for Aeronautical Research and Technology in Europe

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Nomenclature and Acronyms

Acronym HIRM HIRM+ HIRM+RIDE HWEM INTA LF-PUM LFT LGM LMI LTI MCS NFE NLP NLR ODE ODE-PUM ONERA PCS PIO PS PUM QG RIDE SA SAAB SQP TMW UBOR UCAM UGM ULES UST

Definition High Incidence Research Model High Incidence Research Model with parameter uncertainty HIRM+ with RIDE controller Harrier Wide Envelope Model Instituto Nacional de T´ecnica Aeroespacial (National Institute of Aerospace Technology) Linear Fractional - Parameter Uncertainty Model Linear Fractional Transformation Lower Gain Margin Linear Matrix Inequality Linear Time Invariant Multilevel Co-ordinate Search Number of Functional Evaluations Nonlinear Programming (problem) Nationaal Lucht- en Ruimtevaartlaboratorium (National Aerospace Laboratory) Ordinary Differential Equation Ordinary Differential Equation - Parameter Uncertainty Model ´ Office National d’Etudes et de Recherches A´erospatiales (National Office of Aerospace Studies and Research) Parametric Continuation Solver Pilot-in-the-loop Oscillation Pattern Search Parameter Uncertainty Model QinetiQ Group Robust Inverse Dynamics Estimation Simulated Annealing Saab AB Sequential Quadratic Programming The MathWorks Universit´e Bordeaux University of Cambridge Upper Gain Margin University of Leicester Universit¨ at Stuttgart

A.2 List of Standard Symbols

A.2 List of Standard Symbols Symbol aij A bij B cij b cbar C CD CL CX CY CZ Cl Cm Cn Clβ Cnr Cnβ Cmq Cmα CmδT S dij D D f Fl (M, ∆) Fu (M, ∆) g h Glong Ix Ixy Ixz Iy Iyz Iz m

Definition Element of the state matrix A State matrix Element of the input matrix B Input matrix Element of the output matrix C Wingspan (m) Mean aerodynamic chord (m) Output matrix Coefficient of drag Coefficient of lift Coefficient of axial force Coefficient of side force Coefficient of normal force Coefficient of rolling moment Coefficient of pitching moment Coefficient of yawing moment Rolling moment coefficient derivative with respect to β Yawing moment coefficient derivative with respect to r Yawing moment coefficient derivative with respect to β Pitching moment coefficient derivative with respect to q Pitching moment coefficient derivative with respect to α Pitching moment coefficient derivative with respect to δ T S Element of the direct transmission matrix D Direct transmission matrix Domain of eigenvalues in the complex plane Frequency (Hz) Lower linear fractional transformation matrix Upper linear fractional transformation matrix Acceleration due to gravity (m/s2 ) Altitude (feet) FCS gain for longitudinal stick (deg/s/mm) x body axis moment of inertia (kg m2 ) x-y body axis product of inertia (kg m2 ) x-z body axis product of inertia (kg m2 ) y body axis moment of inertia (kg m2 ) y-z body axis product of inertia (kg m2 ) z body moment of inertia (kg m2 ) Aircraft total mass (kg)

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Nomenclature and Acronyms

Symbol M M nx ny nz p pdem q qbar qdem R r s S t u VT x, y, z x Xcg y Ycg Zcg

Definition Mach number LFT system matrix Load factor along x-axis (g) Load factor along y-axis (g) Load factor along z-axis (g) Body-axis roll rate (deg/s) Demanded roll rate (deg/s) Body-axis pitch rate (deg/s) Dynamic pressure (kN/m2 ) Demanded pitch rate (deg/s) Real numbers space Body-axis yaw rate (deg/s) Complex variable in Laplace operator Wing planform area (m2 ) Time (s) Input vector Total velocity (m/s) Earth axes positions (m) State vector Centre of gravity location along x-axis Output Vector Centre of gravity location along y-axis Centre of gravity location along z-axis

A.3 List of Greek Symbols Symbol α β β dem γ ∆ δ δA δν δ CD δ CS δ SA δ SE δT δT S

Definition Angle of attack (deg) Angle of sideslip (deg) Demanded sideslip (deg) Flight path angle (deg) LFT uncertainty matrix Vector of system parameters Aileron deflection (deg) ν-gap metric Differential canard deflection (deg) Symmetric canard deflection (deg) Lateral stick deflection (mm) Longitudinal stick deflection (mm) Tailplane deflection (deg) Symmetric tailplane deflection (deg)

A.4 List of Subscripts

Symbol δT D δR µ φ Φ ϕi λ λi λi (A) π Π θ ρ ρ σ σ ω ω Ω ψ

465

Definition Differential tailplane deflection (deg) Rudder deflection (deg) Structured singular value Bank angle (deg) Phase angle (deg) Phase offset for multi-loop stability margin (deg) Free parameter (continuation parameter) Generic i-th eigenvalue Spectrum of matrix A Point of uncertainty space Uncertainty set Pitch angle (deg) Density of air (kg/m3 ) Stability degree Real part of complex eigenvalue Maximum singular value Reduction factor on aerodynamic uncertainty range Frequency (rad/sec) Frequency range Heading angle (deg)

A.4 List of Subscripts Subscript dem sensor trim Unc

Definition Demanded value (e.g., qdem is the demanded value of q) Sensor measurement value (e.g., ψsensor is the sensed value of ψ) Trim value of a variable (e.g., αtrim is the trimmed value of α) Uncertain parameter (e.g., ClβU nc is the uncertainty parameter representing variation in Clβ )

Index


-margin, 58 – optimally weighted, 62 – weighted, 59
scaled -margin – calibration, 328 – definition, 62 µ-analysis, 242 µ-analysis based clearance, 285 – nz as an additional uncertainty, 274 – altitude as variability, 297 – analysis cycle, 309 – AoA as an additional uncertainty, 271 – comparison of results, 307 – computational effort, 301 – continuous AoA covering, 307 – LFT modelling, 280 – LFT validation, 293 – Mach number as variability, 297 – multi-loop analysis, 306 – multivariable root locus method, 270 – Nichols exclusion zone approximation, 289 – parametric interdependency, 295 – short period approximation, 270 – stability margin criterion, 300 ν-gap clearance approach – analysis cycle, 317 ν-gap metric, 64, 66 nz -limit exceedance criterion, 163, 421 absolute amplitude criterion, 160 aerodynamic fitting, 201, 203 AoA-limit exceedance criterion, 163, 421 average phase rate criterion, 160 baseline solution, 249

– nz -limit exceedance criterion, 257 – absolute amplitude criterion, 255 – AoA-limit exceedance criterion, 257 – average phase rate criterion, 255 – stability margin criterion, 250 – unstable eigenvalues criterion, 252 bifurcation analysis – application, 91 – background, 89, 93 – bifurcation diagrams, 92 – numerical continuation, 92 bifurcation and continuation method – clearance analysis cycle, 98, 359 – clearance implementation, 96, 355 – clearance results, 376 – comparison with baseline clearance, 372 – computation times, 376 – conventional implementation, 95 – local bifurcation runs, 365 – nonlinear sensitivity analysis, 365, 370, 372 – selection of worst-case uncertainties, 365, 367, 370, 372, 376, 380 – worst-case assumptions, 102 clearance – block clearance, 17 – flying quality levels, 27 – frequency domain criteria, 27 – handling analysis, 26 – hardware tolerances, 23 – limitations, 25, 32 – linear handling criteria, 15, 27 – linear stability margins, 15 – manned simulation, 29 – nonlinear analysis, 27 – nonlinear simulation, 437, 444

468

Index

– off-line simulation, 28 – response after failure, 29 – stability analysis, 25 – time domain criteria, 27 – visualisation tools, 31 clearance models – flying qualities, 144 – HIRM+, 141 – linearisation, 148 – nonlinear model, 15, 141 – pertubation model, 15 – RIDE, 141 – trimming, 142 compensation parameters, 208 conservatism, 207 D-stability, 79 – robustness, 79 effect of uncertainties, 18 eigenvalue criterion, 48 flat systems, 226 generalised stability margin, 58 gridding based clearance, 108 HIRM+ – actuator dynamics, 130 – aerodynamics, 127 – automated model code generation, 132 – control surface deflections limits, 136 – controls and gust inputs, 125 – discontinuities in model, 317 – dynamics model, 121 – engine dynamics, 129 – flight envelope, 152, 387, 410 – linearisation, 148 – load factor limits, 135 – mass characteristics, 125 – measurements and evaluation outputs, 124 – object model, 123 – sensor dynamics, 131 – simulation, 133 – trimming, 134 – uncertain parameters, 153

industrial evaluation – clearance criteria, 434 – conservatism, 435 – effort, 435, 439 – evaluators’ background, 435 – generality, 435 – reliability, 435, 443 – stability margin, 436 LFT model, 439, 440, 444 LFT modelling, 197 – affine uncertainty representations, 198 – gridding, 198 – LFR-toolbox, 198 – LFT validation, 206 – min-max method, 208 – order reduction, 206 – partial differentiation, 206 – symbolic equations of motion, 201 – symbolic LFT generation, 206 LFT-based uncertainty modelling – compensation parameters, 216 – min-max approach, 213 – nonlinearity compensation, 215 – trends and bands approach, 214 linear fractional transformation (LFT), 38, 169, 237 – affine dependency, 179 – approximation, 189 – equivalence, 183 – extremal values, 190 – input/output equivalence, 183 – Kalman decomposition, 187 – minimality, 185 – Morton’s approach, 179 – normalization, 175 – object-oriented realization, 178 – operations, 178 – order reduction, 183 – polynomial dependency, 181 – rational dependency, 181 – realization, 176 – similarity, 183 – system similarity, 185 – tree decomposition, 181 multi-loop gain/phase offsets, 62

Index Nyquist stability theorem, 39 optimisation based clearance, 385 – absolute amplitude, 402 – average phase rate, 402 – distance function, 385, 389, 398, 403 – stability margin, 389 – unstable eigenvalues, 397 optimisation methods, 112, 387 – COBYLA, 115, 387 – genetic algorithm, 115, 387 – multilevel coordinate search, 116 – pattern search, 114, 387 – projected gradient quasi-Newton, 114, 387 – simulated annealing, 115, 387 – software, 113 – SQP, 113, 387 – trust-region, 115 optimisation software, 387, 388 – architecture, 389 – COBYLA, 388 – genetic algorithm, 388 – pattern search, 388 – projected gradient quasi-Newton, 388 – SQP, 388 optimisation-based clearance, 107, 415 – analysis cycle, 110 – distance function, 107, 416 – genetic algorithm, 418 – multilevel coordinate search, 418 – nonlinear analysis, 415 – SQP, 418

469

parametric uncertainty model, 201, 236 polynomial-based clearance, 76 – adaptive grid generation, 82, 336 – affine dependence, 78 – multiaffine dependence, 80 – multivariate dependence, 80 – region shape computation, 336 – unstable eigenvalues criterion, 333 – worst case areas, 338, 348 polytopic set covering, 80 RIDE, 141 – control laws, 149 – linearization, 149 Riemann sphere, 66 robust inverse dynamic estimation (RIDE), 141 robust performance, 39 skew µ, 298 small gain theorem, 38 stability margin criterion, 44, 155 – multi-loop analysis, 156, 314 – single-loop analysis, 156 star product, 174 structured singular value (µ), 39 – computation, 41 tangent plane, 69 unstable eigenvalues criterion, 158 worst case margins, 207