Actuator Saturation Control (Control Engineering, 12)

ACTUATOR SATURATION CONTROL CONTROL ENGINEERING A Series of Reference Bool~ and Textbool~ Editor NEIL MUNRO, PH.D., D...

Author: Vikram Kapila

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ACTUATOR SATURATION CONTROL

CONTROL ENGINEERING A Series of Reference Bool~ and Textbool~ Editor NEIL MUNRO, PH.D., D.Sc. Professor AppliedControl Engineering Universityof ManchesterInstitute of Scienceand Technology Manchester, United Kingdom

1. NonlinearControl of Electric Machinery,DarrenM. Dawson,Jun Hu, and TimothyC. Burg 2. Computational Intelligence in Control Engineering,RobertE. King 3. Quantitative FeedbackTheory: Fundamentalsand Applications, Constantine H. Houpisand StevenJ. Rasmussen 4. Self-LearningControl of Finite MarkovChains,A. S. Poznyak,K. Najim, and E. G6mez-Ramirez 5. RobustControl and Filtering for Time-DelaySystems,MagdiS. Mahmoud 6. Classical FeedbackControl: With MATLAB, Boris J. Lurie and Paul J. Enright 7. OptimalControl of Singularly PerturbedLinear Systemsand Applications: High-AccuracyTechniques, Zoran GajMand Myo-TaegLim 8. Engineering SystemDynamics: A Unified Graph-CenteredApproach, Forbes T. Brown 9. AdvancedProcessIdentification and Control, EnsoIkonen and Kaddour Najim 10. ModernControl Engineering, P. N. Paraskevopoulos 11. Sliding ModeControl in Engineering,edited by Wilfrid Perruquetti and JeanPierre Barbot 12. Actuator Saturation Control, edited by VikramKapila and Karolos M. Grigoriadis Additional Volumesin Preparation

ACTUATOR SATURATION CONTROL

edited by Vikram Kapila Polytechnic University Brooklyn, NewYork

Karolos M. Grigoriadis University of Houston Houston, Texas

MARCEL

MARCELDEKKER, INC. DEKKER

NEW YORK. BASEL

ISBN: 0-8247-0751-6 This bookis printed on acid-free paper. Headquarters Marcel Dekker,Inc. 270 Madison Avenue, NewYork, NY10016 tel: 212-696-9000;fax: 212-685-4540 Eastern HemisphereDistribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001Basel, Switzerland tel: 41-61-261-8482;fax: 41-61-261-8896 World Wide Web http://www.dekker.com Thepublisher offers discounts on this bookwhenordered in bulk quantities. For moreinformation, write to Special Sales/Professional Marketingat the headquartersaddress above. Copyright© 2002 by MarcelDekker, Inc. All Rights Reserved. Neither this book nor any part maybe reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying,microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Currentprinting (last dig~t): 10987654321 PRINTED IN THE UNITED STATES OF AMERICA

Series Introduction Many textbooks have been written on control engineering, describing new techniques for controlling systems, or new and better ways of mathematically formulating existing methods to solve the everincreasing complex problems faced by practicing engineers. However, few of these books fully address the applications aspects of control engineering. It is the intention of this new series to redress this situation. The series will stress applications issues, and not just the mathematics of control engineering. It will provide texts that present not only both new and well-established techniques, but also detailed examples of the application of these methods to the solution of realworld problems. The authors will be drawn from both the academic world and the relevant applications sectors. There are already many exciting examples of the application of control techniques in the established fields of electrical, mechanical (including aerospace), and cliemical engineering. Wehave only to look around in today’s highly automated society to see the use of advanced robotics techniques in the manufacturing industries; the use of automated control and navigation systems in air and surface transport systems; the increasing use of intelligent control systems in the manyartifacts available to the domestic consumer market; and the reliable supply of water, gas, and electrical power to the domestic consumer and to industry. However, there are currently many challenging problems that could benefit from wider exposure to the applicability of control methodologies, and the systematic systems-oriented basis inherent in the application of control techniques. This series presents books that draw on expertise from both the academic world and the applications domains, and will be useful not only as academically recommended course texts but also as handbooks for practitioners in many applications domains. Actuator Saturation Control is another outstanding entry to Dekker’s Control Engineering series. Neil M~zro

iii

Preface

All real-world applications of feedback control involve control actuators with amplitnde and rate limitations. In particular, any physical electromecha;lical device can provide only a limited force, torque, stroke, flow capacity, or linear/angular rate. The control design techniques that ignore these actuator limits ~nay cause undesirable transient response, degrade the closed-loop performance, and may even cause closed-loop instability. For example, in advanced tactical fighter aircraft with high maneuverability requirements, actuator amplitude and rate saturation in the control surfaces may cause pilot-induced oscillations leading to degraded flight performance or even catastrophic failure. Thus, actuator saturation constitutes a fundamental limitation of ~nanylinear (aa]d even nonlinear) control design techniques and has attracted the attention of nuinerous researchers, especially in the last decade. In prior research, the control saturation problem has been examinedvia. the extensions of optimal control theory, anti-windup compensation, supervisory error governor approach, Riccati and Lyapunov-based local and semi-global stabilization, and bounded-real, positive-real, and absolute stabilization fra~mworks. This prior research literature and the currently developing research directions provide a rich variety of techniques to account for actuator saturation. Furthermore, tremendous strides are currently being made to advance the saturation control design techniques to address important issues of performance degradation, disturbance attenuation, robustness to uncertainty/time delays, domain of attraction estimation, and control rate saturation. The scope of this edited volume includes advanced analysis and synthesis methodologies for systems with actuator saturation, an area of intense current research activity. This volumecovers so~ne of the sig~ificant research advtmcementsmade in this field over the pabst decade. It emphasizes the issue of rigorous, non-conservative, mathematical fornmlations of actuator saturation control along with the development of efficient computational algorithms for this class of problems. The volumeis intended for researchers and graduate students in engineering and applied mathematics with interest in control systems analysis and design. This edited vohune provides a unified forum to address various novel aspects of actuator saturation control. The contributors of this edited volume include some nationally and internationally recogmized researchers whohave

V

vi

Preface

made or continue to make significant contributions to this important field of research in our discipline. Below we highlight the key issues addressed by each contributor. Chapter I by Barbu et al. considers the design of anti-windup control for linear systems with exponentially unstable modes in the presence of input magnitude and rate saturation. The chapter builds on prior work by these authors on uniting local and global controllers. Specifically, the anti-windup design of this chapter enables exponentially unstable satm’ated linear systems to per~brmsatisfactorily in a large operating region. In addition, the chapter provides sufficient conditions for this class of systems to achieve local performance and global stability. Finally, via a mammalflight control example involving an unstable aircraft with saturating actuator.% it illustrates the efficacy of the proposed control design methodologyin facilitating aggressive maneuverswhile preserving stability. Chapter 2 by Eun et al. focuses on selecting the actuator saturation level for small performance degradation in linear de.signs. A nS~el application of a general stochastic linearization methodology, which approximates the I . nonlinearity ~vith a quasi-linear gain, is brought to bear on this saturation problem. Specifically, to determine the allowable actuator saturation level, standard deviations of performa~ce and control in the presence of saturation are obtained using stochastic linearization. The resulting expression ~br the allowable actuator saturation level is shownto be a function of performance degradation, a positive real number based on the Nyquist plot of the linear part of the system, and the standard deviation of controller output. Numerical examples show that by choosing performaz~ce degradation of 10 percent, the actuator saturation level is a weakfi~nction of a system intrinsic parameter, v/z., the positive real number based on the Nyquist plot of the linear part of the system. Chapter 3 by" Hu et al. is motivated by the issue of asymmetric aetuators, a problem of considerable practical concern. In previous research, the authors studied the problem of null controllable regions and stabilizability of exponentially unstable linear systems in the presence of actuator satnration. However, this earlier attempt was restricted to symmetric actuator saturation and hence excluded a large class of real-world problems with asymmetric az;tuator saturation. This chapter addresses the characterization of null controllable regions and stabilization on the null controllable region, for linear, exponentially unstable syste~ns with asynunetrically saturating actuators. First, it is shown that the trajectories produced by extremal control inputs of linear low-order systems have explicit reachable boundaries. Next, under certain conditions, a closed-tra.jectory is demonstrated to be the boundary of the domain of attraction under saturated

Preface

vii

linear state feedback. Finally, it is proven that the domain of attraction of second order anti-stable systems under the influence of linear quadratic control can be enlarged arbitrarily close to the null controllability region by using high gain feedback. Chapter 4 by Iwasaki and Fu is concerned with regional H2 performance synthesis of dynamic output feedback controllers for linear time-invariant systems subject to known bounds on control input magnitude. In order to guarantee closed-loop stability and H2 performance, this chapter utilizes the circle and linear analysis techniques. Whereasthe circle analysis is applicable to a state space region in which the actuator may saturate, the linear analysis is restricted to a state space region in which the saturation is not activated. It is shownthat the circle criterion based control design does not enhance the domain of performance for a specified performance level ~is-a-vis the linear design. Finally, since the performance overbound is inherently conservative, it is illustrated that the circle criterion based control design can indeed lead to improved performance vis-a-vis the linear design. Both fixed-gain and switching control design are addressed. Chapter 5 by Jabbari employs a linear parameter varying (LPV) approach to handle the inevitable limitations in actuator capacity in a disturbance attenuation setting. The chapter begins by converting a saturating control problem to an unconstrained LPVproblem. Next, a fixed Lyapunov function based approach is considered to address an output feedback control design problem for polytopic LPVsyste~n. To overcome the conservatism of LPVcontrol designs based on fixed Lyapunov functior~, a parameterdependent LPVcontrol methodolog3z is presented. It is shown that the LPVcontrol design fi’amework is capable of handling input magnitude and rate saturation. A scheduling control design approach to deal with actuator saturation is also considered. Twonmnerical examples illustrate the effectiveness of the proposed control methodologies. Chapter 6 by Pan and Kapila is focused on the control of discrete-time systems with actuator saturation. It is noted that a majority of the previolm research effort in the literature has focused on the control of continuous-time systems with control signal saturation. Nevertheless, in actual practical applications of feedback control, it is the overwhelmingtrend to implement controllers digitally. Thus, this chapter develops linear matrix inequality (LMI) fbrmnlations for the state feedback and dynamic, output feedback control designs for discrete-time systems with si~mfltaneous actuator amplitude and rate saturation, l~rtherrnore, it provides a direct methodology to determine the stability multipliers that are essential for reducing the conservatism of the weighted circle criterion-based saturation control design. The chapter closes ~vith two illustrative numerical examples which

viii

Preface

demonstrate tile efficacy of the proposed control design framework. Chapter 7 by Pare et al. addresses the design of feedback controllers for local stabilization aald local performance synthesis of saturated feedback systems. In particular, the chapter formulates optimal control designs [br saturated feedback systems by considering three different performance objectives: region of attraction, disturbance rejection, and £2-gain. The. Popov stability theory and a sector model of the saturation nonlinearity axe brought to bear on these optimal control design problems. The bilinear matrix inequality (BMI) and LMIoptinfization frameworks are exploited to characterize the resulting optimal control laws. Commerciallyavailable LMIsoftware facilitates efficient numerical computation of the controller matrices. A linearized inverted pendulumexa~nple illustratez the proposed local £2-gain design. Chapter 8 by Saberi et al. focuses on output regulation of linear systems in the presence of state arid input constraints. A recently developed novel nonlinear operator captures the simultaneo~ma~nplitude and rate constrahlts on system states and input. The notion of a constraint output is developed to handle both the state and input constraints. A taxonomy of constraints is developed to characterize conditions under which various constraint output regulation problems are solvable. Low-gain arid low-high gain control designs including a scheduled low-gain control design are de-. veloped for linear systems with amplitude and rate saturating actuators. Finally, output regulation problems in the presence of right invertible and non-right-invertible constraints are also considered. Chapter 9 by Soroush and Daoutidis begins by surveying the notions of directionality mid windup and recent directionality and windup compensation sche~nes that account for and negate the degrading influence of constrained actuators. The principal focus of the chapter is on stability a~ld perfor~mmceissues for input-constrained multi-input multi-output (MIMO) nonlinear systems subject to directionality and integrator ~vindup. In particular, the chapter poses the optimal directionality compensation problem as a finite-time horizon, state dependent, constrained quadratic optimization problem with an objective to minimize the distance between the output of the unsaturated plant with an ideal controller and the output of the saturated plant with directionality compensator. Simulation results for a MIMO linear time invariant system and a nonlinear bioreactor subject to input constrMnts illustrate that the optimal directionMity compensation improves system performmlce vis-a-vis traditional clipping and direction preservation algorithms. Finally, the chapter proposes an input-output linearizing control atgorith~n with integral action and optimal directionality compensation to handle input-constrained MIMOnonlinear systems

Preface

ix

affected by integrator windup. This windup compensation methodology is illustrated to be effective on a simulated nonlinear chemical reactor. Chapter 10 by Tarbouriech and Garcia develops Riccati- and LMIbased approaches to design robust output feedback controllers for uncertain systems with position and rate bounded actuators. The proposed controllers ensure robust stability and performance in the presence of normbounded, time-varying parametric uncertainty. In addition, this control design methodologyis applicable to local stabilization of open-loop unstable systems. It is noted that in this chapter, the authors present yet another novel approach, viz., polytopic representation of saturation nonlinearities, to address the actuator saturation problem. Twonumerical examples illustrate the efficacy of the proposed saturation control designs. Chapter 11 by Wuand Grigoriadis addresses the problem of feedback control design in the presence of actuator amplitude saturation. Specifically, by exploiting the LPVdesign framework, this chapter develops a systematic anti-windup control design methodology for systems with actuator saturation. In contrast to the conventional two-step anti-windup design approaches, the proposed scheme involving induced/:2 gain control schedules the parameter-varying controller by using a saturation indicator parameter. The LPVcontrol law is characterized via LMIs that can be solved efficiently using interior-point optimization algorithms. The resulting gain-scheduled controller is nonlinear in general and wouldlead to graceful performance degradation in the presence of actuator saturation nonlinearities and linear performance recovery. An aircraft longitudinal dynamics control problem with two input saturation nonlinearities is used to demonstrate the effectiveness of the proposed LPVanti-windup scheme. Webelieve that this edited volume is a unique addition to the growing literature on actuator saturation control, in that it provides coverage to competing actuator saturation control methodologies in a single volume. Furthermore, it includes major new control paradigms proposed within the last two to three years for actuator saturation control. Several common themes emerge in these 11 chapters. Specifically, actuator amplitude and rate saturation control is considered in Chapters 1, 5, 6, 8, and 10. LMIbased tools for actuator saturation control are employed in Chapters 4, 5, 6, 7, 10, and 11. Furthermore, an LPVapproach is used to handle input saturation in Chapters 5 and 11. Finally, scheduled/switching control designs for saturating systems are treated in Chapters 4, 5, and 8. Wethank all the authors who made this volume possible by their contributions and by providing timely revisions. Wealso thank the anonymous reviewers who reviewed an early version of this manuscript and provided valuable feedback. Wethank B. J. Clark, Executive Acquisitions Editor,

x

Preface

Marcel Dekker, Inc., who encouraged this project from its inception. Last but not least, we thank Dana Bigelow, Production Editor, Marcel Dekker, Inc., who patiently worked with us to ensure timely completion of this endeavor.

Vikram Kapila Karolos M. Gri.qo~adis

Contents

Preface

V

Contributors

xvii

Anti-windup for Exponentially Unstable Linear Systems with Rate and Magnitude Input Limits C. Barbu, R. Reginatto, A.R. Teel, and L. Zaccarian 1.1. Introduction ........................... 1.2. The Anti-windup Construction ................ 1.2.1. Problem Statement ................... Compensator ............ 1.2.2. The Anti-windup 1.2.3. Main Result ....................... 1.3. Anti-windup Design for an Unstable Aircraft ........ 1.3.1. Aircraft Model and Design Goals ........... 1.3.2. Selection of the Operating Region .......... 1.3.3. The Nominal Controller ................ 1.4. Simulations ............................ 1.5. Conclusions ........................... 1.6. Proof of the Main Result ................... References ............................... 2

1 1 4 4 6 8 14 14 16 18 21 22 24 28

Selecting the Level of Actuator Saturation for Small Performance Degradation of Linear Designs Y. Eun, C. GSk~ek, P.T. Kabamba, and S.M. Meerkov 33 2.1. Introduction ........................... 33 2.2. Problem Formulation ....................... 35 2.3. Main Result ........................... 37 2.4. Examples ............................. 38 2.5. Conclusions ........................... 41 2.6. Appendix ............................ 41 References ............................... 44

xi

xii

Contents Null Controllability and Stabilization of Linear Systems Subject to Asymmetric Actuator Saturation T. Hu, A. N. Pitsillides, and Z. Lin 47 3.1. Introduction ........................... 47 3.2. Preliminaries and Notation .................. 49 Null Controllable Regions ................... 3.3. 51 3.3.1. General Description of Null Controllable Regions . . 3.3.2. Systems with Only Real Eigenvalues ......... 3.4,

Second Order Anti-stable

Systems ..........

3.5.2. Higher Order Systems with Two Exponentially stable Poles .......................

67 Un-

3.6. Conclusions ........................... References ............................... Regional 7Y2 Performance Synthesis T. Iwasaki and M. Fu 4.1. Introduction ........................... 4.2. Analysis ............................. 4.2.1. A General Framework ................. 4.2.2. Applications--Linear and Circle Analyses ...... 4.3.

Synthesis

............................

4.3.1. Problem Formulation and a Critical Observation . . 4.3.2. Proof of Theorem 4.1 ................. 4.3.3. Fixed-gain Control ................... 4.3.4. Switching 4.4.

54

3.3.3. Systems with Complex Eigenvalues .......... 56 Domainof Attraction under Saturated Linear State Feedback 58 Semiglobal Stabilization on the Null Controllable Region.. 67 3.5.1.

4

51

Control

...................

74 75 75

77 77 78 78 80 83 83 86 87 92

Design Examples ........................ 4.4.1. Switching Control with Linear Analysis .......

95 95

4.4.2. Switching Control with Circle Analysis ........

96

4.4.3. Fixed Gain Control with Accelerated Convergence

100 102

4.5. Further References

Discussion ....................... ...............................

105

xiii

Contents 5

Disturbance Attenuation with Bounded Actuators: Approach F. Jabbari 5.1. Introduction ........................... .......................... 5.2. Preliminaries

An LPV

5.3. Parameter-independent Lyapunov Functions ......... 5.4. Parameter-dependent Compensators and Lyapunov Functions Example ....................... 5.5. Numerical 5.6. Rate Bounds .......................... State Feedback Case ......... 5.7. Scheduled Controllers: 5.7.1. Obtaining the Controller ................ case: Constant Q ............... 5.7.2. Special 5.7.3. A Simple Example ................... 5.8. Conclusion ........................... References ............................... LMI-Based Control of Discrete-Time Systems with Actuator Amplitude and Rate Nonlinearities H. Pan and V. Kapila 6.1. Introduction ........................... 6.2. State Feedback Control of Discrete-Time Systems with Actuator Amplitude and Rate Nonlinearities . .. ........ 6.3. State Feedback Controller Synthesis for Discrete-Time Systems with Actuator Amplitude and Rate Nonlinearities . . 6.4. Dynamic Output Feedback Control of Discrete-Time Systems with Actuator Amplitude and Rate Nonlinearities . . 6.5. DynamicOutput Feedback Controller Synthesis for DiscreteTime Systems with Actuator Amplitude and Rate Nonlinearities ............................... 6.6. Illustrative Numerical Examples ............... 6.7. Conclusion ........................... References ...............................

109 109 111 113 117 118 119 122 126 128 129 131 131

135 135 137 141 146

148 154 157 159

xiv

Contents Robust Control Design for Systems with Saturating linearities T. Pare, H. Hindi, and J. How 7.1. Introduction ........................... 7.2. Problems of Local Control Design ................ 7.3. 7.4. 7.5.

The Design Approach ............... System Model ..........................

’.

Non163 163 167 170

.....

171 173 174

Design Algorithms ....................... 7.5.1. Stability Region (SR) ................. Rejection (DR) .............. 7.5.2. Disturbance

175 177 178 179

7.5.3. Local £2-Gain (EG) .................. 7.5.4. Controller Reconstruction ............... 7.5.5. Optimization Algorithms ............... 7.6. £2-Gain Control Example ................... 7.7. Conclusions ........................... ............................ 7.8. Appendix 7.8.1. Preliminaries ...................... Region of Convergence Design ............ 7.8.2. 7.8.3. Local Disturbance Rejection Design ......... Design ................. 7.8.4. Local £2-Gain References . 8

.

.

179 181 182 182 183 184 184 184

............................

Output Regulation of Linear Plants Subject to State Input Constraints A. Saberi, A.A. Stoorvogel, G. Shi, and P. Sannuti

and

8.1. Introduction ........................... 8.2. System Model and Primary Assumptions .......... 8.3. A Model for Actuator Constraints .............. 8.4. Statements of Problems .................... ......... 8.5. Taxonomy of Constraints ¯ .......... 8.6. Low-gain and Low-high Gain Design for Linear Systems with Actuators Subject to Both Amplitude and Rate Constraints 8.6.1. Static Low-gain State Feedback ............ 8.6.2. A New Version of Low-gain Design .......... 8.6.3. A New Low-high Gain Design .............

189 190 191 194 197 200 202 203 205 207

xv

Contents 8.6.4.

Scheduled

Low-gain

Design ..............

8.7. Main Results for Right-invertible Constraints ........ 8.7.1. Results .......................... 8.7.2. Proofs of Theorems .................. 8.8. Output Regulation with Non-right-invertible Constraints.. 8.9. Tracking Problem with Non-minimumPhase Constraints 8.10. Conclusions ........................... References ............................... 9

208 209 210 212 219 221 223 224

Optimal Windup and Directionality Compensation in InputConstrained Nonlinear Systems 227 M. Soroush and P. Daoutidis 227 9.1. Introduction ........................... 9.2. Directionality and Windup .................. 228 9.2.1. Directionality ...................... 228 Windup ......................... 230 9.2.2. of this Chapter ............. 231 9.2.3. Organization 9.3.

Optimal Directionality Compensation .... " ........ 9.3.1. Scope .......................... ...................... 9.3.2. Directionality Optimal Directionality Compensation ........ 9.3.3.

231 231 232

232 9.3.4. Application to Two Plants ............... 236 9.4. Windup Compensation ..................... 240 9.4.1. Scope .......................... 240 9.5. Nonlinear Controller Design . . .° ............... 240 9.5.1. Application to a Nonlinear Chemical Reactor .... 242 References .............................. . 244 10 Output Feedback Compensators for Linear Position and Rate Bounded Actuators S. Tarbouriech and G. Garcia 10.1. Introduction ........................... 10.2. Problem Statement ....................... 10.2.1. Nomenclature ...................... 10.2.2.

Problem

Statement

...................

Systems

with 247 247 249 249 249

xvi

Contents 10.3. Mathematical Preliminaries 10.4. Control Strategy via Riccati

.................. Equations ............

10.5. Control Strategy via Matrix Inequalities ........... 10.6. Illustrative Examples ...................... 10.7. Concluding Remarks ...................... References ...................... : ........ 11 Actuator Saturation Control via Linear Parameter-Varying Control Methods F. Wu and K.M. Grigoriadis 11.1. Introduction ........................... 11.2. LPV System Analysis and Control Synthesis ......... 11.2.1. Induced £2 Norm Analysis .............. 11.2.2. LPV Controller Synthesis ............... 11.3. LPV Anti-Windup Control Design ............... 11.4. Application to a Flight Control Problem ........... 11.4.1. Single Quadratic Lyapunov Function Case ...... 11.4.2. Parameter-Dependent Lyapunov Function Case . . . 11.5. Conclusions ........................... References ............................... Index

256 259 264 267 269 270

273 273 276 277 277 281 285 288 291 293 295 299

Contributors

C. Barbu

University of California, Santa Barbara, California

P. Daoutidis

University of Minnesota, Minneapolis, Minnesota

Y. Eun

University of Michigan, Ann Arbor, Michigan

M.Fu

University of Newcastle, Newcastle, Australia

G. Garcia

Laboratoire d’Analyse et d’Architecture des Syst~mes du C.N.R.S., Toulouse, France

C. G6k~ek

University of Michigan, Ann Arbor, Michigan

K.M. Grigoriadis

University of Houston, Houston, Texas

H. Hindi

Stanford University, Stanford, California

J. How

Massachusetts Institute Massachusetts

T. Hu

University of Virginia, Charlottesville, Virginia

T. Iwasaki

University of Virginia, Charlottesville, Virginia

F. Jabbari

University of California, Irvine, California

P.T.

University of Michigan, Ann Arbor, Michigan

Kabamba

of Technology, Cambridge,

V. Kapila

Polytechnic University, Brooklyn, NewYork

Z. Lin

University of Virginia, Charlottesville, Virginia

S.M. Meerkov

University of Michigan, Ann Arbor, Michigan

H. Pan

Polytechnic University, Brooklyn, NewYork

T. Pare

Malibu Networks, Campbell, California

A.N. Pitsillides

University of Virginia, Charlottesville, Virginia

R. Reginatto

University of California, Santa Barbara, California

A. Saberi

Washington State University,

xvii

Pullman, Washington

xviii

Contributors

P. Sannuti

Rutgers University, Piscataway, NewJersey

G. Shi

Washington State University,

M. Soroush

Drexel University, Philadelphia, Pennsylvania

A.A. Stoorvogel

Eindhoven University of Technology, Eindhoven, and Delft University of Technology, Delft, the Netherlands

S. Tarbouriech

Laboratoire d’Analyse et d’Architecture des Syst~mes du C.N.P~.S., Toulouse, France

A.R. Teel

University of California, Santa Barbara, California

F. Wu

North Carolina State University, Raleigh, North Carolina

L. Zaccarian

University of California, Santa Barbara, California

Pullman, Washington

ACTUATOR SATURATION CONTROL

Chapter 1 Anti-windup for Exponentially Unstable Linear Systems with Rate and Magnitude Input Limits C. Barbu,

R. Reginatto,

A.R.

Teel,

and

L. Zaccarian University

1.1.

of California,

Santa

Barbara,

California

Introduction

Virtually all control actuation devices are subject to magnitude and/or rate limits and this typically leads to degradation of the nominal performance and even to instability. Historically, this phenomenonhas been called "windup"and it has been addressed since the 1950’s (see, e.g., [25]). To deal with the "windup" phenomenon, "anti-windup" constructions correspond to introducing control modifications when the system saturates, aiming to prevent instability and performance degradations. Early developments of anti-windu p employed ad-hoc methods (see, e.g., [6, 9, 18] and surveys in [1,19,30]). In the late 1980’s, the increasing complexityof control systems led to the necessity for more rigorous solutions to the anti-windup problem (see, e.g., [8]) and in the last decade new approaches have been proposed with the aim of allowing for general designs with stability and performanceguarantees [12, 17, 29, 31,36, 39, 42, 47]. In manyapplications, actuator magnitude saturation is one of the main sources of performance limitation. On the other hand, rate saturation is

2

Barbu et al.

particularly problematic in someapplications, such as modernflight control systems, where it has been shownto contribute to the onset of pilot-induced oscillations (PIO) and has been the’ cause of manyairplane crashes [7, 16, 33]. The combination of magnitude and rate limits is, in general, a very challenging problem and has been considered less in the anti-windup framework. Someresults have been obtained for specific applications [2, 27,40], and someresults can be adapted to this problem(see, e.g., [12, 15, 36]). Additional results on stabilization of systems with inputs bounded in magnitude and rate can be found in [11,23, 24, 37, 38]. On the other hand, these last results don’t directly address the anti-windup problem, where the performance induced by a nominal predesigned controller needs to be recovered by means of the anti-windup design. Additional concerns arise when the plant contains exponentially unstable modes. In this case, the operating region for the closed loop system has to be restricted in the directions of the unstable modes, a fact that is especially important when large state excursion is required as in tracking problems with large reference inputs. This problem has been addressed in the literature, especially in the discrete-time case, in the context of the reference governor approach [12, 15, 26,27] (see also [14, 35]). In [15] and [12], the reference of the closed-loop system is modified to guarantee invariance of output admissible sets, corresponding to the control signal remaining within certain limits (see [13] for details), and set-point regulation for feasible references. Additional results, following a more general approach labeled "measurement governor" are given in [36]. The main drawback of these approaches is that the output admissible sets are controller dependent and the results hold only for initial co~fditions in these sets; in particular, in most applications, the more aggressive the predesigned controller is, the smaller the operating region becomes. This limitation is even more severe when disturbances are taken into account; for example, nothing can be guaranteed whenimpulsive disturbances propel the state of the system out of the output admissible set. In the continuous-time setting, invariant sets independent of the nominal design are exploited in [39]; for unstable systems, in [27] and [39], the system is allowed to reach the saturation limits during the transients, but the reference value is constrained to be within the steady-state feasibility limits at all times. Morerecent results [2,3,26] allow the reference to exceed the steady-state feasibility limits during transients. In recent years, amongother approaches, a number of results on antiwindup for linear systems have been achieved by addressing the problem with the aim of blending a local controller that guarantees a certain desired performance, but only local stability, with a global controller that guarantees stability disregarding the local performance. The combination of these

Anti-windup for Exponentially Unstable Linear Systems

3

two ingredients (according to the approach first proposed in [43]) is attained by augmenting the local design with extra dynamics in a scheme that retains the local controller whentrajectories are small enough and activates the global controller whentrajectories becometoo large, thus requiring its stabilizing action. Such an approach has been specialized for anti-windup designs for linear systems [42, 45] and has been shownto be successful in a number of case studies [21,40, 41,44, 46]. The main advantage in adopting the local/global scheme for anti-windup synthesis is that, by identifying the local design with a (typically linear) controller designed disregarding the input limitation, the corresponding unsaturated closed-loop behavior can be recovered (as long as it is attainable within the input constraints) on the saturated system by means of an extra (typically nonlinear) stabilizing controller (the global controller) designed without any performance requirement. This decoupled design greatly simplifies, in some cases, the synthesis of the nonlinear controller for the saturated plant. In this chapter, the uniting technique introduced in the companionpapets [42, 43] is revisited to design anti-windup compensationfor linear systerns with exponentially unstable modesin a non-local way. In particular, we address the problem of guaranteeing a large operating region for linear systems with exponentially unstable modes(thus improving the local design givsn in [42]) and give sufficient conditions for achieving local performance and global stability with large operating regions. Preliminary results in this direction are published in [3]. As comparedto the results in [12,15, 36], we want to guarantee stability and performance recovery in a region that is not dependent on the nominal controller design. To this aim, instead of focusing our attention on forward invariant regions for the nominal closed-loop system, we consider the nullcontrollability region of the saturated plant and modify the trajectories of the nominal closed-loop system only when they hit the boundaries of (a conservative estimate of) this last region. The resulting anti-windup design is appealing in the sense that the resulting operating region is typically obtained by shrinking the null-controllability region.of the saturated system; 1 since the null-controllability region is unboundedin the marginally unstable directions, 2 it can be extended to infinity in these directions; whereas, in the exponentially unstable directions it needs to be bounded. Subsequently, ,as an example, the proposed scheme is applied to the linearized short-period longitudinal dynamics of an unstable fighter air1Shrinkingthe null-controllability region is desirableto allowa robustnessmargin towarddisturbancesandto avoidthe stickinesseffect described,e.g., in [27]. 2Resultson null-controllabilityof linear systemswithbounded controlscanbe found, e.g., in [22,34].

4

Barbue~ al.

craft subject to rate and magnitude limits on the elevator deflection. The anti-windup design applied to this unstable linear system allows to achieve prototypical military specifications for small to moderate pitch rate pilot commands,while guaranteeing aircraft stability for all pitch rate pilot commands.Due to the large operating region achieved by the anti-windup scheme, the controlled aircraft allows the pilot to maneuveraggressively via large pitch rates during transients. 1.2. 1.2.1.

The

Anti-windup

Problem

Construction

Statement

Consider a linear system with exponentially unstable modeshaving state x E Rn, control input 5 E Rm, measurable output y ~ a p, and performance output z ~ Rq. Let the state x be partitioned as z =: ~ Rn, where the vector Zu ~ Rn" contains all of the exponentially unstable states and zs ~ Rn° contains all of the other states. The state space representation of the system, consistent with the partition of z, is:

Linear plant

z y { 5c

=

Ax+BS:

= =

Cz x + Dz 5 Cyx+DyS,

As [ 0

A12 Au]

x+

5 [Bu] Bs

(1.1)

where all the eigenvalues of Au have strictly positive real part (As can possibly have eigenvalues on the imaginary axis). For system (1.1), assume a (possibly nonlinear) dynamic controller been previously designed to achieve certain performance specifications in the case where the input is not limited. Let this controller (called "nominal controller") be given in the form: Nominal controller

{ &c g(Xc, uc, uc, r), r) Yc -~ = k(xc,

(1.2)

where xc ~ RTM is the controller state, r ~ Rq is the reference input, and uc E Rp, Yc ~ RTM are its input and output, respectively. For the sake of generality, we allow the nominal controller to be nonlinear, although it frequently turns out to be linear. Weassume that the design of the nominal controller (1.2) is such that the closed loop system (1.1), (1.2) with the feedback interconnection 5 -- Yc, uc = y,

(1.3)

An~i-windup for Exponen~iMly Unstable Linear Systems

5

is well-posed (i.e., solutions exist and are unique) and internally stable, and provides asymptotic set-point regulation of the performance output, lim

z~ = r.

Wealso assumethat, for each constant reference r, there exists an equilibrium (x*, x~) for (1.1), (1.2), (1.3), that is globally asymptotically and we define (x*, y~) =: E(r),

(1.4)

as the corresponding state-input pair. Notice that the internal stability assumptionimplies that the plant (1.1) is stabilizable and detectable. Throughout the chapter we refer to the closed-loop system (1.1), (1.2), (1.3) as "nominal clos ed-loop syst em". Weaddress the problem that arises when the actuators’ response is limited both in magnitude and rate. The rate and magnitude saturation effect can be modeled(similarly as in [40] and [2]) by augmentingthe plant dynamics with extra states 5 E Rmsatisfying the equation: ~= Rsgn(Msat(-~)

-5),

(1.5)

where the functions sgn(-) and sat(.) are the standard decentralized TM sign and saturation functions, Mand R are positive numbers, and u E R is the input to the actuators before saturation. Since the design of the nominal controller disregards the magnitude and rate limits, instability can arise if that controller is connected in feedback with the actual plant (1.1), (1.5), especially because the plant contains exponentially unstable modes. On the other hand, by assumption, the performance induced by the nominal controller is desired for the actual plant (1.1), (1.2) and should be recovered whenever possible. Thus, our antiwindup design problem is to accommodatethe requirements of respecting as muchas possible the performance induced by the local controller, while guaranteeing stability of the closed-loop system in the presence of magnitude and rate limits, without restricting the magnitude of the reference signal a priori. In the next sections we recall the state of the art for the particular anti-windup approach initiated in [43] and make further contributions to that design methodologyespecially suited for MIMO exponentially unstable linear systems subject to magnitude and rate limits.

6 1.2.2.

Barbuet al. The Anti-windup

Compensator

In recent years, a number of results on anti-windup design for linear systems have been achieved following the guidelines in [43]. The underlying strategy is to augment t.he nominal controller with the dynamical system (called anti-windup compensator)

Anti-windup

(1.6)

compensator

where ~ = [~s T ~uT]T E RTM × Rn~’, v = [vT~ vT2]T E Rm × Rp, and xu, Yc as in equations (1.1), (1.2), and to consider the system resulting from (1.1), (1.2), (1.5), (1.6) with the interconnection conditions u=yc+vl,

uc=y+v~.

(1.7)

The anti-windup design described in this chapter relies on the availability for measurement of the exponentially unstable modes, although to provide such information full state measurement might be required. Nevertheless, if the state of the plant is not available for measurementand the disturbances are small, a fast observer can be used. Throughoutthe chapter, we will refer to the system (1.1), (1.2), (1.5), (1.6), (1.7) as "ant i-windup clos ed-loop syst em". Figure 1 sh ows the block diagram of the anti-windup closed-loop system, which can be recognized as a natural extension of [39] for the case when the substate x~ is available for measurementand both magnitude and rate limits are present. Z

Aircraft.

Figure 1: Block diagram of the anti-windup scheme. Someof the critical issues that arise in the design of the anti-windup compensatorare briefly discussed in the following.

Anti-windup for Exponentially Unstable Linear Systems

7

Exponentially unstable systems. A basic issue arising with exponentially unstable plants is that global asymptotic stability cannot be achieved, because the null-controllability region of the plant is boundedin the directions of the exponentially unstable modes. Hence, the results are non-global and the goal is to obtain a large operating region for the closed-loop system without significantly sacrificing performance. The results in [42] apply to exponentially unstable linear systems but only for the solution of the local anti-windup problem, thus not computing explicitly the operating region and possibly resulting in conservative designs. Morerecently, based on [43], a more explicit construction for exponentially unstable plants with only magnitude saturation was given in [39]. Magnitude and rate saturation. The early anti-windup developments illustrate the importance of magnitude saturation in control applications. On the other hand, rate saturation plays a similar role in terms of the effects introduced in the system. For instance, in flight control problems, it has been remarkedin [4, 27, 28] howthe instabilities and/or performance ¯ losses due to windupare generated more frequently by the rate limits than by the magnitude limits. The combination of magnitude and rate limits or even general state constraints is a more challenging problem and has been addressed more in the discrete-time setting (see, for instance, [12, 26, 27]) than in the continuous-time case. The case of both magnitude and rate saturation is addressed in continuous time in [43] and applied to asymptotically stable plants in [40] and [41]. Reference values. Usually (see, e.g., [12, 39]), whenthe plant contains exponentially unstable modes, the reference signal is not allowed to take large values, although these wouldgenerate (at least for a limited amountof time) feasible trajectories for the saturated system. In [26], the problemof allowing large references during transients has been addressed in the context of the reference governor. As pointed out in [26], by allowing the reference to be arbitrarily large, better transient performance for the closed-loop system may be achieved. In [2], arbitrarily large references are allowed during the transients for a particular exponentially unstable plant. It is shown there that the performance of the saturated system is improved adding this extra degree of freedom (nevertheless, due to boundedness of the null-controllability region, the reference cannot be arbitrarily large at the steady state). The main contribution of the approach described in this chapter is in the fact that the resulting anti-windup compensation allows for arbitrarily large references (at least during the transients) for exponentially unstable linear systems when both rate andmagnitude saturation are present at the plant’s input.

8

Barbu et al.

1.2.3.

Main Result

Given a nominal controller and a plant with input magnitude and. rate saturation, in this section we give a design algorithm that, on the basis of a desired operating region for the closed-loop system, and for a given stabilizing static feedback that satisfies certain assumptions, provides an anti-windup compensatorthat achieves stability and allows arbitrarily large references for the saturated system, guaranteeing restricted regulation for any reference outside the operating region. The following definition will be useful in the rest of the chapter. Definition 1.1. The null-controllability region 12 for system (1.1), (1.5) is the subset V C Rn x Rm of the state space such that for any initial condition in ]2, there exists a measurable function u : R>o-~ R that drives the state of the system asymptotically to the origin. Remark 1.1. A desirable property of the closed loop system is to have an operating region as large as possible. However,it is not always desirable to get very close to the boundaryof the null-controllability region. Indeed, assumethat there exists a locally Lipschitz controller that renders the nullcontrollability region forward invariant. Then, necessarily, the boundaryof the null-controllability region is an invariant set and, by continuity of solutions with respect to initial conditions on compacttime intervals, the closer the plant state gets to this boundary, the longer it will take to moveaway from it. Werefer to this behavior as the "stickiness effect" .3 It is desirable then to define an "anti-sticking coefficient" and tune the anti-windup compensatorusing a conservative estimate of the null-controllability region, which guarantees that the trajectories of the system stay far enough from the boundaryof the null-controllability region, thus improving the resulting performance. Wefirst specify a region 4/g C Rn" × Rm where we want the exponentially unstable modesand the inputs of the closed-loop system to operate (accordingly to anti-sticking requirements and/or performance specifications). Wespecify this region to be a compactset. Then, we assume that a stabilizing static nonlinear state feedback 7 is given that guarantees the first or both of the following properties to hold: 1. positive invariance of the set Lt for the plant with input magnitude and rate saturation. 3Thiseffect has beennoticedin a numberof applications(see, e.g., 4Thenull-controllabilityregionis bounded onlyin the subspaceof the exponentially unstablemodes[22], so weonlyneedto specifythe operatingregionin that subspace.

Anti-windup for Exponentially Unstable Linear Systems

9

2. convergence to a set-point in After/A and 3’ are chosen, the last ingredient for the design of the antiwindup compensator is the policy to follow when the closed-loop system is driven by a reference whose steady state value corresponds to an infeasible equilibrium for the saturated system (namely, a value r corresponding to state-input pair (x*~,x*u, 5*) = E(r) such that (x~, 5*) ~/2). To this aim, a function 7) that mapsthe infeasible set-point to a feasible one will be defined. A typical choice for P is to "project" the infeasible set-point to a feasible point that is, in somesense, "close" to the infeasible one. However, the reference limiting action achieved by 7) is not used during the transients but only at the steady state. This strategy allows to completely recover, on the saturated system, the nominal responses (even to infeasible references) for the maximaltime interval allowable within the specified operating region and the saturation limits. The following statements formally define the requirements described above.

Definition 1.2. Define the equilibrium manifold £ C Rn x Rm as the set of all the state-input pairs (x, 5) of the linear system (1.1) associated with an equilibrium of the nominalclosed-loop system, 5 i.e. (with reference to equation (1.4)), £ := {(x, 5) ¯ n xRm: ~r ¯ R q s.t . (x, 5) = E(r )} . (1.8) Let the pair/3, 5c be such that ~ is a compact strict subset of 5/and R’~" x Rm-~ [0, 1] is a continuous function satisfying 6 c (Xu,5)¯9 /3(xu,5) := 0, 1,if if(xu,5) ¯ --~.

(1.9)

Let 9vu be the projection of ~- in the xu direction, i.e.,

:= {xu ¯ R : ¯ s.t. (xu, 5)

(1.10)

The role played by/3 and 5r is to guarantee that the nominal performance is preserved only when (Xu, 5) ¯ .T. Outside 5c, the anti-windup scheme modifies the nominal control action to guarantee forward invariance of b/. A possible choice for the function fl, whenff is a given compact 5In general, £ is a subset of the set of the equilibria for the open-loop plant. 6Given a set ~4, denote with .4 c the complement of ~l and with ~ the closure of ~4.

10

Barbu et al.

strict subset of 5/, is 7

fl(xu,5)

dist~:(xu, 5) } := min 1, ~i~@dist~(z)

Wenowformally state the requirements on the functions 3‘ and P. Property 1. Given 5/ E Rn~’ × Rm, let £, ~-, and 9cu be as in Definition 1.2. The function P : Rn" -~ 2Fu is a continuous function such that (1.11) The continuously differentiable function 7 : Rn" x Rm × Rn" × Rn × Rm -~ Rm is such that, for each value r ~ Rq and the corresponding state-input pair (~:;, 2~, ~) E(r), and fo r any in itial co ndition (xs(O), xu(O), 5(0)) e Rn.~ ×/~, the following properties are satisfied: 1. for anyx u-* ~ Rn" m, and 5, ~ E R 3‘(xu, 5, Xu, O, Yc) = Yc , 2. for any choice of the asymptotically vanishing functions el(t), e3(t) and e4(t), the feedback control law for system (1.1),

e2(t),

u = 7(xu, ~, P(2~ + e~), z - 2" + e2, ~)~ + e3) + ~(x~,, 5)e4 (1.12) is such that 5/is positively invariant and all trajectories are bounded. 3. the feedback control law (1.13) guarantees asymptotic stability of an equilibrium ( x* .~, z* 5*) ~ ~’°x 5c for system (1.1), (1.5), with x~ = :P(5:~) and with region of attraction including R~ × 5/; Remark 1.2. In this work we do not pursue a general construction for 3’ satisfying the requirements of Property 1. Weremark, however, that it is muchsimpler than the design of the whole anti-windup co~npensator; as a matter of fact, the design specific~/tions for ~, do not entail performance 7Wedefinethe distanceof a point p froma set .4 as distA(p) := zigfA[p - z[.

Anti-windup for Exponentially Unstable Linear Systems

11

requirements and are mainly related to the behavior of the closed-loop system on the boundary of/~/. Moreover, different choices of/4 facilitate the design of ~/ by exploiting the strong relation between them. Many results on set invariance and (robust) stabilization of constrained linear systems available in the literature can be used for the design of "~ for a specific problem[5]. Remark 1.3. In the special case when the plant has no poles on the imaginary axis, since all the modes of xs are asymptotically stable, the boufldedness of the xs componentof the state is guaranteed by the boundedness of the input. ~ Hence, item 2 of Property 1 relaxes to the only requirement that the control law (1.12) renders/4 forward invariant (thus implicitly keeping the xu states bounded). In turn, since the function evaluates to zero on the boundary of/4, this is equivalent to asking that the simplified control law u ---- ~f(xu, 5, T~(~C~~l(t)), x -- ~* T ~2(t), ~ + ~3(t )), (1.1 makesN forward invariant for any asymptotically vanishing functions 5~ (t), ¢~(t) and Now,assumethere exists a state feedback ~l(Xu, ~, xu, yc), designed on the basis of the exponentially unstable componentsxu of the plant, that: 1. s~tisfies ~(Xu, ~, xu, Yc) = Yc for all xu ~ R~ and ~, Yc ~ Rm; 2. guarantees forward invariance of 3. stabilizes an equilibrium (x~, ~*) in ~, with region of attraction containing Then the following choice for ~ guarantees items 2 and 3 of Property 1 to be satisfied: s ~(x~, ~, ~(~), x - ~*, ~) = ~(x~, ~, ~(~), + ~(x~, ~) 72(x - 2"), (1.15) where 72 (satisfying 72(0) = 0) performs a stabilizing action on the modes of the system 9 whenever 3 ¢ 0. The corresponding expression for equation (1.14) becomes:

=

6,

+

+

+ Z(x.,

-

+

8Item1 of Property1 can be e~ily satisfi~ by ~signinga suitable allocation of the plant’s input to the function7~. 9Thefunction 72 can be chosento be identically zero. However,whenslow mod~ are pr~entin the plant, it can significantly improvethe performance of the anti-windup design.

12

Barb~leta,!.

which makes/4 forward invariant because the second term is zero on the boundaryof/4 and, by the equation (1.11), the third argument of 3’1 belongs to 5cu for any value of ~1. Hence, positive invariance of/4 is guaranteed for any value of c3 by the three assumptions on 71 listed above. Based on the definition equations can be chosen as

of P, ~3 and % the anti-windup compensator

~ = A~+B(5-yc) vl = a(xu, 5, xu - ~u, ~, Yc) -

(1.16a) (1.16b)

v2

(1.16c)

= -Cy

~ - Dy (5

- Yc),

where ~ E Rn, and the function c~ : Rn~, × Rra × Rn~,. × Rn × Rrn m -~ R is defined as

~(x~,5, ~, ~, yc):= 7(x~,5, ~(~),~, yc)+ Z(~,5) (~ -7(n~,~, n~, 0, ~)). and the interconnection conditions are given by (1.7). Remark 1.4. In the trivial case when the plant has no exponentially unstable modes (this means that nu = 0 and the matrix Au is empty), the null-controllability region of the system is the whole state space (see, e.g., [34]). Since the substate x~ is emptyin this case, the functions/3 and :P have no meaning and the anti-windup output equations (1.16b), (1.17) simplify to Vl

= a(Xu, 5, Xu-~u,~,yc) -- Yc = 70(5, ~, Yc) +Yc-7o(yc,O, Yc)- Yc = ~(~),

(1.18)

where we have chosen ~/0(5, (, y~) = ~(~) Yc(in dependently of thefirs argument 5), with ~(0) = 0, so that item 1 of Property 1 (specialized droppingthe first two argumentsof 7 that are emptyin this case) is trivially satisfied. Items 2 and 3 of Property 1 can be satisfied by designing the function ~(.) on the basis of results on stabilization of linear systems with rate and magnitude bounded inputs such as [11,23, 24, 37, 38]. A complete anti-windup solution along these lines for systems with inputs limited onl:y in magnitudeis given in [42]. The following theorem establishes that the anti-windup compensator given above guarantees stability of the anti-windup closed-loop system for any reference, reproduces any trajectory of the nominal system that does

Anti-windup for Exponentially Unstable Linear Systems

13

not hit the saturation limits or the operating region boundaries and guarantees convergence of the performance output z to a point that coincides with the reference wheneverit is feasible. Theorem1.1. Let 2(t), ~c(t), and 5(t) represent state, control, and performance output generated by the nominal closed-loop system, starting from the initial condition (~(0), £%(0)) = (~:0, ~c0). If Property then the anti-windup closed-loop system (1.1), (1.2), (1.5), (1.7), (1.17) is such that, for (x(0), xc(O)) = (~0, ~:c0), 1. If ~(0) = 0, 5(0) = ~c(0), and there exists a compact set positive constants Mo, Ro, such that, for the nominal closed-loop system, lo

(5su(t),~]c(t))r) E9r0Cint(9 Vt>_0 then z(t) = 5(t), Vt 0; 2. if the initial conditions satisfy (xu(0), 5(0)) E/g, (xu(t ), 5(t)) /~, Vt _> 0 and all the trajectories are bounded; 3. if the initial conditions satisfy (x~(0), 5(0)) e/g t_..co(xslim

-

(t), ~:u(t), 2c(t))(xs,~*xu,-* ~)

(1.19)

then: ~ X x~, ~*,5", z*). lim (xs(t), xu(t), x~(t), [(t), 5(t), = ( s, P(~),

t-*co

Proof. See Section 1.6. An interpretation of the three results in Theorem1.1 is in order. Item 1 states that, if the anti-windup compensator is appropriately initialized, and the reference signal is sufficiently small (namely, if it keeps the system within ~- and does not cause the input to saturate), the anti-windup closed-loop system will perform identically to the nominal closed-loop system. The statement in item 2 conveys the requirement that the trajectory (x(t), 5(t)) of the anti-windup closed-loop system never leaves the operating region 5/. This statement is completed with item 3 which gives the desired convergence properties for the anti-windup closed-loop system. If 10Denoteby int(5 r) the interior of the set .~’.

14

Barbu et al.

the trajectory of the nominal closed-loop system converges to a point in the trajectory of the anti-windup closed-loop system converges to the same point; however, different transient behavior should be expected due to the presence of the actuator limits. Onthe other hand, if the steady-state value of the nominal closed-loop system is outside ~’, the same convergence property is not feasible for the anti-windup closed-loop system. In this case, the anti-windup closed-loop system converges to a point which is close to the nominal steady-state value in the sense of the projection function P. The peculiarity of the general structure given in equation (1.16) is the fact that, with the coordinate transformation X := x - ~, the antiwindupclosed-loop system in the (X, xc, x, 5) coordinates is the cascade two subsystems: the (X, xc) subsystem, exactly reproducing the dynamics of the nominal closed-loop system, and the (x, 5) subsystem, taking into account the effects of the saturation nonlinearity on the plant dynamics:

(x, xc)

subsystem (x, 5) subsystem

1.3.

{ {

~

=

AX+Byc

~ = g(x~,x, r) y~ = k(x~,X,r)

(1.20)

~: = Ax+B~ ~ = Rsgn(Msat (-~) - 5) u = (~(xu, 5, Xu, x -X, y~).

(1.21)

Anti-windup Design for an Unstable Aircraft

Magnitude and rate saturation are two of the most frequently encountered nonlinearities in modern flight control. As an example of the antiwindup design synthesized in Section 1.2, we focus on the short-period, longitudinal dynamics of a prototypical unstable fighter aircraft subject to rate and magnitude limits on the elevator deflection. For,this systern, large pitch rates requested by the pilot may not be achievable while maintaining stability of the aircraft. Based on the results in Theorem1.1, an anti-windup compensatoris designed for the unstable aircraft that achieves prototypical military specifications for small to moderate pitch rate pilot commands,guarantees aircraft stability for all pitch rate pilot commands and allows the pilot to maneuveraggressively. 1.3.1.

Aircraft

Model and Design Goals

According to the experimental data in [32], the linearized short-period longitudinal dynamics of the McDonnellDouglas Tailless AdvancedFighter Aircraft (TAFA)model at a dynamic pressure of 450 psf (corresponding

Anti-windup for Exponentially Unstable Linear Systems

15

to a specific trim flight condition), are described by the following linear system: ~:=

0

= =:

6 -2 Az+B5

z+

~ = Rsgn[Msat(~)-5],

8 (1.23)

where the variable q represents the body axis pitch rate and a and 5 are, respectively, the deviation of the angle of attack and of the elevator deflection angle from the trim flight condition. The magnitude and rate limits of the elevator deflection are quantified by Mand R, respectively. In this example, the maximal(deviation of the) elevator deflection angle is limited between ±20 deg (M = 0.35) and the maximal elevator deflection rate is limited between ±40 deg/sec (R = 0.7). Note that system (1.22) can diagonalized via a suitable coordinate transformation to obtain ~s = As xs + bs 5 24 = Au xu + bu 5,

(1.24a) (1.24b)

where As = -4 corresponds.to an exponentially stable mode and Au = 1 corresponds to an exponentially unstable mode. Our control problem is to design a (dynamic) feedback with inputs (a, q, ~) and pitch rate pilot command qd so that, for any trim flight condition (i.e., for any choice of the dynamicpressure) the closed-loop satisfies the following properties: 1. For small to moderate pitch rate commands,the pitch rate response satisfies a prototypical military specification; here, based on [20], we take q(s) 1.4s + q~(s) 2 +1. 5s + 1

(1.25)

Moreover, this response is recovered asymptotically after large commands. 2. The aircraft

is BIBSstable 11 from the pilot commandinput qd.

3. The aircraft is highly maneuverable;i.e., large pitch rates are attained by the control scheme. llA systemis BIBS(Bounded Input Bounded State) stable if the state response any boundedinput is boundedas well.

16

Barbu ei al.

1.3.2.

Selection

of the Operating

Region

In this section we study the structural limitations of the saturated system (1.22), (1.23) (or, equivalently, of system (1.24), (1.23)). In particular, we define and explicitly compute the maximalstability region achievable within the actuator saturation limits. Based on this, we give a selection for the operating region/g introduced in Section 1.2.3. Wefirst computethe null-controllability region ~;. In particular, first note that ~ = R x ~2p, where )~p C R2 is the projection of )2 on the (xu, 5) plane. Now, consider the limitations due to magnitude saturation and observe that, by equation (1.24b), any initial condition outside the set )2M := {(xu, 5): Ix~l b~M/)~u} (t he boundary of )2M corresponds to the horizontal dotted lines in Figure 2) generates a nonconverging trajectory because &uXu>_ 0 for all times and IXuol ~ O. On the other hand, if there is no limitation on the control input rate (namely, R -* ~x~), a simple proportional controller (5 -- -Kxu, K sufficiently large) can drive to zero any trajectory with initial conditions (xu(0), 5(0)) E )?M. It follows .-Vp C ~,~M

Null-controllability region

%.... ~:-_::..... :NM ............................................................

-1

-2

-3

-,; -; Figure 2: The sets

;

; ,;

5[deg]

~M,

~)P,

~,

~" and

Anti-windup for Exponentially Unstable Linear Systems

17

Whenthe effects of rate saturation are considered, Yp can be defined as the set "~p ::

{(Xu0,

50) E JIM : ~U(’)

: ~9(t;

,

50, ~t(t )) e ~M

Vt 0},

(1. 26

where ¢(t; Xuo, 5o, u(t)) denotes the trajectory of system (1.24b), (1.23) starting at (xu(O), 5(0)) = (xu 0, 5o) and with a measurable input function u(.). Althoughequation (1.26) characterizes the region Pp, this definition implicit, and thus not of practical utility. However,due to the structure of the model (1.24b), (1.23), the boundaries "~pcan be comp uted expl icitly and they correspond to the dashed lines in Figure 2 (the explicit equations " are not included here due to space constraints). Note that, while the interior of ~p is weakly forward invariant (namely, there exists at least one selection of the input u(.) that makes it forward invariant), the complement of Yp is strongly forward invariant (namely, regardless of the input u(-), trajectories never leave this set). Hence, trajectories leave the interior of Yp, they cannot return to the interior of "~p regardless of the control action through the input u. As already .pointed out in Remark1.1, this fact plays an important role in the control design. Indeed, by continuity of solutions with respect to initial conditions on compact time intervals, if the trajectories get close to the boundary of Fp, they will take a long time to move away from this boundary, thus exhibiting an undesiredstickiness effect (see, e.g., [27]). To avoid this phenomenon, we choose an operating region L/ that is strictly smaller than the null-controlla.bility region. In particular, b/is chosen as the region ~2p that wouldbe obtained if the magnitudeand rate limits were 80%of their actual values (see the light shaded area in Figure 2), union with two rectangular regions in the upper left and lower right corners (corresponding to the dark shaded areas in Figure 2). A natural choice for r is then a contraction of the set L/sufficiently close to/d (corresponding to the dash-dotted lines in Figure 2). The diagonal set corresponding to the "stars" represents the projection ~u of the set ~ on the (xu, 6) plane and corresponds to the set of all the equilibria that the input u can induce on the open-loop system. Based on this choice for the sets/g and 9v, the following result provides a function 7 that satisfies items 1 and 2 of Property 1, thereby guaranteeing by Theorem1.1, the effectiveness of the anti-windup construction. Theorem 1.2. Given the sets/A and 3r represented function

in Figure 2, the

1 7(xu, x~) := -~u (AuXu + (xu - x~)),

(1.27)

18

Barbuet ail.

satisfies items 1 and 2 of Property 1 for the system (1.22), (1.23). Proof. Item 1 easily follows from the definition (1.8) (note that, equation (1.24b), the equality in (1.8) corresponds to ~ x~ + b~ 5 = bu To prove item 2, note that the set {5 : lal -- M}on the boundary of /~ (corresponding to the dotted vertical lines in Figure 2) is related to the structural limits on the input magnitude, and input saturation prevents trajectories from leaving b/ through these vertical boundaries. Hence, a trajectory could leave the set b/only by crossing the horizontal boundaries, which correspond to the solid curves delimiting/~ from above and below in Figure 2. However,by equation (1.24b), trajectories cannot leave b/from the flat horizontal boundaries in the upper left and lower right corners, because on those boundaries, xu must be nonincreasing in norm. Moreover, by the presence of the discontinuous dynamics (1.23), given any set-point x* the control law (1.27) corresponds to a line that separates the set /~ in two regions. In the upper region, the argument M (sat(~)) - 5 of the function in (1.23) is negative and the input rate is -R; whereas, in the lower region, the input rate is +R. The line defining the two regions passes through the equilibrium (x~ -~ x~,) and its slope is equal to - b~ can be easily verified that, with this slope, for any m~, E ~’u, the boundary of/g from above (respectively, from, below) on the right (respectively, the left) of the equilibrium manifold, is completely contained in the upper region (respectively, in the lower region), namelyit ensures that the input rate is maximal on these boundaries. From this, by construction, we conelude that bt is positively invariant. [] 1.3.3.

The Nominal Controller

The nominal controller block in the scheme of Figure 1 (corresponding to equations (1.2)) is constituted by a linear controller designed on basis of the unsaturated plant (that induces the desired linear closed-loop transfer function (1.25) when the pilot commandis small enough) and by nonlinear dynamic command limiting block (that acts like a nonlinear filter guaranteeing performance recovery when the pilot commandis infeasible). The nominal controller can be designed on the basis of the unsaturated (linear) plant dynamics. This is possible due to the presence of the anti.windup compensator (1.6) that ensures that the anti-windup closed-loop

Anti-windup for Exponentially Unstable Linear Systems

19

system is equivalent to the cascade structure (1.20), (1.21). Within context, the nominal controller is to be understood as a nonlinear modification of a linear controller that anticipates anti-windup compensation. This modification only occurs when the nominal trajectory crosses the operating region boundaries, thus becominginfeasible for the saturated plant. Hence, although the nominal controller is designed disregarding saturation, it modifies the linear response only at times when it wouldn’t be feasible for the anti-windup closed-loop system. This preserves the linearity of the small signal response of the compensated system and the requirement that the anti-windup compensation does not modify feasible trajectories. In other words, the decoupling properties associated with the cascade structure (1.20), (1.21) allow us to design the anti-windup compensation following a two-step procedure. In the first step, nonlinear modifications of the linear controller dynamics are performed on the basis of the unsatUrated plant, and, in the second step, this nonlinear nominal response is recovered for the saturated plant by means of the anti-windup compensator (1.6). A global controller corresponding to the function 7 will guarantee BIBSstability of the resulting anti-windup closed-loop system, as long as the nonlinear nominal closed-loop system arising from the first step of the construction is BIBSstable too. The nonlinear nominal closed-loop system is represented in Figure 3 as the interconnection between the linear controller and the linear unsaturated plant through a nonlinear command limiting block. Dynamic

q] .......................... :up ~- ~_-~ - - - i ~p I Unsaturated

COllllllalldLimiting

[--~H~,

Rate Saturation ~

[~ w Y~’~)

[~~

, Li~war I ] C~ltroller I .............................

14

Aircraft

I --~ I- ’1 ~ ]~ I

~-~ ~ Iraqi

Figure 3: Nominalclosed-loop system with nominal controller

] structure.

Design of the Linear Controller The linear controller in Figure 3 is constituted by an inner stabilizing static feedback, an outer dynamic feedback and a dynamic feed-forward action. Based on the fact that the linearized aircraft dynamicsare minimum phase, the inner stabilizing feedback is chosen as: Ks=-~

1 [ 6 -2

],

20

Barbu et al.

and the zero dynamics (that becomeunobservable) are asymptotically" stable. Once the inner loop has been closed, the plant is transformed into an integrator and any desired closed-loop transfer function can be obtained by choosing appropriately the feedback and feed-forward dynamic elements Cib(S) and Cii(s), respectively. In particular, to obtain the closed-loop transfer function (1.25) from ~d to ~, the two dynamic elements have been chosen as: 1.4s+ 1 1 1.bs÷ 1 (1.28) VII(s)’1.bs+l’ CIb(S)’-8 Dynamic Command Limiting On the basis of Theorem 1.1, the BIBS stability of the anti-windup closed-loop system designed in Section 1.2.3 is guaranteed for any nominal controller that stabilizes the unsaturated plant. To this aim, the dynamic commandlimiting block in Figure 3 is not necessary. However, implementing a linear controller such as the one described above without any commandlimiting action could lead to poor performance when the pilot commandis large enough to drive the unsaturated system outside the operating region/~. Indeed, the equilibria corresponding to such commandsare infeasible for the anti-windup closed-loop system and limiting the steadystate pilot commandis important to avoid steady-state differences between these equilibria and the ones achieved by the nominal closed-loop system in Figure 3. Following the above reasoning, it is straightforward that the dynamic commandlimiter is to be designed with the goal of keeping the nominal trajectory "close" to the operating region by adding a feed-forward action exclusively whenthe trajectory is not in ~; in this way, any maneuverthat stays within the operating region (namely, any feasible maneuver) is not modified by the command limiter, but infeasible trajectories are not allowed to movetoo far from the operating region itself. Given the magnitude saturation limit M, define qM = 2Mas the maximumfeasible steady-state pitch rate commandfor the saturated plant (1.22). Consider the function ~L(Xu, 5) : R × R --~ [0, 1] satisfying equation (1.9) but not necessarily equal to ~(xu, 5). ~2 The dynamic command 12It has been verified everywhere. In particular, and to select l~L(Xu,5):=

O, if { 1, if

that good performance is ~chieved when ~L(Xu, 5) <_ B(xu, a possible choice is to define a set $’~ such that 9v C ~ C Lt (xu,5) (xu,~)EiP

f~(xu,5)::

-~. O, if (xu,5)~L( { 1, if (xu,~)E.Tz~

Anti-windup for Exponentially Unstable Linear Systems

21

limiting block can be chosen as ~d = qd + ~L(Zu, 5) (SatqM(qd) --

(1.29)

where SatqM (’) is the scalar saturation function with saturation level qM. The effect of the commandlimiting strategy in equation (1.29) is that the pilot commandis unmodified until the trajectory reaches the operating region boundaries. Once such boundaries are approached, the nominal response is not feasible and the commandis limited (according to the the phenomenaoutlined in [16], to prevent the pilot from getting confused, a perceptive action could be implemented on the stick to inform the pilot about the feasibility of his command). Basedon the exponential stability of the linear closed-loop system, since I-~dl <-- Iqctl, then the solutions to the nominalclosed-loop systemin Figure 3 are uniformly bounded. Namely, this closed-loop is BIBSstable. Since the plant (1.22) does not have poles on the imaginary axis, then the forward invariance of/4 implies boundedness of all trajectories and the result in Theorem 1.1 can be easily extended to prove the BIBS stability of the overall anti-windup closed-loop system.

1.4.

Simulations

In this section we test the anti-windup construction described in Section 1.3 using a numberof different pitch rate commands. For small pitch rate commands,since the system operates in the unconstrained region, the nominal controller enforces the augmentedaircraft performance and perfect reference tracking. In this situation, the anti-windup controller performs a very small compensation on the system. Figure 4 represents the response to a doublet pitch rate commandof amplitude qd ~- 25 deg/s (dotted line). The anti-windup response (solid line) exhibits differences with the nominal response (dashed line) until control input is saturated (either in rate or magnitude), thus showingthat the maximalcontrol effort is exploited to recover the nominal trajectory. Steady-state performance recovery is not achievable for large pitch rate commandsdue to magnitude and rate saturation. This is illustrated in Figure 5 which shows the pitch rate response of the anti-windup closedloop system, compared to the response of the nominal closed-loop system for a doublet pitch rate commandof 50 deg/s. In this case, the antiwindupsystem exploits all the available control effort to enforce restricted tracking of the nominal trajectory (perfect tracking is not possible), while guaranteeing stability of the closed-loop. The dynamic commandlimiting

22

Barbu et al.

20

-20 5 Time[s]

6

7

8

9

10

5 Time[s]

6

7

8

9

10

5 Time[s]

6

7

8

9

10

20

-10 -20 -3:0

1

2

3

4

4O

20 o

-40 1

2

3

4

Figure 4: Pitch rate response of the anti-windup system (solid) and of the unsaturated system (dashed) to a 25 deg/s doublet pitch rate command (dotted line). action (which preserves as much as possible the pilot command)allows reach, during the transient, pitch rate values that would be infeasible at the steady state (see the two bumps in the upper plot). This property significantly improves the maneuverability of the aircraft because it allows the pilot to change the aircraft pitch angle at highly increased rates, as compared to static commandlimiting strategies.

1.5.

Conclusions

In this chapter, we presented an anti-windup scheme for exponentially unstable linear systems with inputs subject to both magnitude and rate constraints. The proposed anti-windup strategy extends previous results

Anti-windup for Exponentially Unstable Linear Systems

0

1

2

1

2

1

2

3

4

23

5 Time[s]

6

7

8

5 Time[s]

6

7

8

5 Time[s]

6

7

8

9

10

9

10

40

-20 -40 -600 4O

"E -20 -40 0

3

4

9

10

Figure .5: Pitch rate response of the anti-windup system (solid) and the unsaturated system (dash-dotted) to a 50 deg/s doublet pitch rate command(dotted line).

by allowing arbitrarily large references, accounting for rate and magnitude limits, and allowing large operating regions. The design of the anti-windup controller relies on the design of a static stabilizing state feedback for the plant with magnitude and rate limits; hence, within the proposed scheme, several existing results on robust stabilization of constrained linear systems can be brought to bear on anti-windup design problems. Wealso applied the design technique to the manual flight control of an open-loop unstable aircraft to allow aggressive maneuverswhile preserving stability. The resulting anti-windup controller is shownto allow high pitch rates and a fast response to the pilot stick commandsby employing the maximal effort of the constrained actuators.

24

Barbu et al.

1.6.

Proof of the Main Result

To prove Theorem1.1, we appeal to the following result. Lemma1.1. Let ~/ : R --~ R be an absolutely continuous function satisfying [rl(t)l _< M, Vt _> 0 and [/~(t)[ _< R for almost all t in [0, oc). initial value problem ~ E R SGN (Msat (~)-

~), ~(0) =

(1.30)

where SGN(x)

1, -1, [-1,1]

if if if

x>0 x <0 x=0.

(1.31)

has the unique solution ~(t) = ~?(t) on [0, Proof. Since SGN(x) is an upper semicontinuous set valued map and is nonempty, compact, and convex for each x E R, by [10, Theorem 1, page 77], there exists at least one solution and since the right hand side is uniformly bounded, each solution is maximally defined on [0, c~). Let be an arbitrary solution of (1.30) defined on [0, oc) and define e Then, e(0) = 0 and, from equation (1.30), ~ e -RSGN(e) - h for a lmost a ll t in [0, c~),

(1.32)

and, since IS(t)] _< R, then for almost all t e [0, ~), 1 d(ee) 2 dt

= e (-nsgn(e)-//) <_-nlel +nlel =0,

where the single-valued function sgn(x) is the sign of x for x ~ 0 and arbitrarily defined at x = 0. Thus, e2(0) -= 0 and d(d-~t~ <_ 0 for almost all times. By standard comparison theorems, the result follows. [] Proof of Theorem 1.1. Define X := x-~, Y := y+v2, z~ := Cz ~ + Dz (5 - Yc), :=z - z~and r ewrite the a nti- windup closed-loop

Anti-windup for Exponentially Unstable Linear Systems

25

system equations in the new coordinates (X, xc, x, 5):

2 = (X,

subsystem

Y Z =

AX +Byc Cy X + Dy Yc Cz X + Dz yc

(1.33)

g(xc,Y,r) Yc

(x, ~)

subsystem

~ =

Ax + B6 Rsgn (M sat (~) -

7(x~,5, ~(x~),~- x, yo) fl(x~,6)(~-r(x~, ~, x~,o, ~)).

(1.34)

Notice that the system in this form has a cascade structure where the (X, Xc) subsystem feeds the (x, 6) subsystem. Item 1. Let the initial conditions of the anti-windup closed loop system be given by x(0) = i(0) Xc(0) = ~c(0) 6(0) = ~)~(0) ~(0) =

(1.353) (1.35b) (1.35c) (1.35d)

Thus, (X(0), x~(0)) = (2(0), ~(0)) and, since the equations the (X, Xc) subsystem (1.33)are coincident with the ones describing the nominal closed-loop system, by uniqueness of solutions for the unsaturated closed-loop system it follows that X(t) = c?(t), Z(t) = ~(t), yc(t) = ~)~(t),

Vt >_O,

(1.363)

Vt > 0, Yt >_ 0.

(1.36b) (1.36c)

Weonly need to prove that, under the assumptions of this item, (x(t), 6(t)) = (X(t), yc(t)) is the unique 13 solution of the (x, 6) subsystem(1.34). From this fact, it immediately follows that z~(t) = 0, Vt _> 0, and thus

4t) = ~(t), > 0. To prove this fact we invoke Lemma1.1 after deriving bounds on lut and I/t I. First notice that P(Xu(t)) = X~(t) = 2~(t), Vt >_

(1.37)

13It is clear that it is a solution, but we cannot directly assert the uniqueness since the right hand side of (1.34) is not Lipschitz.

26

Barbuet al.

which follows from the assumption (2u(t), ~]c(t)) ¯ jr, Vt >_ to gether with equation (1.36a) and property (1.11) of P(.). Moreover, if (xu, 5) ¯ ~-, from compactnessof jr, by the state equation of system (1.1) and equation (1.5), there exists L > 0 such that

~ = Rsgn(Msat(~)-5)

(1.38)

-
Defining A :-- infwe~o dist0y(w) and the maximaltime T := AlL, since jr0 is a compactstrict subset of jr, by inequality (1.38) and continuity solutions with respect to time, we can state that for any to >_ O,

(xu(to),5(to))¯ jro ~(xu(t),5(t))¯ ~:,vt ¯ [to, (1.39 where T does not depend on to. From(1.35a) we obtain xu(O) = 5":u(0), which combinedwith (1.35c) the theorem assumption that (~Cu(t),ft(t)) ¯ jro, yiel ds (xu(O), 5(0)) ¯ jr0. Thus, from (1.39), we obtain that (x~,(t), 5(t)) ¯ .,~, [0, T], an d by using (1.36c), the definition of/3, and (1.37), the input to the saturation block can be computed as

u(t) =~c(t)~(~(t), ~,x~( t), ~(t) - x(t), f~c ~(xu(t),~,x~(t),o,~3~), vt ¯ [o, T]

(1.40)

Appealing to compactness of ~" and differentiability of % there exist continuous nonnegative and nondecreasing functions p~, i = 1, 2, 3, such that p~(0) = p2(0) = 0, and for all t ¯ [0, (1.41) Since x(0) = X(0), 5(0) = ~c(0) and by continuity of the functions p~, i = 1,2, 3, for each e > 0, there exists T~ > 0, independentof the initial conditions, such that 15 max {pl([x(t)

X(t)l), pe(Ix(t) - X(t)l) + w¯ [0,

14ThenormI " I denotesthe standardEuclideannorm. 15RecMl that ~ ---- x - X.

Anti-windup for Exponentially Unstable Linear Systems

27

Thus, recalling the theorem assumptions I~lc(t)l < Mo < M, I~c(t)l < Ro < R, picking e = min{M-M0,R-R0}, equations (1.41) can be bounded by lu(t)l <_ M, liz(t)l <_R, Vt [0, T~], which, by Lem ma 1.1imply that

(x(t), a(~))=(~(t), ~c(t)),vt ¯ is the unique solution of the (x, a) subsystem (1.34) for the initial condition (1.35a) and (1.35c) in the interval [0, Tel. Hence, (x~(T~), 5(T~)) (2u(Te), ~lc(Te)) ~ and t he s ame argument can b e re peated to ex ten d the solution (x(t), 5(t)) over [Te, 2Tel, and so .on, up to time T. Since T is uniform in time, the solution can be extended to infinity and the item is proven. Item 2. Since the dynamic equations (1.33) of the (X, xc) subsystem coincide with the dynamic equations of the nominal closed-loop system and since, by assumption,(&(t), 5:c(t)) -~ (5:*, 5:~),

x(t) = ~*- ~(t) x,(t)= ~.-* + yc(t) = Ye ca(t), where e, (t), e2(t) and e3(t) are asymptotically vanishing signals, and limt~ 9c(t). Since, by item 1 of Property 1, ~(xu, Ye, Xu, O, Yc) = Ye for all x~ and y~, then, by continuity of 7, e4(t) := ~(t) ~(Xu(t), ~( t), X~ (t), O, ~(t )) ~ 0 and u can be written as

3(x~(t), a(t)) e4(e).

(1.42)

From item 2 of Property 1, it follows that for any initial condition (xu(O), 5(0)) e (xu (t), 5(t )) ~ U , Vt and(x (t ), 5 (t)) is bounded; consequently, all states are boundedand thus item 2 is proven. Item 3. Since e~(t), e2(t), e3(t) and g4(t) in equation (1.42) are ing, item 2 of Property 1 guarantees boundedness of the trajectories. Due to the theorem assumption (1.19) and equations (1.36),

~im(z,(t), x~(t), ~(t)) ( ,, ~,

0.43)

Based on standard results on cascaded systems applied to the cascaded structure (1.33), (1.34), the convergence property (1.43) of the first system (1.33) and the boundedness of trajectories of the second subsystem

28

Barbuet al.,

(1.34), together with the forward invariance of b/and the stability assumption in item 3 of Property 1, is sufficient to guarantee the convergence. property (recall that ~ = x - X): lim (xs(t), xu(t), Xc(t), ~(t), 5(t)) s x~, x’c, ~*,5"). Moreover, by equation (1.17) and the properties of lim 6(t) = tli~lll~7(Xu(t),

5(t), P(Y~),

- ,Yc),

hence, by item 3 of Property 1, x~ = P(2~), thus completing the proof.

References [1] K.J. _~strSm and L. Rundqwist. Integrator Windup and Howto Avoid It, in: Proceedings of the American Control Conference, volume 2., Pittsburgh (PA), USA(1989) 1693-1698. [2] C. Barbu, R. Reginatto, A.R. Teel, and L. Zaccarian. Anti-windup Design for Manual Flight Control, in: American Control Conference, San Diego (CA), USA(1999) 3186-90. [3] C. Barbu, R. Reginatto, A.R. Teel, and L. Zaccarian. Anti-windup for Exponentially Unstable Linear Systems with Inputs Limited in Magni-tude and Rate, in: American Control Conference, Chicago (IL), USA. (2000) 1230-1234. [4] 3.M. Berg, K.D. Hammett, C.A. Schwartz, and S.S. Banda. An Analy-. ¯ sis of the Destabilizing Effect of Daisy Chained Rate-limited Actuators, IEEE Trans. on Control Systems Technology, 4(2) (1996) 171-176. [5] F. Blanchini. Set Invariance in Control- A Survey, Automatica, 35(11) (1999) 1747-1768. [6] J. Debelle. A Control Structure Based upon Process Models, Journal A, 20(2) (1979) 71-81. [7] M. A. Dornheim. Report Pinpoints Factors Leading to the YF-22 Crash, Aviation Week and Space Technology, (November 1992) pages; 53-54. [8] J.C. Doyle, R.S. Smith, and D.F. Enns. Control of Plants with Input; Saturation Nonlinearities, in: ACC, Minneapolis (MN), USA(1987) 1034-39. [9] H.A. Fertik and C.W.Ross. Direct Digital Control Algorithm with Anti-windup Feature, ISA Transactions, 6(4) (1967) 317-328.

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Anti-windup for Exponentially Unstable Linear Systems

[10] A.F. Filippov. Differential Equations with Discontinuous Righthand Sides, Kluwer AcademicPublishers, (1988). [11] R. Freeman and L. Praly. Integrator Backstepping for Bounded Controls and Control Rates, IEEE Trans. Aut. Cont., 43(2) (1998) 262. [12] E.G. Gilbert, I. Kolmanovsky,and K.T. Tan. Discrete-time Reference Governors and the Nonlinear Control of Systems with State and Control Constraints, Internat. Y. Robust Nonlinear Control, 5(5) (1995) 487-504. [13] E.G. Gilbert and K.T. Tan. Linear Systems with State and Control Constraints: The Theory and Application of Maximal Output Admissible Sets, IEEE Trans. Aut. Cont., 36(9) (1991) 1008-1020. [14] G.C. Goodwin, S.F. Graebe, and W. Levin. Internal Model Control of Linear Systems with Saturating Actuators, in: 2nd ECC, Groningen, the Netherlands (1993) 1072-1077. [15] Td. Graettinger and B.H. Krogh. On the Computation of Reference Signal Constraints for Guaranteed Tracking Performance, Automatica, 28(6) (1992) 1125-41. [16] Why the Gripen Crashed, page 11.

Aerospace America,

(February

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AntiAuto-

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Week and Space

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Anti-windup for Exponentially Unstable Linear Systems

31

[36] J.S. Shamma. Anti-windup via Constrained Regulation with Observers, Systems and Control Letters, 40 (2000) 1869-1883. [37] J.M. Shewchun and E. Feron. High Performance Bounded Control of Systems Subject to Input and Input Rate Constraints, in: Proc. of AIAA GNCConference, NewOrleans (LA), USA(1997) 770-779. [38] A.R. Teel. A Nonlinear Small Gain Theoremfor the Analysis of Control Systems with Saturation, IEEE Trans. Aut. Cont., 41(9) (1996) 12561270. [39] A.R. Teel. Anti-windup for Exponentially Unstable Linear Systems, Int. J. Robust and Nonlinear Control, 9 (1999) 701-716. [40] A.R. Teel and J.B. Buffington. Anti-windup for an F-16’s Daisy Chain Control Allocator, in: Proc. of AIAA GNCConference, NewOrleans (LA), USA(1997) 748-754. [41] A.R. Teel, O.E. Kaiser, and R.M. Murray. Uniting Local and Global Controllers for the Caltech Ducted Fan, in: Proc. of the American Control Conference, volume 3, Albuquerque (NM), USA(1997) -1543. [42] A.R. Teel and N. Kapoor. The/:2 Anti-Windup Problem: Its Definition and Solution, in: Proc. ~th ECC,Brussels, Belgium (1997). [43] A.R. Teel and N. Kapoor. Uniting Local and Global Controllers, Proc. ~th ECC,Brussels, Belgium (1997).

in:

[44] L. Zaccarian and A.R. Teel. A Benchmark Example for Anti-windup Synthesis in Active Vibration Isolation Tasks and an/:2 Anti-windup Solution, EuropeanJournal of Control, 6(5) (2001). [45] L. Zaccarian and A.R. Teel. Nonlinear /:2 Anti-windup Design: An LMI-based Approach, in: Nonlinear control systems design symposium (NOLCOS),Saint-Petersburg, Russia (2001). [46] L. Zaccarian, A.R. Teel, and J. Marcinkowski. Anti-windup for An Active Vibration Isolation Device: Theory and Experiments, in: Proceedings of the American Control Conference, Chicago (IL), USA(2000). [47] A. Zheng, M. V. Kothare, and M. Morari. Anti-windup Design for Internal ModelControl, Int. J. of Control, 60(5) (1994) 1015-1024.

Chapter 2 Selecting the Level of Actuator Saturation for Small Performance Degradation of Linear Designs Y. Eun, S.M.

P.T.

Kabamba,

and

Meerkov

University

2.1.

C. G~ik~ek,

of Michigan,

Ann Arbor,

Michigan

Introduction

In practice, control systems are often designed using linear techniques. In reality, control systems often (or, perhaps, always) include saturating actuators. The question arises: Howlarge should the level of saturation be so that the performance, predicted by linear design, does not degrade too much? In this paper this problem is addressed in the framework of the disturbance rejection problem. The scenario considered is as follows: The disturbance is a Gaussian random process and the performance is measured by the standard deviation (SD) of the output.

The linear design (i.e., the design under the assumption that no saturation takes place) results in the output SD, denoted by Oz.

33

34

Eun et al. ¯ The saturation is described by (see Figure 1) u8 = a sat (~),

(2.1)

where a represents the level of saturation.

U

Figure 1: Saturating actuator. ¯ The tolerable e > 0, i.e.,

performance degradation is characterized by a number

ai _~ (I + e)(rz, (2.2) ’where (r2 is the SDof the output, 5, of the closed loop system with the controller obtained by the above-mentioned linear design and with saturating actuator (2.1). Within this scenario, this paper showsthat (2.2) is satisfied if the level of actuator saturation is selected according to a >_~(e, r)~u,

(2.3)

where au is the SDof the signal at the output of the controller in the linear design (which can be calculated using a standard Lyapunovequation), r a positive real number (which depends on the linear part of the system and can be evaluated using its Nyquist plot), and ~ is a function given by 2r+(l+e) ~(e,r) ---- ~/~(1 + e)erf -1 (i~_~)-~.~])].

(2.4)

Moreover,it turns out that B is relatively insensitive to r and for a wide range of r’s, ~(0.1, r) ~ 2. (2.5) Thus, the following rule-of-thumb can be formulated: If a >_ 2au, the performance degradation of the linear design due to actuator saturation will be less then 10~.

Selecting the Level of Actuator Saturation

35

Figure 2: Linear system. To derive these results, Section 2.2 below provides the problem formulation and describes the approach to its solution. This approach is based on the method of stochastic linearization, which is an approximate methodfor analysis of nonlinear systems with randominputs, [4], and which is similar to the method of describing functions [1]. Section 2.3 presents the main results. Section 2.4 describes examples, and in Section 2.5 conclusions are formulated. The proofs are given in the Appendix. 2.2.

Problem

Formulation

Consider the SISOlinear system shownin Figure 2, where Pi, i = 1,2, 3, is the plant, C the controller, F the coloring filter, and w, u, y, and z, are standard white noise, control signal, feedback signal, and controlled output, respectively. Suppose that C is designed using a linear design technique so that the system of Figure 2 is asymptotically stable and transfer functions FP"1-~ and I+P1P2C FP~C are strictly proper. Assume also that the goal of l+P1P2C control is disturbance rejection, quantified by the standard deviation (SD) of z. Underthese assumptions, it is easy to see that FPa 2

FP2C 2

(2.6)

where L = P1P2C. Assumenowthat the above controller is used for the same pl~-nt but-witha saturating actuator. The resulting nonlinear system is shownin Figure 3, where all signals are denoted by the same symbols as in Figure 2 but with " ~ " Assume further that this system remains globally asymptotically stable. One wouldlike to comparethe SDof z with that of 5. Unfortunately, direct calculation of a~ is a formidable task. Therefore, a simplification is necessary. In [3] such a simplification has been developed, based on the

36

Eune.t ~l.

Figure 3: System with saturating

actuator.

method of stochastic linearization [1, 4]. According to this method, the nonlinear system of Figure 3 is replaced by a system, shownin Figure 4, where all signals are denoted by the same symbol as in Figure 2 and 3 but with ..... . The quasi-linear gain, N*, of Figure 4, is calculated according

to FP~.C 2 c~ *) 1T~L = .~/~ erf-l(N where the error function is defined by

(2.7)

err(x/= ~x e-t~et" The SDof 5 of Figure 4 is given by

I

FP3

a~= 1-~L 2"

(2.9)

Remark2.1. It is reported in [1] and [2] that (2.7) may have multiple solutions. In such situations, stochastic linearization is not applicable.

Figure 4: Stochastically linearized system.

37

Selecting the Level of Actuator Saturation

Re[L(jw)] 1

0

Figure 5: Disk D(r).

Therefore, it is assumed throughout this paper that (2.7) has a unique solution. It has been indicated in [3] and elsewhere (see, for instance [1,2, 4, 5], that the accuracy with which a~ approximates a~ is good (typically, within 10%); however, no rigorous results in this regard are available. In this paper, we use a~ in order to evaluate the degradation of Crz due to actuator saturation. Assumethat the acceptable level of performance degradation is defined by ae <_ (1 + e)az, (2.10) where e > 0. The problem considered in this paper is as follows: Given e > O, find the level of saturation of the actuator, ~, so that

< (1 + e)~z. 2.3.

Main Result

Let D(r) denote the closed disk in C with radius r, centered at (-r-l, (see Figure 5). Let/3(e, r) be defined ~(s,r) = ] . x/~(1 + e)erf-’ Introduce the following assumptions:

(2r+(l+e) \ (f~_~)T2;~-l)

j0)

(2.11)

38

Eunet al. 2.8 2.6 2.4 2.2 2 1.8 0

2

4

6

8

10

Figure 6: Function fl(e, r). (A1) The closed loop system of Figure 3 with w =0 is globally asymptotically stable. (A2) Transfer proper.

functions FP3 and F~,~c 1--£Z-, where L = PIP2C, are strictly

(A3) Equation (2.7) has a unique solution Theorem2.1. Let (A1)-(A3) hold, e be the tolerable level of performance degradation, and r be such that the Nyquist plot of the loop gain L lies entirely outside of D(r). Then, 0 <_ a~ _< (1 + e)~rz

(2.12)

if Proof. See the Appendix. Figure 6 illustrates the behavior of fl(e,r) with e = 0.1 and e = 0.05, for a wide range of r. Obviously, ~(e, r) is not very sensitive with respect to r for r > 1. In particular, it is close to 2 for all r > 1, if e -- 0.1. This justifies the rule-of-thumb given in the Introduction. 2.4.

Examples

Example 2.1. Consider the feedback system with P-controller shown in Figure 7. Using the Popov criterion, one can easily check that this

Selecting the Level of Actuator Saturation

39

Figure ’~: System of Example 2.1. system is asymptotically stable. If no saturation takes place, ~r~. = 1.1238 and a~,, = 1.4142. To select a level of saturation, (~, that results in less than 60 10%perfornmnce degradation, ¯we draw the Nyquist plot of L = .~(s+~)(s+10) and determine the largest disk D(r) such that L(jw) lies entirely in its exterior. It turns out that r = 4.2, as shownin Figure 8. Thus~ according ~o Theorem 2.1, ~ >_ ~(0.1~4.2)~ = 1.9 × 1.4142 = 2.~8~ guarantees

that

the degradation

(2.13)

of perform~mce is at most 10%. With

Figure 8: Nyquist plot and D(r) for Example 2.1.

40

Eun et al.

Table 1: System performance and accuracy for Example 2.1. ~ ~ Simulation conditions as = 0.01s, t~ = 50s, t2 = 106s, ode5 1.2252 9.02 % 2.13 % = 0.1s, tl = 104s, t2 = 106s, ode5 1.2120 7.85 % 1.03% a = 2.687, the SDof the output is ors = 1.1996, which is larger than ~r~ by 6.74%. To obtain c~s, we simulated the system of Figure 7 using MATLAB Simulink and evaluated ors as follows: as =

~2dt.

(2.14)

The results are summarizedin Table 1, where t.~ is the si~nulation sampling time in seconds and "ode5" is the differential equation solving method. Since, the open loop poles are 0, -2 and -10, and the closed loop poles are -10.65, and -0.674+j2.276, the sampling time t.~ = 0.1s is small enough to obtain reliable simulation results. Twocases with t., = 0.1s and t.~ = 0.01s are provided. Results in Table 1 show that the level of a selected indeed ensures not mo~’e than 10% of per~brmance degradation. Also note that the accuracy of as as an approximation of as is about 2%. Example 2.2. Consider the feedback control system with PI-controtler shown in Figure 9. Here again, using the Popov criterion, one can easily check that this system is asymptotically stable. Without the saturation, a~ = 0.0211 and a~, = 0.4432. Again, a~ssume that 10% degradation of llsq-6 ~r.~ from ~rz is acceptable. The Nyquist plot of L = .~(~+2.~+~) lies entirely outside of D(80). Therefore, a >_ ,(3(0.1, 80)a,~, = 1.86 x 0.4432 = 0.8244 achieves the desired guarantee. With a = 0.8244, ors = 0.0224, which is

Z

Figure 9: System of Example 2.2.

Selecting the Level of Actuator Saturation

41

Table 2: System performance and .accuracy in Example 2.2. ~ °~-°~ Simulation conditions ors 0.0231 9.48 % 3.12% ts = 0.1s, tl = 300s, t2 = 10~s, ode5 ts = 0.01s, tl = 300s, t2 = 6 x 105s, ode5 0.0232 9.95 % 3.57% larger than ~rz by 6.16%. The results of MATLAB simulations to evaluate a~ are given in Table 2. Since, the open loop poles are -2, -0.5 and -1 + j2.2360, and the closed loop poles are -2, -0.5, -0.366, and -0.82 + j3.967, again, ts = 0.1s is small enoughto obtain reliable simulation results. Twocases with ts = 0.1s and ts = 0.01s are provided. Thus, both examples illustrate the effectiveness of the proposed method for selecting an actuator saturation level.

2.5.

Conclusions

This paper provides a simple method for selecting a level of actuator saturation that leads to small performance degradation of linear design. It maybe useful for control engineers whouse linear techniques for designing controllers that are to be implemented with saturating actuators. Along with providing a formula based on the Nyquist plot of the linear part of the system, we give a simple rule-of-thumb, which states that if the level of actuator saturation is at least twice larger than the standard deviation of the controller output in the linear case, the performance under saturating actuator degrades not more than 10%.

2.6.

Appendix

To prove Theorem 2.1, the following Lemmais needed. LemmaA.1. Let assumptions (A1) and (A2) hold and r be such the Nyquist plot of the loop gain L = PIP2Clies entirely outside of D(r). Then F P3 2 2r 1 1 ~ < (2r+l)N-1 ~rz, vge(~-~--~, FP2C 2r 1 ~ 2 < (2r+l)N-1 ~ru, YNe(~--~,l]. we prove the existence [[oflq-NLll2’ FP~ ~+NL EPiC 2 and~+i l I+NL~ N E (~-~+1 1]. The first two norms exist if FP~ and ~ are ~ I+NL I+NL

Proof. First, for all

42

Eun et al.

1 proper for all N E (~-i, 1]. The asympFPa and ~ are asymptotically totic stability follows from the fact that 1-~ I+L stable (due to (A1)) and the number of encirclement of -1 + j0 L(j ~) and NL(jw) are the same (due to the fact that L(jw) is outside of D(r) and ~1< N _< 1). The strict properness of these two transfer functions FP3 12 and l EPiC I+NL12 exist for all N E (~¥~, 1 1]. follows from (A2). Thus, I+NL

asymptotically stable and strictly

Since, as it follows from the above,~1+5 is also asymptotically stable 1 1]. I exists for all N ~ (y~-~, and, obviously, proper, I ~+L ~+NL ~ Next, we show that 1

FP3 2 1 +L ~2 <- I+N.L ~az’ ~

<-

I+NL

au,

VNe(2-~-~’N

~

~,

1 ],

(A.1)

(~-~,

Indeed, (A.1) can be obtained as follows

(A.3) Inequality (A.2) is proved similarly. In turn, ~+L ’ is bounded as follow: I+NL[~, for each N ~ (2-~+1 1], I+L =

m~ax 1 1 + ReL(jw) + jImL(jw) + NReL(ja~) + jNImL(jw) l+x+jy max xTjy(~D(r) 1 + Nx +

¯

(A.4)

For each positive c # ~-, the level set {(x,y) I~+Nz+jNyl+Z+~U ] = c} is a circle given by c2N(N-1)~ 2 y2 {c(1 - N) 2 (A.5)

Selecting the Level of Actuator Saturation

43

It is possible to show that the level set (x,y) l+gx+jNy l+x+.~y = c} is not entirely contained in D(r) for all c small enough, that there exists a unique c = c* such that the circle (A.5) is tangent to and contained in D(r), and that for all c > c*, the level set is strictly contained in D(r). This c* is an upper-boundfor the right-hand side of (A.4), i.e., 1 + x + jy c*. max < x+jy~D(r) 1 + Nx + jNy

(A.6)

The condition of tangency is: (2r + 1) i.e., c*’ =

c2N(N- 1) _ c(N- 1) 21 - c2N 12’-- e2N

2r (2r + 1)N-

(A.7)

1 VN ~ (~, 1]. Ar~-i

(A.8)

Therefore, I+L I+NL

< 2r - ~ (2r+l)N-l’

1 VN E (~-~-~,

1],

(A.9)

which, along with (A.1) and (A.2), completes the proof. Proof of Theorem2.1. First,

we show that a >_ f~(e, r)cru implies

FP2C 2 < a 1 ÷ NIL ’-- X/~ erf-l(Ni) where 1 2r + (1 + e) N1= (1 + e)(2r + > 2r +-----~"

(A.10)

(A.11)

Indeed, rewriting a >_ 13(e, r)au using (A.11) and (2.11) yields

-1 = + )erf

2r+(l+e)

= x/-~(1 ÷ e)erf-l(Ni)au.

(A.12)

Then, from LemmaA.1, (A.11) and (A.12), we obtain FP2C112 < 2r ’ -l(Ni) 1 + NiL - (2r + 1)Ni - 1 ~u = (1 e) au < V~erf

(A.13)

44

Eun et al.

Next, we show that the quasi-linear gain N* of the system of Figure 4 corresponding to a given c~ > ~(e, r)au exists and, moreover, N* _> N1. Indeed N* is defined by ~ ~TL ~ = x/~ erf-l(N*)" 1FP2C 2

(A.14)

For N = 1, due to the fact that erf-1(1) = co, we can write: c~ FP2C -~ ~ 2 v/~ err-l(1)

- O. iA.15)

a Since, FP_2_~C I+NL 2 is continuous in N, from (A.10) and (A.15’t," x~ erf-l(N) we conclude that there exists N* _> N1 satisfying (A.14). Finally, using LemmaA.1, we show that a~ _< (1 + e)az: F Pa I~-~-*L

2r ~ -< (2r+l)N*-I 2r - (2r+l)Nl-1 = (1 + e) az.

ffz (A.16)

This completes the proof. References

[1] A. Gelb and W. Vander Velde. Multiple-input Describing Functions and Nonlinear System Design, NewYork: McGraw-Hill, (1968). [2] C. GSk~ek. Disturbance Rejection and Reference Tracking in Control Systems with Saturating Actuators. PhD thesis, The University of Michigan, 2000. [3] C. GSkqek, P. Kabamba, and S. Meerkov. Disturbance Rejection in Control Systems with Saturating Actuators, Nonlinear Anal., 40 (2000) 213-226. [4] J. Roberts and P. Spanos. RandomVariation and Statistical tion, NewYork: John Wiley and Sons, (1990).

Lineariza-

Selecting

the Level of Actuator Saturation

45

Approach to Opti[5] W. Wonham and W. Cashman. A Computational mal Control of Stochastic Saturating Systems, Int. J. Contr.~ 10(1) (1969) 77-98.

Chapter 3 Null Controllability and Stabilization of Linear Systems Subject to 1 Asymmetric Actuator Saturation T. Hu, A. N. Pitsillides, University

3.1.

of Virginia,

and Z. Lin Charlottesville,

Virginia

Introduction

Weconsider the problem of controlling exponentially unstable linear systems subject to asymmetric actuator saturation. This control problem involves basic issues such as characterization of the null controllable region by boundedcontrols and stabilizability on the null controllable region. These issues have been focuses of study of and are now well-addressed for linear systems that are not exponentially unstable. For example, it is wellknown[10,11] that such systems are globally null controllable with bounded controls as long as they are controllable in the usual linear system sense. ¯ In regard to stabilizability, it is shownin [12] that a linear systemsubject to actuator saturation can be globally asymptotically stabilized by smooth feedback if and only if the system is asymptotically null controllable with bounded controls (ANCBC),which, as shown in [10, 11], is equivalent the system being stabilizable in the usual linear sense and having open loop 1Worksupported in part by the USOffice of Naval Research YoungInvestigator Programunder grant N00014-99-1-0670.

47

48

Huet al.

poles in the closed left-half plane. A nested feedback design technique for designing nonlinear globally asymptotically stabilizing feedback laws proposed in/14] for a chain of integrators and was fully generalized in [13]. The notion of semiglobal asymptotic stabilization on the null control-lable region for linear systems subject to actuator saturation was introduced in [7]. The semiglobal frameworkfor stabilization requires feedback laws that yield a closed-loop system which has an asymptotically stable equilibrium whose domain of attraction includes an a priori given (arbitrarily large) boundedsubset of the null controllable region. In [7], it was shown that, for linear ANCBC systems subject to actuator saturation, one can achieve semiglobal asymptotic stabilization on the null controllable region using linear feedback laws. On the other hand, the counterparts of the above mentioned results for exponentially unstable linear systems are less understood. Recently, we made an attempt to systematically study issues related to null controllable regions and the stabilizability on them of exponentially unstable linear sys-. tems subject to actuator saturation and gave a rather clear understanding of these issues [4]. Specifically, we gave a simple exact description of the null controllable region for a general anti-stable linear system in terms of a set of extremal trajectories of its time-reversed system. Wealso constructed. feedback laws that semiglobally asymptotically stabilize any linear time invariant system with two exponentially unstable poles on its null controllable region. This is in the sense that, for any a priori given set in the interior of the null controllable region, there exists a linear feedback law that yields a closed-loop system which has an asymptotically stable equilibrium whose domain of attraction includes the given set. One critical assumption made in [4] is that the actuator saturation is symmetric. The symmetryof the saturation function to a large degree simplifies the analysis of the closedloop system, it, however, excludes the application of the results to many practical systems. The goal of this chapter is to generalize the results of [4] to the case where the actuator saturation is asymmetric. Wewill first characterize the null controllable region and then study the problem of stabilization. Wetake a similar approach as in [4] to characterize the null controllable region. In studying the problem of stabilization, we found the methods used in [4] to derive the main results not applicable to the asymmetriccase, since the methods rely mainly on the symmetric property of the saturation function. For a planar anti-stable system under a given saturated linear feedback, we showed in [4] that the boundary of the domain of attraction is the unique limit cycle of the closed-loop system. The uniqueness of the limit cycle was established on the symmetric property of the vector field

Null Controllability and Stabilization

49

and the trajectories. Wefurther showed that if the gain is increased along the direction of the LQRfeedback, then the domain of attraction can be madearbitrarily close to the null controllable region. This result was also obtained by applying the symmetric property of the trajectories. In this chapter, we propose a quite different approach to solving these problems for the case of asymmetric saturation. In particular, we will construct a Lyapunovfunction from the closed trajectory, and show that under certain condition, the Lyapunovfunction is decreasing within the closed trajectory, thus verifying that the closed trajectory forms the boundary of the domain of attraction. If the state feedback is obtained from the LQR method, then there is a unique closed trajectory (a limit cycle). Wewill also showthat if the gain is increased along the direction of the LQRfeedback, then the domainof attraction can be madearbitrarily close to the null controllable region. This result will be developed by a careful examination of the vector field of the closed-loop system. For higher order systems with two anti-stable modes, we have similar results as in the symmetric case: given any compact subset of the null controllable region, there is a controller (switching between two saturated linear feedback laws) that achieves a domain of attraction which includes the given compactsubset of the null controllable region. 3.2.

Preliminaries

and Notation

Consider a linear system 2(t) = Ax(t) + bu(t),

(3.1)

where x(t) E n i s t he s tate a nd u(t) ER is the cont rol. Give n real numbers u- < 0 and u+ > 0, define /4a := {u: u is measurable and u- <_ u(t) <_ +, Vt ~ R}. (3 .2) A control signal u is said to be admissible if u ~ b/a. In this chapter, we are interested in the control of the system (3.1) by using admissible controls. Our first concern is the set of states that can be steered to the origin by an admissible control. Definition 3.1. A state x0 is said to be null controllable if there exist a T ~ [0, oc) and an admissible control u such that the state trajectory x(t) of the system satisfies z(O) = x0 and x(T) = O. The set of all null controllable states is called the null controllable region of the system and is denoted by C.

50

I-I~zet al. With the above definition,

we have

{

e-Arbu(T)dT

(3.3}

TS[O,~/ Remark 3.1.

For a linear system with multiple inputs, it(t) = Ax(t) + Bu(t),

(3.4)

where u ~ Rm, and ui ~ [u~-,u~+], u~- < 0, u~+ > 0, let Ci be the null controllable region of the system it(t) = Ax(t) + b~ui(t), then it is easy to verify that the null controllable region of the system(3.4) is m

Hence, it is without loss of generality that we consider the single input system (3.1). For simplicity, a linear system and the matrix A are said semistable if all the eigenvalues of A are in the closed left half plane; and anti-stable if all the eigenvalues of A are in the open right half plane. Werecall a fundamental result from the literature

[2,10,11]:

Proposition 3.1. Assumethat (A, b) is controllable. a) If A is semistable, then C = ~. b) If A is anti-stable, the origin.

then C is a bounded convex open set containing

A20 Al~Rnlxnlanti-stableandA2~Rn~xn2 ] c) IrA= [ A10 with semistable,

and b is partitioned

as b~ accordingly,

then

~: C ,= C~ x R where C1 is the null controllable region of the anti-stable system ~:l(t) ---- AlXl +blU(t).

Null Controllabilfty and Stabilization

51

Because of this proposition, we can concentrate on the study of null controllable regions of anti-stable systems. For this kind of systems, ~ = x = - e-A’bu(T)dT:

U e

(3.5)

where ~ denotes the closure of C. Wealso use "0" to denote the boundary of a set. In this chapter, we will derive a methodfor explicitly describing c9Cin Section 3. In the study of null controllable regions we will assume, without loss of generality, that (A, b) is controllable and A is anti-stable. For a general system 2=f(x,u), (3.6) its time-reversed system is (3.7)

~ = -f(z, v). Consider the time-reversed system of (3.1):

(3.8)

i(t) = -Az(t) - bv(t).

Definition 3.2. A state zf is said to be reachable if there exist T E [0, ~c) and an admissible control v such that the state trajectory z(t) of the system (3.8) satisfies z(0) = 0 z(T)= zf. The se t of all reachable states is called the reachable region of the system (3.8) and is denoted 7~. It is knownthat C of (3.1) is the sameas 7~ of (3.8) (see, e.g., [8]). avoid confusion, we will continue to use the notation x, u and C for the original system (3.1), and z, v and 7~ for the time-reversed system (3.8). 3.3.

Null

Controllable

Regions

In Section 3.3.1, we show that the boundary of the null controllable region of a general anti-stable linear system with saturating actuator is composed of a set of extremal trajectories of the time-reversed system. The descriptions of this set are further simplified for systems with only real poles and for systems with complex poles in Sections 3.3.2 and 3.3.3, respectively. 3.3.1.

General Description

of Null Controllable

Regions

Wewill characterize the null controllable region C of the system (3.1) through studying the reachable region 7~ of its time-reversed system (3.8).

52

I-Ia et al. Since A is anti-stable,

we have

{ /0

~ = z = --

e-A~-bv(w)d’r: v

z = -

: v E l~a ¯

Noticing that e Ar = e-A(O-r), we see that a point z in ~ is a state of the time-reversed system (3.8) at t = 0 by applying an admissible control v from -co to 0. Define the asymmetric sign function sgna(. ) as sgna(r ) :=

+ u 0, u-,

r > 0, r = 0, r<0.

Theorem 3.1. 07-~= z=-OO

sgn a

c

T~ is strictly convex. Moreover, for each z* G cgT~, there exists a unique admissible control v* such that -. z* = - /_°cc eArbv*(~-)d~ Proof. This can be proved similarly as Theorem2.3.1 in [41.

(3.10) El

Theorem3.1 says that for z* ~ 0P,., there is a unique admissible control v* satisfying (3.10). From(3.9), this implies v*(t) = sgna(c’eAtb) for some c ~ 0 (such c, Ilcll = 1, may be nonunique, where I1" It is the Euclidean norm). So, if v is an admissible control and there is no c such that v(t) sgna(c’eAtb) for t < 0, then --

eArbv(r)d~- ~ cgP,.

and must be in the interior of 7~. Since sgna(kc~eA~b) = sgna(c~eA~b) for any positive number k, equation (3.9) shows that 07~ can be determined from the surface of a unit ball in Rn. In what follows, we will simplify (3.9) and describe 0R in terms of a set of trajectories of the time-reversed system (3.8).

Null Controllability and Stabilization

53

Denote ~c :=

{V(t)

= sgn a (cleAtb),f5

R: c ~ 0} ,

(3.11)

and for an admissible control v, denote ~P(t, v) :=- e-A(t-r)bv(r)dT.

(3.12)

Since A is anti-stable, the integral in (3.12) exists for all t E R, so qS(t, is well defined. If v(t) = sgna(c’eAtb), then ~(t,v)

=

f:

-e-A(t-~’)bv(~-)&

= _ [,7 eaTb sgna (c’eate~’b) E for all t E R, i.e., ff)(t, v) lies entirely on 07~. Anadmissible control such that ff)(t, v) lies entirely on cqT~is said to extremal andsuch~(t, v) an extremal trajectory. On the other hand, given an admissible control v(t), if there exists no c such that v(t) sgna(c’entb) for al l t _<0, then by Theorem3.1, (I)(0, v) ~ c97Z and must be in the interior of 7Z. the time invariance property of the system, if there exists no c such that v(t) = sgn~(e’entb)for all t _~ to, ~(t, v) must be in the interior of ~ for all t _~ to. Consequently, £c is the set of extremal controls. Definition 3.3. vl, v2 ~ £c are said to be equivalent, denoted by v~ ~ v2, if there exists an h ~ R such that v~(t) = vz(t - forall t ~ R. The following theorem shows that 07~ is covered by a minimal subset of the extremal trajectories. Theorem3.2. Let £~ C £c be such that for every v ~ £~, there exists a unique V1 E £~n such that v ~ v 1. Then

0k = {~(t, v): t E R, v ~ gy}. Proof. For any fixed t ~ R, it follows from (3.9) that 07~ = e-{ A(t-r)b St~o = --

sgna (c’e-AteArb) dT : c ¢

e-A(t-~-)b sgna (c’eA’rb) dr : c 7~ 0

(3.13)

54

Huet al.

i.e., 07~ = {~(t, v) : v ¯ $c}, for any fixed t ¯ R. So 07~ can be, viewed as the set of extremal trajectories at any frozen time. Nowlet t vary, then each point on 07~ movesalong a trajectory but the whole set is invariant. So we can also write 0TO= {~(t, v) : v ¯ ~c, t ~ R}, which is equiw~lent to 0g=

:t

(3.14)

Noting that a shift in time of the contro ! corresponds to the same shift of the state trajectory, we see that, if v, ~ v~, then

And (3.13) follows from the property of 3~. It turns out tha~ for someclasses of systems, £~ can be easily described. For second order systems, $~ contains only one or two elements, so 0g can be covered by no more than two trajectories; and for third order systems, £~ can be described using a parameter that varies within a real intervM. Wewill see later that for systems of different eigenvalue structures, the descriptions of £~ can be quite different. All the results in the following subsections are easy extension of the counterparts in [4], hence the proofs are omitted. 3.3.2.

Systems with Only Real Eigenvalues

It follows from, for example,[8, p. 77], that if A has only real eigenvalues and c ~ 0, then c’eAtb has at most n- 1 zeros. This implies that an extremal control can have at most n- 1 switches. It was shownin [4] that the converse is also true. Theorem3.3. For the system (3.8), values, then

assume that A has only real eigen-

a) an extremal control has at most n - 1 switches; b) any bang-bangcontrol with n- 1 or less switches is an extremal control. By Theorem3.3, the set of extremal controls can be described as follows, ~c = sgna(-t-u) u(t) = { {1,

~, (-1)

(-1)

t ¯ [t~,ti+l),

t

- -} ~x~
55

Null Con~rollabili~ and S~abili~a~ion

where u+ (or u-) denotes a constant control v(t) =- + ( or u -). H ere we allow ti = ti+l (i # 1) and tn-1 = o~, so the above description of £c consists of all bang-bangcontrols with n - 1 or less switches. By setting tl -- 0, we immediately get Sm

$~ ~ sgna(±U):u(t)= { { 1,

(-1) ~, t~[t~,t~+~), (-1) n-l, t e [tn-l,~),

~or each v ~ c , we have v(t) ~- (or ~+) for all t ~ 0. Hence. for t ~ ¯ (t, v) =

e-a(t-~lbv(t)dr

= -A-~b~ - ( or - A-~b~+).

And for t > 0, v(t) is a bang-bang control with ~ - 2 or less switches. Denote z~ = -A-~b~ + and z 2 = -A-~b~ -, then from Theorem a.2 we h ave, Observation 3.3.1. 07~ = c~C is covered by two bunches of trajectories. The first bunch consists of trajectories of (3.8) whenthe initial state is %+and the input is a bang-bangcontrol that starts at t = 0 with v = uand has n- 2 or less switches. The second bunch consists of the trajectories of (3.8) when the initial state is z[ and the input is a bang-bang control that starts at t = 0 with v = u+ and has n - 2 or less switches. Remark3.2. Since the trajectories of the time-reversed system (3.8) and those of the original system are the sameexcept that their directions are opposite, we can also say that 07E = 08 is covered by the trajectories of the original system under the same controls. The fnndamental difference is that it is quite easy to generate the trajectories with the time-reversed system, while it is unrealistic to get the trajectories from the original system. For example, suppose that we have a trajectory of the time-reversed system that starts at z~+ under the control v = u-, then it goes toward z[ since z~- is a stable equilibrium under the control v = u-. On the contrary, if we apply u = u- to the original system with initial condition z~-, we cannot get a reversed trajectory because z~- is an (unstable) equilibrium under the control u = u-. The trajectory of the time-reversed system from z~+ to z~under the control v = u- could be partly recovered by the original system if we knowone point (except z[) on the trajectory. But this is unrealistic.

56

Huet al.

Furthermore, 0T~ can be simply described in terms of the open-loop transition matrix. o97¢ =

- u+)(-1)~e -A(t=td -- sgna (+(--1) n) I A-lb:

:l:(uLi=I

0=t~

~t~...~tn_~

Here, we allow t~ = t~ to include ~z~. For second order systems,

If n = a, then one half of 0~ = 0C can be formed by the trajectories of (a.8) starting from z~ with the control initially being v = ~- and then switching at any time to v = ~+. So the trajectories go toward z 2 at first then turn back toward z~. The other half is formed by the trajectories of (a.8) starting from 2 wigh t he c ontrol i nitially b eing v = ~+andthen switching at any time to v = ~-. So the trajectories go toward z~ at first then turn back toward z 2. That is, 0~=

-~)-/

U~e-atz;-[ 3.3.3.

~-~(~-’)~-e~-/~-~(~-’)~+e’: dO

JO

Jr2

e-a(t-~)bu+dT-[e-n(t-r)bu-dr:

Systems with

Jr2

O~t~t~ O~t~t~.

Complex Eigenvalues

For a system with complex eigenvalues, the set g~ is harder to determine. In what follows, we consider two important cases. Case 1. A ~ R~x~ has a pair of complex eigenvalues a ~j~; a,~ > 0. It can be verified that {sgna(eteAtb):

c¢0}= {sgna(sin(~t+0)):

0~ [0,2~)}.

Null Controllability and Stabilization

57

Hence the set of extremal controls is £c = {v(t)= sgna(sin(~t

+0)), t E R: 0 E [0,2~r)}.

It is easy to see that g~m = {v(t)= sgna(sin(~t)), contains only one element. Denote Tp = 5’ then e-ATp = --e-~TpI. Let z: = (1 - ~_aTp)-I (_~_ + e_aTp~+) A_lb, and Z~= (1-

e-aTP)

-1

(--~+

+e-aTv~-)A-lb.

It can be verified that the extremal trajectory corresponding to v(t) sgn~(sin(3t)) is periodic with period 2Tp: in the first half period it goes from z~ to z~ under the control v = u+ and in the second half period it goes from z) to z~ under the control v = u-. That is, O~ = e-ntz~ - e-n(t-r)bu+d~ U e-Atz{ - e-A(t-r)bu-d7

: t [O, Tp ] : t~ [O,

= ~{e-~tz)-(I-e-~t)A-~bu-:

t ~ [0,

Tp]}.

(3.16)

Case 2. A ~ R3x3 has eigenvalues a ~ j~ and al, with a, ~, a~ > 0. a) a = a~. Then similar to Case 1, g~={v(t)=sgn~(k+sin(~t+0)),t~R:

k~R,

0~[0,2~)}.

Since sgna(k + sin(~t + 0)) is the same gr all k k 1 (or k ~ -1), we

e2 = {v(t)=

+sin(f/t)),

t ~ R: k ~ [-1,1]}.

Each v ~ g~ is periodic with period 2Tp, but the lengths of positive and negative parts vary with k. ~(t, v) can be easily determined from simnlation. A formula can also be derived for ~(t, v).

58

Hu et al.

b) a # c~1. Then

gc = {v(t) =

sgn a (]~le (O~l-(~)t

-~ k2 sin(/~ (]¢1,/¢2)

~- 0)),~ # (0,0),0

E E [0,271")}.

It can be shown that gcTM = {u+,u-} U {v(t)= sgna(sin(~t)) } Ugh3. where u+ (or u-) denotes a constant control v(t) =- + (or u -) a gc~ = {v(t)= sgna (±e(C~l-c~)t+ sin(/3t

+ 0)),t~ R:0~ [0,2~r)}

(3.17) Whenal < c~, for each v ~ gc~3, v(t) = + ( or u -) f or a ll t _<0, the corresponding extremal trajectories stay at z~+ = -A-ibu + (or z[ -A-lbu -) before t = 0. And after some time, they go toward a periodic trajectory since as t goes to infinity, v(t) becomesperiodic; Whenc~1 > (~, for each v ~ $~, v(t) = + (or u -) f or a ll t >_0, andthe corr esponding extremal trajectories start from near periodic and go toward %+or zj. Plotted in Figure 1 are some extremal trajectories on 0R of the timereversed system (3.8) with A=

0 0.8 -2 0 2 0.8

,

B= 1 , 1

u +=1,

u-=-0.5.

For higher order systems, the relative locations of the eigenvalues are more diversified and the analysis will be technically muchmore involved. It can, however, be expected that in the general case, the number of parameters used to describe g~ is n - 2. 3.4.

Domain Linear

of Attraction State Feedback

under

Saturated

Consider the open loop system, !c(t) = Az(t) + bu(t),

(3.18)

with admissible control u E/4a. A saturated linear state feedback is given by u = sata(fx), where f ~ Rlxn is the feedback gain and sata(’) is the

Null Controllability and Stabilization

59

0-

-0.5

Figure 1: Extremal trajectories asymmetric saturation

on cgT~, al < c~.

fnnction ~+,

sato(r)

r,

+, r ~ tt

,e

tt-,

Sucha feedback is said to be stabilizing if A + bf is asymptotically stable. With a saturated linear state feedback applied, the closed loop system is ~(t) = Ax(t) + b sata(fx(t)).

(3.19)

Denote the state transition mapof (3.19) by ¢ : (t, xo) ~ x(t). The domain of attraction S of the equilibrium x = 0 of (3.19) is defined S:={x0ERn:t_~lim¢(t,

x0)=0}.

Obviously, $ must lie within the null controllable region C of the system (3.18). Therefore, a design problem is to choose a state feedback gain that $ is close to C. Werefer to this problem as semiglobal stabilization on the null controllable region. Wewill first deal with anti-stable planar systems, then extend the results to higher order systems with only two anti-stable modes. Consider the

60

Huet al.

system (3.19). Assumethat A E 2×2 i s a nti-stable. F or t he s ymmetric case where u- = -u +, it was shown in [4] that 0S is the unique limit cycle of the system (3.19). This limit cycle is unstable for (3.19) but stable for the time-reversed system of (3.19). So it can be easily obtained by simulating the time-reversed system. However, the method used in [4] to prove the uniqueness of the limit cycle relies on the symmetric property of the vector field. There is no obvious way to generalize the method to the asymmetric case. In this section, we will present a quite different approach to this problem. Actually, we will construct a Lyapunovfunction from the closed trajectory, and show that the Lyapunovfunction decreases in time as long as the trajectory starts from within the closed trajectory. Therefore, the open set enclosed by the closed trajectory is the domainof attraction. Lemma3.1. The origin is the unique equilibrium point of the system (3.19) and there is a closed-trajectory. + Proof. The other two candidate equilibrium points are xe+ = -A-lbu and x[ = -A-lbu -. For x~+ to be an equilibrium, we must have fx+~ ~ u+ so that u -- sata(fX+e) = +. Since A is ant i-stable and (A,b) is controllable, we can assume without loss of generality that -al

a2

,

This implies that if f = [ fl f2 be easily shownthat

al,a2 ~ 0, ] is

b=

stabilizing,

1 " then fl/al < 1. It can

fx+e = -fA-ibu + _= (fl/al)u + +.
F

Proof. Westart the proof by showing some general properties of second order linear autonomous systems, ~(t) = Ax(t),

get(A) ~

(3.20)

Null Controllability and Stabilization DenoteO(t) : /x(t),

r(t) : [[¢(t)[[,

61 then ¢(t) = r(t)[ sin0(t) cos0(t) , and it can

be verified that -

0

(3.21)

sin 0 ]"

Noting that [ cos0

sin0

] -1 0

sinO =0,

equation (3.21) has at most four equilibriums on [0, 2~r), which correspond to the directions of the real eigenvectors of A. If ~(0) is an equilibrium, then O(t) is a constant, and x(t) is a straight line. If 0(0) is not an equilibrium, then 0(t) will never reach an equilibrium at finite time. If it reaches, the trajectory x(t) will intersect with a straight line trajectory, which is impossible. Hence, if x0 is not an eigenvector of A, ~(t) will never be equal to zero, thus ~(t) will be sign definite. This shows that /x(t) is strictly monotonouslyincreasing (or decreasing). Let’s nowconsider the direction angle of the trajectory, /&(t) =: "~(t). Since 2(t) -- AS:(t), by the sameargument, 7(t) also increases (or decreases) monotonically. Weclaim that if the system is asymptotically stable or antistable, then ~(t) and ;~(t) have the same sign, i.e., the trajectories towardthe origin; if the signs of the two eigenvalues are different, then O(t) and ~(t) have opposite signs. This can be simply shownas follows. Rewrite (3.21) as,

~=I_L_x, [ 01 1 lax.

(3.22)

Similarly,

~/= ~ x -1 0

-

f°r It a2x2 is trivial to verify that A’ [_01 01JA=det(A)[

0

AAx.

-01 01]

matrix A. So

~ =det(A) ~ x,[

-01

01]dxIlxll~det(n)0"ll~ll~ -

(3.23)

This showsthat the claim is true. Nowwe can apply the above result to the closed trajectory F. Werefer to Figure 2. Assumethat F goes anti-clockwise and intersects both the lines fx : u+ and fx = u- (the arguments also apply if F only intersects one

62

Huet al.

1

"--4

-3

-2

-1

Figure 2: Illustration

0

1

2

3

4

for the proof of Proposition 3.2.

of the straight lines). Let the intersections be x(0), x(tl), x(t2) and x(t3). From x(0) to x(tl), we have Ax+ bu+, whichcan be rewritt en as d(x - x+e )/dt = A(x - x+e By Lemma3.1, xe + is below the line fx = u+. So from x(0) to x(tl), /Ix(t)- +] increases, an d hence /2 (t) in creases (s ince d is anti-stable). From x(tl) to x(t2), we have ~ = (A + bf)x, so Zx(t) and /k(t) increase (since A + bf is stable). Similarly, /2(t) increases from x(t2) x(t3), and from x(t3) to x(0). It is straightforward to verify that k(t) continuous at x(0), x(tl), x(t2) and x(t3). Hence, /2(t), the direction angle, is monotonically increasing along F. This implies that the region f~ enclosed by P is convex. [] The following theorem shows that under certain condition, trajectory F is the boundary of the domainof attraction.

a closed

Theorem3.4. Let F be a closed-trajectory of the system (3.19). Let. the intersections of F with the line {#A-lb : # E R} be Xbl and Xb2 (see Figure 2). If fXbl, fXb2 ~ [U-, U+], i.e., the two intersections Xbl and xb2 are between the two lines fx = u- and fx = u+, then 0S = F.

63

Null Controllability and Stabilization Proof. Without loss of generality, 1

a2

’

we assume that

al,a2

> O, b =

b2 ’

and f = [ 0 1 ]. Then fx =- u- and fx = u+ are two horizontal lines (see Figure 3). Since A + bf is Hurwitz, we have bl < al, b2 < -32 and that the trajectories go anticlockwise. Denotethe region enclosed by F as Ft. Since Ft contains the origin in its interior, we can define a Minkowskifunctional ~(x) := min {7-> 0: x ~ 7~-~}. (If ~ is symmetric and convex, n(x) is a norm). Clearly, n(x) = 1 for x E F. Since F is a trajectory and the vector field ~ in (3.19) is continuous, Oh(x) Ox exists and is continuous along F. Since ~ is bounded and convex, it follows that oh(x) --57-x ~ 0 for all x E F. Note that ~ is the gradient of the function ~(x), so it is perpendicular to the tangent of the curve F = {x e R2 : n(x) 1}, which is 2. Therefore, cOx J ~=0’

(3.24)

Yx~F.

Define a Lyapunovfunction as V(x) := ½n~(x). It can be verified that for any constant c~ > 0,

=

V(ax)= a V(x),

and Ox ~ ,o~(~) ov(~) Since ore) o~ = ~[x)--~-~, o~ exists follows that

cOy(x) Ox

Ox

"

~nd is continuous for all x ~ R~. It ---

or(x) a ~

(3.25)

and ---~-~x .] 2 = 0, cOx " # 0, Yx ~ r.

(3.26)

Weconclude that for all x ~ ~, along the trajectory of the system (3.19),

k Ox J2

=

kax)

(Ax+ bsata(fx))

< 0.

-

(3.27)

66

Huet al.

straight lines fx = u- and fx = u+. In other words, the two intersections are on F1 and F3. Therefore, k2(xr) # 0 for all xr E F2 U F4. If k2(xr) has the same ’+’ on F~ and F~, then

~(ax~) = ag(a,z~) <0, vae (0,1),z~ ~ (r~ur~)\(rlur3), i.e., ~(x) < 0 for all x in the interior of fls and fla. The next step is showthat this is indeed the case. Actually, by the continuity of k2(Xr) and that k2(x~) ~ 0 for all xr ~ F2 U F4, we only need to find one point on F~, and one point on F4 such that k~(x~) > With the special form of A and f, the line above the origin is + fx = u and the one below the origin is fx = u-. Also a trajectory goes along F anticlockwise. Hence, on F~, there is a point x~ such that ~:r =

0

, dl > 0

and -(~X

,

C1>0.

Cl

Note that the gradient points outward of F. Let zr =

xl + a2x~ + b2u + = So we have x~ + a2x2 = -b2u+. From the stability A+bf=

X2

’

then

0 " of

[0 1 -al+b~ a2 + b2 ]

we also have, b~ < al, b2 < -a2 < 0. Hence x~ + a2x2 > 0. It follows that

= [ o ] -alx2 +

[ c~(x~ +a~z2) >

Similarly, on F4, there is a point xr such that icr = [ d~ ],d2 > O and OV(x~) _ [ 0 ],Ox

-ce c2>0.

In particular, xl -t- a2x2~- b2u-

0 "

Null Controllability and S~abiliza~ion

67

So we have xl + a2x2 -~ -b2u- < 0. Note that b2 < 0 and u- < 0. It follows that

k2(xr)=

] [ xl+ -alx2

]>0.

These show that there exist one point on F2 and one point on F4 such that

k~(zr) > In summaryof the the above analysis, we have ~(x) < 0, for all x the interior of ~2 and ~4, and ~(x) = 0 for all x E ~ ~A ~3. It follows that no trajectory starting from within ~ will approach F = Wenext showthat there exists no closed trajectory within ~. Let E be the line on the commonboundary of ~ and ~. Suppose that there is closed trajectory F1 that intersects E at x0: Note that F1 must enclose the origin. Let the trajectory start at x0, then it goes through ~2, and returns to x0 at some t. Since ~ < 0 in the interior of ~2 and f~4, we must have V(xo, t) < V(xo, 0). This is a contradiction since V is independent of t. Therefore, all the trajectories starting from within ~ will converge to the origin. Since the trajectories do not intersect each other, all the trajectories starting from outside of F will stay outside of it. Wehence conclude that the interior of ~ is the domainof attraction. That is, 05 = O~ = F. [] The condition fXb~, fXb~ ~ [U-,U+] in Theorem3.4 is always true in a special case when the line {#A-ib : p ~ R} is in parallel to the straight lines fx = u- and fx = u+. This is the case if bl -- 0 in the special form of A, b, f in the proof of the theorem. So in this case, any closed-trajectory is the boundary of the domain of attraction. Thus, we can further conclude that there is a unique closed trajectory (and hence a unique limit cycle). In the next section, we will show that if f is designed by the LQR method, then the line {#A-~b : # ~ R} is in parallel to the straight lines fx = u- and fx = u+. Moreover, the domain of attraction $ can be made arbitrarily close to the null controllable region C by simply increasing the feedback gain.

3.5.

Semiglobal Stabilization Controllable Region

3.5.1.

Second Order Anti-stable

on the Null Systems

In this subsection, we continue to assume that A ~ R2×2 is anti-stable and (A, b) is controllable. Wewill show that the domain of attraction of the equilibrium x -- 0 of the closed-loop system (3.19) can be made

68

Huet al.

arbitrarily close to the null controllable region C by judiciously choosing the feedback gain f. To state the main result of this section, we need to introduce the Hausdorff distance. Let X:L, 2~2 be two bounded subsets of Rn. Then their Hausdorff distance is defined as, d(X1,X2) := max where d(Xl,X2):

sup

inf

IIX1--2~211.

XlE2~I x2 E 2d2

Here the vector norm used is arbitrary. Let P be the unique positive definite solution to the following Riccati equation, A’P + PA - Pbb’P = 0. (3.28) Note that this equation is associated with the minimumenergy regulation, i.e., an LQRproblem with cost J = u’(t)u(t)dt. The corresponding minimumenergy state feedback gain is given by f0 =: -b~P. By the infinite gain margin and ~0~ gain reduction margin property of LQRregulators, the origin is a stable equilibrium of the system, ~(t) = Az(t) bsat~(~f0~(t)),

(~.29)

for all k > 0.g. Let S(k) be the domain of attraction of the equilibrium x = 0 of (3.29). The following lemmais a simple generalization of the result of [3]. Lemma 3.2.

Let Um=min{-u-,u+}.

Define

~o = x ~ R2 :x~Px ~ b~Pbj. { 4u~ Then D0 is in the domainof attraction for the system (3.29) for all k > 0.5. Theorem3.5. limk-~ d(,..q(k),

C)

Proof. For simplicity and without loss of generality, we assume that 1 a2

,

al,a2>O,

b=

--

.

Null Controllability and Stabilization

69

Since A is anti-stable and (A, b) is controllable, A, b can always be transformed into this form. Suppose that A has already taken this form and b=[bll’LetV=[-A-lbb2

-b ],

then V is nonsingular

and it can be

verified that V-1AV = A and V-lb= [ 0_11. With this special form of A and b, we have, P=2

a~ 0

a2

,

f0=[0

2a2],

and 1 a2(1-2k)

A- ib=

"

Hence, the line {#A-Ib : # E R} is actually the line z~ = 0 and it is between + u- ) the two lines k fox = u+ and k fox = u- (i.e., x2 u= ~ and x2 = 2a~k for all k > 0.5. Therefore, the condition in Theorem3.4 is satisfied for all k > 0.5 and the closed-loop system has a unique limit cycle which is the boundary of 8(k). Also, by Lemma3.2, the limit cycle always encloses the fixed ellipsoid ~0. To visualize the proof, OC, ~o and O8(k) for somek, are plotted in Figure 4, where the inner closed curve is O8(k) = F, and the outer dashed one is OC. For convenience, we proceed the proof with the time-reversed system of (3.29), ~(t) = -Az(t) - bsata(k foz(t) (3.30) Observe that F is also the unique limit cycle of this system. Recall from (3.15) and (3.16) that c9C is formed by the trajectories of the system 2 = -Az - bv: one from ze+( or zs +) to z~-( or z~-) under the control v = u- and the other from z~- ( or z~-) to + ( orzs+) under the control v = u+. On the other hand, when k is sufficiently large, the limit cycle must have two intersections with each of the lines kfoz = u+ and kfoz = u-. Suppose that the limit cycle trajectory starts at the righthand side intersection with kfoz = u-, goes clockwise and intersects the two lines successively at time tl, t2 and t3 (see the points z(0), z(t~), z(t2) and z(t3) in Figure 4). Wealso note that from z(0) to z(tl), sata (kfoz) = ufor the closed-loop system (3.30) and from z(t2) to z(t3), v = +. By comparing the two closed trajectories F and OC, we see that the proof can be completed by showing that as k -~ oc, z(O), z(t3) -~ ze+(or zs+), z(tl), z(t2) --* z~- ( or

?0

H~Iet al.

Figure 4: Illustration

for the proof of Theorem3.5.

Note that kfoz = 2ka2z2, we can rewrite the closed-loop system (3.30) (3.31) (3.32)

~1 = alz2, ~ = -zl - a2z2 + Sata(2ka2z2). Since the trajectory goes clockwise and by (3.31), we have ~l(tl)

< 0, ~2(tl) > 0~ ;~l(t2)

With the particular uz~=-A-lbu-= ]

(3.33)

> 0, k2(t2) > 0.

form of A, b, we have %+= -A-ibu + =

0

and

[ 0 . Let h = max{Iz21 : z E 0~0}. In the following

proof, we will consider k such that I u~%~-a+~ , u~%-~-~I < ½h. This meansthat the height of the part of ~t0 above (below) the line kfoz = u+ ( kfoz = u-) is greater than ~h. 1 Since $(k) is convex, since it must enclose ~t0 and inside C, it follows from (3.33) that there exists a constant ~? > 0 such that the slope of F at z(t~) and z(t2) satisfy:

Null Controllability and Stabilization

71

Suppose that we draw a line that is tangent to F at z(t2); then by the convexity of $(k), z(tl) must be to the right of this line. This implies that t ~ ~ Zl(t2)--~ ~(t2)~ 2kz i2)--Z2(tl)),

Zl(tl)

Zl(t2) < Zl(tl)

_{_~ ( ~1~,t2,~(Z2(t2)_

~t + -- ~t-

_< zl(tl)

2rlka~ ,

(3.34)

and similarly, z(t~) is to the right of the line tangent to F at z(t~),

+ ~(t~)~z ~ zl(tl)

It

(3.35)

2~ka~

It follows from (3.34) and (3.35) that limk~(z~(t~) - zl(t~)) = ilarly limk~(zl(ta) - z~(0)) = 0. Since limk~ z~(0) z~(t~) limk~ z~(t~) = lim~ z~(ta) = 0, these imply that lim (z(tt)

- z(t~)) = 0, lim (z(ta) z( 0)) =

(3.36)

From (3.33), we also have }2(tl)

= --zl(tl)

a2z 2(tl)

+ u - = - -z l(tl) --

-~- + u- >

It follows that z~(tl)

< u- - 2~"

(3.37)

Nowwe break the proof into two cases. Case 1. A has two real eigenvalues. In this

case,

ze + = 0 and zj =

0 are on the boundary of For the particular structure of A and b, it can be verified that every point in C is to the right of z[ and to the left of z~+. Since z(t~) must be in C, we have zl(tl) > u-. It follows from (3.37) that limk__,~ z~(tl) = u-. With (3.36), we finally have limk_~o z(tl) = limk_~o~z(t2) = z[ and similarly, limk-~ z(0) = lim~ z(t3) =

72

Hu et al.

Case 2. A has a pair of complexeigenvalues a ± jf~. Denote Tp = 7’ then e -ATp = --e-aTpI. First, we claim that as k tl -~ Tp. To prove this claim, we recall some simple facts about a secondorder linear system With a pair of complexeigenvalues, i; = -Av.

(3.38)

For this system, suppose that v(0) ¢ 0, then /v(t) is monotonically creasing (or decreasing). Consider v(tl) = e-Atlv(O). If the trajectory (e-Atv(O): t [0, tl ]} ca n beseparated fro m theorigin with a str ai ght line, then tl < Tp. Nowsuppose 0 < t~ <_Tp. If v(t~) and v(0) are aligned, then we must have t~ = Tp; If v(t~) and v(0) tend to be aligned, then will approach Tp. From z(0) to z(t~), ~ -- -Az-bu-. If we let v -- z-z[, then the part of F from z(0) to z(t~) is a trajectory of (3.38). From Lemma3.1 knowthat z[ does not belong to the half plane kfoz <_ u-, so this part of trajectory is below z~- (the origin in the v coordinate). Hence we must have 0 < tl < Tp. Since z(0) must be the right of~0, we have Zl(0) > 0 sufficiently large k. It follows that IIv(0)ll

-u z2(0) ]I -Zl(0)

is greater

than a constant. So IIv(t~)ll is also greater than a constant. Note that as k -~ co, v2(0), v2(t~) --~ 0. Therefore, v(0) v(tl ) tendto beali gned, so we get limk-~ tl = Tp. Similarly, lima_s(t3 - t2) = Tp. Now we have lime -At~

= lime -A(t~-t~)

= e -AT~ =--e-~T~I.

It follows that, as k -+ (Z(tl)

-- Z:) ÷ -aT~ ( z(0) - - Z

= (e -At~ ÷ e-"T~I) (Z(0)- Z[) --~

(3.39)

and

(z(t~)-zt) + ~-°~(z(t~)-

0

(3.40)

Recall from (3.36) that

z(t~)- z(t~)o,

~(0)- z(t~)-~

the property in (3.40) implies that (z(O) - z+e) + -aTp ( z(tl) - +)-~ O.

(3.41.)

73

Null Controllability and Stabilization Let

~: = (z(0)- ~) + ~-~% (~(t~) It can be solved that

~(t~) ~ (~ - ~-~%)-’(~; - ~-~)+ o(~) + 0(~). ~om (3.39) and (3.41), have lira z(t~) Similarly,

we know that lim~_~ ~l = lim~_~ ~2 = O, so

= - (1 - e -a~)-t (u- - e-~%u+) A-ib =

we have lim~ z(O) = z~. I¢ follows from (3.36) lira z(t~) = z~, lira z(ts) =

-2

-3

--6

:-5

-4

-3

-2

Figure 5: The domains of attraction

-1

0

1

2

under different feedback gains.

Example 3.1. Consider the open-loop system (3.1) with 0.8

0.6

3

’

4 ’

74

Ht~et al.

u- -- -0.5 and u+ = 1. Then we have [0.12

-0.66].

In Figure 5, the boundaries of the domains of attraction corresponding to different f-- kfo, k = 0.50005, 0.6, 0.7, 1, 2, are plotted from the inner to the outer. It is clear from the figure that the domainof attraction becomes larger as k is increased. The outermost dashed closed curve is 0C. 3.5.2.

Higher Poles

Order Systems

with Two Exponentially

Unstable

Consider the following open-loop system

where x = [ x~ x~ ]’, xl E R2,x2 E Rn, A1 ~ R2x2 is anti-stable and A2 ~ Rn is semistable. Assumethat (A, b) is controllable. Denote the null controllable region of the subsystem :~l(t) ---- AlXl(t) ~- blOt(t) as C~, then the null controllable region of (3.42) is C1 ×Rn. Given"~1, ~2 > denote ~1(71) := {7~x~ ~R2: xl ~i}, and

0:

( ~o ~2("~2)

:=

~X2 E an:

IIX211

~ ~2j

When3’~ = 1, ~1(3’~) = ~1 and when 7~ ( 1, ~(7~) lies in the interior of C~. In this section, we will show that given any ~1 < 1 and 7~ > 0, a state feedback can be designed such that ~1(~1) X ~2(~2) is contained in the dpmainof attraction of the equilibrium x = 0 of the closed-loop system. Fore > 0, let

P(e)= F P’(e)

P~(e) ] eR(~+n)x(~+n)

positive definite solution to the ARE A’P + PA - Pbb’P + (~I = 0. Clearly, as e $ 0, P(e) decreases. Hencelim~--.0 P(e) exists. Let P1 be the unique positive definite solution to the ARE ’ =O. A’~P~ + PiA~ - P~b~biP~

(3.43)

Null Controllability and Stabilization

75

Then by the continuity property of the solution of the Riccati equation [15], ~o

0 0 "

Let f(e) := -btP(e). First, consider the domain of attraction equilibrium x = 0 of the following closed-loop system ~(t) = Ax(t) + bsata(f(e)x(t)).

of the (3.44)

Let Um=min{-u-, u+}. It is easy to see that D(~) := { x e R2+n: x’P(~)x ~_

~ ’’ 4u.~/llbP~(¢)ll

}

is contained in the domainof attraction of the equilibrium x = 0 of (3.44) and is an invariant set. Note that if x0 E D(e), then x(t) ~ D(e) and

If(~)x(t)l _<~m forall t >0. That is, x(t)willstayinthelinear region

the closed-loop system, and in D(e). Theorem 3.6. Let f0 = -b~P1. For any 71 < 1 and ~ > 0, there exist k > 0.5 and ~ > 0 such that ~1(Yl) x ~t2(~/~) is contained in the domain attraction of the equilibrium x = 0 of the closed-loop system 2(t) = Ax(t) + bu(t), u(t) = { sata(kfoxl(t)), Sata(f(~)x(t)),

x ~

Proof. Similar to Theorem4.4.1 in [4] and Theorem4.2 in [5]. 3.6.

(3.45) []

Conclusions

In this chapter we have studied the problem of controlling a linear system subject to asymmetric actuator saturation. The null controllable region of such a system is first characterized. Simple feedback laws are constructed to stabilize a system with no more than two exponentially unstable open-loop poles. The feedback law guarantees a domain of attraction that includes any given compactset inside the null controllable region. References [1] J. A1varez, R. Suarez and J. Alvarez. Planar Linear Systems with Single Saturated Feedback, Systems ~ Control Letters, 20 (1993) 319326.

76

I-Iu et al.

[2] O. H£jek. Control Theory in the Plane, Springer-Verlag, (1991). [3] P. -O. Gutman and P. Hagander. A NewDesign of Constrained Controllers for Linear Systems, IEEE Trans. Automat. Contr., 30 (1985) 22-33. [4] T. Hu and Z. Lin. Control Systems with Actuator Saturation: Analy.si,~ and Design, Birkh£user, Boston, (2001). [5] T. Hu, Z. Lin and L. Qiu. Stabilization of Exponentially Unstable Linear Systems with Saturating Actuators, IEEE Transaction on Au-. tomatic Control, to appear. [6] H. Z. Zhalil. Nonlinear Systsems, MacMillan, NewYork, (1992). [7] Z. Lin and A. Saberi. Semiglobal Exponential Stabilization of Linear Systems Subject to ’Input Saturation’ via Linear Feedbacks, Systems and Control Letters, 21 (1993) 225-239. [8] J. Macki and M. Strauss. Introduction to Optimal Control, SpringerVerlag, (1982). [9] A. Saberi, Z. Lin and A. R. Teel. Control of Linear Systems with. Saturating Actuators, IEEE Trans. Automat. Contr., 41 (1996) 368378. [10] W. E. Schmitendorf and B. R. Barmish. Null Controllability of Linear Systems with Constrained Controls, SIAMJ. Control and Optimiza-. tion, 18 (1980) 327-345. [11] E. D. Sontag. An Algebraic Approach to Bounded Controllability Linear Systems, Int. J. Control, 39 (1984) 181-188.

of

[12] E. D. Sontag and H. J. Sussmann. Nonlinear Output Feedback Design for Linear Systems with Saturating Controls, in: Proc. 29th IEEE Conf. on Dec. and Control, (1990) 3414 3416. [13] H. J. Sussmann, E. D. Sontag, and Y. Yang. A General Result on the Stabilization of Linear Systems Using BoundedControls, IEEE Trans. Automat. Contr., 39 (1994) 2411-2425. [14] A. R. Teel. Global Stabilization and Restricted Tracking for Multiple Integrators with Bounded Controls, System and Control Letters, 18 (1992) 165-171. [15] J. C. Willems. Least Squares Stationary Optimal Control and Algebraic Riccati Equations, IEEE Trans. Automat. Contr., 16 (1971) 621-634.

Chapter 4 Regional T/2 Performance Synthesis T. Iwasaki University of Virginia, Charlottesville,

Virginia

M. Fu University of Newcastle, Newcastle, Australia

4.1.

Introduction

Actuator saturation is inevitable in feedbackcontrol systems. If it is ignored in the design, a controller may"wind up" the actuator, possibly resulting in degraded performance or even instability. A classical approach to avoiding such undesirable behaviors is to add an anti-windup compensator to the original controller [1, 7, 11, 13, 14, 21, 25]. This approach has an advantage of providing control engineers with insights, for the role of each control componentis clear. On the other hand, higher performance may be expected if a controller is designed a priori considering the saturation effect. Lin, Saberi and their coworkers (see [16, 17, 23] and the references therein) have developed control design methodsalong this line using Riccati equations as a basic tool. Other Riccati equation approaches include [2, 10, 27]. Recent results also include those developed using the circle and the Popov criteria within the frameworkof linear matrix inequalities (LMIs) [5, 6, 20, 22, 28]. The idea is based on Lyapunovfunctions that are valid in a certain domain of the state space, and is very close in spirit to that of [23] mentionedabove. For more detail and an overview of recent developments, we refer the reader to [15, 26].

77

78

Iwasaki and Fu

This chapter presents some methodsfor designing controllers to achieve a certain ~2 (or linear quadratic) performance. In the design, the troublesome saturation nonlinearity is captured in a specific state space region by a sector-bound condition and the circle criterion is applied to guarantee stability (i.e. convergenceto the origin) and the ~/2 performance. This will be called the circle analysis. Whenthe state space region is restricted to those states that do not activate the saturation nonlinearity (i.e. the linear region), the sector boundreduces to a single line, resulting in a simpler but seemingly more conservative performance bound. This will be called the linear analysis. In [12], it is shownthat (i) the circle analysis can give a better estimate of the domain of attraction than the linear analysis for a given system, but (ii) the former provides no better result than the latter when they are used to design a controller that maximizes the estimated domain of attraction. This chapter first showsa result analogous to this for the case where our main concern is the d~main of ~/2 performance rather than the domain of attraction. Thus, the "optimal" controller within the framework of circle analysis can be designed using simple linear analysis conditions. However, the second half of this chapter shows by numerical examples that the "optimal" controller thus designed maynot be the best in terms of the actual ~2 performance (or others such as settling time and overshoot) due to inherent conservatism of the ~2 performance bound. It is illustrated by an example that the circle criterion can indeed be useful to improve the actual performance over the controller designed via the linear analysis. Weuse the following notation. The set of n × rn real matrices is denoted by ]R’~×m. For a matrix M, MT denotes the transpose. For a vector x, xi is the ith entry of x. For vectors x and y, x > y meansthat xi > y~ for all i, and similarly for x >_ y. For a symmetric matrix X, X > 0 (X _> 0) means that X is positive (semi)definite. For a square matrix Y, He(Y) :-- Y + Finally, a transfer function is denoted by A B

4.2. 4.2.1.

Analysis A General

Framework

Consider the feedback system depicted in Fig. 1, where H(s) is a linear time-invariant (LTI) system given ~ = Ax + lieu,

z = ~x, e = Cx + :Du

(4.1)

Regional 7~2 Performance Synthesis

79

and ¢ : IRTM -~ IRmis a saturation nonlinearity, i.e.

~i (u~>~) -c~ (u~ < -~) where c~ E ]Rmis a given vector with positive entries.

Figure 1: Feedback system with saturation nonlinearity. The set of state vectors A is called a domainof attraction if any state trajectory starting from a point in A converges to the origin as the time goes to infinity. Moreover, the set of state vectors P is called a domainof performance(with level 3’) if it is a domainof attraction and any output in response to x(0) E P has its/22 norm squared less than or equal to 3’. Our first objective is to characterize a domainof performance. The following lemmais the basis for our analysis. A similar idea has been used in the literature on saturating control; see e.g. [6, 22]. Lemma4.1. Consider the nonlinear

system

So=f (x), e -- g(x) where f : ]Rn ~ ]Rn and g : ]R~ --~ ]Rm are continuous functions passing through the origin. Let X be a subset of ]Rn containing the origin. Assume existence and uniqueness of the solution.to ~ = f(x) for any initial state x(0) E X. Suppose there exists a continuously differentiable function V ]R’~ --~ N satisfying, for somepositive constants a, b, and c, allxl] 2 ~ V(x) bl lxll ~, Vxe X, ~f(x)

+ g(x)’g(x)

-c []x[[ 2, Vx~ X

(4.3)

PcX where P:={xe]Rn:

V(x)_
}.

(4.4)

80

Iwasaki and Fu

Then for each nonzero z(0) E P, the resulting response satisfies z(t) E P, V t >_ O, lim x(t) =

f

o° Ile(t)ll2dt < ~/V(x(O)).

Proof. The fact that P is an lnvariant set and that x(t) approaches the origin directly follows from Lemma2 of [12]. From (4.3) and the system equations, we have OV 2 - ~-z Ile(t)ll2 _<-c~llx(t)l] ~(t). Integrating from t = 0 to oc and noting the stability

property, we have

[[e(t)l]~ <_-c~][z(t)ll~dt - (r(0) - V(x(0))) <~V(x(O)).

4.2.2.

Applications--Linear

and Circle

Analyses

Applying Lemma4.1 to our system f(x)

:= Ax ÷ B¢(/Cx), g(x) := Cx ÷

one can obtain characterizations of the domain of performance P. The characterization will depend on the choices of V(x) and X. Weconsider quadratic storage function V(x) := xTPx, elliptic domain of performance P, and polytopic outer region X as follows: P := { x ¯

x:--{xe~:

]Rn:

xTPx <_ 1 }

(4.5)

I~zl_<~ (i=l,...,m)}

where/~i is the ith row of matrix ]C and pi are real scalars to be specified in the analysis. If pi are chosen as pi = c~i, then ¢(/Cx) =/Cx for all x ¯ n and t hus the above analysis becomesvery simple. In this special case, we have the following linear analysis result. Lemrna 4.2. Let a symmetric matrix P and a scalar 3’ > 0 be such. that He

C+~/C

-71

<

0

~Ei < c~P (i = 1,...,m). Then P in (4.5) is a domain of performance with level

(4.6) (4.7)

Regional TI2 Performance Synthesis

81

Proof. The result follows from Lemma4.1 by noting that (4.3) and P C X reduces to (4.6) and (4.7), respectively, where we redefine 7/2 be 3’. [] Wenowconsider the case where Pi _> ai. In this case, condition (4.3) reduces to the following: + ~Du)

(4.8)

u = ¢(/Cx), [/Cix[ _< p~, V i = 1,...,m.

(4.9)

2xTp(Ax+13u)+ ~(Cx +/)u)T(Cx holds for all x ~ 0 and u such that

Note that, if x and u satisfy (4.9), then (u~ -]C~x)(ui - s~lC~x) _< 0, Y i = 1,...,

m

(4.10)

holds where s~ := a~/p~. This is easy to see once we notice the fact that the saturation nonlinearity ¢ will lie in the sector [si, 1] whenits input zi is restricted by lz~l _< p~ (see Fig. 2). Applyingthe circle criterion to the sector-bounded nonlinearity, we have the following:

Figure 2: Sector bound for saturation nonlinearity. Lemma4.3. Let a symmetric matrix P and a scalar 3’ > 0 be given. Suppose there exist diagonal matrices 0 <_ R < I and T > 0 such that He

P(A +13]C) -RTIC

PB 0 -T 0

IC~ICi < p~P (i = 1,...,m)

(4.11)

(4.12) where pi := 6~i/(1 - ri) with ri being the ith diagonal entry of R. Then P in (4.5) is a domainof performance with level 23’.

82

Iwasaki and Fu

Proof. particular, for all x (i = 1,...,

The result follows from Lemma4.1 as discussed above. In applying the S-procedure, a sufficient condition fo~: (4.8) to hold ¢ 0 and w such that (4.9) is given by the existence of ti > rn) satisfying

2x’P(.Ax + Bu) + ¼(Cx + :Du)T(Cx m

ti(u - ix)(u-si : x)
It is straightforward to verify that this condition is equivalent to PA-

He

[

K~TSK TSK

c

PB + KTT -T

0 0

<0

where S and T are the diagonal matrices with si :-- a~/pi and ti on the diagonal, respectively. A congruence transformation of this inequality leads to (4.11) by defining R := I - S and redefining 7/2 to be 7- Finally, it can be verified that (4.12) implies P C X. It should be noted that the circle analysis in Le:nma4.3 reduces exactly to the linear analysis in Lemma4.2 when p -- a. This can be seen once we notice that there exists a (sufficiently large) T > 0 satisfying (4.11) if only if (4.6) holds, because p -- a i~mplies R = By the congruence transformation with diag(P -~, T-1, I) and by the Schur complement, it can be shownthat the conditions in (4.11) and (4.12) are equivalent to

0 ] -RKQ -V 0 < (C + DI~)Q Dv -71

0,

p~ > K:~Q~

(4.13)

where Q := p-1 and V := T-1. Thus, the largest estimate of the domain. of performance is obtained by maximizing det(Q) subject to (4.13) symmetric Q, diagonal V, and diagonal 0 _< R < I. This problem is difficult in general due to the product term R~Qwhich destroys the linearity of (4.13). However,if R is fixed, then the problem becomesa quasi-concave maximization [19] subject to LMIconstraints and thus can be solved efficiently. This property is particularly appealing for the single input case~ for the parameter R becomesscalar and its appropriate value can be found by a line search.

Regional ~2 Performance Synthesis 4.3.

Synthesis

4.3.1.

Problem Formulation

83

and a Critical

Observation

Consider a linear time invariant system ~ = Ax + Bu, e = Cx + Du, y = Mx

(4.14)

where x(t) E 1R" is the state, u(t) ~ m is thecontrol inpu t, e(t) ~ ]Rgis the performance output (e.g., error signal), and y(t) ~ IRk is the measured output. Suppose the actuator has a limited power and we have the following constraint on the magnitude of admissible control input: (4.15)

lu~(t)l < ai, V t >_ O, i = 1,...,m.

Our objective is to develop methods for designing a feedback controller that uses y(t) to generate u(t) satisfying the saturation constraint (4.15) such that the closed-loop system sustains a high ~2 performance in a large region in the state space. Let us formulate the following: Synthesis Problem: Let the plant (4.14), the controller order nc >_ 0, and a desired domain of performance P C IRn (n := n + no) be given. Design controller such that:

(a)The saturation

constraint (4.15) is satisfied;

(b) All the closed-loop states convergeto the origin as the time goes to infinity wheneverthe initial state belongs to P; (c) The worst case ~u performance measure J := sup x(o)~p

Ile(t)ll2dt

(4.16)

is minimized. This problem is difficult, and no exact solution is yet available. Wewill address this problemconservatively, using the analysis results developed in the previous section. Consequently, the desired domain of performance is restricted to the class of ellipsoids specified by (4.5) where P = pT > and n is the dimension of the closed-loop state space, and the controller

84

Iwasaki and Fu

structure will be somecombination of the saturation nonlinearity ¢ and a linear time invariant system. Weconsider two classes of nonlinear controllers that meet the cation (a) of the Synthesis Problem, i.e. the saturation constraint One is given by u = ¢(z), z Ks(s)y

in (4.2) specifi(4.15). (4.17)

and the other is given by u=¢(z),

z = Ka(s) y ]z - u "

(4.18)

The closed-loop system with these controllers are depicted in Fig. 3 where the dashed part is absent and K(s) := Ks(s) for (4.17) while the dashed part is present and K(s) := Ka(S) for (4.18). The additional dashed feedback loop has been suggested in the literature to account for the actuator saturation effect and used as a basis for anti-windup compensation. Hereafter, we shall refer to (4.17) as the directly-saturating (DS) controller to (4.18) as the anti-windup (AW)controller.

Figure 3: Feedback control system. Recall that Lemmas 4.2 and 4.3 provide sufficient conditions for a closedloop system to meet the specifications (a) and (b) in the Synthesis Problem and to satisfy J < 2~/ where J is the worst case performance in (4.16) and ~ is a given scalar. Hence, an approximation of the original Synthesis Problem would be to minimize 7 over the class of DS or AWcontrollers subject to the conditions given by either Lemma4.2 or Lemma4.3. Thus we have four possible problem formulations. Table 1 shows the four formulations, where each of ~/’s denotes the optimal performance bound achievable by the corresponding problem formulation. Clearly, the class of AWcontrollers is larger than the class of DS controllers, and the sufficient condition in Lemma4.3 is no more conservative than that in Lemma4.2. Hence we immediately see the following

Regional ~2 Performance Synthesis

85

Table 1: Possible problem formnlations. Directly-saturating Anti-windup Control (4.18) control (4.17) Linear Analysis ~/~s ~/~a Lemma 4.2 Circle Criterion %s %a Lemma 4.3 relations:

{

’~s

~ ’~a

~ ~ca

It is tempting to expect that strict inequalities hold in the above relations. Surprisingly, however, it turns out that the achievable performance bounds are all equal: and hence neither the circle criterion nor the anti-windup structure improves the achievable guaranteed performance within the control synthesis frameworkdiscussed here. The following formally states this result. Theorem4.1. Consider the plant P(s) in (4.14) and the control systern in Fig. 3. Fix the controller order nc to be any nonnegative integer. Let a desired domain of performance P in (4.5) and a desired performance bound ~/> 0 be given. Suppose there exists an AWcontroller (4.18) such that the corresponding closed-loop transfer function H(s) in Fig. 1 is partially strictly proper from u to z and satisfies the condition in Lemma4.3 (the circle criterion). Thenthere exists a DScontroller (~4.17) such that corresponding closed-loop system satisfies the condition in Lemma4.2 (the linear analysis). Consequently, we have 7~s = %a. Moreover, one such DS controller is given by (4.17) with Ks(s) being the transfer function from y to u in Fig. 4.

Figure 4: Ks(s) of the DScontroller.

86

Iwasaki and Fu

4.3.2.

Proof

of Theorem4.1

Let P = pT E IRnxn, ? E IR, and a state space realization of the transfer function Ka(S) Ka(s)

= C~ D~ = C~ Dal

with state vector xc(t) ~ ~0¢ be given, where Ba and D~ are partitioned compatibly with the dimensions of the two input vectors in (4.18). If we view the closed-loop system in Fig. 3 with K(s) := K~(s) as a special case of the general feedback system in Fig. 1, we see that H(s) is determined by P(s) and K~(s). Denote this particular transfer function by Ha(s). The analysis results in Section 4.2 require that the upper block of Ha(s) is strictly proper. It can be verified that I

0

and hence D~ must be ~ero. A state space realization given by

Ha(s)

(Ba~ + B~2Da~)M A~ + Ba2Ca DalM Ca C 0

of H~(s) is then

=:

0 D

~a

0

Ca

where the closed-loop state vector is x := [ xvxcv ]v. Suppose there exist diagonM matrices 0 ~ R < I and T > 0 such that (4.11) ~nd (4.12) hold. Ks(s) be d efi ned as t he mapping from y to u in Fig. 4 with its state space reMization

= where the state vector is xc. D~no~ethe corresponding closed-loop transfer function by Hs(s). I~s state space realization is given by

Hs(s)

BsM DsM C

As Cs 0

0 0 D

=:

/Cs 0 Cs Ds

,

x=

x xc

(4.19)

Weshowthat this Hs(s) satisfies linear analysis conditions (4.6) and (4.7).

,

87

Regional ~2 Performance Synthesis Noting that condition (4.12) with ~ := ~a implies (in fact is equivalent to) (4.7) K~:= K:s. Also note that satisfaction of (4.11) by Ha(s) implies He ~ N

~

-RT~ -T 0 [ P(Aa + BaK’a) PBa 1

He [ P(Aa+Ba(I-R)~a) Ca + ~a(I - R)Ea

N< 0

0 ] < 0 -TI

where N :=

.

T-1B~P 0

I

It is tedious but straightforward to verify that Aa + Ba(I - R)~a = .As + Ba~a, Ca + l)a(I

-- R)£a = Ca + l?s£s.

Hencewe see that Ha(s) satisfies (4.6) as well. This completes the proof. 4.3.3.

Fixed-gain

Control

In view of Theorem4.1, the circle criterion (Lemma4.3) does not help improve the ~2 performance bound for the closed-loop system when designing a controller with actuator saturation. Hence we use the linear analysis result (Lemma4.2) as a basis for developing a control design method. The following provides a necessary and sufficient condition for existence of an output feedback controller that yields a closed-loop system satisfying the linear analysis condition. Theorem 4.2. Consider the feedback system given by the plant (4.14) and the DS controller (4.17). Suppose there exist symmetric matrices and Y and matrices F, K and L such that

X I LiM

He

XA+FM C+DLM

He

CY+DK

I MT L~T] Y Kg > 0, (i

0 ] -71 <0

(4.20)

<0

(4.21)

-7I

= 1,...,

rn)

(4.22)

88

Iwasaki and Fu

where Li and K~ are the controller (4.17) with

ith

of L and K, respectively.

row

/(8(s):=

Then. the (4.23)

Ds

[ AsC~ DsBS ] := I Z XB1-11N-XAY F] [-YO I K L MY T N := -(A + BLM) - (C + DLM)T(CY + DK)/(2~/), := X - y- -1 has the following properties: For any initial closed-loop state vector x := [ xTxcT ]~ (wherex and xc are the plant and the controller state, respectively) satisfying

×(0)

×(0) ×do) ][ x z

the control input never saturates (i.e. u -- z in (4.17)), the state trajectory converges to the origin, and

f0

< 2%

Proof. With the controller in (4.17), the closed-loop system in Fig. can be described by Fig. 1 with H(s) given by Hs(s) in (4.19) where Bs, Cs and Ds are the state space matrices of Ks(s). We8how that the conditions in Lemma4.2 with this Hs(s) can be equivalently transformed to (4.20)-(4.22) by standard change of variables and parameter elimination. It can be shown(see e.g. [24]) that if there exists a controller (4.17) some order satisfying conditions (4.6) and (4.7) for someP, then there exist8 a controller of the same order a8 the plant satisfying the same conditions for another P. Moreover, the state coordinates of such a controller can always be chosen (see e.g. [9,18]) such that the conditions are satisfied with P of the following structure:

Let F :=

y

_y

,

-1. Y:-- (x - z)

Then we have the following identity:

0 I 0 0 0 I

C~+:DflC~

~= F

XA + FM N A + BLM AY + BK C + DLM CY + DK LM K

Regional 7~2 Performance Synthesis

89

where we used the following change of variables [18, 24] K L

:=

0

0

+

0

I

Cs Ds

MY

I

"

It is then easy to verify that (4.7) and (4.22) are related by the congruence transformation by F and the Schur complement. Similarly, by congruence transformation with diag(F, I), (4.6) can be equivalently transformed O ] He

A + BLM AY + BK 0 + DLM CY +N DK -~/I [ CXA+FM

<0.

It can further be shownthat this condition is equivalent to (4.20) and (4.21) by eliminating N using the projection lemma[3, 8]. [] In Theorem4.2, the domain of performance is characterized by (4.24). Note that the region of plant initial state x(0) to yield the desired 7~2 performance is dependent upon the choice of the initial controller state Xc(0). Rewriting (4.24)

×(of(x- z)×(o)+ (×(o)

+ _< 1,

we see that the best choice of xc(0) to maximizethe plant state domain performance is given by xc(0) = -x(0), in which case the 7-/2 performance is guaranteed whenever ×(0)TY-I×(0) _< Thus the plant state domain of performance can be maximized within our frameworkby solving the following quasi-convex optimization problem: max det(Y) subject to (4.20)-(4.22) X,Y,F,K,L

It can be shown that optimal choices of X and F are given by X := aXo and F := aFo for sufficiently large ~r > 0 where Xo and Fo are any matrices satisfying He(XoA + FoM)< and Xo> 0. With the se cho ices of X and F, condition (4.20) is always satisfied and condition (4.22) reduces [Ki Y K~ ] >c~i 0, (i=l,...,m). Hence the above problem becomes max det(Y) subject to (4.21) and (4.25). Y,K

(4.25)

90

Iwasaki and Fu

It turns oflt that this is exactly the same problem as that for the state feedback synthesis. This makes sense because it means that if the initial plant state is knownthen the output feedback with an observer achieves the same plant domainof performance as that achievable by state feedback. Whenthe initial plant state is not known,we maysimply choose the zero initial controller state xc(0) = 0. In this case, the domainof performance is characterized by x(0)~X×(0)_< Therefore, the domain with the largest volume is obtained by minimizing det(X) subject to (4.20)-(4.22). This is not a quasi-convex optimization problem and is difficult to solve. On may replace det(X) by tr(X) approximate the volume, in which ease the problem becomes convex. Letting 3’ be arbitrarily large, we have the following regional stabilization result that has been obtained in [12]. Corollary 4.1. Consider the feedback system given by the plant (4.1,1) and the DS controller (4.17). Suppose there exist symmetric matrices and Y and matrices F, K and L such that He(XA + FM) < 0

(4.26)

He(AY + BK) < 0

(4.27)

and (4.22) hold. Then the controller (4.17) with (4.23) has the followiug properties: For any initial closed-loop state vector satisfying (4.24), the control input never saturates, and the state trajectory converges to the origin. Wecan give an alternative proof of the semi-global stabilization result in [15] using Corollary 4.1. That is, assumingthat all the eigenvalues of A are in the closed left half plane and that those eigenvalues on the imaginary axis, if any, are simple1, one can show that IIXII can be made arbitrarily small, indicating that the initial plant state region, with guaranteed state convergenceto the origin, can be arbitrarily large. Let Xo > 0, Yo >-0, Fo and Ko be such that (4.26) and (4.27) hold. Such matrices exist if and only if (A, B, C) is a stabilizable and detectable triple. By the assumption on the purely imaginary eigenvalues, there exists W> 0 such that AW + WAT <_ O. 1 The assumptionon the purely imaginary eigenvalues is not required in the semi-globalstabilization result of [15] but our proof belowrequires it.

91

Regional 7-12 Performance Synthesis Wecan assume that W > Xo without loss of generality properly. Then it can readily be verified that X:=eXo,

F := eFo,

Y := W/e + Yo,

by scaling

W

K := Ko, L=O

satisfy conditions (4.26), (4.27) and (4.22) for sufficiently small e > 0. IIXII can be arbitrarily close to zero, the domainof attraction for the plant state can be arbitrarily large, assumingzero initial controller state. Wenow consider the state feedback problem. The result basically follows from Theorem4.2 by letting M-- I. Corollary 4.2. Let Q, K and 3’ be such that He CQ + DK --/I

<0,

2 K~ s a

> 0.

-1 be the state feedback controller. Let u = Kx with K: := KQ initial state satisfies x(O)TQ-Ix(O) _< 1.

(4.28) Suppose the (4.29)

Thenwe have [ui(t)l _< a~ for all t >_ 0 and

fo~Ile(t)ll~dt<23‘. Proof. It is easy to verify that the closed-loop system with the indicated state feedback controller satisfy conditions (4.6) and (4.7) A := A, B := B, C := C, 7P:=D,

tC := KQ-~,

p := Q-1.

In general, there exist Q and K satisfying (4.28) for any fixed value of 7 > 0, provided (A, B) is stabilizable. This can be seen as follows. Stabilizability of (A, B) implies existence of Qo and Ko such that He(AQo BKo) < 0 and Qo > 0. Since the second condition in (4.28) is equivalent to c~Q > K~Ki, there exists a sufficiently small e > 0 such that Q := eQo and K := eKo satisfy this condition. Now,for these Q and K, there exists a sufficiently large 3’ > 0 such that the first condition in (4.28) holds. Thus we have shownthat there is a triple (Q, K, 3’) satisfying (4.28) and Q whenever (A, B) is stabilizable. Let (Qo, Ko, %) be redefined to be one such triple. For 3‘ > 3’o, it is clear that (Qo, Ko, 3‘) satisfies (4.28). Onthe other hand, if 3‘ < 3‘o, then it can be verified that (crQo, aKo, 3") satisfies (4.28) where ~r := 3’/7o. In general, the more stringent the performance

Iwasaki and ~ requirement (i.e. smaller 7), the smaller the domainof performance (i.e. smaller Q). A state feedback controller that ma~ximizesthe domain of performance for a given level 7 can be designed as follows: Fix y > 0 and solve the following: max det(Q) subject to (4.28). Q,K

This is a quasi-concave maximization problem which can be solved efficlearly. Once we find a solution (Q,K), a control gain is calculated as -1. ]~ = KQ 4.3.4.

Switching

Control

The performance of a fixed-gain state feedback controller can be improved by introducing a switching logic structure into the controller. The basic idea is as follows [4, 27]: Prepare a set of feedback gains K~ (k == 0,..., q) such that a certain performance is guaranteed by the fixed-gain control law u = ¢(/~x) in the state space domainP~ c n. The fi rst ga in ~0 is designed to yield a sufficiently large domain of performance P0 to cover the possible region of initial states. The other gains are determined so that the resulting do~nains of performances are nested: Pk+l

cPk (k=0,...,q)

where Pq+l := { 0 }. The control gains are switched in accordance With the following logic: u=¢(lCkX)

(whenxEPkandxCPk+l)

(k=0,...,q).

(4.30)

This strategy improves performance by successively switching the control gain from a low gain to a high gain as the state gets closer to the origin. An important question is: Howmuch is the performance improved by switching? If the gains are designed within the framework of Lemma4.1, each gain /Ck satisfies (4.3) for some c > 0, Vk and ~’k, and guarantees (without switching) the performance bound

fo

~ Ile(t)ll2dt

< ~kVa(x(O))

whenever x(0) E Pa. Now, suppose that the gains are switched in ac-be the time instants when cordance with (4.30). Let tk (k = 1,...,q) the switchings occur and define to := 0, tq+~ := oc, and xk := X(tk)

Regional ~ Performance Synthesis (k = 0,...,q

93

÷ 1). Then, from (4.3), q

II~(~)ll~d~ k=0

~

k=0

If in particular Vk(x) is given by a quadratic function Vk(x) := xTPkxwith

p~ =~:~> 0, q

< ~ ~k(x~P~xk- x~+lp~x~+l) k=O q

= ~/ox~oPoxo÷ ~_,(~/~ - ~/~-lx~P~-lxk). For each k = 1,... ,q, by definition x~Pkxk---- 1 holds and hence the value of X~kP~_lx~is bounded below by m~n{ xTP~_ix : xTPkx = 1 } = Amin(P[IP~_~) where ~min(’) denotes the minimumeigenvalue. Thus we have the following ~2 performance bound for the switching controller: ¢x~

q

f0II~(~)ll~d~< ~o+~-~(~

- "~k-1)~min(p~lpk-1))

(4.31)

k:l

whenever x(0) ~ P0. Specializing the general idea to the linear analysis case, we have the following result. Weconsider for brevity the single input case only. Theorem 4.3. Consider the system in (4.14) with M = I and m = Suppose that matrices Q~ = Q~ and K~, and scalars 3’~ and #k satisfy He CQ~ +DKa --"/kI for k = 0,...,

Kk c~ ~ > 0,

(4.32)

q, and "/~ ÷ #~ Qk-~ < Qk < Qk-~ (4.33)

for k -- 1,...,

q. Then the switching controller

u = ](:kx (when xTQ~x <_ 1 < xrQ-~_ix) (k = O,...,q)

94

IwasakiandF~

with ]Ck :---- KkQ,-~1 and -1 := ocI yields [u(t)[ _< a for all t _> 0 and Qq+l

Ile(t)ll2dt < 2

(4.34)

wheneverx(0)TQ~-lx(0) _< Proof. The result basically follows from the preceding argument and the fixed-gain state feedback synthesis result of Corollary 4.2 with the change of variables Qk := P~-I and Kk := ]C~Q}. The additional constraint Qk < Qk-1 in (4.33) is imposed to guarantee the nesting property Pk C Pk-1. The performance bound can be shown as follows. The second term on the right hand side of (4.31) is bounded above by --#k if and only if _ ,~ [,,~--i/2~~--I/2\ "/k "]- #k < "~’k-1 minlfl~k_ 1 ~,~k~,~k_l )

or equivalently,

(~k + p~)Q~-~ < ~k-,Q~. Redefining 7k/2 and #k/2 to be 7~ and #k, respectively, we have the first inequality in (4.33) and the performance level is boundedas in (4.34). In view of Theorem4.3, the switching mechanismin the controller improves the closed-loop performance bound by 2# := 2 ~=~ Pk, for the performance bound with the fixed-gain controller u = ]Cox is 2"~0. The best switching controller within this frameworkresults when70 - # is minimized over the variables Qa, Kk, "/k and #~ (k = 0,..., q) subject to the constraints in Theorem4.3. This problem is nonconvexand it is difficult to compute the globally optimal solution. Hence, we propose a successive convex optimization to find a reasonable switching controller as follows. SwitchingControl DesignAlgorithm: 1. Design an initial feedback gain ]Co as follows. Fik ~ and maximize det(Q) over K, Q and ~ subject to (4.28). Let the optimizers be ~. Initialize k to be k -- 1. Qo, and ~’0 and define ]C0 := KoQ-~ 2. Find /C} as follows. Fix Kk_~, Q~, and 7}-1, and maximize #k over Kk, Qk, and q,~ subject to constraints (4.32) and (4.33). 3. If k -- q then stop where q is the numberof switchings specified in advance. Otherwise let k +- k + 1 and go to 2. The switching controller thus obtained does not optimize the overall performance. However, the gain ]C~ is chosen so that the performance bound

Regional 7-12 Performance Synthesis

95

of the switching controller consisting of ]c0,. ¯., ]ck is optimized for given gains ]Co,... ]Ck-1. With this compromise, each gain ]Ck can be obtained by solving the convex optimization problem defined in Step 2. One can develop a similar design algorithm using the circle criterion instead of the linear analysis as has been done above. From Theorem4.1 and its proof, however,it readily follows that the use of circle criterion does not improvethe performanceboundin (4.34). It will be illustrated later, contrast, that the circle criterion indeed improves the actual performance for somecases.

4.4.

Design Examples

4.4.1.

Switching Control with Linear Analysis

Weuse the design condition derived from the linear analysis (Theorem 4.3) to design a switching state feedback controller. The following example illustrates the design procedure and the benefit of the switching strategy. Example 4.1. Consider the system given by (4.14) with A:= 0.10 0.10

’

B:=

1 ’

C:=[ 1 0], D:=0, M=I.

Weconsider the Synthesis Problem with a = 1 and design a switching state feedback controller based on the Switching Control Design Algorithm with q = 4. Wehave chosen ~’0 = 50 and designed the initial feedback gain ]Co as described in Step 1 of the algorithm. Wethen successively computed the gains K:I through ]C4 following Step 2. The results are as follows: 7o ~/1 72 = ~3 "Y4

50.0000 16.2234 5.3730 1.7991 0.6063 /Co ]C, ]C2 =

,

#1 #2 P3 P4

=

1.9348 0.6954 0.2447 ’ 0.0850

-0.2467 -0.1497 -0.3615 -0.1795 -0.5405 -0.2179 -0.8170 -0.2667 ]C3 -1.2433 -0.3279 ]C4 The domainof performance for each value of 3’i is plotted as an ellipse in the x~-x2 plane in Fig. 5. Note that the ellipses are nested, as specified.

96

Iwasaki and ~ 10 5

~ o -5 -10 -6

-4

-2

0

2

X 1

Figure 5: Domainsof performance and state trajectories. Wecompare the performance of the switching controller (SWC) with that of the fixed-gain controller (FGC)u = K:0x. First of all, the guaranteed performance bound of the SWCis 94.08 as opposed to 100 guaranteed by the FGC,whenever the initial state is within the ellipse xtQ~lx <_ 1. The state trajectories and the time responses are shownin Figs. 5 and 6, respectively, for the case x(0) = [ 5 - 9 IT, where the thick curve is for the SWCand the thin curve is for the FGC. Wesee that the SWCperforms much better than the FGCI although the improvement of the guaranteed performance is only 6%. 1

N

8

-2

10

20

30 Time

40

50

Figure 6: Initial

4.4.2.

10

20

30 Time

40

50

state responses.

Switching Control with Circle Analysis

Recall that Theorem4.1 states that the circle criterion does not help in the synthesis of saturating control in the sense that it does not improve (i.e. enlarge) the achievable domainof performance for a given performance level % Thus, the controller designed in Example 4.1 is the best within

Regional ~2 Performance Synthesis

97

our framework in terms of the guaranteed 7~2 performance. However, the performance bound given by our analysis is conservative and hence Theorem 4.1 does not eliminate the possibility that the circle analysis may produce a controller with better actual performance. The purpose of this section is to show.byan examplethat this is indeed the case and to illustrate howthe circle criterion can be useful for improving the performance. To this end, let us first present a switching control synthesis condition based on the circle analysis, which parallels the result in Theorem4.3 derived from the linear analysis. Theorem 4.4. Consider the system in (4.14) with M= I and rn = Suppose that matrices Kk, symmetric matrices Qk > 0, diagonal matrices Vk > 0 and scalars 7~, #~ and 0 _~ r < 1 satisfy He

--rKk --Vk 0 CQk + DK~ DV~ -7~I

for k = 0,...,q,

Q~ K~

< 0,

Kk p2

>0

(4.35)

where p := a/(1 - r), and 7~ + #~Qk-1 < Qa < Q~-I

(4.36)

7k-1

for k = 1,..., u =¢(z),

q. Then the switching controller z = ~ax

-1 x) (k--0,...,q) (when xT Q~ -1 x_~ l<x T Q~+I

1 and Q~I := ccI yields Iz(t)l with ~k := KaQ-~

~_ for al l t >_0, and

k=l

whenever x(O)VQ-~Ix(O) < 1. Proof. The result directly follows from condition (4.13) with a change of variable K := EQ and from the argument for switching control design presented in Section 4.3.4. [] Example 4.2. Consider the system treated in Example 4.1. Wewill design state feedback switching controllers using the circle criterion summarized in Theorem4.4. The design steps are parallel to Switching Control Design Algorithm and are as follows. First fix r (and hence p) to be some value in the interval 0 < r < 1. Initialize k as k = 0. The numerical problem

98

Iwasaki and Fh

for designing the initial gain/C0 is to maximizethe domainof performance det(Qk) for a fixed value of performance’ level %, over the variables Q~ and K~, subject to constraints (4.35). This problem is a quasi-concave max!tmization whichcan be solved efficiently. The initial gain is then obtained a~s 1. Wethen increment k and go on to calculate additional gain ]Co :-- KoQ~ 1 ]Ck := KkQ~ by maximizing #k over Kk, Qk, "~ and #k subject to (4.35) and (4.36). Repeat this last step for k = 1,... ,q where q is the number switchings. From Theorem4.1 and its proof, we knowthat the optimal performance bound is obtained when r = 0 (i.e. p = a = 1) so that the circle criterion (4.35) reduces to the linear analysis condition (4.32). Therefore, we should always let p -- 1 to optimize the performance bound. However, as we show below, a "good" design mayresult whenp > 1. In particular, it will be seea that increase in p does not substantially affect the domains of performance but yet the control gain is very muchinfluenced so that the actual output response can be improved. Fix p to be either p = 2 or p = 10, and follow the design procedure outlined above. The results are found to be 7o 71 "~2 = "~3 ")’4

50.0000 16.3991 5.4522 1.8300 0.6178

,

#1 #2 #3 #4

=

1.9014 0.6792 0.2388 ’ 0.0830

for the p ---- 2 case (the values of "~k and ].t k for the p = 10 case are similar), and K:0 2C1 1C2 ](~3

/C4

=

(Case: -0.4890 -0.7053 -1.0453 -1.5720 -2.3843

p = 2) -0.3018 -0.3635 -0.4420 -0.5412 -0.6656

,

(Case: p = -2.3963 -1.5302 -3.4303 -1.8405 -5.0704 -2.2372 -7.6041 -2.7386 -11.5061 -3.3676

and the corresponding domains of performance are plotted as ellipses in Fig. 7. The ellipses for both p = 2 and p = 10 are almost identical to those obtained in Example 4.1 via linear analysis (i.e. p = 1), although the values of det(Q}) are slightly smaller. Similarly, the values of 7~ and #k are found insensitive to p, and the resulting performance bounds are 94.20 when p = 2) and 94.40 when p = 10. The control gains, on the other hand, are heavily dependent upon the value of p. In particular, it seems that larger p yields higher gain in general.

Regional ~2 Performance Synthesis

99

10 5

~ o -5 -10 -6

-4

-2

0

2

4

X 1

Figure 7: Domainsof performance and state trajectories

(circle criterion).

Using these gains we run simulations to obtain initial state responses with the same initial condition as in Example4.1. The results are shownin Figs. 7 and 8 wherethe thick curves are the responses for p -- 2 and the thin curves are for p = 10. Wesee that the response for the p = 2 case is actually better than the optimal ~2 bound response obtained in Example 4.1 in the sense that it has no overshoot with shorter settling time. This indicates that circle criterion can improve actual performance although it does not help to improve theoretically guaranteed T/2 performance bound (it is in fact slightly worse). Finally, when p = 10, the control gain is higher and the input u hits the saturation bound more often, but the output response is worse than the case p -- 2. The purpose of showingthis worse case is to illustrate that p can be used as a tuning parameter for "better" performance by adjusting the degree of saturation.

1

-2

10

20

30 Time

40

50

10

20

30 Time

40

Figure 8: Initial state responses (circle criterion).

5O

100 4.4.3.

Iwasaki and Fixed Gain Control with Accelerated

Convergence

Wehave seen in Example 4.2 that the circle criterion can be used to heuristically improve the actual performance of the "theoretically optimal" controller designed in Example 4.1. In this section, we present another heuristic methodto design a fixed-gain (without switching) state feedback controller that outperforms that in Example 4.1. To this end, we use the following synthesis condition obtained by modifying Corollary 4.2 to accelerate the convergence of the output signal. Lemma4.4. Fix/~ _> 0 and let Q, K and ~ be such that CQ + DK

-~,I

< 0,

Let u = KQ-lx be the state feedback controller. satisfies x.(O)TQ-ix(O)_< 1.

Ki ~i 2

> 0.

Suppose the initial

(4.37) state (4.381)

Thenwe have lui(t)l ~_ ~i for all t _> 0 and i -- 1 .... ,m, and

fo

~ {{ef~te(t){l 2dr< 2"/x(O)~Q - ~-x(O).

Proof. Note that condition (4.37) is obtained by applying the condition in (4.28) to the new system obtained by replacing A by A +/31. The result simply follows from the well knownfact: the initial state response of the modified system is given by zz(t) -= (C-t- DK)e(A+BK+~I)tx(O)= ef~t(C + DK)e(A+BK)tx(O) where z(t) is the response of the original system. Example 4.3. Consider the system given in Example 4.1. To acceler-ate the convergence, fix/3 > 0 and maximize det(Q) over Q and K subject to (4.37). In view of the previous examples, we choose ~/= 50. For various values of/3 > 0, we solved the quasi-concave maximization problem and calculated the corresponding state feedback gains. For each gain, we estimate the domain of performance to guarantee the performance bound ~f = 50 for the original system (/3 = 0) using the circle criterion (Lemma4.3). appropriate change of variables as in (4.13), the problemreduces to a quasi-concave maximization plus a line search over the "degree-of-saturation" parameter p. After some trial and error, we found that/3 = 0.2 gives the domain of performance whose size is nearly the same as the largest ellipse

Regional 7-12 Performance Synthesis

101

in Fig. 5. The feedback gain and the domain of performance for this case are given by K=[-0.6708-0.4171

],

29.8150 -44.6171 Q= -44.6171 108.8536

where the performance domainxTQ-lx _< 1 is plotted as the larger (dashed) ellipse in Fig. 9. The optimal value of p that yielded this Q is p = 2.7858. Note that the two straight lines correspond to K:x = ± 1 and thus the control input does not saturate if and only if the state is in the region betweenthe lines. The smaller (solid) ellipse in Fig. 9 indicates the guaranteed domain of performance weighted by fl = 0.2. 10 5

~ o -5 -10

-6

Figure 9: Domainof performance and state trajectories

(fixed gain).

1

-1 0

10

20 30 Time

40

50

10

20 30 Time

40

Figure 10: Initial state responses (fixed gain). Initial

state responses are obtained for x(0) = ~1 or ~2 where

5O

102

Iwasaki and Fu

and plotted in Figs. 9 and 10. Note that the case x(0) = 41 has been treated in the previous examples.Wesee that the effective use of the circle criterion yielded a fixed gain controller that outperforms the switching controller designed in Example 4.1. Note, however, that the switching controller was systematically obtained without design iterations while the fixed-gain controller required a heuristic parameter tuning of ft. Finally, we remark that the fixed gain controller does saturate for certain initial conditions as shown by the plots for the case x(0) = 42, in contrast with the fact that any controller, fixed or switched, designed by using Corollary 4.2 or Theorem4.3 wouldnot saturate for any initial conditions within the domain of performance. 4.5.

Further

Discussion

In the preceding analysis and synthesis, we assumed that the part of transfer function H(s) from u to z in Fig. 1 is strictly proper to simplify the argument. In this section we show that this assumption can be made without loss of generality when the high frequency gain in question is diagonal. Below, we consider for simplicity the case where the saturation nonlinearity has the unity bound, i.e. a~ = 1 in (4.2). Consider the mappingfrom 4 to u in Fig. 11 (left), u = ¢(4 + au)

(4.39)

where 4, u E ]Rm, G ~ ]R re×m, and ¢ is the saturation nonlinearity defined in (4.2). This feedback loop is said to be well posed if, for each 4, there exits a unique u satisfying (4.39).

Figure 11: Saturation with algebraic loop. Lemma4.5. Let G ~ ]R mxm be a diagonal matrix and ¢ be the saturation nonlinearity defined in (4.2) with c~i = 1. The feedback loop Fig. 11 (left) is well posed if and only if F := I - G > 0, in which case, the mappingfrom 4 to u defined by Fig. 11 (left) is identical to that in Fig. (right), that is,

u=¢(4+Gu)

,~

Regional 7~2 Performance Synthesis

103

Proof. Weprove the result only for the case where u and ~ are scalars. The general case where u and ~ are vectors directly follows because the matrix G and the function ¢ are both diagonal. If G >_ 1, then the equation u = ¢(~ + Gu) is satisfied by both u = 1 and u = -1 when ~ is zero. Thus the mapping from ~ to u is not uniquely defined at ~ = 0, and we conclude that G < 1 is a necessary condition for well-posedness. Below, we show that this condition is also sufficient, by explicitly constructing the mapping. Suppose G < 1. Weclaim that if u = ¢(~ + Gu) holds then ~>_I-G [~I-
¢* ~ ~

u=l, u=(1-a)-l~, u=-l,

from which the result follows directly. Consider the first equivalence. If u -- 1, then u---1

~

~+Gu>_l

~ ~>_I-G.

To show the converse, suppose ~ _> 1 - G but u ~ 1. Then ~+Gu >_ 1 -G+Gu = (1- G)(1 -u) +u > u >_ where we noted that (1 - G)(1 - u) > 0 due to G < 1, u ~ 1 and u _< Also note that u < 1 implies ~ ÷ Gu < 1. Consequently, -l<~+Gu
~

u=~÷Gu>u

which is a contradiction. Thus u = 1 must be true whenever ~ _> 1 - G. The third equivalence can be shownsimilarly. Finally, consider the second equivalence. If u = (1 - G)-I~, then

I0 -G)-1¢1-< 1 since u is the output of the nonlinearity ¢. Clearly, this condition is equivalent to [~[ _< 1 - G and thus we have "~=." To showthe converse, suppose [~[ _< 1-G holds. If~÷Gu > 1, then u -- 1 and 1 -G < ~ which contradicts the supposition. Similarly, if ~ ÷ Gu < -1, then u = -1 and ~ < G - 1, which is again a contradiction. Hence we must have [~ ÷ Gu[ <_ 1, implying that u = ~ + Gu or u = (1 - G)-I~ as claimed. Lemma4.6. Consider the controller with anti-windup compensation depicted in Fig. 12 where ¢ is the saturation nonlinearity in (4.2) with ai 1 and K(s) is a transfer function with the following state space realization K(s)=

C D1 D2 ( A BI B~

104

Iwasaki and t~

Supposethat D2 is diagonal and the feedback loop in Fig. 12 is well posed, i.e. the output u is uniquely determined by the input y. Then D2 < I and the mappingfrom y to u in Fig. 12 with the above K(s) is identical to the mapping from y to u in Fig. 12 with K(s) replaced by /go(s)=

( CA D1 B1

-1 )B2(I --~D2) --

’

Therefore, allowing for nonzero D2 in the anti-windup controller does not enlarge the class of controllers unless D2 has nonzero off-diagonal entries.

Figure 12: Controller with anti-windup compensation. Proof. Let us explicitly write downthe equations describing the mapping from y to u in Fig. 12 as follows: :i: = Ax + Bly + B~(z - u), z = Cx + D~y + D~(z - u), u=¢(z). Welloposednessrequires that I - D2 is invertible, equation can be solved for z as

in which case the second

z = (I - D2)-l(zo - D2u), := Cx q- Dly. Substituting this expression for u = ¢(z) we have U ---- ¢((I -- D2)-l(zo D2u)). From Lemma4.5, u is uniquely determined from Zo if and only if (D~-I)-ID2
or

D2
holds. In this case, u is given by u = ¢((I - (02 I) -102)-1(I -

D2)-lzo)

= ¢(Zo).

Finally, noting that ~c = Ax + B~y + B2((I - D2)-l(zo - D2u) = Ax + B~y + B2(I- D2)-~(Zo-

Regional ~2 Performance Synthesis

105

we conclude the result. [] WhenD2 = 0 in the anti-windup controller in Fig. 12, well-posedness of the control law is automatically guaranteed. WhenD2 is diagonal, the feedback loop is well posed if and only if D2 < I. The above result shows that the feed-through term D2 can be set to zero without loss of generality whenspecifying the class of controllers to be designed, provided that D2is diagonal. In other words, the nonzero diagonal D2 term does not contribute to improve the achievable performance. Note, however, that this does not eliminate the possibility that a nonzero D2 term of general structure may indeed improve the performance.

References [1] C. Edwards and I. Postlethwaite. An Anti-windup Schemewith Closedloop Stability Considerations, Automatica, 35 (1999) 761-765. [2] M. Fu. Linear Quadratic Control with Input Saturation, Control Workshop, (Newcastle, December, 2000).

Proc. Robust

[3] P. Gahinet and P. Apkarian. A Linear Matrix Inequality Approach to T/~ Control, Int. J. Robust Nonlin. Contr., 4 (1994) 421-448. [4] D. Henrion, G. Garcia, and S. Tarbouriech. Piecewise-linear Robust Control of Systems with Input Constraints, European J. Contr., 5(1) (1999) 157-166. [5] D. Henrion, S. Tarbouriech, and G. Garcia. Output Feedback Robust Stabilization of Uncertain Linear Systems with Saturating Controls: An LMI Approach, IEEE Trans. Auto. Contr., 44(11) (1999) 22302237. [6] H. Hindi and S. Boyd. Analysis of Linear Systems with Saturation using Convex Optimization, Proc. IEEE Conf. Decision Contr., pages 903-908. [7] P. Hippe and C. Wurmthaler. Systematic Closed-loop Design in the Presence of Input Saturations, Automatica, 35 (1999) 689-695. [8] T. Iwasaki and R. E. Skelton. All Controllers for the General Ha Control Problem: LMI Existence Conditions and State Space Formulas, Automatica, 30(8) (1994) 1307-1317.

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[9] T. Iwasaki and R. E. Skelton. On the Observer-based Structure c.f Covariance Controllers, Sys. Contr. Lett., 22 (1994) 17-25. Controller Design for [10] V. Kapila and W. Haddad. Fixed-structure Systems with Actuator Amplitude and Rate Non-linearities, Int. J. Contr., 73(6) (2000) 520-530. [11] N. Kapoor, A. Teel, and P. Daoutidis. An Anti-windup Design for Linear Systems with Input Saturation, Automatica, 34(5) (1998) 574. [12] T. Kiyamaand T. Iwasaki. On the Use of Multi-loop Circle Criterion for Saturating Control Synthesis, Sys. Contr. Lett., 41 (2000) 105-114. [13] M. Kothare, P. Campo, and M. Morari. A Unified Framework for the Study of Anti-windup Designs, Automatica, 30(12) (1994) 1869-1883. [14] M. V. Kothare and M. Morari. Multiplier Theory for Stability Analysis of Anti-windup Control Systems, Automatica, 35 (1999) 917-928. [15] Z. Lin. LowGain Feedback, Springer, (1999). [16] Z. Lin and A. Saberi. Semi-global Exponential Stabilization of Linear Systems Subject to ’Input Saturation’ via Linear Feedbacks, Syst. Contr. Lett., 21(3) (1993) 225-239. [17] Z. Lin, A. Saberi, and A. Stoorvogel. Semiglobal Stabilization of Linear Discrete-time Systems Subject to Input Saturation via Linear Feed: back --.An ARE-based approach, IEEE Trans. Auto. Contr., 41(8) (1996) 1203-1207. [18] I. Masubuchi, A. Ohara, and N. Suda. LMI-basedController Synthesis: A Unified Formulations and Solution, Int. J. Robust and Nonlinear Contr., 8 (1998) 669-686. [19] Yu. Nesterov and A. Nemirovsky. Interior-point Polynomial Methods in Convex Programming, SIAMStudies in Applied Mathematics, (1994). [20] T. Nguyen and F. Jabbari. Output Feedback Controllers for Disturbance Attenuation with Actuator Amplitude and Rate Saturation, Proc. American Contr. Conf., pages 1997-2001.

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107

[21] Y. Peng, D. Vran~id, R. Hanus, and S. S. R. Weller. Anti-windup Designs for Multivariable Controllers, Automatica, 34(12) (1998) 15591565. [22] C. Pittet, S. Tarbouriech, and C. Burgat. Stability Regions for Linear Systems with Saturating Controls via Circle and Popov Criteria, Proc. IEEE Conf. Decision Contr., pages 4518-4523. [23] A. Saberi, Z. Lin, and A. Teel. Control of Linear Systems with Saturating Actuators, IEEE Trans. Auto. Contr., 41(3) (1996) 368-378. [24] C. Scherer. Robust Generalized H2 Control for Uncertain and LPV Systems with General Scalings, IEEE Conf. Decision Contr., pages 3970-3975. [25] J. Shamma.Anti-windup via Constrained Regulation with Observers, Proc. American Contr. Conf., pages 2481-2485. [261 S. Tarbouriech and G. Garcia (Eds). Control of Uncertain Systems with BoundedInputs, Springer, (1997). [27] G. Wredenhagen and P. Belanger. Piecewise Linear LQ Control for Systems with Input Constraints, Automatica, 30(3) (1994) 403-416. [28] F. Wu, K. Grigoriadis, and A. Packard. Anti-windup Controller Design using Linear Parameter-varying Control Methods, Int. J. Contr., 73(12) (2000) 1104-1114.

Chapter 5 Disturbance Attenuation with Bounded Actuators: An LPV Approach F. Jabbari University

5.1.

of California,

Irvine,

California

Introduction

In this chapter, the linear parameter varying (LPV) approach is used to deal with the inevitable limitations in actuator capacity. By modelingsaturation as a section (0,1) nonlinearity or as an uncertain parameter with the same bounds, a variety of results from absolute stability or robust control can be applied. During the last few years, exploiting recent progress in several areas, a host of results have been obtained in design of LTI compensators for system with bounded actuators ([6, 7,14,19, 30, 35], amongmany). More recently, the LPVstructure has become quite popular. Due to the availability of the input commandand the actual input, the unknown/uncertain parameters can be obtained/cMculated online ([27, 31, 37, 38]) so that the most recent, and least conser~}ative, LPVtechniques can be used for compensator design. The motivation for work presented here has been earthquake engineering applications in which a few large capacity actuators are used to counter the effects of disturbances (i.e., ground motion in an earthquake episode), where disturbances typically have a large range of magnitudes. The system

109

110

Jabbar.i

itself is almost always stable, due to structural damping. As a result, the.’ focus is on performance, primarily that of establishing guarantees for the; L2 gain from the disturbances to the controlled outputs, though other mea-sures are possible-after some standard modifications. The main goal is to reduce the conservatism inherent in manyof the techniques that are com.monly used in saturation problems. The salient features of the technique presented here include explicit incorporation of the saturation nonlinearity and dependence of the performance guarantees on the actuator capacity (i.e., larger actuators leading to better performance guarantees). This feature can be useful in the design stage for trade-off regarding the actuator size and performance. Furthermore, resulting controllers have higher gains, to better use the available actuator capacity. The main goal here is to develop control methods that exploit the progress made in the LPVliterature. Initially, the most commonform of the LPVproblem; i.e, with a constant Lyapunov matrix, is discussed. While it is well knownthat this leads to rather conservative results, the ease of solution and implementation has proven to be an important advantage. Once the basic framework is set up, the extension to parameter-dependent Lyapunovmatrices is addressed. Several issues regarding computation and implementation of such controllers are discussed. Next, the issue of actuator rate bounds are discussed. Again, some of the natural connections with LPVresults will be discussed. Finally, to further reduce the conservatism, a scheduling approach for dealing with actuator saturation is presented in which the scheduling parameter is related to the saturation constraint. Due to rather complicated notations needed, this last result is presented for special cases (generalizations are available in references). For most of this chapter, the system itself is assumed to be described with a linear parameter varying model, with knownbounds for the actuators (and possibly knownbounds on the actuator rate limits). It is further assumed that only the system matrix is parameter varying and the remaining matrices are constant. Similarly, affine dependence on the parameter is assumed. Finally, we start with only one disturbance and zero initial conditions. These assumptions are used to make the presentation less cluttered, and can be eliminated relatively simply. The more general cases will be discussed separately. Additionally, the results here depend on several assumptions that are reasonably general and are met in most applications. These include ¯ standard decentralized saturation (i.e., [SAT(u)]i = sat(u~), sat(u~) = ui if luil _< Ulim~and sat(u~) = sgn(u~)u~im~ if lull > Ulirn~). ¯ Actuator capacity, Ulim~, is knownfor each actuator (and in the rate

Disturbance Attenuation with Bounded Actuators saturation case, the rate bound/~lim~

111

is known).

¯ An estimate for Wma~= m~xtlw(t)l is available. If the disturbance is a nwvector, then it is assumedthat an estimate is available for maxtlwTwl (which is no more that nwllwll~) ¯ The direct feedthrough terms for the plant and compensator (i.e., D11,D22and Dc) are zero.

5.2.

Preliminaries Consider the following LPVsystem with zero initial

conditions:

{

:~ - - A(p(t))x Bl VO + B2SAT(u) z : 01x+D12SAT(u) y = C2x+D2~w.

(5.1)

Note that in the control output, the saturated value of the input is used. This is due to the fact that often the penalty is placed on absolute acceleration of the system, for which the actual force applied is needed. If D12u is used instead, the development below becomes a bit simpler. Following standard treatments, A(p) is assumed to depend continuously on p(t). As mentioned earlier, we start with assuming affine dependence and discuss more general cases later. For simplicity, p(t) ~ 8 is assumed to be continuously differentiable, since its time derivative is often needed in the controller when parameter varying Lyapunov functions are used. In manycases this continuity condition can be relaxed-as long as the solution to (5.6) exists and is well defined (e.g., continuous with boundedderivative or simply measurable). Vector p(t) ~ ~s is assumed to belong to a compact set

= : < <_ w = 1,2, ...s}

(5.2)

with knownbounds p~ and ~. The corresponding vertex set is defined as A~×= {p: p~ = p~ or p~ = ~ Vi = 1, 2, ...s}.

(5.3)

The 2s elements in Avex will be denoted by p3, j = 1,..., 28. Anyp ~ A can be expressed as a convex combination of the vertex values s2

j=l

112

Jab bari

for some Aj, where Aj > 0 for all j = 1,...,2 s and E~21)’J = 1. parameter-dependent matrix, H(p), evaluated at pJ E Avex will be denoted. by a superscript j, i.e, HJ = H(p/). A matrix H(p), which depends affinely on p, can be computedfrom the values of its vertices as in (5.4), i.e. H(p) Westart with replacing each sat(u/) term by Pi(t)u~(t) where

sat(ui(t)) ui(t), (pi(t) = ti(t )

=

0)

(5.5)

which yields B2SAT(u) B2~(t)u,

~( t) = di ag{p~(t)}

resulting in - - A(p(t))x + Blw + B2~(p(t))u Clx + D~2tg(p(t))u C~x + D2~w.

(5.6)

Since p~ is a time varying parameter which can be measured on-line (by comparing the commandand the saturated value), we can combine it with p. It lacks the C~ continuity that is assumed for p unless the saturation is replaced by another smooth function approximating it. In certain cases however, by using a combination of anti-windup and LPVapproaches, the problem of solving the constrained LPVproblem becomesno more difficult than that of the unconstrained case while resulting in certain conceptual and computational advantages. For performance, we use the boundedreal matrix inequality to establish the L2 gain from the disturbance to the controlled output. Simultaneously, the reachability inequality (see [1,5]) is used to obtain an estimate for the reachable set of the closed loop system (see [27, 28] for details). Oncethe reachable set is established, bounds for the maximum control effort can be obtained. Next, by allowing the command to the i th actuator to be d~ x ~tlim~ in the reachable set, high gain controllers are obtained. The resulting p~ will be bounded by by 1 and ~~,¯ i.e, controllers discussed here will be designed such that 1 p(t) ~ P = {p(t) : ~ _< p~(t) _< 1 Vi = 1, 2,... n~}

(5.7)

instead of the traditional boundsof (0, 1). Withoutthe reachable set bound, the lower limit would be zero, which corresponds to the open loop. In such

Disturbance Attenuation

with Bounded Actuators

113

a case, the open loop would necessarily be required to be stable and the best L2 gain would the one from the open loop. Lastly, we denote the vertex set associated with (5.7) 1 Pvex = {p: pi =" ~ or pi = 1 Vi = 1,2, ...nu}.

(5.8)

In the rest of this chapter, often the explicit dependenceof ~(p(t)) on t is dropped to simplify the presentation. The results here have a great deal of.similarity to the high gain form of the ’low gain and high gain’ structure used in [23, 29] and several related papers by the same authors. The results here use the general B1 matrix, deal with disturbance attenuation and rely on a multi-objective approach (e.g., [8]) to obtain performance guarantees that are dependent of the actuator capacity and/or worst cases disturbance. Nonzeroinitial conditions within the ellipsoid of reachability are trivially addressed. While(local) stability can be studied and established with the techniques presented here, techniques focused on the stability issue may be more appropriate if enlargement of the region of attraction is the mainfocus ( [6, 23]).

5.3.

Parameter-independent Functions

Lyapunov

First, we discuss the most commonform of LPVcontroller design that relies on constant Lyapunovfunctions. The results for parameter dependent Lyapunovfunctions will be discussed in the next section, using a similar notation and approach. Consider the following structure for the compensator (with the schematic of Figure 1): kc = [Ac(p) It

+M(p)]Xc

+Bey+ NSAT(u)

= Ccxc.

(5.9)

The constant matrix N is used to eliminate the need for a compensator dynamic matrix of the form Ac(p,p). This simplifies the on-line calculations and will becomehelpful in other cases. The L2 gain or estimate for the reachable set for the closed loop is then established along standard techniques ( [16, 17]), in which it is assumed, WLOG, that the underlying Lyapunovmatrix and its inverse have the following structure =

X S -1

+X

’

=

-S

S

with Y = S + X-1 > 0. Using ~T ~_ (X T XcT ) for the closed loop states, V = FcTQ-I~is used as the I~yapunov function. The L2 gain is established

114

Jabbari

by guaranteeing ~ + zTz - ~wTw ~__ O. Standard manipulations and transformations (e.g., see [16,17]) leads to the inequality (5.11) below. Next, 2 < 0 implies that ($ : £,TQ-I~ <Wma - Wm~x) 2 x } is an overbound ? + for the reachable set. Again following the same procedure leads to (5.12) below. The saturation is addressed by allowing the i th actuator command(i.e., the i th entry of Ccxc) to be equal or less than 6i x Ulim, within the reachable set ((5.13)). This, in turn, establishes the bounds for pi, which leads p E P. Due to affine dependence on p and p, checking only the corners of Avexx Pvex is sufficient. Wethen have: W

~

Z

.

y

P(p)

U

K(p)

Figure 1: Polytopic LPVsystem with output feedback. Theorem 5.1. Consider (5.6) where A(pJ) is denoted Aj, and a given set of 5~’s. The there exist constant matrices X where j = 1 .... ,2 s, and F~ (i = 1,2,...n~) for some a > 0 such that the following inequalities are feasible

YB1 + GD~ B1 -7~I 0

xcT~ + FT¢(p)DT~2 0 --I

< O, j = 1,2,...2 ~, p E Pvex

(5.11)

Disturbance Attenuation

115

with Bounded Actuators

YAj T ÷ AjTy ÷ c~Y ÷ GC2 ÷ C~G Aj ÷ LjT q- c~I GT BT~ y + DT~I YB1 + GD~I

AJT + LJ ÷ ~I

_< 0, j = 1,2,...2 s, p E ~Ovex

(5.12)

-od Y I Opo,, ~=. I X

0 T F~

>0,

(5.13)

] where ~b2~ = AJX + XAjT + B2K~(p)F + FTK~(r)BT2¯ Then the parameterdependent output feedback controller of the form given in (5.9), with M N Cci Be At(p)

= 0 = (Y-X-t)-IYB2 = X-l" Fi = -(Y = (Y- X-1)-I[YA(p)

guarantees local quadratic stability

(5.14) +GC2- L(p)X-1], with L~-gain of Yoo from w to z~.

The ab(~ve results depend on Wmaxand Ulim, aS seen in (5.13), so that larger Ulim or smaller Wmax relaxes the constraint in (5.13), which in turn yields lower 3’~. Also, 5i is an arbitrary positive scalar greater than one, and it is used to maximizethe utilization of the available control capability. The system performance can be improved by using large 5i (see simulation results below). As 5 increases, so does size of Pvex, which decreases the L2 gain guarantee, though this is often only a modest increase due to the ’matched’ structure in which p(t)’ enters (5.6). The open loop need not be stable. In problems with unstable open loop, for small values of Ulim there might be no feasible solution (see [27] for an illustrative example). Furthermore, from the proof (see [28] for example) is clear that as long as the initial conditions are in ~cTQ-I~,the state does not leave the reachable set, hence the local quadratic stability property, in the absence of external disturbance. The reachable set thus becomes the region of attraction. Naturally, if enlargement of domainof attraction is the main goal, results of [23], for example, which are keyed to (semi) global stabilization might be more appropriate. Note that when there are nonzero

116

Jabbari

initial conditions, the standard modification for the definition of the L2 gain, incorporating the initial conditions, will be needed (i.e., f~ zTzdl~, <_ 2 ¯~/oo f~ wTwdt+ xTo Q-lxo) The term NSAT(u) in (5.9) is similar to the anti-windup terms used in traditional approaches (see [34] and its references). This term could be absorbed into L (i.e., L(p,p)) which would lead to Ac(p,p). The (2,1) and (1,2) terms of the inequalities (5.11) and (5.12) have a collection nonlinear terms that are bunched together as L; i.e., the (2,1) term is the form A(p) T + YA(p)X - SBcC2X + YB2qd(p)CcX - SAcX. The term NSAT(u)eliminates the p-dependent term and thus the need include Pi in the interpolation. At a minimum,this simplifies the implementation of the controller since it reduces the numberof matrix variables Lj’s and reduces the complexity of the required interpolation. In someinstances, this concept can lead to additional benefits. For example, consider the following for M(p) in (5..9) M(p) = (Y - X-l) -’ + Yd(p). With this M(p) and N = S-IYB2, the nonlinear -SBcC2X

(5.15)

(2,1) terms become

- SAcX

which allow letting L = -SBcC2X - SA~X = constant. This eliminates the large numberof Lj’s as well any need for interpolation on line, since in that case one can use A~ = (Y - X-1)-I[GC~ - LX-1],

(5.16)

with the same N, Co, and B¢ as in (5.14). In special cases, such arrangements can be used without any loss of generality. Consider the case where w is measured on-line (e.g., earthquake applications, Tokomaks). In such cases, it is. reasonably simple to show that one can use a compensator of the form fcc = A(p)xc B2SAT(u) + B~w + L( y - Cox - D2~w), u = Fx where F is from the state feedback problem (which is convex and easy to solve). It is straightforward, though tedious, that a sufficient condition for the closed loop to have the same L2 gain as the state feedback is existence of R > 0 and G such that

Disturbance Attenuation with Bounded Actuators

117

Since this implies existence of a constant L, under these conditions the search with a constant L will be feasible, which simplifies computation and implementation of the controller. Remark 5.1. If the Lyapunov functions were parameter varying however, or if the dependence of the system matrices on p were higher order, checking the corners of Avex would not be enough for feasibility for all of A. Different techniques for solving these cases include gridding the parameter space [36], the multi-convexity concept [12], and general (i.e., higher order polynomial) techniques [2, 33]. Parameter dependent B1, B2 and however,do not alter the overall complexity significantly.

5.4.

Parameter-dependent Lyapunov Functions

Compensators

and

If matrix Q in (5.10) is parameter independent, the computational effort is relatively minor, though the results are knownto be quite conservative, since they allow arbitrarily fast time variations in p ( see e.g., [3,11,12, 36]). Use of parameter dependent Q, however, requires additional continuity conditions and information on the bounds (and values) of For this section, we further assume p to be continuously differentiable and the parameter variation rate to be bounded by a set of nonnegative numbers #~ I fii I_< tti Vi= 1, 2, ...s. Naturally, we do not have C1 continuity for p(t). However,as discussed below, we can eliminate this difficulty by using the anti-windup form discussed earlier. Westart by allowing the parameter-dependent controller depend on both p and ~. The following is then obtained along the standard path ( [33,36]). As a result, no proof is necessary and the modifications needed for Theorem5.1 is relatively straightforward. For example the (1,1) element of (5.11) becomes

Y(p)(p)A(p)

8 ÷ AT(p)Y(p) ÷ G(p)C2 ÷ C~GT(p) ÷ ~ ±(#i-~p~)

while G, F and n(p) are replaced with G(p), F(p) and L(p, ~), respectively. The notation ~’~1 +(’) used here implies that every combination of ÷(.) and -(.) should be included in the inequalities. Typically, X(p) and Y(p) are assumed to be affine functions of p, though more general forms can be implemented.

118

Jabbari

Often, in the state feedback problem, ~ is not used by the controller (though its bounds are used in designing it). In the output feedback case, use of parameter-dependent Lyapunov function for LPVsystems typically results in controllers that depend on both p and ~. While the bounds on maybe reasonable to obtain or be assumed a priori, its direct measurement on-line is more problematic. This is more of a problem with respect to p, since it is not C1. Use of the N term in the compensator thus reduces the complexity (both in solving and implementing of the controller). It also eliminates the need to introduce new approximate variables for p. There are several ways in ~vhich dependence on ~5 can be avoided. Since ~5 appears due to derivative of X(p), a commonapproach in avoiding direct use of ~5 is to assume a constant X (or a constant Y in the dual formulation). In general, the conservatism associated with such an assumption is not known. In certain cases, however, it can be shown that output feedback compensators can be designed that do not rely on ~, without resulting in any conservatism - it the sense that they have the same stability and L2 gain properties as the state feedback problem. These include; i) when w is directly measured, ii) when [(AN + -~I, is left invertible and minimumphase, where AN,DA are obtained from A(p) = AN + DAH(p)EA. The proofs are quite tedious (and similar to those in [17]) and do not warrant a detailed treatment. If suffices to say that in both cases, one can showthat once the full state problem is solved, there exists constant Be, and constant matrix S = Y(p) -X-1 (p) such that the sufficient conditions for the L2 gain are satisfied. As a result, one can search for a constant G, and L(p) (not dependent on ~5) and implement the following Bc Ac(p)

= -S-1G = S-I[Y(p)A(p)

+ GC2 + (YB2F(p) - L(p))X-I(p),

(5.17)

where Cc is from the state feedback (or X(p)-IF), along with the same N and Mmatrices as in Theorem 5.1. 5.5.

Numerical

Example

Consider the lateral axis dynamics of the L-1011 aircraft. The state space representation for the L-1011 aircraft, with states associated with the yawrate, side slip angle, bank angle and roll rate, is given by: -2.98 0.93+ r(t) -0.21 -0.99 0 0 0.39 -5.555

0 -0.034 0.035 -0.0011 0 1 -1.89 0

, B2 =

-0.032 0 0 -1.6

Disturbance Attenuation

119

with Bounded Actuators

and B~T = [0 0 0 1], where I r(t) I_< 1. The uncertainty for this system can be modeled as AA = DAr(t)EA where DA=[

1 0 0 o]T

andEA=

[ 0 1 0 0].

The controlled output is chosen to be the roll rate, i.e. C1 = [0 0 0 1] C2=

00 01 10 0o] ’

(5.18)

which means the side slip and bank angle can be measured. The parameterdependent controller in Theorem 5.1 can be obtained by using the LMI Control Toolbox. For %o = 1, Ulim = 2 and Wraa~ = 5, and for different gains, i.e. 5 = 20 and 5 = 500, the LMIs of Theorem 5.1 are found to be feasible with c~ = 0.4. For simulation purpose, the parameter is

r(t) =cos(t). Figure 2 compares simulation results for different gains, 5 = 20 and 6 = 500. System performance for the high gain case, i.e. 6 = 500, is better because it is able to take advantage of the available control. For the linear controller, which is low gain and corresponds to 6=1, the resulting control input is small and the system performance is similar to that of the open loop case. 5.6.

Rate

Bounds

Often, the limit on the rate of the actuator output is a critical design limitation, which can lead to major, even catastrophic, system degradation and failure [4]. The most commonapproach seems to be assuming a first order model for the dynamics of the actuator, which leads to a representation of the form shown in Figure 3 (see [28] for the LTI version). The model for the state of the actuator v, in Figure 3 is thus de = A(p)x+B~w+B2v

= satr(:(satp(u)-

(5.19)

wherev E ~-~n~,with the constraint I~)¢(t)l <_ hL~m,~ K: ~ blockdiag{~,~,...,~n~,}.

(5.20)

The subscript ’p’ (as in Satp) is used for magnitudesaturation, while ’r’ used for rate (i.e., satr), with limits/q~m, for the i th actuator. Wedefine ri(t) A=satr(~i(satp(u~) -- v~)) with r~(t) = 1 if satp(u~) - v~ = 0 (5.21)

(satp(u )

120

Jabbari Disturbance profile

X4(delta=20) 0.6 0.41 0.2!

oi -2

-0.2

-4

-0.4

-6

0

5

10

5

X4(delta=500 andopen loop(dashed)) 0.6

10

U(delta=500)

0.4 0.2 0 -0.2

-1t..... -2P ...... 5

10

0

5

10

Figure 2: Plots of disturbance, x4 and control input. or equivalently

{

~: = A(p)x

+ Blw + B2v,

-

=

(5.22)

In general, the magnitude saturation box can be placed before or after the rate limiter loop. While the two are not identical, they can be handled similarly. Here, we focus on the arrangement shown in Figure 3. For this arrangement, have can the following claim (see [28]) Claim I: [Satp(U~) - v~[ _< 2Ulim~. Whenthe rate limiter is active; i.e., ~lim~ < 2NiUlim~, parameter ri defined in (5.21), satisfies [" ?’tlim’ ’ ri(t) ~2t~Ulim~

1].

(5.23)

Disturbance Attenuation

with Bounded Actuators

satp

121

satr

Figure 3: Schematic for magnitude and rate saturation¯ Similar to the previous development, the vector r(t) -- Jr1 belongs to the compact set 74 = {r(t)

Ulim~ < ri(t) : 2gittlim, -

< 1 Vi = 1, 2,... -

nu}

T ¯ ..rn~]

(5.24)

which has the vertex set 74vex ---- {r(t):

i =It lim--------k-~ 2t~i?~limi

orrf = I Vi-- --

1, 2,¯¯¯nu}.

(5¯25)

Next, defining q2r(t) = diag{ri(t)} and noting the diagonal structure q~p(t), ~r(t) and K:, the augmentedstate space model becomes

Zl’

~-

~12

+ Dl2¢p(t)u

(5¯26)

where 2T = Ix T vT ], etc. In this case, the elimination of nonconvexity due to the saturation nonlinearity requires the availability of r~ which in turn can be calculated from the actual input to the plant, i.e., v in Figure 3. If this signal is available, a structure similar to (5.9) leads to a tractable search for the compensator (quite similar to the unconstrained case). this signal is not available, the problem becomes a system in which only a subset of parameters are measured and techniques similar to [20] can be applied. In both cases, the details are along those presented in [20, 28] and thus not repeated here. Generalizing this idea is relatively straightforward. Instead of the first

122

Jabbari

order model, a general model of form ~v = Avxv U

~ v

+ Bvv

(5.27)

Cvx

for the actuator dynamics can be used, in which v is the control command to the actuator and xv is the state of the actuator. The bounds can be incorporated by I~ti(t)[ ~//,max,, I/til ~ ~max~. (5.28) The augmented state is then formed and the constraints in (5.28) can incorporated similar to the development discussed earlier, by conditions on the size of appropriate variables in the reachable ellipse (i.e., similar to (5.13)). (see [21] for all relevant details). This generalization, however, loses certain advantages of the first order model, such as the Claim above and the natural bounds on ri in (5.24). If the actuator limitations are incorporated through (5.28), then the resulting problemwill be a convexsearch, but the controllers will be of the low gain variety and quite conservative. To increase the effectiveness, the same principle as before can be used, i.e., letting ui and/or/t~ be 5i times the physical limits. The choice depends on the specifics of the system, e.g., which limitation is least likely to be violated. In general, when 5i is used for both, the guaranteed performancelevels deteriorate significantly [281]. 5.7.

Scheduled Case

Controllers:

State

Feedback

In this section, we address the issue of conservatism that is faced in many of the techniques used for saturation, namely designs being based on the worst case description of the disturbance. Often, a conservative estimate of the peak disturbance is used in design, while the actual disturbance is considerably smaller. Furthermore, designs often are keyed to the worst possible response of the system, whereas a typical disturbance might result in a considerably more benign system response. Here, we rely on concepts of parameter dependent Lyapunov functions and parameter dependent performance measures (see [9, 22] for a recent exampleof each), as well as scheduling of the controller. Scheduled controllers have been used before (e.g., the early work of [13, 26] as well as the more recent work of [10,15, 24, 34, 37], amongothers). The basic approach here is to use state dependent ellipsoids (Lyapunovlevels sets) to obtain a scheduling parameter that is used to adjust the controller. As the state become larger, smaller gains are used. Smaller ellipsoids allow higher gains, thus better performance. As a result, the guaranteed performance bound (i.e., L2 gain

Disturbance Attenuation

with Bounded Actuators

123

bound) is parameter dependent as well. The choice of controller is thus function of the system response and not any a priori estimate of the worst cases disturbance. As before, we are interested in having the performance be an explicit function of the actuator capacity. To simplify the presentation, we focus on the state feedback problem, and drop the dependence of the system matrices on p. Including the dependence on p is almost immediate, though at the cost of significant notational clutter. The system is thus the following

{

:~ = Ax+Blw+B2u z = Clx+D~2u

(5.29)

Throughoutthis section, we use the following for an ellipsoid 2

(5.30)

$(P,~/,) = {x: xTpx < Ulira } -- y,~

where P = pT > 0 is a matrix, 7, is a positive real number and Ulim is a constant. The basic idea is to have a number(or family) of nested ellipsoids corresponding to different ~/,’s, with inner ellipsoids correspond to larger 7,. For example, consider a set of positive scalars 7,1 > 7,2 > ... > "7,i > ¯ .. > ~,(m-1) ~/*m and th e corresponding set of nested ell ipsoids

~Y*i

for someQ > 0. Clearly, for a constant Q,wehave $(Q- 1, "~*i ) C ~(Q-- 1, ,~,j if i < j. It is easy to show that if Q(r) > 0 and F(r) satisfy the following matrix inequality [

Q(F~i

r

FT/r)

]

>0

(5.31)

then for a~iy r~ = r, the controller K(r~) = F(r~)Q-~(r~) will maintain the control input belowsaturation limit inside the ellipsoid $(Q- ~ (rl), rl). Moreover, if we put a condition on P(r) = Q(r) -1, so that -~-_ dR > O, then the ellipsoid $(Q-l(r), r) will decrease in size as r increases in magnitude for any two positive numbers r~, r2 with r~ > r~, we have $(Q-l(r~), r~) $(Q-~(r2),r2). If the inequality (5.31) is valid for a range of r, then have a continuous set of nested ellipsoids for that range of r. By making Q(r) and F(r) satisfy an appropriate matrix inequality, along with (5.31), we can obtain a specific performance guarantee for each ellipsoid. The

124

Jabbari

performance guarantee for the higher r will be better, due to the fact that in smaller ellipsoids the constraint on the control gain is relaxed, allowing higher control gain and lower ~/(r). Weseek the ellipsoid, from the family of ellipsoids ~(Q-l(r), r), gives the tightest bound on the state of the system. To accommodatethis, let i 5 be given by the following 2

15(t) = max{r¯ [rmin,rmax] ¯ xT(t)Q-l(r)x(t) < uli-~-m -- r2}

¯

(5.32)

Then, at any time t, $(Q-l(ib(t)), ifi(t)) will give the smallest possible soid (in the family of $(Q-~(r), r)) containing the state of the system; i.e. for all rl _< i5, all ellipsoids of the form $(Q-~(rl), r~) will also contain state of the system and for r2 > iS(t), x ¢ $(Q-~(r2), r2). Inside ellipsoid $(Q-1(/5(t)), ib(t)), the controller K(p(t)) will be chosen so that the control input remains below the saturation limit. Since the state Of the system is continuous and Q- 1 (r) P(r) is als o a c ontinuous function of r, 15(t) defined by the relation (5.32) will be a continuousfunction of time. Note that the closed loop is nowa function of the parameter i 5, which itself is a function of the state. Here parameter ifi is used (consistent with earlier results) to denote the parameters that enter the system due to saturation, and not the traditional parameters (i.e., p) that often represent general nonlinear terms. Due to technical reasons, we change controllers to the one corresponding to bigger ellipsoids when the states grow but we do not switch back to higher gains corresponding to smaller ellipsoids when the state enters smaller ellipsoids. As a result, at any time t, we will implement the controller associated with the smallest value of/~ in the interval [0, t), though can increase with time if the state returns to smaller ellipsoids. The benefit, and cost, of this assumption becomes more clear later in this section. We thus define our parameter p to be p(t) = mint>~>0p(a)

(5.33)

This parameter may not be differentiable, though is continuous. Since we will need to take derivative of this parameter, we can define a function ~(t)

= ~- ~-P(h)’

1 >> ~- > 0

(~.a4)

which is C~ and the difference between p and/5 can be made arbitrarily small for small enough ~-. Due to continuity of Q on r and use of strict inequality, we can use/5(t) in the proofs, but p in implementation.

Disturbance Attenuation

with Bounded Actuators

125

Wecan now state the following result Theorem5.2. Consider the full state problem, with zero initial conditions and given Ulim, and Wmax.Consider parameter r where 0 < rmin ~ r < rmax with rmin : u-zm~ If there exists a C1 matrix Q(r) > 0 with dQ< 0, and F(r) such that the following inequalities are satisfied for some dr a>OandT(r) >0, AQ(r) + q(r)A T + B2F(r) + FT(r)B~ 0

C1Q(r) D12F(r)

Q(r)c FT( )DT2 ] 0 -I rQ(r) F(r)

(5.35) ] <0

FT(r) ] >0 r

(5.36)

and AQ(rmin)

Q(rmin)A T + o~Q + B2F(rmin) +

BT

FT(rmin)BT2

B~ ]<0

Then, the controller F(p(t))Q-~ (p(t)) x(t), where p is from (5.33), guarantees that the state does not leave {x xTQ-l(rmin)X <_ W2max} and the L2 gain is less than 7(1), where 1 is the smallest value of p used over time; i.e., l = mint{p(t)}. Ordinarily, there would be a term involving ib in the (1,1) element (5.35), similar to the parameter dependent Lyapunovfunction results previous section. The main problem is that estimating bounds for/5 is quite difficult and requires a great deal of information on the characteristics of the disturbance and the systems response to it. This is practically infeasible. The assumption regarding the sign of/5 allows this term to be dropped, as the cost of someconservatism (see [18] for details). Remark5.2. It is easy to see that the developmenthere can be used for multi-input case in a straightforward fashion. For example, by scaling the columns of B2 the same value of UlimCanbe used for all actuators, which means no changes are necessary in the basic definitions of the ellipsoids. There will be a separate inequality of the form in (5.36) for each actuator.

126 5.7.1.

dabba~’i Obtaining the Controller

To obtain K(r) and Q(r), the following linear spline functions will be used (quite similar to the spline fnnctions used in [25] for the LPVproblem). For a range of a parameter x E~ [Xl,Xm] ~vith Xl < x2 < ...xk < ... < xm, and m symmetric matrices Q1, Q2,... Qm, function Q(x) is defined as Q(x)

:

Qk 2v (x

- Xk)

-

(Qk+l

- Qk)

(5.38)

where xk ~ x ~ xa+~,l < k < m- 1, and 1 fx+~/2 Qs(x) = ~ ~=~/2 Q(h)dh,

(5.39)

for some appropriately chosen I. It is relatively easy to show that i) if Q~ > 0, so are Q(x) and Qs(x), ii) if Q~+I - Qk > 0, then ~Q(x) and ~Q8(x) are positive definite, and iii) and Q(x) and Q~(x) can be made arbitrarily close by choosing l small enough(see [18]). As seen below, Qs(x) is used in the proofs of stability as the Lyapunov matrix, but the control law is based on Q(x) only. Since the technique is based on strict inequalities, and ~he error between Q and Qz can be made arbitrarily small (without affecting the controller), Q~ and Q can be used interchangeably (e.g, Qs for proofs, and Q for computation and implementation). All relevant details can be found in [18,32]. Similarly, h)r r k ~ r ~ rk+l, we will use F(r)

±_-

= F~ + ±~, (Fk+i -- F~) ~rk+l - rk)

=

+

k k+l-

k)

(5.40)

-

and the controller will be of the form ~(~) = F(~)Q-~(~) but the inverse of the Lyapunovfunction will be the smooth version of Q in (5.39)). The following set of inequalities constitute sufficient conditions for the inequalities in Theorem5.2: Westart with a set of ri, i = 1,... ,N such that r~ = rmin ~ ~w and ri+~ > ri, for some large Cmax. Then we have 1 H~ <0 , Hk + ~(Ea + E[) <0, l < k < N ~ (5.35)

holds

(5.42)

Disturbance Attenuation

127

with Bounded Actuators

1 Mk >0 , Mk + [(Na + N[) > O 1
Hk=

rmax],

where

AQa + QkAT + B2F~ + F~ B2

~

B1 QaC~ + F[DT~

-~

o

0

-I

CIQ~ + D~2F~

Ek=

(5.43)

A(Qa+~ - Q~) + B2(G+~ 0 Cl(Qk+l - Qk) + D12(F~+l -

o

o 7 0 0

0

Mk= F~ r~

__

Nk = ½[rk(Qk+l - Q~¢) 0+ Qk(rk+l

rk)]

T

l
_/5-;

½[~+~-

Furthermore, the condition ~ < 0 can be satisfied Qk+~-Qk
F;+l

]

through (5.44)

The numerical algorithm thus consists of solving for Qa, F~, and "),a such that (5.37), (5.42), (5.43), and (5.44) hold for some a > 0 while minimizing N E~=~ ~, ~ > 0.

Remark 5.3. As before, it can be seen that 7i+~ _4 3q, wlog. The number of segments increase the computational burden significantly more for this case. Note, however, that Q does not need to be changed at every ra and can be kept constant over several segments (by setting Qd = Qj+~= ¯ ¯ .). See the examplebelow for motivation. The case of linear independence of parameters is the special case of setting all intermediate Qk’S in terms of any two (e.g, Q1 and QN).

128 5.7.2.

Jabbari Special

case: Constant Q

Often implementingthe scheduled controller discussed above is difficult, due to the need to obtain p on=line. If F is scheduled but Q is constant then one can store a finite number of gains and obtain a substantially reduced complexity controller (at a cost of some conservatism). For this, we start with a set of positive scalars 3’.1 > 7.2 > ... > 7.~ > ... > ~’*(m-1) 7*m, where 7*m= u_z~_. Next, a family of nested ellipsoids £(Q-1,7.~ ) is constructed. For each ellipsoid, a gain Kk = FkQ-1 is obtained such that guarantees an L2 gain for that ellipsoid (Tk) while not violating the saturation limit. It is thus relatively easy to obtain a controller of the form u = K(x)x according to K(x)=FiQ -1 if

Pi-1 < xTQ-lx=q(x)

<_pi,

i=l,2,...,m

(5.45)

where we define p0 = 0 (i.e, 7,0 = oc), flm+l ---- OO (i.e, 7.m+1= 0) with Q and Fi obtained from the modified set of inequalities of the main results of the previous section. For example, instead of (5.35), there will be inequalities of the form

AQ+QAT

+B2F~+F[B~ T B~ C10. + D~2F~

BI QCT~ -7~ 0

+F~D~ 0 -I

< 0 (5.46)

for k = 1, 2,... m, with Fo = F~ and Fm+l= Fro. Similarly, (5.36) would be replaced by rn inequalities in the form of

Lastly, (5.37) is used (with a constant Q and Fmused for Fr,...). It is easy to showthat this controller guarantees that the state does not leave £(Q-1,7*m) and L2 gain is less than 7t, where l corresponds to the largest index i of F~’s that is used (i.e., l = maxt{i : K(x) = F~Q-1}). Note that this is obtained while minimizingCmi~= Y~j=~fljTy, ~3~ > O. As shown in [32], this technique can be used to obtain scheduled controllers with reasonable performance, while minimizing the complications associated with scheduling. Furthermore, the weights ~, can be used m emphasize the performance in different ellipsoids. Remark5.4. The controller in (5.45) is discontinuous. To avoid difficulties associated with such a controller, we can use a continuous controller

Disturbance Attenuation

with Bounded Actuators

129

that is based on the F~’s above. For example, consider, k(x) = (Pi - q(x) Fi_~ P~ --

P~-I

q(X) -- Pi-1- ~,’~P: )(~)-I,

P~ - P~-I

(5.47)

where Pi-~ < xTQ-~x = q(x) <_ p~. The proof needs only trivial alterations. The controller in (5.47) results slightly higher gains and lower y, though its mainbenefit is its continuity. 5.7.3.

A Simple Example

Consider the simple system. A = -10 with B2T = [0 6] and

C1 :

-10

1]

(5.48)

[5 0].

0.4

0.2

Figure 4: Performance (7~ vs 7*) plot for the scheduled controller.

Figure 4, showsthe ~/~ vs 7. graphs for the this example. Recall that ~,. establishes the Size of the largest ellipsoid the state state vector enters (i.e, (5.30)). The segment length and the frequency with which Q is changed differ in two cases. In both the cases, the range of r is taken to be [1,

130

Jabbari

0.8

800

0.6 0.4 0,2

JV c~ 300

200

-0.4

1 O0 --0.80

2

4

Time

6

8

10

00

~

~

6

8

10

Figure 5: The disturbance record (left) and the change in p (right).

4° f

1000 800 600

__e o --400 --20 --600 --30

--400

2

4

6

2

4

Time

6

8

10

Figure 6: Controlled output and the input command.

726]. The figure corresponds to the case whenthe segments are of length 10 throughout. Three Q are used: the first change in Q is after first 10

Disturbance Attenuation

with Bounded Actuators

131

segments, the second change is after 50 more segments. Note that the graph tends to flatten out when Q is not changed. The dotted curve corresponds to the ideal case where the actual Wmaxis knownand a constant controller is designed for this particular Wmax (see [27, 28] for details). If one increases the frequency in which Q is changed (say to once every 5 to to segments), the two curves becomealmost indistinguishable. Next, we apply the disturbance record shown on the left hand side of Figure 5, normalized for a peak of Wmax= 1000. The actuator capacity is assumed to be Ulim ~-~ 1000. The results discussed earlier are used to implement a scheduled controller. For brevity, we only provide a sample of the details (see [32] for more information). The scheduling parameter p is shownin the plot on the right hand side of Figure 5. The resulting controlled output is shownin the left plot in Figure 6. The dotted line is the response of the high-gain technique in [27]. The control commandis shown on the right hand side of Figure 6. Note that the commandreaches the saturation limit on two occasions, and yields performance improvement over fixed controllers. This performance improvement becomes more pronouncedas the order of the system increases.

5.8.

Conclusion

Control techniques for dealing with actuator saturation are discussed that take advantage of recent progress in linear parameter varying control design. The saturation nonlinearity is handled through different techniques that simplify computation and implementation of the controller, for systems described with general linear parameter varying models. The main objectives include obtaining desirable performance guarantees that are functions of the actuator capacity.

References [1] J. Abedor, K. Nagpal, and K. Poolla. A Linear Matrix Inequality Approach to Peak-to-Peak Gain Minimization, International Journal of Robust and Nonlinear Control, 6 (1996) 899-927. [2] F. Amato, M. Corless, M. Mattei, and R. Setola. A Multi-variable Stability Margin in the Presence of Time-varying, BoundedRate Gains International journal of robust and nonlinear control, 7 (1997) 127-143. [3] G. S. Becker. Quadratic Stability and Performance of Linear Parameter Dependent S~stems, Ph. D. Dissertation, Department of Mechanical Engineering, University of California, Berkeley (1993). [4] J. M. Berg, K.D. Hammet,C. A. Schwartz, and S. S. Banda. An Analy-

132

Jabbad sis of the Destabilizing Effect of Daisy Chained Rate Limited Actuators, IEEE Transactions on Control System Technology, 4 (1996),17].176.

[5] S. Boyd, L. E1 Ghaoui, E. Feron, and V. Balakrishnan. Linear Matr:ix Inequalities in System and Control Theory, SIAMBooks, Philadelphia (1994). [6] C. Burgat, and S. Tarbouriech. Intelligent Anti-windup for Systems with Input Magnitude Saturation, Int. J, Robust Nonlin Cont., 8 ((1998) 1085-1100. [7] P. J. Campo, and M. Morari. Robust Control of Processes Subject to Saturation Nonlinearities, Computers Chem. Engng, 14 (1990) 343358. [8] M. Chilali, P. Gahinet, and C. Scherer. Multi-Objective OutputFeedback Control via LMIOptimization, Proceedings of International Federation of Automatic Control, D (1996) 249-254. [9] E. Feron, P. Apkarian, and P. Gahinet. "Analysis and Synthesis of Robust Control Control Systems via Parameter-Dependent Lyapunov Functions", IEEE Trans. on AC, 41 (1996) 1041-1046. [10] J. M. Shewchun, and E. Feron. High Performance Bounded Control Proceedings of 97-ACC, Albuquerque, NM(1997) 3250-3254. [11] P. Gahinet, and P. Apkarian. A Linear Matrix Inequality Approach to H~ Control, Int. J. of Robust and Nonlinear Control, 4 (1994) 421-448. [12] P. Gahinet, P. Apkarian, and M. Chilali. Affine Parameter-Dependent Lyapunov Functions for Real Parametric Uncertainty, IEEE Trans. on Auto. Control, 41 (1996) 436-442. [13] P. O. Gutman, and P. Hagander. A NewDesign of Constrained Controllers for Linear Systems, IEEE Trans. on Auto. Control, 30 (1985) 22-33. [14] W. M. Haddad, and V. Kapila. Anti-windup Controllers for Systems with Input Nonlinearities J. of Guidance, Control and Dynamics, 6 (1996) 1387-1390. [15] D. Henrion, G. Garcia, and S. Tarbouriech. Piecewise-linear Robust Control of Systems with Input Constraints, Proceedings of the 98ACC,Phil., PA(1998) 3545-3549. [16] T. Iwasaki, and R. E. Skelton. All Controllers for the General H~Co,~trol Problem: LMI Existence Conditions and State Space Formulas, Automatica, 30 (1994) 1307-1317.

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with Bounded Actuators

133

[17] F. Jabbari. Output Feedback Controllers for Systems with Structured Uncertainty, IEEE Transactions on Automatic Control, 42 (1997) 715719. [18] F. Jabbari. "Scheduled Controllers for Disturbance Attenuation of Systems with BoundedInputs," Submitted for publication. [19] V. Kapila, and W. M. Haddad. Actuator Amplitude Saturation Control for Systems with Exogenous Disturbances, Proceedings of the American Control Conference (ACC-98), Philadelphia, PA (1998) 1468-1472. [20] i. E. KSse, and F. Jabbari. Control of LPVSystems with Partly Measured Parameters, IEEE Transactions on Automatic Control, 44 (1999) 658-663. [21] i. E. KSse, and F. Jabbari. Control of Systems with Actuator Amplitude and Rate Constraints, Proceedings of American Control Conference (ACC-01), Arlington, VA, (2001). [22] H. L. Lee, and M. Spillman. A Parameter-Dependent Performance Criteria for Linear Parameter-Varying Systems, Proceedings of CDC (1997) 984-989. [23] Z. Lin, and A. Saberi. A Semi-Global Low-and-HighGain Design Technique for Linear Systems with Input Saturation - Stabilization and Disturbance Rejection, Int. Your. of Robust and Nonlinear Control, 5 (1995) 381-398. [24] Z. Lin. Global Control of Linear Systems with Saturating Actuators, Automatica, 34 (1998) 897-905. [25] I. Masubuchi, A. Kume, and E. Shimemura. Spline-Type Solution to Parameter-Dependent LMI’s, Proceedings of CDC(1998). [26] A. Megretski. L2 BIBOOutput Feedback Stabilization with Saturated Control, 13th IFACWorld Congress, D (1996) 435-440. [27] T. Nguyen, and F. Jabbari. Disturbance Attenuation for Systems with Input Saturation: An LMI Approach, IEEE Transactions on Automatic Control, 44 (1999) 852-858. [28] T. Nguyen, and F. Jabbari. Output Feedback Controllers for Disturbance Attenuation with actuator Amplitude and Rate Saturation, Automatica, 36 (2000) 1339-46. [29] A. Saberi, Z. Lin, and A. R. Teel. Control of Linear Systems with Saturating Actuators, IEEE Transactions on Automatic Control, 41 (1996) 368-378.

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[30] M. M. Seron, G. C. Goodwin,and S. F. Graebe. Control System Desi~in Issues for Unstable Linear Systems with Saturated Inputs, IEE Proc. Control Theory Appl., 142 (1995) 368-378. [31] G. Scorletti, J. P. Flocher, and L. E1 Ghaoui. ConvexOutput Feedback Control Designs with Input Saturation, In 2nd IFAC Symposiumon Robust Control Desgin, June (1997). [32] S. Srivastava, and F. Jabbari. Scheduled Controller for Disturbance Attenuation of Systems with Bounded Inputs, Proceedings of American Control Conference, ACC00), Chicago, IL, (2000) 735-739 [33] J. Yu, and A. Sideris. Ho~ Control with Parametric Lyapunov Functions, Systems and Control letters, 30 (1997) 57-69. [34] A. R. Teel. Linear Systems with Input Nonlinearities: Global Stabilization by Scheduling a Family of H~-TypeControllers, Int. Journal of Robust and Nonlinear Control, 5 (1995) 399-411. [35] F. Tyan, and D. S. Bernstein. Anti-Windup Compensator Synthesis for Systems with Saturation Actuators, Int. Jour. of Robust and Nonlinear Control, 5 (1995) 521-537. [36] F. Wu. Control of Linear Parameter Varying Systems, Ph.D. Dissertation, Department of Engineering-Mechanical Engineering, University of California, Berkeley (1995). [37] F. Wu., K. M. Grigoriadis, and A. Packard. Anti-windup Controller Synthesis via Linear Parameter-varying Control Design Methods, Proceedings of the 1998 American Control Conference, ACC-98,Philadelphia, PA (1998) 343-347. [38] F. Wu, and K. M. Grigoriadis. LPV-based Control of Systems witch Amplitude and Rate Actuator Saturation Constraints, Proceedings of the 1999 American Control Conference, ACC-99, San Diego, CA(1999) 3191-3195.

Chapter 6 LMI-Based Control of Discrete-Time Systems with Actuator Amplitude Iand Rate Nonlinearities H. Pan and V. Kapila Polytechnic

6.1.

University,

Brooklyn~ New York

Introduction

A commonassumption in manycontrol designs is that the system actuators can generate the necessary level of control effort for closed-loop stabilization and performance. However, most actuators have physical constraints that limit the control amplitude and rate simultaneously. In fact, an Air Force investigation [5] identified control surface rate saturation as the cause of the crash of the YF-22Aprototype fighter aircraft. Investigations have also concluded[19] that large pilot inputs in pitch and roll led to rate saturation in the elevons causing the crash of the Swedish Gripen prototype fighter aircraft. Therefore, the need for control schemes which ensure stability and performance despite the presence of amplitude and rate-limited control efforts is fairly evident. The control problem for linear systems with actuator amplitude satu1Researchsupported in part by the Air Force Research Lab/VAAD, WPAFB, OH, under IPA: Visiting Faculty Grant, the NASA/New YorkSpaceGrant Consortiumunder Grant32310-5891,and the MechanicalEngineeringDepartment,PolytechnicUniversity.

135

136

Pan and Kapila

ration has been a topic of considerable interest over the past several years. For continuous-time systems, an extensive literature is devoted to it (see, e.g., [3, 10, 12, 21, 25] and the numerousreferences therein). In additiol], for continuous-time systems, the control design problem with simultaneous actuator amplitude and rate saturation has recently received significant attention [15,17,18, 20, 26, 27]. Since most physical processes evolve naturally in continuous time, it is not surprising that the bulk of the actuator saturation control theory has been developed for continuous-time systems. Nevertheless, it is the overwhelmingtrend to implementcontrollers digitally. The references that address the actuator amplitude saturation control issue for discrete-time systems include [13,14, 16, 22]. In particular, for discretetime systems, Riccati equation-based global and semi-global stabilization techniques for actuator amplitude saturation control have been developed in [16, 22]. In addition, the application of an anti-windup actuator saturation control frameworkto discrete-time systems is given in [14]. In a recent paper [13], a Riccati equation-based global and local static, output feedback control design frameworkfor discrete-time systems with time-varying, sector-bounded, input nonlinearities was developed. Unfortunately, ho~vever, in contrast to the continuous-time systems, the problem of stabilizing discrete-time systems in the presence of control signal amplitude and rate saturation has received scant attention. In this chapter, we develop a linear matrix inequality (LMI) [1] formulation for state feedback and dynamic, output feedback control of discretetime systems with actuator amplitude and rate nonlinearities. The LMI formulation for the aforementioned problem is motivated by a desire to provide a simple and numerically tractable approach to actuator amplitude and rate saturation control. Specifically, since the LMI-basedfeasibility and optimization problems are convex (for which commercial software is available, e.g., [9]), the proposed actuator amplitude and rate saturation ¯ control scheme can be effortlessly implemented. In addition, we provide a direct approach to finding the stability multipliers that are paramount to reducing the inherent conservatism of the weighted circle criterion-based saturation control design. This chapter is organized as follows. Section 6.2 provides a statement of the state feedback stabilization problem and a sufficient condition for state feedback stabilization while Section 6.3 contains the controller synthesis. Section 6.4 states the output feedback stabilization problem with the controller synthesis being provided in Section 6.5. Twoillustrative numerical examples are given in Section 6.6. Finally, concluding remarks are presented in Section 6.7.

Amplitude and Rate Saturation Control

137

Nomenclature ()~, ()-~,tr 0 I~,O~

-

_

~r, Nr, pr Z1

~ Z2,

-

Z1 < Z2

n,l,m,p,d,

-

nc, n~,5 -

x, u, y, z, xc A,B,C

-

Kx, Iiu

-

Ac, Bc, Cc El, E~u, Ez. q

-

M1, M2, M

-

M~.,M~,M.,H~ -

H

6.2°

real numbers,r × s real matrices, ]Rrx 1 transpose, inverse, trace r × r identity matrix, r x r zero matrix r × r symmetric, nonnegative-definite, positive-definite matrices Z2 _ Z1 ~ Nr,

z2

_ Z1 ~ pr;z1

,Z2

~ Sr

0<e 0 u (control amplitude) or v (control rate) m x m diagonal matrices diag[Mlu, M~v], diag[M2u, M2v], diag[Mu, diag[Hu, Hv]

State Feedback Control of Discrete-Time Systems with Actuator Amplitude and Rate Nonlinearities

In this section, we introduce the stabilization problem for discretetime, linear, dynamic systems with actuators containing a set ¯ of timevarying, conic sector nonlinearities. The goal of the problem is to determine an optimal, state feedback controller that stabilizes a given discrete-time, linear, dynamic system with actuator amplitude and rate nonlinearities Cq(q(k), k) E (where q r efers to control amplitude, u, or control rat e, v) and minimizes a quadratic performance criterion involving weighted state and control variables. The structure of ¯ is specified later in this section. These objectives are addressed by developing a nonlinear matrix inequality (NMI)that guarantees global asymptotic stability of the closed-loop system for all Cq(q(k), k) ~ andprovides a gu ar anteed bound on t he quadratic performance criterion. State Feedback Stabilization Problem. Given the nth-order, stabilizable, discrete-time plant with actuator amplitude nonlinearities

138

Pan and Kapiia

Cu(u(k),k) x(k+l)

Ax (k)-B¢~(u(k),k),

x( 0)=x0, k~

A;, (6

where u(k) ~ ]~m, determine a state feedback controller k-1

u(k) = u(O) + E Cv(v(s),

s),

(6.2)

v(k) = Kxx(k) + K~u(k),

(6.3)

where v(k) e Rmand Cv(v(k), k) ~ ~, k _> 0, that satisfies the following design criteria i) the zero solution x(k) =- oftheclosed-loop system (6.1)(6.3) is globally asymptotically stable for all Cq(q(k), k).~ (b, k ~ A/’, ii) the following quadratic, performance functional is minimized ~= sup EzT(k)z(k), Cq(’,’) ~ ~ k=o z(k) ~= Elx(k) + E2uu(k) + E2vv(k),

(6.4)

J(K)

p. z e R

(6.5)

Note that the feedback interconnection of (6.1)-(6.3) is shownin Figure 1 with actuator amplitude nonlinearities ¢~(u, k) and actuator rate nonlinearities Cv(v, k). The input amplitude and rate model shown in Figure 1 represents a software rate limiter that ensures that no rate commandsare sent to the actuators that are beyondtheir specified limits.

Figure 1: State feedback control with actuator amplitude and rate nonlinearities. To characterize the class ~ of time-varying, sector-bounded, memoryless actuator amplitude and rate nonlinearities the following definitions are needed. Let Mlq, M~q~ ]~m×mbe given diagonal matrices such that M~q= diag(Mlq~,..., M~q,~ ), M~q= diag(M~,..., M~q~), and Mq ~= M2q- M.tq is positive-definite with diagonal entries Mq,, i -= 1,..., m. Next, we define

139

Amplitude and Rate Saturation Control the set of allowable nonlinearities Cq(., .) ¢ =~ {¢q :Nm xN’-~]R m :Mlq~q~ <_¢q~(q,k)qi i = 1,...,m, q e ]Rm, k e Af}.

<_M2q~q~, (6.6)

Now, we provide a closed-loop NMIthat guarantees global asymptotic stability of the closed-loop system (6.1)-(6.3) for all actuator amplitude and rate nonlinearities Cq(., .) ~ gb. First, we decomposethe nonlinearity Cq(’, ") into linear and nonlinear parts so that Cq(q(k), k) = ~gqs (q(k), Mlqq(k). In this case, the closed-loop system (6.1)-(6.3) has a state-space representation

5:(0) = 5:0, k e N’, (6.7) (6.8)

2(k+1)

~(~) = where

= ~(~) , = MIvKx

MlvKu

= ~(~) + Lm ’

v;s(a(k),k) ¢~s(~(k)’k) ="[ ¢~s(V(~),~) = Om -Ira

’

=

k

K~

g~ ¯

In addition, the performance variable z(k) is given by z(k) =/)5:(k), where ~, A__ ~_,~ + E2vK, /~1 _-~ [El E2u], and K ~ [ Kx Ku]. Note that the transformed actuator amplitude and rate nonlinearities Cqs(’, ") belong the set ~s given by Rmx~Rm:O~qs~(q,k)q~ ~s =~ {¢qs: i=l,...,m, q~R ~, k~}.

q~qi,M2 qieR, (6.9)

In order to reduce conservatism within the synthesis framework presented below, we introduce a constant, diagonal, positive-definite scaling matrix Hq ~ ~mxmthat preserves the structure of the nonlinearities [12]. The following result provides the foundation for our state feedback controller synthesis framework. For the statement of this result, we define the notation -~. Ro ~ 2HM Theorem 6.1. Let 2m x 2m diagonal matrices M~ and M2 be given such that M~- M1is positive-definite. In addition, let an m x na matrix K and a scalar e, 0 < e < 1, be given. Suppose there exists a 2m x 2m diagonal, positive-definite matrix H and an na x na positive-definite matrix P satisfying <0" 6"lO)_~+~TP~ ( (H~--~Tp~)T ~TP~-P+eP+~T~H~-~TP~ [

Pan and Kapila

140

Then the function V(~:) ~Tp,~ is a Lyapunov fun ction tha t gua rantees that the zero solution ~(k) -- 0 of the closed-loop system(6.1)-(6.3) is globally asymptotically stable for all actuator nonlinearities Cq(., .) E ~. Furthermore, the performance functional (6.4) satisfies the bound J(5:0, K)

Proof. First note that since P is positive-definite, it follows that the Lyapunovfunction candidate V(~) ,~ Tp.~, ~ ~ ~n,~, is positive-definite for all ~ ¢ 0. The corresponding Lyapunovdifference along the trajectories ~(k), k ~ N’, of the closed-loop system (6.7), (6.8), is given

s(fi(k),k) (6.11)

k

Now, adding and subtracting 2~Ts((z(k),k)HM-l~)s(~(k),k) and 2~sT(~(k), k)H~5:(k), k ~ Af, to and from (6.11) and collecting terms yield

~s(a(k),

k)

HO - ~Tp~

" [~s(~(k),~(k) ]j + 2~ k ~ ~.

_~ +

(~(k), k)H[M-l~s(~(k), k)~(k)] (6.12)

Finally, using (6.10) we obtain AV(~(k))

< --~T(k)[eP ~T~]~(k) +2~(fi(k),k)H[M-l~s(~(k),k)

- fi(k)],

k e ~. (6.13)

Since eP is positive-definite, ~T~ i8 nonnegative-definite, and 2~/H[M-~¢s- ~] ~ 0 for all ¢q~ (., .) e ~s, it follows that AV(~(k)) < 0, k ~ ~, and V(.) is a Lyapunov function for the closed-loop system (6.7), (6.8). Hence, the zero solution ~(k) ~ 0 of the closed-loop system (6.7), (6.8) is globally asymptotically stable for all Cqs(’, ~ ~s. In a ddition, since, with Cqs (q(k), k) Cq(q(k), k) - M~qq(k), (6. 7), (6. 8) is equivalent to (6.1)-(6.3), global asymptotic stability of the closed-loop system (6.1)-(6.3) is immediate. " Next, to showthat the performance functional (6.4) satisfies the bound J(~o, K) < V(~0), note that (6.13) implies AV(~(k))

< --~T(k)~T~(k).

(6.14)

Amplitude and Rate Saturation Control

141

Now, summing(6.14) over k ¯ N" yields

sup

<-

Cq(.,.)e¢

(6.15)

Next, since ~(k) --~ 0 as k -~ co, where ~(k), k ¯ Af, satisfies obtain

(6.7),

J(~o,K) < V(5:(O))- V(5: (k))

= v( 0) = 2oTp2o.

(6.16)

Note that J(~0, K) < 20Tp20 = tr P2020T, which has the same form as T the H~ cost appearing in the standard LQRtheory. Hence, we replace ~o~o by DD~, where D ¯ Rno×a, and proceed by determining the controller ~ = tr DTPD.Next, in the spirit of [13], gains that minimize tr PDD ~Y(P, K) __Atr DTpDcan be interpreted as an auxiliary cost. Theorem6.1 provides an efficient computational approach for closedloop stability analysis when the controller K, scalar e, 0 < e < 1, and the sector-bounds M1, M2for input nonlinearities be(’, ’) ¯ ~P are given. Specifically, since (6.10) is an LMIin the variables H and P, one can efficiently determine the feasibility of (6.10) to establish the asymptotic stability of (6.1)-(6.3). In this chapter, however, we focus on extending Theorem 6.1 to design stabilizing feedback controllers for systems with actuator amplitude and rate saturation nonlinearities. Before proceeding, observe that (6.10) is an NMIsince it contains product terms involving and P, H. 6.3,

State Feedback Controller Synthesis for Discrete-Time Systems with Actuator Amplitude and Rate Nonlinearities

In this section, we present the main theorem characterizing state feedback controllers for discrete-time systems with actuator amplitude and ~ate nonlinearities. In order to state this result, throughout this section, we assume that a scalar e, 0 < e < 1, is given. In addition, we assume that M1 and M2are given 2m× 2mdiagonal matrices such that M2- M~is positivedefinite. For conveniencein stating this result, we define the notation =

~-

Omxn ’

=

-Ira

142

Pan and Kapila

so that f~ ~= YAYT - YBMluZT T, + ZZ T. ~ = WK ~- XZ (6.17)

~ = A + ZMlvK, [3 = YBXT T, - ZW

In addition,wedefinethe notationR=~ p-1 , S =~ KR,and/:/ = ~ H-1 for arbitrary P E ]?n~, K E ]I~ m×n~, and H E p2m. Theorem 6.2. Let 2m x 2m diagonal matrices M1 and M2 be given such that M2-M1 is positive-definite. Furthermore, let scalar e, 0 < e < 1, be given. Suppose there exists an na × na positive-definite matrix R, an m x na matrix S, and a 2m × 2m positive-definite matrix/:/satisfying

I

Zll"

Z12 ]

Z22

(6.18)

< 0,

where T (WS

-k

Zll=/~

xzTR)

On~× 2rn

_2M-~/:/

Z12 z~

-0.5R

On~>:p

= [ %×,~o

-I~

Z22/%

In addition, let P, K, and H be given by -~ P ,= R

-~, K=SR

H =/:/--1.

(6.19)

Then P, K, and H satisfy (6.10) and the zero solution x(k) =- ofthe feedback interconnection of discrete-time, linear, dynamic system with input nonlinearity Cq(., .) ~ ~ given by (6.1)-(6.3) is globally asymptotically stable for all Cq(., .) ~ ~. Finally, for all Cq(., .) ~ ~, the auxiliary ~(P, K) satisfies ,~(P,K)

< tr

(6.20)

where Q ~ ]I ~d is such that the LMI variable R E P~ satisfying additionally satisfies DT ] > O’D

(6.18)

R (6.21)

[ Q

]

"

Amplitude and Rate Saturation Control

143

Proof. First, form ~(6.10)~"T, with ~ =~ diag (R,/:/),

to obtain

~ (ws + xzrR - fI~TR"~)

(ws + xzrR

-2M-l[-I

+ [-I~TR-I[~[I

(6.22)

where ~ ~ -~R + Z MIvS and/~ =~ ~IR + E2vS. Now, note that (6.22) can be equivalently written as T (WS + xzTR) - 2M- 1/:/+

(WS + xzTR)

+E~TR-~Z2 + E~R-~E1< 0,

(6.23)

whereE1 = A [~ 0,~×2m]andE~_- A ithR[0n~ l>0, -/~/:/].Next, w it follows that (Fq - E2)TR-I(Fq- F~2) _> 0. Thus, the last two terms (6.23) can be bounded as E~TR-1E2+ E2TR-~E~_< E~TR-~E~+ E2TR-1E2. Now, for given scalar e, 0 < e < 1, and 2m × 2rn diagonal matrices M; and M~such that M2- M~is positive-definite, the existence of R S ~ ]~m×n~, and /?/ ~ ]~m satisfying^(6.23) can be guaranteed by the existence of R ~ pn~, S ~ ~m×~,and H ~ ]~2m satisfying (WS + XZTR) T ] <0.

2~ITR- I ~ - R + eR + ~T ~ (WS + XZTR)

-2M-~[t

+

(6.24) By a repeated application of the Schur Complement[1] on (6.24), it follows that (6.24) is equivalent to (6.18). This proves that the existence R ~ ]?’~, S ¯ ]~m×n~,and ~ ~ F~msatisfying (6.18) is sufficient for the existence of P ~ ~n~,, K ~ ~mx~, and H ~ ~2m satisfying (6.10). Hence, for the case whenCq(., .) ~ ~, (6.18), (6.19) provide a sufficient condition for the global asymptotic stability of (6.1)-(6.3). Next, to show that ~(P, K) satisfies inequality

the bound (6.20), we consider

Q > DTpD,

(6.25)

which yields (6.20). Using the Schur Complement[1] and (6.19), it follows that the existence of Q ¯ ]~d and P ¯ l~n~ satisfying (6.25) is equivalent

144

Pan and K~pila

the existence ofQ E ]pd and R E I~n~ satisfying (6.21). Thus, for all actuator amplitude and rate nonlinearities Cq(., .) ~ ~, to minimizethe performance bound (6.20), we consider the LMI minimization problem: minimize tr subject

to R ~ pn~,

/~ E p2m, Q E pd,

and S ~ ~mxn~ satis~ing

(6.18)

and (6.21). [] Next, we define ~b C ~ and consider the case such that the input nonlinearity is time-invariant, i.e., ~q(q, k) = ~q(q), and ~q(q) is contained in ¯ for a finite range of its argument q as expressed below Cq ~ ~b ~ {¢q:

~m ~ ~m : Mlq~q~

~ Cq~(q)qi

~ M2q~q~,

i = 1,...,m},

~i ~

qi ~ ~’ (6.26)

where ~ < 0 and ~i > 0, i = 1,..., m, are given and correspond to the lower and upper limits, respectively, of q~. In this case, since Ca(.) ~ ¢ holds only locally, it follows that the results of Theorem6.1 and Theorem6.2 will also be valid only locally. Next, we present a frameworkfor constructing an est~ate of the domainof attraction for local stabilization of systems with actuator amplitude and rate nonlinearities. Thus, for i ~ {1,..., 2m}, we define ~ ~ max [-~,~] and X~ ~ {~ e Rn~ : ~ ~ ~}, where ~ is the ith row of ~, ~i and ~i, i ~ {1,..., 2m}, correspond to the upper and lower limits, respectively, of gi. In addition, we define X ~~i=1 :m Xi and g ~{~ e R~ : ~Tpi < 1/% ~ > 0}, where P ~ F~- satisfies (6.10). Now, defined by ~A ~ {~ e ~n~ : ~Tp~ <

1/% 7 > 0},

(6.27)

is an estimate of domainof attraction if ~ c X. Next, using Lemma 2 of [7], it follows that ~ C X if and only if

7ui is satisfied, where Li is the ith row of a 2mx 2m identity matrN. Since (6.28) is an LMIin the variables P, K, and ~, the feasibility of (6.28) be efficiently determined using convex numerical algorithms. In addition, minimizing7 while seeking the feasibility of (6.28) enlarges the estimate the domain of attraction. Finally, note that by forming ~(6.28)~ 7, where ~ ~ diag (R, 1), and using (6.17) and (6.19), (6.28) can be equivalently written as

Amplitude and Rate Saturation Control

145

Remark 6.1. A key application of Theorem 6.2 isthe case in which Cq (q) represents a vector of time-invariant, actuator amplitude and rate saturation nonlinearities. Specifically, let Cq(q(k)) = [¢ql(ql(k)),..., Cq,,(qm(k))] T, where Cq~(qi(k)), k >_ O, {1,. ..,m}, is c har acterized by

=

Iq ( )l

Cq~(qi(k)) = sgn( q~(k)), [q~( k)l > aq ,. (6.30 Now, suppose there exists a 2m x 2m diagonal, positive-definite matrix H and an na x na positive-definite matrix P satisfying (6.10) with given mxna matrixK, scalar e, 0 < e < 1, M~q = 0, and M2q = I. Then, with q replaced by u and v, (6.6) captures control amplitude saturation and control rate saturation, respectively. In this case, since Cq(.) E Theorem6.2 can be used to guarantee global asymptotic stability of the closed-loop system (6.1)-(6.3) for alt Cq(.) satisfying (6.3o). Alternatively, suppose there exists a 2mx 2m diagonal, positive-definite matrix H and an na x na positive-definite matrix P satisfying (6.10) with given m x matrix K, scalar e, 0 < e < 1, Mlq > 0, and M2q = I >_ Mlq > 0. Then, with q replaced by u and v, (6.26) captures control amplitude saturation and control rate saturation, respectively. In particular, with Mlq > 0, take -qi --~ --qi -Mlqiaq’ , i =1, ... ,m,in (6.26).In this case,sinceCq(.)~ Theorem6.2 and (6.27) can be used to guarantee local asymptotic stability of the closed-loop system (6.1)-(6.3) for all Cq(.) satisfying (6.30) guaranteed domain of attraction. Next, we present a numerical algorithm for the state feedback control of discrete-time systems with actuator amplitude and rate nonlinearities. The LMI formulation of Theorem 6.2 is used to exploit the computational advantage afforded by the convex formulation of LMI-based optimization problems. The basic structure of the numerical algorithm used is given below. Algorithm 6.1. To design a static, state feedback controller for discretetime systems with time-varying, sector-bounded amplitude and rate nonlinearities, carry out the following procedure: Step 1.

Begin with some initial values of M~u, M2u, Mlv, and M2v. In case ¢(., .) G ~, minimizetr Q subject to LMIs(6.18) and (6.21) in variables R G JPn", /2/ ~ ~2m, Q E ]~d, and S ~ 1Rmxn~. In case, ¢(.) G Ob, solve the problem of minimizing #7 + (1 - #) tr Q, # e [0, 1],

(6.31)

146

Pan and Kapil~ subject to LMIs(6.18), (6.21), and (6.29) in variables t:I E ]~2rn, Q E ]~d, S E ]~mxn.., and ? > 0.

Step 2.

Compute P, K, and H using (6.19). Now, vary Mlu, M2u, Mlv, and M2vto represent larger sector nonlinearities, then repea~t the above procedure (step 1) until feasible solutions are found for the target values of MI,, M2u, Ml,, and M2v, or until no feasible solution is found.

Note that (6.31) involves a convex combination of two scalar costs. varying # ~ [0, 1], (6.31) can be viewed as a scalar representation of multi-objective cost (see, e.g., [11] and the references therein). By setting # = 1, we obtain th~ problem of maximizing an estimate of the domain of attraction without regard to the performance bound (6.20). Alternatively, setting # = 0, disregards the problem of enlarging estimate of the domain of attraction and minimizes the performance upper bound (6.20). The practical value of this formulation is the case # ~ (0, 1) in which the optimization problem involves a trade-off between the performance bound (6.20) and a measure of the domain of attraction estimate 1/% 6.4.

Dynamic Output Feedback Control of Discrete-Time Systems with Actuator Amplitude and Rate Nonlinearities

In this section, we introduce the problem of dynamic, output feedback control of discrete-time, linear systems with actuator amplitude and rate nonlinearities. The goal of the problem is to determine a strictly proper, optimal, dynamic compensator (Ac, Be, Co) that stabilizes a given discretetime, linear, dynamic system with actuator amplitude and rate nonlinearities Cq(q(k), k) ~ andminimizes a qu adratic perf ormance crit erion involving weighted state and control variables. These objectives are addressed by developing an NMIthat guarantees the global asymptotic stability of the closed-loop system and provides a guaranteed bound on the quadratic performance criterion. Dynamic Output Feedback Stabilization Problem. Given" the nt~-order, stabilizable and detectable, discrete-time plant with actuator amplitude nonlinearities ¢~(u(k), x(k + 1) = gx(k) - B¢~,(u(k), k), y(k) = Cx(k),

x(0) -- x0,

k ¯ Af, (6.32) (6.33)

whereu( k ) e ]~m, y(k ) ~ determine an ntch-order, lin ear, tim e-invariant,

Amplitude and Rate Saturation Control

147

dynamic compensator xc(k + 1) = Acxc(k) + Bey(k),

~(k)= cexc(k),

(6.34)

(6.35)

k-1

u(k) = u(O) + ~ ev(V(S),

(6.36)

that satisfies the following design criteria i) the zero solution of the closedloop system(6.32)-(6.36) is globally asymptotically stable for eq(q(k), k) E ~5, k E Af, and ii) the following quadratic performance functional is minimized J(Ae,

Bc,Ce)

su p ~z T(k)z(k). eq(’,’) ~ ~’k=o

(6.37)

Next, we provide an NMIthat guarantees global asymptotic stability of the closed-loop system (6.32)-(6.36) for all eq(., .) ~ ~. Note that feedback interconnection of (6.32)-(6.36) can be represented as shown Figure 2. Using a similar procedure as in Section 6.2, the closed-loop system is given by

~(k+ 1) = A~(~) - ~)~s(a(~), ~(k) = ~(~),

&(0) = 5:0, k ~ N’, (6.38) (6.39)

where

In addition, the performance variable z(k) is given by z(k) =/~:~(k), where ~ ~

[

/~1

E2vCc

].

The following result provides the foundation for our dynamic, output feedback compensation framework. For the statement of this result, we define the notation/~ Theorem 6.3. Let 2m x 2m diagonal matrices M~ and M2 be given such that M2- M~is positive-definite. In addition, let (Ae, Be, Ce) and scalar e, 0 < ~ < 1, be given. Suppose there exist a 2mx 2mdiagonal,

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Pan and Kapila"

Figure 2: Dynamic output feedback control with actuator amplitude and rate nonlinearities. positive-definite

matrix H and an ~ x ~ positive-definite matrix/5 satisfying

<0" (6"40)_Ro [ ~Tpl, p + ~/5+/~HO -- ~Tpl (H~--~Tpl)T] ~-~Tp~

Then the function V(9) : }TP9 is a Lyapunov function that guarantees that the zero solution 9(k) : 0 of the closed-loop system (6.32)-(6.36) globally asymptotically stable for all actuator amplitude and rate nonlinearities Cq(., .) E ~. Furthermore, the performance functional (6.37) satisfies the bound J(~o, Ac, Be, Cc) < V(~0). Proof. The proof is analogous to the proof of Theorem 6.1 with the closed-loop dynamics(6.7), (6.8) replaced by (6.38), (6.39). [] Note that, as in Section 6.2, it follows that J(5:0, Ac, Next, J(~0, Ac, Be, Co) < tr ~0T/5~0: tr /5~0~:0T, which has the same form as the H2 cost appearing in ,the standard LQGtheory. Hence, we replaceS:°~°Tbyb~T’where/)/x [= BoDeD1 ] Dl E Nn~Xd, D~ E Nlxd, an d, D~D~> 0, and proceed by determining the controller the auxiliary cost ~7(/5, Ac, Bc, Co) ~ tr 6.5.

gains that minimize

Dynamic Output Feedback Controller Synthesis for Discrete-Time Systems Actuator Amplitude and Rate Nonlinearities

with

In this section, we present our main theorem characterizing dynamic, output feedback controllers for discrete-time systems with actuator amplitude and rate nonlinearities. In order to state this result, as in Section 6.3, we assume that a scalar e, 0 < e < 1, and a 2mx 2m diagonal, positivedefinite matrix H are given. In addition, we assume that M1and M2are

Amplitude and Rate Saturation Control

149

given 2mx 2m diagonal matrices such that M2- M1is positive-definite. For the remainder of this section, we assume that nc = na. For convenience in stating the main result of this section, recall the definitions of W, X, Y, Z and define the notation

: 0m×n ~m ,Ba=

0m--~m ,Ca= [02m×nX],

so that T BcCY

’

A¢

wcc]. (6.41)

Ono x 2m ’

Next, without loss of generality, consider the following partitioning of and ~-1

where/~, ~ E ]?n‘‘. In addition, we define =

~r T 0~

’

=

On~ /l:/T

.(6.43)

Using/5/5-1 = In, it now follows that /5II1 = II2.

(6.44)

Witha slight modification of [4,8], we define a changeof controller variables as follows AK ~= 2~lAci~ T + i~IBcCyT~ + ~ZM~vCcI~T + ~Aa~, r. BK ~ ~Bc, Cg ~Cc~ (6.45) By defining the variables a

=

T-

(6.46)

aS in [4, 8], Weobtain the identities A~

~Aa + BKCY T ’

~Ba ’

~ = [ c~+wc~ Ca ], ~= [ k~+E,.C~ ~ ], D = ~D~ --k

BKD2 ’

In,,

~ "

Pan and Kapila

150

O,/~, D, and t 5 are aitine

Before proceeding, note that the variables fi,,/~, in (/~, :~, AK, BK, CK). Finally, we define Oip_lO~T, i ~ {1,...,

m}

, i e {m+1,...,~m} V8 ~-- min {min(~ +, ~-)}, i=l~...~2m DA=~

{~R~:V(~)
~<0~<~,_

_ .(6.48)

where 0~ is the ith row of 0, ~ ~ ~ + ~M~0, and ~ e R~xe, ff > 0, satisfies (6.40) for a given compensator(A~, Theorem 6.4. Let 2m x 2m diagonal matrices M~ and M~ be given such that M2-M~is positive-definite. Furthermore, let a 2mx 2mdiagonal, positive-defipit~ matrix H and a scalar e, 0 < e < 1, be given. Suppose there exist R, S ~ ~n~ and (AK, B~, Cg) satisfying 211

212

]

(6.49)

215 222 < 0, where -(1 - e)15 211 ~

=

A

-0.515

HO

02m×e

0p×~

-Ip

OT H ]

O~

~T

-Ro

fit 02~×~

"

In addition, let P and (Ac, Be, Ce) be given by /3 = Yi2ii71, -T, Ae = ~-I[AK - BKCY~ - [~ZMI,~CK - ~tAa~]I~ Be = ~-IBK, I~Cc T. = CK

(6.50 (6.51) (6.52) (6.53)

Then/5 and (Ac, Be, Cc) satisfy (6.40) and the zdro solution 2(k) = 0 of the feedback interconnection of linear system with input amplitude and rate

Amplitude and Rate Saturation Control

151

nonlinearities Cq(., .) ¯ 9 given by (6.32)-(6.36) is globally asymptotically stable for all input amplitude and rate nonlinearities Cq(., .) E 9. In addition, if Ca(.) ¯ 95 then the zero solution ~(k) -= 0 of the closed-loop system (6.32)-(6.36) is locally asymptotically stable ~)A defi ned by (6.48) is a subset of the domainof attraction Of the closed-loop system. Finally, for all ~)q(.,-) ¯ 9, the auxiliary cost if(/5, Ac, Be, Ce) satisfies if(P,

Ae, Be, Ce) < tr Q,

where Q ¯ pd is such that the LMIvariable/~, (6.49) additionally satisfy

(6.54)

~ ¯ pna and BKsatisfying

(6.55) [{~DT] D

Proof. First,

> /~

0.

note that (6.40) can be equivalently written ~TH] -k ~2 + ~/5~’l < O’ (6.56)H~

[ 2‘T~2‘-- [9 + e[~ + -Ro [~ ~-

~T/5~

where ~l =a [ 2‘ 0~×m ] and ~__a [ 0~ -/~ ]. Next, with/5 > 0, it follows that the last two terms in (6.56) can be bounded as ~T/5~ ~T/3~ <_ ~T/5~l + ~/5~2. Now, for given scalar e, 0 < e < 1, 2m × 2m diagonal, positive-definite matrix H and 2m × 2m diagonal matrices M~ and M2such that M2- M1is positive-definite, the existence of/5 ¯ ]?e and (Ae, Be, Cc) satisfying (6.56) can be guaranteed by the existence of/5 and (Ae, Be, Ce) satisfying < 0. (6.57) H~

-R0 +

By a repeated application of the Schur Complement[1] on (6.57), it now follows that the existence of/5 ¯ l? e and (Ae, Be, Ce) satisfying (6.57) equivalent to the existence of/5 ¯ F~ and (Ae, Be, Cc) satisfying

< o. (6.58)

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Pan and Kapila

Next, it follows by forming T(6.58)TT, where T 2 diag ( H~T, Yi~T, I2m, HT~, Ip ), using (6.41)-(6.46), and after some algebraic manipulation that existence of/5 ~ l?~ and (Ac, Be, Co) satisfying (6.58) is equivalent to existence of/~, ~ ~ FTM and (An, B/~, C/~) satisfying (6.49). Specifically, the aforementioned equivalency statement, the necessity part of the proof is a direct consequence of Lemma 4.2 of [4] while the sufficiency part ibllows by construction. See Theorem .4.3 of [4] for a similar proof. This ^ ^ proves that the existence of R, S ff ]?n~ and (AK, BK, C~c) satisfying (6.49) is sufficient for the existence of/5 E ]?e an~ (A~c, Be, Co) satisfying (6.40). Note that (6.49) is an LMIin’the variables R, S, AK, B~, CK. In addition, note that having determined the feasibility of (6.49) for /~, ~ E Fn~ a~nd (AI~, BK, CK), one can construct/5, Ac, Be, Cc using (6.50), (6.53). Specifically, with/5/5-1 = I~ and/5 > 0, it follows that 2~//~ -T = In~ - ~ is nonsing~lar. Now,as in [4, 8], it follows that it is always possible to compute square, nonsingular matrices ~ and/~, for example, via the Schur decomposition, singular-value decomposition, etc. Next, it follows that II~ and 112 defined by (6.43) are nonsingular. Now,(6.50), (6.53) are obtained from (6.44)~, (6.45); For further details concerning the numerical computation of Mand N, see [4, 8]. As in Section 6.3, it now follows that for the case whenCq(., .) ~ (I), (6.49)-(6.53) provide a sufficient condition global asymptotic stability of the zero solution :~(k) ~- 0 of the closed-loop system (6.32)-(6.36). In addition, for the case whenCq ~ (I)b, (6.49)-(6..53) provide a sufficient condition for local asymptotic stability of the zero solution 2(k) -- 0 of the closed-loop system (6.32)-(6.36), and 7:)A defiued by (6.48) is a subset of the domainof attraction of the closed-loop system. For further details concerning the construction of a subset of the domain of attraction for the closed-loop system (6.32)-(6.36) ~)A defi ned by (6.48), see [12]. Next, to showthat ~7(/5, Ac, Bc, Cc) satisfies the bound (6.54), we consider the inequality (~

>

~T/5~,

(6.59)

which yields (6.54). Using the Schur Complement[1] and (6.46), it follows that the existence of ~ ~ ]?d, /5 ~ p~ and B~ satisfying (6.59) is equivalent to the existence of (~ ~ pd, /~, ~ ~ pn~, and BKsatisfying (6.55). Thus, for all Cq ~ ¢, to minimize the performance bound (6.54), we consider the LMI minimization problem: minimize tr Q subject to and (AK, BK, CK) satisfying (6.49) and (6.55). Remark 6.2. It is important to note that the estimate of the domain of attraction :DAgiven by (6.48) for the closed-loop system (6.32)-(6.36) predicated on open Lyapunovsurfaces. See [12] for further details.

153

Amplitude and Rate Saturation Control

l=temark 6.3. The conservatism of the bound ~TA~t2 + Q2TA~I~ Q~TA~I+ ~t~TAf~2,(with A > 0) used in the proofs of Theorem6.2 and Theorem 6.4 can be reduced by considering (aft1 --a-lf~2)TA(a~tl -c~-1f~2) which yields ~ITA~2+ f~2~A~l _< a2~ITA~I + a-2~2TA~2, where a is any scalar. It is also possible to introduce scaling matrices instead of the scalar a for further reduction in the conservatism of the above bound. However, note that introduction of scaling variables (e.g., a) gives rise to additional parameters in the matrix inequalities (6.18) and (6.49) and converts to NMIs. This leads to additional computational complexity and requires iteration between the controller determination and the scaling parameter determination. Remark 6.4. Note that even though at the beginning of this section we set nc = na, analogous to [4], a similar procedure as in Theorem6.4 can be used for the design of a reduced-order controller. See [4, 24] for further details. For the dynamic, output feedback control problem, following the approach of [6] and [23], we decomposethe problem of feasible stability multiplier determination (matrix H) and optimal control design (matrices Ac, Be, and Cc) into two LMIsubproblems. This enables us to exploit the computational advantage afforded by the convex formulation of the LMI-based feasibility and optimization problems. The basic structure of the numerical algorithm used is as follows. Algorithm 6.2. To design a dynamic, output feedback controller for discrete-time systems with time-varying, sector-bounded amplitude and rate nonlinearities, carry out the following procedure: Step 1.

Obtain an initial stabilizing controller for Om×nOm

’

Ira

, [ C On~×m

using, e.g., the linear quadratic Gaussian scheme. Step 2.

Beginning with some initial values of M~u, M2~, Mlv, and M~v and the current controller gains, solve the feasibility problem involving LMI(6.40) in variables ~ ~ P~ and H G p2m.

Step 3.

With matrix H obtained in step 2, minimize tr Q subject to LMIs (6.49) and (6.55) in variables ~, pna, Q ~ pd AKe Nn~xn~, BK ~ Nn~xl, and CK ~ Nmxn~. For ¢(.) ~b , us e ~A defined by (6.48) to obtain a subset of the domainof attraction of the closed-loop system.

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Pan and Kapila

Step 4.

6.6.

Compute P, Ac, Bc, and Cc using (6.50), (6.51), (6.52), (6.53). Now,vary Mlu, M2u, MIv, and M2vto represent larger sector nonlinearities, then repeat the above procedure (steps 2, 3) until feasible solutions are found for the target values of M~u,M2u, Mlv, and M2v, or until no feasible solution is found.

Illustrative

Numerical Examples

In this section, we provide two illustrative numerical examplesto demonstrate the proposed framework for designing actuator amplitude and rate saturation controllers. The resulting controllers are tested using the actuator amplitude and rate saturation model given in Figure 3, where Urs(k), k _> 0, denotes an amplitude and rate saturated control signal.

Figure 3: Amplitude and rate saturating

actuator model.

Example 6.1. Consider an open-loop unstable, x(k

+ 1) = 0 -1.02

1 ¢(u(k)),x(0)

discrete-time

system

= x0, k G Af,

k--1

u(k) = u(O) ~-~.¢,(v(s),s),

(6.61)

where the amplitude saturation nonlinearity Cu(U(k)), k E Af, is given by (6.26) with q = u, i = 1, ~nd au~ = 10 and the rate saturation nonlinearity ¢~(v(k)), k ~ ~, is given by (6.26) with q = v, i = 1, and a~ = performance wriable z in (6.5) is given z(k)

= 0 0 x(k) 0 0

v(k).

(6.62)

Next, we choose the design parameters D = diag(I2, 0), p = 0.2, and the target M2u = M2v = 1.0 and Mlu = M~v ~ 0.81, respectively. For this design data, we use Algorithm 6.1 to design a state feedback controller.

Amplftude and Rate Saturatfon Control

155

Finally, we computean LQRcontroller with the state and control weighting = EThEl and R2 =E2,E2, matrices R1 T , with E1 and E2v as given in (6.62). To illustrate the closed-loop behavior of the system let x0 = [ -10 5 ] T and u(0) ---- 0. For the controller designed using Algorithm6.1, the guaranteed domain of attraction is computed via the stability analysis subproblem (minimization of "~ subject to the LMIconditions (6.10) and (6.28)) and is given by /:)A = {x : V(x) < 1.3439 × 101°}. The corresponding [ 13.2190 -5.2696 ] Note that stability multiplier is given by H = -5.2696 18.0610 " V(xo) = 3.8899 × 101° so that x0 ~ I~A. However, it can be seen from Figure 4 that the controller designed by Algorithm 6.1 results in an asymptotically stable system while the LQRcontroller in the presence of an input amplitude and rate saturation nonlinearities drives the closed-loop system response to a nonzero equilibrium. Figure 5 provides a comparison of u(k), k E A/’, for the nominal LQRcontroller and Cu(u(k)), k E A/’, for the saturated LQRcontroller and the Algorithm 6.1 designs. In addition, Figure 5 provides a comparison of Aurs(k), k ~ Af, i.e., the saturated control rate, for the saturated LQGcontroller and the Algorithm 6.1 designs.

121 Theorem 7.2 Saturated LQR Nominal LQR

-I;

5 10 S~mplingnumber

15

Figure 4: Comparison of LQR, saturated (state response): Example6.1.

Theorem 7.2 Saturated LQR Nominal LQR

1° I

-6,)

5 10 Samplingnumber

15

LQR, and Theorem 6.2 designs

156

Pan and Kapila

~

8~

Theomrn 7.2 Saturated t.QF Nomina] LQR I

-6} -8~ 5 10 Samplingnumber

15

0

5

10 Samplingnumber

15

Figure 5: Comparison of LQR, saturated LQR, and Theorem 6.2 designs (control amplitude and rate): Example 6.1.

Example 6.2. Consider a valve-control system [2] in which the valve is constructed with a spring on the "flapper" so that if power is removed the valve closes. The control input is a torque applied to the flapper. The dynamics of the valve with control amplitude saturation is

1 Cu(U(t)),

-10-1 y(t) = [1 Discretization yields

0Ix(t).

(6.63) (6.64)

of the above dynamics with sampling period Ts = 0.5 sec

-2.4940-0.1173 x(0) = x0, k e Af,

x(k)-

0.2494

Cu(u(k)), (6.65)

1 0 Ix(k),

(6.66)

where the amplitude saturation nonlinearity Cu(u(k)), k EAf, is given by (6.26) with q = u, i = 1, and aul = 5. Now,assuming that the actuator is rate constrained, we have k-1

u(k) = u(O)+~-~.¢v(v(s),s),

(6.67)

where the rate saturation nonlinearity Cv(v(k)), k ~ is g iven by ( 6.26) with q = v, i -- 1, and avl = 1.5. The performance variable z in (6.5)

157

Amplitude and Rate Saturation Control given by

z(k)

Next, we select

0 0 x(k)+ 0 0

the design parameters

0 v(k). 0.01

(6.68)

I2 01×2 02 ] and D2 = D1 = 01×2

[01×3 0.01] and the target M2u = M2v = 1.0 and Mlu = Mlv = 0.84. We initialize our iterative design procedure with an LQGcontroller corresponding to the state weighting matrix R1 = diag(0.01,02), control weighting matrix R2 = 0.0001, plant noise intensity V1 = "D1DT~,and measurement noise intensity V2 = D2D~.For this design data, we use Algorithm 6.2 to design a dynamic, output feedback controller. To illustrate the closed-loop behavior of the system let x0 -- [ 0.01 2 ] T, u(0) = 0, and xc(O) = 03×1. For the controller designed using Algorithm 6.2, the guaranteed domain of attraction is computed via (6.48) and given by :DA = {2 : V(2) < 9.6132}. Note that V(20) = 1.1842 so 20 E :DA. It can be seen from Figure 6 that the controller designed by Algorithm 6.2 results in a satisfactory closed-loop response in the presence of control amplitude and rate saturation constraints. In comparison, an LQGcontroller for (6.65), (6.66), with the state weighting matrix diag(0.01,0), control weighting matrix R2 = 0.0001, plant noise intensity V1 = I2, measurement noise intensity V2 = 0.0001, and xc(0) = 02×1, the presence of input amplitude and rate saturation constraints drives the closed-loop system response to a nonzero equilibrium. Figure 7 provides a comparison of u(k), k e Af, for the nominal LQGcontroller and ¢u(u(k)), k ~ Af, for the saturated LQGcontroller and the Algorithm 6.2 designs. In addition, Figure 7 provides a comparison of Aurs(k), k ~ Af, i.e., the saturated control rate, for the saturated LQGcontroller and the Algorithm 6.2 designs.

6.7.

Conclusion

In this chapter, we developed a state feedback and a dynamic, output feedback control design methodology for discrete-time systems with timevarying, sector-bounded, actuator amplitude and rate nonlinearities using weighted circle criterion. Our results are directly applicable to systems with amplitude and rate saturating actuators and provide guaranteed domains of attraction. The technical difficulties associated with the design inequalities involving nonlinear terms in the decision variables were nullified by

158

Pan and KapiJa

developing LMI-based sufficient conditions for actuator saturation control. Furthermore, this design framework offers flexibility to obtain a trade-off between the closed-loop performance and the closed-loop stability regior,. A numerical algorithm was developed to exploit the computational advantage afforded by the convex formulation of the LMI conditions. Finally, the effectiveness of the design approach was demonstrated via two numerical examples.

0.6 0.5

Theorem7.4 Saturated LQG Nominal LQG

0.4 0.3 0.2 0.1 o -o.1 -0.2 -o.3 -0.~)4~

5

10 15 Sampling number

Figure 6: Comparison of LQG, saturated (output response): Example 6.2

20

25

LQG, and Theorem 6.4 designs

1.5 ---

1.5

Saturated LOG 1

1

,"’..

~ - ....................... 0.5

,’

0.5

-o.4 -1.-i!

V 5

10 1’5 Samplingnumber

2’0

25

5

10 15 Samplingnumber

20

Figure 7: Comparison of LQG, saturated LQG, and Theorem 6.4 designs (control amplitude and rate): Example 6.2

25

Amplitude and Rate Saturation Control

159

References [1] S. Boyd, L. E1-Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory, SIAM,(1994). [2] J. B. Burl. Linear Optimal Control, Addison Wesley Longman, Inc., (1998). [3] P. J. Campo, M. Morari, and C. N. Nett. Multivariable Anti-Windup and Bumpless Transfer: A General Theory, in: Proc. of American Control Conf., Pittusburgh, PA (1989) 1706-1711. [4] M. Chilali and P. Gahinet. H~ Design with Pole Placement Constraints: An LMI Approach, IEEE Trans. Automat. Control, 41 (1996) 358-367. [5] M. A. Dornheim. Report Pinpoints Factors Leading to YF-22 Crash, Aviation Week and’ Space Technology, pages 53-54. [6] E. Feron, P. Apkarian, and P. Gahinet. Analysis and Synthesis of Robust Control Systems via Parameter-Dependent Lyapunov Functions, IEEE Trans. Automat. Control, 41 (1996) 1041-1046. [7] A. Fischman, J. M. G. da Silva Jr., L. Dugard, J. M. Dion, and S. Tarbouriech. Dynamic Output Feedback under State and Control Constraints, in: EuropeanControl Conf., Brussels, Belgium(1997). [8] P. Gahinet. Explicit Controller Formulas for LMI-BasedH~ Synthesis, Automatica, 32 (1996) 1007-1014. [9] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali. Toolbox, The MathWorksInc., (1995).

LMI Control

[10] P. O. Gutman and P. Hagander. A NewDesign of Constrained Controllers for Linear Systems, IEEE Trans. Automat. Control, 30 (1985) 22-33. On the Gap Between H2 and [11] W. M. Haddad and D. S. Bernstein. Entropy Performance Measures in H~ Control Design, Systems and Control Letters, 14 (1990) 113-120. [12] W. M. Haddadand V. Kapila. Fixed-Architecture Controller Synthesis for Systems with Input-Output Time-Varying Nonlinearities, Int. J. Robust and Nonlinear Contr., 7 (1997) 675-710. [13] W. M. Haddad and V. Kapila. Static Output Feedback Controllers for Continuous-Time and Discrete-Time Systems with Input-Output Nonlinearities, EuropeanJ. Contr., 4 (1998) 22-31.

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[14] R. Hanus, M. Kinnaert, and J. L. Henrotte. Conditioning Technique: A general Anti-windup and Bumpless Transfer Method, Automatica, 23 (1987) 729-739. [15] R. A. Hess and S. A. Snell. Flight Control System Design with Rate Saturating Actuators, AIAA J. Guid., Contr., and Dyn., 20 (1997) 90-96. [16] P. Hou, A. Saberi, Z. Lin, and P. Sannuti. Simultaneous External and Internal Stabilization for Continuous and Discrete-Time Critically Unstable Linear Systems with Saturating Actuators, in: Proc. of American Control Conf., Albuquerque, NM(1997) 1292-1295. [17] P. Kapasouris and M. Athans. Control Systems with Rate and Magnitude Saturation for Neutrally Stable Open Loop Systems, in: Proc. IEEE Conf. on Dec. and Control, Honolulu, HI (1990) 3404-3409. [18] V. Kapila and W. M. Haddad. Fixed-Structure Controller Design for Systems with Actuator Amplitude and Rate Nonlinearities, in: Proc. IEEE Conf. on Dec. and Control, Tampa, FL (1998) 909-914. [19] J. M. Lenorovitz. Gripen Control Problems Resolved Through InFlight, Ground Simulations, Aviation Week and Space Technology, pages 74-75. [20] Z. Lin. Semi-Global Stabilization of Linear Systems with Position and Rate Limited Actuators, Systems and Control Letters, 30 (1997) 1-11. [21] Z. Lin and A. Saberi and A. R. Teel. Simultaneous Lp Stabilization and Internal Stabilization of Linear Systems Subject to Input Saturation: State Feedback Case, in: Proc. IEEE Conf. on Dec. and Control, Orlando, FL (1994) 3808-3813. [22] A. Mantri, A. Saberi, Z. Lin, and A. A. Stoorvogel. Output regulation of linear discrete-time systems subject to input saturation, in: Proe. IEEE Conf. on Dec. and Control, NewOrleans, LA (1995) 957-962. [23] T. E. Par~ and J. P. How. Robust H~ Controller Design for Systems with Hysteresis Nonlinearities, in: Proc. IEEE Conf. on Dec. and Control, Tampa, FL (1998) 4057-4062. [24] S. Scherer, P. Gahinet, and M. Chilali. Multiobjective OutputFeedback Control via LMI Optimization, IEEE Trans. Automat. Control, 42 (1997) 896-911. [25] F. Tyan and D. S. Bernstein. Anti-Windup Compensator Synthesis for Systems with Saturation Actuators, Int. J. Robust and Nonlinear Contr., 5 (1995) 521-537.

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[26] F. Tyan and D. S. Bernstein. Dynamic Output Feedback Compensation for Systems with Independent Amplitude and Rate Saturations, Int. Journal of Control, 67 (1997) 89-116. [27] C. Zhang and R. J. Evans. Rate Constrained Adaptive Control, Int. Journal of Control, 48 1988) 2179-2187.

Chapter 7 Robust Control Design for Systems with Saturating Nonlinearities T. Pare Malibu Networks, Campbell, California H. Hindi Stanford University, ’ Stanford, California J.

How

Massachusetts Institute

of Technology, Cambridge,

Massachusetts

7.1.

Introduction

Whena linear control system is operated in conditions which cause actuator saturation, it can exhibit nonlinear behavior such as local stability, finite disturbance rejection, and performance degradation. In this chapter we provide some tools, based on the Popov Criterion, which allow the designer of a control system to explore the following simple idea: does robustifying a linear controller to sector boundednonlinearities at the input improve its performance when operating in saturation? The idea is based on the well knownobservation that when the input to the saturation element is known to be bounded by some r > 0, then the saturator can be modeled locally

163

164

Pareet ai’. ~ sat(q) 1.

s.1

Figure 1: Saturation locally sector bounded as parameterized by r.

as a sector boundednonlinearity, see Figure 1. Perhaps the most fundamental analysis of a system with saturation is a prediction of regions of attraction. Early work applied absolute stability theory to address this question by isolating the nonlinearity, and casting the problem in the Lur’e-Postnikov framework [4, 36, 37]. As is typical in these works, a Lyapunovfunction is used to define regions in the state space in which energy is guaranteed to decrease, while the nonlinearity is bounded by a prescribed amplitude. With these conditions satisfied, it is then shownthat initial conditions in these regions will result in state trajectories that converge to the origin. Whenthe input to the nonlinearity remains within certain bounds, that is if Iq(t)l < r, Vt >_ 0, the saturation can then be treated as a memorylessnonlinearity, locally sector-bounded in 1 Using this observation, the early analysis was later [/~, 1], where/~-- 1 - ~. extended to produce better stability region estimates by utilizing the circle stability criteria [7,18]. 1 Morerecently, the local stability analysis for multiple nonlinearities using various Lyapunovcriteria, such as circle, Popov, piecewise quadratic, and parameter varying, was formulated in an LMI setting [9-12,14, 27,39]. The flexibility provided by the LMIframeworkallowed for a generalization of the stability guarantee to include performance as measured by external disturbance rejection and local £2-gain. These latter results, in particular those in [10, 11, 27] provide the analytical basis for the local control design algorithms described in this chapter. The practical importance of control design for systems using bounded input keeps this an active area of research [34]. Various linear, piecewise linear and nonlinear methods can be found throughout the literature. A Lyapunovapproach has been used to construct norflinear approximations to 1Seealso [15, pp. 407-419]for a detailed scalar example.

Robust Control for Systems with Saturation

165

linear, time-optimal control laws [33], while Kamarovproposed a nonlinear technique based upon an inner estimation of attainable ellipsoidal sets by solving a matrix differential equation [17]. Althoughthese nonlinear techniques explore the limits of performance possible using bounded control, their application is limited to relatively simple systems, and by the common assumption of state feedback. Low-and-high gain control (LHG)is another state feedback technique with a simple implementation, approaching the problem by first designing a low gain controller so as to avoid actuator saturation and thus widening the region of attraction for the system. Because the low-gain controller is intentionally conservative, resulting in sluggish transient performance, the design is then augmentedwith a high gain outer loop that can help speed up system response [29]. Piecewise linear LQcontrol (PLC)[38] is a less conservative state feedback technique in which the feedback gains are increased in a piecewise fashion as the state approaches the origin. While providing improved response, PLC lacks robustness to large uncertainties and the ability to reject large bounded disturbances. Recently, a state feedback design framework combining the LHGand PLC techniques in order to achieve a desirable mix of the robustness and performance offered by each was proposed in Ref. [21]. The PLC and LHGdesigns represent some of the latest attempts at achieving closed loop performance with bounded control. In other recent work, state feedback compensation for global stabilization with eigenvalue assignment and local/:2-gain performance has been achieved based on the solution to an algebraic Riccati equation (ARE)[31,32], see also [13]. AREbased approaches have also been applied to the more general output feedback case by utilizing state observers [19, 20], and to local LQGdesign by solving coupled Riccati equations [35]. As with these later approaches, this chapter introduces synthesis algorithms for output feedback control aimed at specific local performancecriteria. In particular, the new design methods build on the recent local saturation analysis in which the authors formulate bounds for regions of attraction, disturbance rejection and local £:2-gain performance, all based on the Popov stability criterion. The analysis is generalized to treat the multiple nonlinearity case, and because the results are posed in terms of LMIs, the bounds are readily computed by solving semidefinite~ programs[11,27]. Synthesis of local regions of attraction via state feedback based on the circle criterion was introduced in [27] and later extended to the case of output feedback [25, 28]. In these works, the stability region was defined in state space by the ellipse,

$p={x

[xTpx
P=pT>0

}.

(7.1)

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Pare et al.

Controllers were computed that attempt to maximize £p, by minimizing the trace of P (this minimizes the sum of the inverse squares of the semi-axes of £p, which tends to make the semi-axes large). The resulting controllers guaranteed that any initial condition x0 E ~p resulted in a stable trajectory, in the sense that x(t) ~ O. In [25, 28], the controllers were dynamic, of the same order as the given linear plant, and computed using an LM[ approach, The resulting stability regions are not invariant, but have a property referred to as pseudo-invariance [6], which means that state trajectories originating in the region mayexit but will eventually return as the state converges. Interestingly, a recent result [16] shows that this approach is futile! This is because the largest stability region (in the trace sense) derived from the linear analysis that considers the behavior of the states in the linear (nonsaturated) region only, turns out to be the same as the largest region based on. the nonlinear analysis using the multi-loop circle criterion. Thus the circle criterion does not help to increase the size of the region of attraction (in the trace minimization sense) in saturating control synthesis when compared with that resulting from the linear analysis. This chapter presents the analogous local stability design method based on the less conservative Popov criteria, using the analysis from [10, and builds this basic approach into two new design algorithms that yield dynamic compensation for closed loop robustness or local £2-gain perfor.mance for systems with actuator saturation [23]. In the first case, the controllers are designed to reduce the sensitivity of the closed loop system to external disturbances. This is accomplished with designs that guarantee stability in the presence of all external disturbances, w(t), that are bounded by a given level of energy, Ctrnax (i.e., Ilwl12 <_amaz). The secondalgorithm solves for controllers that attempt to minimizethe £2-gain across a performance channel. In general, the local performance and external disturbance rejection problems have competing objectives. That is, controllers with the best £2-gains may correspond to designs that will only tolerate relatively small external disturbances. In cases where both robustness and performance are critical, the analysis and design algorithms can be used in an iterative fashion to arrive at an optimal combination of the two objectives. This trade-off is illustrated with a simple numerical exampleat the end of the chapter.

2Applying Popovcriteria leads to regions of attraction definedby a constraint with both quadratic and nonquadraticcomponents and thus generalizes the region defined in (7.1).

Robust Control for Systems with Saturation

167

sat(’)

P

~

W

Z

U

Figure 2: Control system with saturation nonlinearity. The system within inner dashed line is original plant H augmentedwith lowpass filter L, while outer dashed-region is closed loop system. 7.2.

Problems

of Local

Control

Design

A typical control system that includes bounded, or saturating control inputs is depicted in the standard three block frameworkshownin Fig. 2. In the diagram sat(.) is the unit saturation function, H is an LTI plant that is to be controlled and K is an LTI controller that is to be designed. Note that the linear plant includes the block L, which is some strictly proper very high bandwidth low pass system included for technical reasons discussed later. Using standard robust control notation, the signal w is the disturbance input, and z is the performance variable. Systems operating in saturation can exhibit nonlinear behavior such as local stability, finite disturbance rejection, and performance degradation. These effects are analyzed and quantified in [10,11] with bounds that are computable by solving a convex optimization over a set of linear matrix inequalities. The analysis is continued here with the consideration of three basic engineering design problems. To simplify the development, the case of a single saturation nonlinearity is considered, although the results are readily extended to the MIMOcase. Let Umaxbe the maximumavailable actuator level (subsequently assumed normalized to 1) and let ~x/-~max bound the energy of input disturbances w that can be rejected by the system starting with

Pareet ai!.

168 zero initial conditions:

~ sup{~ I I1~11~ < ~, x(0)0, ,~%x(t)0},

(7.2)

where x is the overall state of the linear plant (consisting of H and L) and the controller K. Similarly, define the E2-gain (energy gain from w to z) 322 of the closed loop system, starting from a zero state as: ’~22 ’~ ~

[[zll2

sup

(7.3)

~(0)= Finally, let ~ be a region in the state space of H of initial conditions froln which the closed loop system is guaranteed to be brought back to zero with the finite actuator authority in the absence of disturbance:

v = { x ¯ Rn I x(0)¯ 9, =0 ~ x(t) -~0, ~st -~ ~ }. (7. Nowconsider following design problems, where we are given a set of design specifications and asked to design a controller K that achieves one of the following objectives:

(SR): (DR):

(EC):

Given

: H,

Um~x=l,w=0

Find

: K which maximizes

Given

: H, Uma×= 1,. 722 -< ~spec

Find

: K which maximizes

Given

: H, ~ffmax

Find

: K which minimizes 722

: 1, O~max ~> O~spec

In the (SR) case, we are asked to find a controller which maximizes,in some sense, the stability region; in (DR) we seek a controller which optimizes the closed loop disturbance rejection, while maintaining an £2-gain less than some value of 7s~ec; whereas in (EG) we require a controller which minimizes energy gain, while maintaining a disturbance rejection bound of at least O~spec.

It might appear as though we have created an artificial dependence between disturbance rejection and £~-gain in problems (DR) and (EG), since each of these is stated in terms of both C~ma~and 722, while in the global definitions (7.2) and (7.3), the two seem unrelated. In practice, however,

169

Robust Control for Systems with Saturation

solving for the optimal (EG) controller without requiring amax >_ O~spec can lead to closed loop systems that are not sufficiently robust to external disturbances. Of course, in the extreme case, an unconstrained (EG) controller designed by ignoring the effects of saturation, altogether will achieve the highest performance, but can result in a closed loop system highly sensitive to external disturbances. Similarly, controllers that provide good disturbance rejection do so by maintaining stability despite large levels of actuator saturation. 3 This relationship motivates the need for a systematic approach that enables the designer to strike a balance between these two objectives. A natural choice for a design variable then becomesthe saturation parameter, r, depicted in Figs. 1 and 2, that can be used to control the relative amount of nonlinear operation that a system will encounter when acted on by external disturbances, or when starting from some initial condition. The designer should expect better (EG) performance if operation is maintained "close to linear" by setting the parameter r not much larger than 14; while better disturbance rejection (DR) could be achieved by allowing higher values of r. This motivates the following local versions of the global metrics given in (7.2, 7.3, and 7.4) [11] that have explicit dependenceon r. First, consider the r-level local stability region :Dr as the maximum volumeset of initial conditions in the state space of H, for which the control never exceeds r, in the absence of disturbance, defined by, :Dr= xERn x(O)

E:Dr’w=-O~

Iq(t)l<_r,

Vt>_0

"

Similarly, the r-level local disturbance rejection is defined by the largest disturbance that results in r-bounded control response: a~nax zx sup{a I Ilwll~ _< ~, x(O) -- O, lim x(t) = O, Ilqll~ -< r}. (7.6)

Note that from (7.6), it follows that computinga~nax is equivalent to computing the r-level local E~-to-Eo~-g~infrom w to q: sup

"

-=

(7.7)

¯ (o)= 3This relationship example in §7.6 4Note that

between the (DR) and (EG) designs

r < 1 means that

the system

operates

is illustrated

linearly.

with a simple

170

Pare ¢~ ~.

and finally, the associated r-level local/:2-gain from w to z is defined as:

snp Ilzll Ilwll O~max < r Ilwll z(0)=

(7.8)

Note thatarea xr increases monotonically with r. Hence as r increases, the supremumin (7.7) and (7.8) is taken over larger sets. It follows that for r given closed loop system, °~max, r 7oe2 and 7~2 are all monotonically increasing functions of r, which tend to their global values for large r. Therefore, they will always produce less conservative bounds than their global coun.terparts. Thus it seems reasonable to recast problems (SR), (DR), (EG) in terms ofO~max,r7~2r and 7~’2. Note that, in contrast to the global case, there is nowa definite relationship amongO~max,r V~2,r V~2, and r.

7.3.

The Design

Approach

The local definitions above make it possible to talk about the local stability and performance of linear systems with saturation in a ~precise. way. Unfortunately, at this time, the exact computationof ~Dr,C~ma and xr , V~2for general linear systems with saturation such as Fig. 2 operating with r > 1 is still an open problem, and the same goes for the synthesis problems above. Therefore, in our actual design procedures for problems (SR), (DR), and (EG), ~wewill make use of estimates of these objects. Specifically, rm compute: Dr, an inner approximation to Dr;~max^r, a ax, lower boundon a (and hence an upper bound on 7~2, ~’~2), and ~2, an upper bound /:2-gain 7~2. These estimates are computed using LMI/BMItechniques, by applying the Popov criterion to the r-level sector model of the system which will be described in the following sections. Consider the design of a controller for problem (EG). Strictly speaking, this problem is a mixed/:2-gain and E2-to-E~-gain optimization problem. Such mixed norm multiobjective problems are, in general, not very easy to solve, even in the linear case. So one reasonable approach would be to simply start by trying a linear design, i.e., assuming that (EG) can solved with a controller that does not saturate, i.e., r = 1. One can then solve the single objective controller synthesis problem of computing a K which minimizes the ~/~-norm of the closed loop system using standard techniques. The closed loop system with K can then be postanalyzed to ensure that theO~ma O~spec constraint is met. If this is the case, then xl ~ the problem is solved. (Note that when the system operates linearly, it is possible to compute exactly ama×~ , 7~ and a maximumvolume ellipsoid contained in ~D~.)

Robust Control for Systems with Saturation

171

If theOZma constraint is not satisfied, then the r = 1 saturation xl ~ O~spec level disturbance rejection bound is too small. At this point, one might ask if it would be possible to increase C~m~xby allowing the system to saturate slightly, while possibly trading off some £2 performance. One can try increasing r to a value slightly greater than 1, and redoing the synthesis, this time computing a controller K which minimizes ~2 using BMIsynthesis. Equation (7.7) showsthat if the relative increase in the gain r for the new controller is smaller than the relative increase in r, then c~ax will increase. Then the closed loop system can once again be post^ are checked(conservatively) checking If not, then r can analyzed: LMItechniquesbyare usedifO~ma to computeamax^r constraints x r ~ O~spec ’ and the be increased once more and the process of BMIsynthesis and LMI postanalysis can be repeated, until either the LMI’s used in the computations becomeinfeasible, or the specification is achieved. Similar BMI-synthesis/LMI-postanalysis design methods can be proposed for (SR) and (DR). Such methods, while seemingly crude, can often do a good job at tuning an initial controller to satisfy somedesired specifications. A numerical example in §7.6 is used to demonstrate this design approach.

7.4.

System Model

Wewill now describe the models that we use for the components of Fig. 2. The linear plant H(s) shown is given by the dynamics:

H

:~p = z = y

:

Apxp Cpzxp

-~ + ~-

CpyXp

BpwW + Bpup DpzwW -~ Dpzup DpywW,

(7.9)

where it is assumed the matrix A may have unstable eigenvalues, and that the system is both observable and controllable. The control u is assumed to be filtered with the high bandwidth, lowpass network L(s): ~L

[~

&L

=

q

= L.

ALXL CLX

BLU

(7.10)

Withoutthis filter, the control signal feedthrough to the nonlinearity would significantly complicate the controller elimination in the synthesis in §7.5. The filter output q is subject to saturation p----sat(q).

(7.11)

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Pareet al.

dzn(.)

w

Y

Figure 3: Transformed system with deadzone nonlinearity. Transformation converts saturation into deadzonenonlinearity, and creates feedthrough path from lowpass system L(s) to linear plant H(s).

Wewill consider the design and analysis of proper, linear controllers K of the form ice = Acxc + Bey (7.12) K { u = Ccxc + Dcy. Following the analysis in Ref. [11], define the deadzone nonlinearity dzn(.) sat(.) + dun(.) =

(7.13)

and apply the loop transformation p = dzn(q) = -(sat(q)

-

(7.14)

which transforms the system in Fig. 2 to that in Fig. 3, which is the nom.inal closed loop system (i.e., with no saturation) perturbed by the dzn(.) nonlinearityfi The corresponding open loop plant G : (p, w, u) H (q, z, 5See also [2, pp. 230-31] for another analysis of saturation transformation which converts sat(.) into a dzn(.) nonlinearity.

employing this loop

Robust Control for Systems with Saturation

173

shownby the inner dashed line in Fig. 3 is defined by

G 8

Ap 0 0 Cpz

BpuCL AL CL DpzuCL

Cpy

0

Cq Cz

0 -Dpzu

0 0 Dpzw 0

(7.15)

Dqp Dqw Dqu Dzp D~w Dzu

This modelis used in the synthesis phase of the design procedure. Similarly, the closed loop plant, shownby the outer dashed line in Fig. 3, ~ : (/~, w) (q, z) An BcCy Cq C13

BuCc Ac 0 0

~ip

BcDyw 0 Dzw

(7.16)

J~ qp ~zp All ~-~ A+BuDcCy,M14 ~-~ Bw +BuDcDy~o,and C13 ~-~ Cz +CqDzp. This model is used in the analysis phase of the design procedure. The level r relates to the transformed system, Fig. 3, as the limit on the input q to the dzn nonlinearity. Nowif the input signal is limited such that [q(t)[ _< r, Vt, then the dzn is guaranteed to act as a sector bounded nonlinearity, in sector[0,~r], where ~r --- (1 - ~), as depicted in Fig. 4. Normalizing the sector to [0, 1] then results in a multiplication of the correspondingmatrices, Bp, Dqp, Dzp, Dyp, of system (7.15) by a factor of/~r, and similarly for the closed loop matrices defined in (7.16). In the next section, these scaled transformed matrices will be designated with superscript r (i.e., ~, etc.). where

7.5.

Design

Algorithms

Because the dzn(.) function is a sector bounded, memorylessnonlinearity, the system, Fig. 3, is in a form suitable for Popovanalysis [3]. In this section we present the analysis based on the Popov criterion to estimate

174

Pare et al. dzn(q) 1

q

p

Figure 4: Deadzone nonlinearity

and saturation

parameter r.

the stability regions, disturbance rejection capability, and local £2-gain performance for the closed loop system, and provide the corresponding (SR), (DR) and (EG) design algorithms. Note that each case is based on an sumed r-level being maintained, so that the local sector model described above is satisfied (depicted in Fig. 4). The (SR), (DR) and (EG) and design solutions presented in the following sections extend the LMI local analysis and synthesis work in [1,11, 25], and are based on the Popov Lyapunovfunction of the closed loop system state x: V(x) = xTDx + 2 E "~ i~l

dzn(cr) da,

(7.17)

JO

with ~ = ~T > 0, Ai > 0, and where ~q,i denotes the ith row Of the system output matrix, ~q. For brevity, we present only a summaryof results; the proofs are omitted but can be found in [10, 23]. 7.5.1.

Stability

Region (SR)

In this case, we set the disturbance w -- 0, and consider the simple objective of designing a controller K that maximizesthe region of attraction for the system (7.16). Adapting the general analysis presented in [11, 27] to the system 17.16) we have the following theorem, which defines a region of attraction/)r for the closed loop system in terms of a level set of the Lyapunov function (7.17). Here a level set is simply the region in state space which satisfies the inequality levsY(x) -= { x e Rn I V(x) _< e }.

(7.18)

Theorem7.1. An r-level region of attraction 7~r is given by the invariant set le~lV, where V is the Popov function obtained by solving the

Robust Control t~or Systems with Saturation

175

following convex optimization problem in the variables/5 = h = diag(/kl,..., Anq), and T = diag(tl,... ,tnq) [10,11]: minimize subject to

r eq, 1 >

Le, A>0, /5 j_o, T>0,

P>0,

/ for i = 1,...,

/

<0,

nq where the closed loop system matrices are defined in (7.16).

The corresponding control design is posed by the following corollary. Corollary 7.1. A controller which maximizes the region of attraction /)r defined by a set of matrices satisfying Thm. 7.5.1 is parameterized by the matrices R, S that solve the following optimization problem [23, 24]: minimize subject to (7.20)

for i = 1,...,nq, where MI~ : B~ + RATC~A + RC~T, N~2 = SB~ + ATC~A + C~T, and U± = ACqB~

±

D~p T ±’

with U± being the orthogonal complement of U, and the open loop system matrices defined in (7.15). 7.5.2.

Disturbance

Rejection

(DR)

The local disturbance rejection problem can be presented in a similar way. The theorem below computes an upper bound on the level of disturbance energy that a closed loop system can tolerate before instability for

176

Pare et al.

a given saturation level r. This theorem is then followed with the corre,sponding corollary for control design in terms of a semidefinite prograra solution. Theorem7.2. For system (7.16), an r-level disturbance rejection bound, area×, can be computedas [10, 11] r2

+ 2Au r ~(a) fo

&~nax = max

da,

(7.22)

wherefore~ch~ ~ [0,1],(P~,A~) is theoptimal valueof thefollowing convex semidefinite program in thevariables s~, minimize(1 - p)s~+ ps2 subject to

~ P>0,

~>0,

t

~

0

s>0,

M~ ~T ~ B~P

.

0,

(7.23) -2T

~T ~T

h

AC~B~ -I

~ O,

Corollary 7.2. A controller that maximizes the disturbance rejection is parameterized by the matrices R, S that solve the following optimization problem [23, 2@ minimize (1 subject to

--

p)S1 "~ #82

CI

A

R >_ 0,

> 0

I S >0, A>O, s>O, AR ÷ RAT M12 [ Bw

ATS

+ AA

N5 for i = 1,..., U±=

?~q,

-2T[

U.< 0

S B~ NI~ TA -I B~T Cq (’)~3 -2T

V±<0,

(7.24)

where ACqBu ± 0

, V± = I

Dv~w 0

± 0, I

(7.25)

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RoSust Control for Systems wfth Saturation

and N12, M12are defined as in (7.20).

An extension of Theorem 7.2 and the corresponding design corollary to the case of multiple nonlinearities is possible using the analytical extension given in [10, 11]. Wenext consider the case with an output performance variable z(t). 7.5.3.

Local £2-Gain

(EG)

Extending the disturbance rejection case to the local £2-gain design can be accomplishedby considering the small gain dissipation inequality [3]:

v(x(t)) <

_ zrz,

(7.26)

where the storage quantity V(x(t)) is taken to be the Lyapunov function (7.17). Of course, the resulting controller will simply minimize the gain from disturbance w to performance variable z. Closed loop stability will still be subject to the absolute size of the disturbance (in an £2 sense), and can be checked using the disturbance rejection (DR) test (7.23).

Theorem7.3. For system (7.16), whenever [Iw[12 _0,

A>O, T>O,

PSw ~ MI~ -b C~ Dzp ~r T -r -r T D D 2T Adq.~w ÷ (Dzp) Dzw _< O, ( zp) ~T ~T ~T ~r I DzwDzw- ~/~r BwC~ A ÷ DzwDzp ] ~T ~r

~T ~ ~T BwP + DzwCz

where M~2is defined as in (7.23).

Corollary 7.3. A controller K~,r that minimizes the £:2-gain bound is parameterized by the matrices R, S which solve the following optimization

178

Pare et al.

problem [23, minimize subject to

S >0, A>0, T>0, AR + RA T M12 -2T

TA BwTCq CzR D~p ATs + SA SBw BwT S -~I N~ ACqB~ Cz Dzw

U±
1/~ <0,

(7.2s) where M,2 =/3~ + RATCTqA + RCTq T, NI3 = SB~ + ATCTqA + CTq T, and the outer perpendicular matrices are

(7.29)

The identity matrices appearing above in (7.29) for U± and V± have dimension nw + n~ and np + nz, respectively.

7.5.4.

Controller

Reconstruction

To obtain the controller K from a pair (R, S) that solves one of the optimization problems requires the solution of a feasibility problem: M~/,r

+ UKV T + VKTU T < 0

(7.30)

where U, V are complementsof the corresponding matrices above in (7.21, 7.25, and 7.29). The matrix M~,r is a constant matrix involving the open loop system matrices and the specified performance and saturation levels. The form of the matrices My,r, U and V are easily derived by isolating terms involving the controller K, which is a step necessary prior to application of the Elimination Lemma[3, 30]. Optimization determines the matrices (R, S, A, T) and parameters r, 7 which then, through the use of the CompletionLemma[22,30], completely define the feasibility problem (7.30). Details of this elimination/completion technique are well documentedin [5] and omitted here for brevity. Instead, the final matrices necessary to solve

Robust Control for Systems with Saturation

179

Figure 5: Inverted pendulum with disturbance.

(7.30) for the (SR), (DR) and (EG) problems are detailed sections 7.8.2, 7.8.3 and 7.8.4, respectively.

7.5.5.

Optimization

in

Algorithms

The optimization problemsin §7.5.1-7.5.3 are bilinear in the multipliers (A, T) and the Lyapunovvariables (R, S). This problem is well knownto nonconvex, and is typically solved by successive optimization over the two sets of variables. This technique is referred to as BMIsynthesis [1,8, 26], and is applicable to the local designs described above. The successive iteration algorithms described in detail in [23] are readily applied to the (SR), (DR) and (EG) problems. In particular, the reduced order Popov/7-(~ algorithm in [23] was used directly to solve the local/:2-gain problemdescribed below.

7.6. -\

~:2-Gain

Control

Example

Here we consider the linearized inverted pendulumproblem, depicted in Fig. 5, with control force u input and angle 0 ot/tput, and dynamics O/f = 1 s--~, with k = 0.1. The disturbance is a force w, entering the system in the same way as the control. The performance z = [WIO W2u]T is a weighted combination of angle and control effort, with W1= (0.1s + 1)/(s +

180

Pare et al. 4.5

3.5

3

2

1.5

1

0.5

Figure 6: Performance and disturbance level dependence on r.

and W2 = W~-1, and the lowpass set at L = 1/(s/200+ 1). Using BMIsynthesis algorithm based on Corollary 7.5.3, Z;2-gain controllers were designed for this system using saturation levels ranging from r = 1.1 to r = 3.0, in increments of 0.10. For each r value, the algorithm was initialized with a controller designed by extending the Circle criterion methodin [25] to optimize the given performance. The BMIalgorithm for this example typically converged to a minimumafter 13 iterations, and yielded a new controller, based on the Popov analysis, that improved performance by about 20%over that based on the Circle criterion. The performance, @~2,of the new controllers is shownfor each r value in Fig. 6. Performancedegrades as r increases, with ;/~ increasing from 1.7 to 4.5 as r ranges from 1.1 to 3.0. This trend should be expected since higher values of r correspond to larger sector widths, which allows the possibility of more nonlinear control behavior. However, widening the sector generally improves the disturbance rejection capability. The disturbance energy bound for this system, &~ computed using Theorem 7.5.2, increases by max, 66%over the full range of r (see Fig. 6). So while performancedoes degrade, designs using higher values of r yield closed loop systems that can tolerate

Robust Control for Systems with Saturation

181

4.5

4

3,5

2.5

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1/~a~ Figure 7: Disturbance rejection vs./;2-gain performance trade-off.

larger disturbances. The trade-off between performance and disturbance rejection is depicted more clearly with a graph of ~/~2 vs. 1/&[naxin Fig. 7. Systems with good performance (low values of x/2~) are predicted 6 to have relatively poor disturbance rejection (high values of 1/&[nax). Conversely, designs with improved disturbance rejection are necessarily penalized with degraded performance. Of course, if disturbance rejection is critical and performanceis not an issue, then the controller should be designed directly using a BMIalgorithm based on Corollary 7.5.2. 7.7.

Conclusions

This chapter detailed control synthesis for systems with actuators that are subject to saturation. Optimal control designs were considered for three different performanceobjectives: region of attraction, disturbance rejection and/:e-gain. Each is formulated as a solution to an LMI/BMIoptimization problem. 6Remember: &[naxis a lowerboundon the actual disturbancerejection C~[nax.

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The design algorithms are based on the Popov stability criterion, and employ a sector model of the saturation nonlinearity. The BMIimplemenrations are efficiently solved using available LMIsoftware. For a simple numerical examplethe algorithms typically converged after 13 iterations and produced Popov controllers that achieved a 20%improvement in £2-gain performance comparedto controllers designed based on the Circle criterion. One limitation in the £2-gain design case is that each closed loop system must be postanalyzed to determine the worst case disturbance levels the system can tolerate while achieving the optimal disturbance attenuation. As a practical consideration, it may be desired to trade off performance against disturbance energy levels in order to accomplish the final design. Control design using this approach was explored with a numerical example. Finally, the BMIsolutions for each of the local control designs are presented in a form consistent with the reduced order designs detailed in [23]. This means that the algorithms that solve these problems can be structured to provide reduced order solutions as well. Of course order reduction is desirable when controller complexity needs to be avoided; reduced order algorithms will allow the designer to trade off the local performanceagainst controller order.

7.8.

Appendix

7.8.1.

Preliminaries

As mentioned in §7.5.4, g~ven a solution to one of the BMIproblems, consisting of the set of matrices (R, S, A, T), the corresponding optimal controller can be recovered by finding a K(s) satisfying M~(A, T) +

Ut(V T ~-

VKTUT < O.

(7.31)

The matrices comprising this LMIare described here. This procedure is taken from [23]. It automatically produces a reduced order controller, if possible. First, given the matrix pair R, S, the quadratic stability matrix is computed Q = R- S-*, (7.32) and decomposedusing the singular value deeomposition as [W, E, WT] = svd(Q).

(7.33)

The controller order is then given by nc, the number of nonzero singular values of Q. The columns of.W corresponding to the nc most significant

Robust Control for Systems with Saturation

183

singular values are then selected,

=[wl,..., wno]

(7.34)

and the reduced order matrix of singular values Zr = diag(~rl,..., allows the formation of the reduced order, stability matrix

crnc) then

Next, define the following t-subscripted matrices which have the reduced order controller dimension, and are given as

with Cq,t

= [ Cq 0 ], Cz,t

(7.37)

= [ Cz 0 ],

and Bt= Dl,t 7.8.2.

I

0

,

Ct=

Cy

0

’

= [ 0 Dz~ ], D2,t = D~w "

Region of Convergence Design

The corresponding feasibility LMIthat must be solved using the optimal solution to the region of convergence problem (7.2(}) has the form (7.31). In this case the matrix Mfl~ is a function of the parameter fir used to sector bound the dzn(.) nonlinearity, as detailed in §7.4 and Fig. 4, and is given by: M~(A,T) = [ At~ + ~ATt ~Bp,t + ATt CTq,tA + CTq,tT -2T (G

’

(7.40)

and the outer matrices are given as

U= [

ACq,tBtl , and V = f ~T ] .

(7.41)

The stability matrix ~ is formedas in (7.35), and the t-subscripted’matrices are the same as above in (7.36-7.38). However, care must be taken to use the matrices augmentedwith the lowpass filter, and incorporating the sector parameter fir as defined by the system (7.15).

184 7.8.3.

Pare et al. Local Disturbance

Rejection

Design

The corresponding feasibility LMIthat must be solved using the optiraal solution to the disturbance rejection problem (7.24) again has the form (7.31). In this case the matrix At~ + gAT (JBp,t + ATt C~,t A + C~,tT ~Bw,t ~ ACq,tBw,tJ , M~.(A, T) = (.)~T -2T 2 T T -I B~,tC~,tA

[

(7.42) and the outer matrices are U= ACq,tB¢ 0

, andV=

.

(7.43)

D~,t

where again, all t-subscripted matrices are defined in §7.8.1 and the system matrices are the augmented;¢ersions, as discussed in §7.4. 7.8.4.

Local /22-Gain Design

For the local/22 optimal compensation, reconstruction requires solution of the feasibility LMI(7.31), where

M5~(A,T, 7)

At~ +~, A Tt (.)T~ T ~ B~,tS

C,,t

T A -b Cq,t T T ~ Bp,t + At TCq,t ~Bw,t -2T A Cq,t B~,t T T -’~ Z,~, Bw,tC~,tA D,p D~w

T C

D~)~ D~T~ --’)’In,

(7.44)

is nowalso a function of the performance metric 7. Similarly, the outer matrices for the corresponding feasibility are given as:

U=

~Bt ACq,tBt 0

0

’

and V =

~T DT2,t 0

(7.45)

Again, the various system matrices here are defined according to the design model described in §7.4.

References [1] D. Banjerdpongchai and J. P. How. LMI~Synthesis of Parametric Robust 7-/~ Controllers, in: Proc. American Control Conf., volume 1 (1997) 1655-1660.

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[2] S. P. Boyd and C. H. Barratt. Linear Controller Design: Limits of Performance, Prentice-Hall, (1991). [3] S. Boyd, L. E1 Ghaoui, E. Feron, and V. Balakrishan. Linear Matrix Inequalities in System and Control Theory, SIAM,(1994). [4] E. J. Davison and E. M. Kurak. A Computational Method for Determining Quadratic Lyapunov Functions for Nonlinear Systems, Automatica, 7 (1971) 627-636. [5] P. Gahinet. Explicit Controller Formulas for LMI-Based~o¢ Synthesis, Automatica, 32 (1996) 1007-1014. [6] L. E1 Ghaoui and J. P. Folcher. Multiobjective Robust Control of LTI Systems with Unstructured Perturbations, Syst. Cont. Let., 28 (1996) 23-30. [7] A. H. Glattfelder and W. Schaufelberger. Stability Analysis of Single Loop Control Systems with Saturation and Antireset-windup Circuits, IEEE Trans. on Automat. Contr., AC-28 (12) (1983) 1074-1081. [8] K. C Goh, J. H. Ly, L. Turand, and M. G. Safonov. #/K,~-Synthesis via Bilinear Matrix Inequalities, in: Proc. IEEE Conference on Decision and Control (1994) 2032-2037. and Control of [9] A. Hassibi and S. Boyd. Quadratic Stabilization Piecewise-linear Systems, in: Proc. of the American Control Conference, Philadelphia, PA(1998) 3659-64. [10] H. Hindi. Local Analysis of Perturbed Linear Systems with Application to Saturating Control Systems. PhDthesis, Department of Electrical Engineering, Stanford University, 2000. [11] H. Hindi and S. Boyd. Analysis of Linear Systems with Saturation using Convex Optimization, in: Proc. IEEE Conf. on Dec. and Control, volume 1 (1998) 903-908. [12] M. Johansson and A. Rantzer. Computation of Piecewise Quadratic Lyapunov Functions for Hybrid Systems, IEEE Trans. on Automat. Contr., 43(4) (1998) 555 -559. [13] V. Kapila. Robust Fixed-Structure Control of Uncertain Systems with Input-Output Nonlinearities. PhDthesis, Georgia Institute of Technology, 1996. [14] V. Kapila and H. Pan. Control of Discrete-time Systems with Actuator Nonlinearities: An LMI Approach, in: Proc. IEEE Conf. on Dec. and Control, volume 2 (1999) 1419 -1420. [15] H. K. Khalil. Nonlinear Systems, Prentice-Hall, Secondedition, (1996).

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[16] T. Kiyamaand T. Iwasaki. On the Use of Multi-loop Circle Criterion for Saturating Control synthesis, in: Proc. of the American Control Conference, volume 2 (2000) 1225-1229. [17] V. A. Komarov. Design of Constrained Controls for Nonautonomous Linear Systems, Automation and Remote Control, 10 (1995) 128.01286. [18] R. L. Kosut. Design of Linear Systems with Saturating Linear Control and Bounded States, IEEE Trans. on Automat. Contr., AC-28(1) (19S3) 121-124. [19] Z. Lin. Semi-Global Stabilization of Linear Systems of Linear Systems with Position and Rate Limited Actuators, Systems and Control Letters, 30 (1997) 1-11. [20] Z. Lin. Semi-Global Stabilization of Discrete-Time Linear Systems with Position and Rate Limited Actuators, in: IEEE Conference on Decision and Control (1998). [21] Z. Lin, M. Pacher, S. Banda, and Y. Shamash. Stabilizing Feedback Design for Linear Systems with Rate Limited Actuators. in: Control of Uncertain Systems with Bounded Inputs, number 227 in Lecture Notes in Control and Information Sciences, pages 173 186. Springer-Verlag, (1997). [22] A. Packard, K. Zhou, P. Pandey, and G. Becker. A Collection of Robust Control Problems Leading to LMI’s, in: Proc. IEEE Conf. on Dec. and Control, volume 1. (1991) 1245-1250. [23] T.. E. Par& Analysis and Control of Nonlinear Systems. PhDthesis, Department of Mechanical Engineering, Stanford University, 2000. [24] T. E. Par~, H. Hindi, and J. P. How. Local Control Design for Systems with Saturating Actuators, in: Proc. of the American Control Conference (1999) 3211-3215. [25] T. E. Par~, H. Hindi, J. P. How,and S. P. Boyd. Synthesizing Stability Regions for Systems with Saturating Actuators, in: Proc. IEEE Conf. on Dec. and Control (1998) 1981-1982. [26] T. E. Par~ and J. P. How. Algorithms for Reduced Order Robust Ho~ Control, in: Proc. Conf. Decision and Control (1999) 1863-1868. [27] C. Pittet, S. Tarbouriech, and C. Burgat. Stability Regions for Linear Systems with Saturating Controls Via Circle and Popov Criteria, in: Proc. IEEE Conf. on Dec. and Control (1997). [28] C. Pittet, S. Tarbouriech, and C. Burgat. Output Feedback Synthesis via the Circle Criterion for Linear Systems Subject to Saturating

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Inputs, in: Proc. IEEE Conf. on Dec. and Control, volume 1 (1998) 401-406. [29] A. Saberi, Z. Lin, and A. R. Teel. Control of Linear Systems Subject to Input Saturation, IEEE Trans. on Automat.’ Contr., ACo41(1996) 368-378. [30] R. E. Skelton, T. Iwasaki, and K. Grigoriadis. A Unified Algebraic Approach to Linear Control Design, Taylor and Francis, (1998). [31] R. Su~rez, J. A1varez-Ramirez, and J. Solis-Daun. Linear Systems with BoundedInputs: Global Stabilization with Eigenvalue Placement, Int. Journal of Control, 26 (1998) 303-333. [32] R. SuSrez, J. Alvarez-Ramirez, M. Sznaier, and C. Ibarra-Valdez. /22Disturbance Attenuation for Linear Systems with Bounded Controls: An ARE-Based Approach. in: Control of Uncertain Systems with Bounded Inputs, number 227 in Lecture Notes in Control and Information Sciences, pages 25-38. Springer-Verlag, (1997). [33] R. Su£rez, J. Solis-Daun, and J. Alvarez. Stabilization of Linear Control Systems by Means of Bounded Continuous Nonlinear Feedback Control, Systems and Control Letters, 23 (1994) 403-410. [34] S. Tarbouriech and G. Garcia, editors. Control of Uncertain Systems with Bounded Inputs, number 227 in Lecture Notes in Control and Information Sciences, Springer-Verlag, (1997). [35] F. Tyan and D. S. Bernstein. Dynamic Output Feedback Compensation for Systems with Input Saturation. in: Control of Uncertain Systems with Bounded Inputs, number 227 in Lecture Notes in Control and Information Sciences, pages 129-164. Springer-Verlag, (1997). [36] J. A. Walker and N. H. McClamroch. Finite Regions of Attraction for the Problem of Lur’e, The International Journal of Control, 4(1) (1967) 331-336. [37] S. Weissenberger. Application of Results from Absolute Stability Theory to the Computation of Finite Stability Domains, IEEE Trans. on Automat. Contr., AC-13 (1968) 124-125. [38] G. F. Wredenhagen and P. R. Belanger. Piecewise-linear LQ Control for Systems with Input Constraints, Automatica, 30 (1994) 403-416. [39] F. Wu and K. Grigoriadis. LPV-based Control of Systems with Amplitude and Rate Actuator Saturation Constraints, in: Proceedings of the 1999 American Control Conference, volume 5 (1999) 3191-3195.

Chapter 8 Output Regulation of Linear Plants Subject to State and Input Constraints A. Saberi Washington State

University,

Pullman,

Washington

A.A. Stoorvogel Eindhoven University Delft University

of Technology,

Eindhoven,

of Technology, Delft,

and

the Netherlands

G. Shi Washington State

University,

Pullman,

Washington

P. Sannuti Rutgers University,

Piscataway,

New Jersey

*The work of A. Saberi and G. Shi is supported by the NSF grant ECS0000475.

189

190

8.1.

Saberi et al.

Introduction

Tracking reference signals of knownfrequencies and rejecting disturbances of knownfrequencies is a classical problemknownas the output regulatiol.~ problem, and it lies at the core of multivariable control literature. During the past ten years, research on the control of linear systems with constraints has achieved great progress in different directions, amongwhich the study of regulation problems for systems with magnitude and rate constraints occupies a significant part. A recent book [15] formulates and studies a numberof problems on different facets of output regulation of linear systems with or without constraints on input. Input constraints mainly arise frora physical limitation of actuators. Besides actuator constraints, constraints on one or more of states often exist. One example of state constraints results from the linearization procedure in that a linear model is valid only for a certain region of state and control space. In process control, state and control constraints arise from the economic necessity of operating the plant near the boundary of a feasible region. In connection with safety issues, state and control constraints are a major concern in manyplants. In certain possibly hazardous syst~ems, such as a nuclear power plant, certain safety limits on somevariables are strictly enforced. The violations of such predetermined safety measures may cause system malfunction or damage. This implies that magnitude constraints or bounds on states must be taken as integral parts of any control system design. Having studied during the last decade several aspects of control de,sign problems for linear systems with magnitude and rate constraints on control variables, during the last two years the research thrust of the first two authors and the last author and their students has broadened to i~tclude magnitude and rate constraints on control variables as well as state variables. In connection with stabilization, whenever amplitude and rate constraints on both state as well as input variables exist, a taxonomyof all possible constraints is introduced, and several fundamental results on global, semiglobal, and regional stabilization were developed in [11]. It has become evident that the taxonomy of constraints developed there plays a dominant role in every type of constraint control problem including output regulation which is the main focus of this chapter. Our previous work on output regulation [5] was concerned with only amplitude constraints but not rate constraints. This chapter is a continuation of [5] and focuses on so called constrained semiglobal and global output regulation problems with constraints on both amplitude and rate of state as well as control variables. Wenote that recently in [20] a novel model which takes into account the amplitude and rate constraints on the input

Output Regulation with Constraints

191

was introduced. The basic idea is to incorporate a deadbeat type operator in the controller so that control signals do not overload the actuators during a transient period. This chapter investigates the applicability of the same basic idea to the current output regulation problem where we have amplitude and rate constraints on both the input as well as state variables. Most of the notations in this chapter are standard. They will be clear when we first introduce them. Welet C, C+, C- and C° denote respectively the entire complex plane, the open right-half complexplane, the open lefthalf complex plane, and the imaginary axis.

8.2.

System Model and Primary Assumptions

Consider the following continuous-time system model for output regulation:

E:

5: = Ax + Bu + Ew, x ¯ ]~n, (V = Sw sw ¯ ]R y ¯ ~g y = Cyx + Dyw, p z =Czx+Dzu z¯R r. e =C~x+D~w e¯]R

U

¯ ]~m (8.1)

In this system model, x, u, and y are respectively the state, input, and measured output of the plant. The second equation describes the exosystern with state w. The exosystem ~nodels the class of reference signals and/or exogenousdisturbances. The plant is subject to the effect of external disturbances represented by Ewwhile the measurements are perturbed by external disturbances D~w. Obviously, those plant and measurement disturbances need to be rejected. On the other hand, the controlled output C~x should track the reference signal -Dewand the resulting tracking error is denoted by e. The constraint output z is subject to the constraints, z(t) ¯ ,5, ~(t) ¯ for all t > 0

(8.2)

where $ and 7- are two convex sets, named amplitude-constraint set and rate-constraint set respectively. Note that this is the most general way of modeling the constraints. The configurati(m of output regulation is shown in Figure 1. In general the output regulation problem for arbitrary constraint sets $ and 7- is hard to solve. The most restrictive assumption we make is that 0 should be an interior point of these two sets. Wemake the following primary assumption throughout the chapter. Assumption 8.1. The convex sets $ .and T satisfy the following:

192

Saberiet al.. Dew

w

Controller

Figure 1: Configuration of output regulation. 1. Both the sets $ and 7- are closed and contain 0 as an interior Point.. 2. $ n T is bounded. 3. The matrices Cz and Dz satisfy decompositions:

CT~Dz= 0. Moreover, we have the

S = ($NimCz) + (SnimDz), T = imCz + (T AimDz). Remark 8.1. Note that item 3 of this assumption is just a mathematical representation of the fact that the constraint on z should be consistent with the state and input constraints, which are always imposed separately on a real control system. Morespecifically, in this modelim C~reflects the state constraints and im Dz reflects the input constraints. In view of item 3, we only consider rate constraints on the input. This is natural since rate constraints on states can be directly converted into amplitude constraints by using the state equation. Our objective here is three-fold: Firstly, to achieve internal stability, that is, if we disconnect the exosystem (w -= 0) the closed-loop is asymptotically stable in either the global or semiglobal sense. Secondly, to achieve output regulation, i.e. whenthe exosystemis present, we have e(t) --~ 0 t -~ ~ for a certain set of initial conditions of the closed-loop system. Thirdly, to meet the output constraints, that is, for two a priori given convex sets $ E RP and T E Rp satisfying Assumption 8.1, we have z(t) ~ and ~(t) ~ 2r for all t _> As is well known,even in the study of classical output regulation problems for linear systems without constraints, certain standard assumptions are often required for output regulation (see [15] for more details). Such assumptions are also required for constrained output regulation, and are stated below.

Output Regulation with Constraints Assumption 8.2. There exist matrices H and F satisfying regulator equation: [IS = AII + BF + E 0 =CeII + De.

193 the so-called

(8.3)

Assumption 8.3. The matrix S has all its eigenvalues in the closed right half plane. Assumption 8.4. The pair (A, B) is stabilizable. Assumption 8.5.

The pair

is observable. Assumption8.2 is necessary for the solvability of the classical output regulation problems having no constraints. Assumption8.3 is without loss of generality, because asymptotically stable modesin the exosystem do not need regulation. Assumption8.4 is necessary for stabilization. Finally, Assumption 8.5 is only needed in the case of tneasurement feedback. Obviously it is only necessary that (Cy, A) is detectable. There exists a standard reduction technique after which the pair (8.4) is detectable (see [15]). On other hand, we can relax the assumption from observability to detectability but this requires somehighly technical conditions related to the shape of the constraint sets in order to guarantee that the unobservable dynamics do not violate the constraints. Before closing this section, we emphasizethat the rate constraint ~(~) 7- must be enforced for all t 2 0, including t = 0. This requirement brings technical difficulties in the regulator design. Because of the rate constraint at t = 0, it is not possible in general to switch on a controller at time t = 0 without precautions, otherwise it might violate the rate constraint. For this reason, we have the obligation to protect the system by imposing some sort of filtering mechanismso that the rate constraint is not overloaded during the transient part of the control process. This is implemented by introducing a deadbeat type operator in the controller which always produces control signals that satisfy the constraints for all t > 0. In the next section we review such an operator that has been successfully applied in our previous works in connection with the rate saturation of only actuators.

194

8.3.

Saberi et

A Model for Actuator Constraints

Whenevera linear system is subject to amplitude and rate saturations of the actuator, it raises challenges to controller design, mainly due to the dynamic rate constraint, which must be satisfied during the whole control process. An operator-type model for the amplitude and rate constraints was first introduced in [20] in order to simplify the design of a stabilizing controller while at the same time utilizing most of the available results in low-high gain design methodologies. Such an operator-type model is referred to as a constraint operator, and is further explored and applied to problemsbeyond stabilization in [17,19,21]. The basic idea of the constraint operator is depicted in Figure 2, where it is shownthat the control signal generated by the controller passes through the constraint operator before it gets to the actuator. The constraint operator (denoted by as,q) guarantees that the actuator constraints are always satisfied. The main design effort is therefore not to guarantee that the constraints are satisfied but to avoid instabilities due to this nonlinear element. As discussed in detail in [20], an important key point in utilizing the constraint operator is to view it as a part of the controller: The constraint operator turned out to be very successful in our earlier works. Since we are concerned with both amplitude and rate constraints (although in a much more general framework including both input and state constraints), it is natural to also incorporate the constraint operator we used earlier. The modeling of such an operator is not trivial, as can be seen from our previous works. The purpose of this section is to review the constraint operator and clarify several points that are relevant to the current problem setting. Wewill also generalize the operator to arbitrar:y constraint sets instead of only hypercubes as in our previous work. Constraint

~

u

~ Actuator

Plant

~-~

[ Controller Figure 2: Incorporating the constraints in the controller. Roughly speaking, the objective of the constraint operator is to do the following. Given any multivariate signal u, the output signal from the oper-

Output Regulation with Constraints

195

ator, denoted here as ua -- as,~-(u), satisfies the amplitude limit Ua(t) and the rate constraint iZa(t) ¯ 7- for all t _> 0, where $ and 7, are compact and convex sets which contain 0 in the interior. Wefirst introduce some notation. 65 and ~T denote the boundary of and 7, respectively. Givena closed convexset 7¢, it is well knownthat given any point x there exists a unique point t~ ¯ ~ that minimizes IIx - Yll over all y ¯ T4.. The mappingfrom x to ~ is hence well defined and is sometimes referred to as the nonlinear projection on the set T¢ with notation 1-In. Next, given a closed convex set T¢ containing 0 in the interior, it is well knownthat given any point x there exists a unique 2 ¯ 7¢ such that ~ = ~x with A > 0 which minimizes IIx - 5:11. The mapping from x to 5: is again well defined and denoted by ~n. Let 1-Is and ~- denote those operators associated to the sets S and T respectively. For implementation purpose, we need an accurate model for the operator as,=r. The model is basically a deadbeat filter. Werequire the continuoustime operator ~rs,~r(u) to have the following properties: ¯ For any continuously differentiable u(t), as,~’(u)(t) is differentiable and as,~-(u)(t) and ~ as,7-(u)(t) ¯ T for all t >_ 0. ¯ If u(t) ¢ as,z(u)(t) ~T.

then either

as,~-(u)(t) ¯ or ~t as, r(u)(t) ¯

In [20], the sets $ and 7" are hypercubes. Wewouldlike to stress that the extension to general complex sets creates some crucial differences. First note that in the special case that S and T are circles, a differential gamehas been introduced in the literature (see [2-4, 8]) which is commonlyreferred to as "the lion and the man" and stresses some of the issues involved in the extension from hypercubes, which can be reduced to 1-dimensional problems, to the case of circles, which are intrinsically 2-dimensional. This differential game is a pursuit/evasion game where a lion tries to catch a man while both are constrained to a circle and have bounded velocity. In our case ~rs,~-(u) is the "lion" trying to catch the "man" u. Obviously the "man" u has a very large velocity the "lion" aS,~r(u) will never catch the man. Define

{u¯ c1 : u(t)¯ s, u(t)¯ 7, forall >0}. In the case that $ and 7" were hypercubes it can be easily shown that if u ¯ Cs,~-, then there exists a T > 0 such that crS,T(u)(t) = u(t) for all t > T using a properly designed as,~-. In other words there exists a T such that independent of initial conditions the lion catches the man after at most T seconds. In the case that $ and 7" are circles it has been shown

196

Saberiet a,l.

in the aforementioned references that the lion might never catch the man (independent of which strategy it is using) even though the lion can get arbitrarily close. Also, as we shall see soon, achieving a finite capture time T is possible if we restrict both u and d to the interior of constraint sets S and %r. In these papers there is also an optimal strategy for the lion to catch the man which trivially generalizes to arbitrary convex sets $ and ~ (although the strategy might not be optimal any more). This strategy can be easily described: first the lion movesto the center of the circle. Then the liou movesoutward towards the manwhile staying on the line between the origin and the man. A state space model for an operator with the listed properties must be described in a functional differential framework. Even for the case that ,~ and %r are hypercubes, it was argued in [20] that we cannot describe Ua := crs,:r(u ) via a standard differential equation. Instead we used a limiting argument. Weagain use a similar limiting argument in this chapter. W~e will not use the strategy for differential gamesoutlined above but a simpler strategy which is easily seen to have the desired properties (but it is not an optimal strategy with respect to somedifferential game). Let x~ (t, u) be the unique solution of the following differential equation

=

-

x (0)

where ~ > 0, and u is any measurable input signal. defined by a~,z-(u).(t)

(S.5) Then as,:r(u)(t)

li m x~(t,u).

(8.6)

Similarly as in [20] it can be shown that a$~(u)(t) and (if T is bounded) Lipschitz-continuous with the property that for any t, s E

-

e (t

It is important to note that in the discrete-time case, as described in [20], we can have an exact state space realization of the operator as,~- but in the continuous-time case the exact modelresults from a limiting process, which is not feasible in practice. However,from a practical point of view, we can choose a sufficiently large A to get an approximate state space realization (8.5). Note that xx(t,u) for any ~ satisfies both our amplitude and rate constraints. The operators defined above has a deadbeat property. If u(t) = 0 for t > t~ then as,~-(u)(t) = u(t) for t > el +2mwith m> 0 a smallest number

Output Regulation wi~h Constraints

197

such that $ c m:r. Moreover, if u is differentiable and such that for some p E (0, 1) and for all t > tl we have u(t) ~ p$ and/t(t) ~ p:T, then after a finite transition time T we have as,~(u)(t ) = u(t) for all t > tl + T. However,from the theory of differential games, we knowthat in worst case T ~ oc as p-~ 1. Wenote that the constraint operator defined above is a dynamic object and therefore it has its owninitial conditions. It is easily seen that the future behavior of as,~r(u) is uniquely determined by u(t) for t > 0 and cr,~,~r(u)(0). Since ~r~,~-(u)(0) ~ $ we can define a state A’s for th is dynamic object with Xs = $. This notion will show up when we state our problem formulations later, because the operator will be an integral part of the closed-loop system and its dynamicsis part of the closed-loop dynamics. As a part of our notation, we shall refer to the state of the operator as xs, the initial condition as xs°, and the state space as 8.4.

Statements

of

Problems

Weare mainly interested here in the output regulation problems in the global and semiglobal setting which will be defined precisely soon based on the descriptions of the system models presented earlier. Wefirst note that due to the output constraints z E $, the initial states of the system cannot be arbitrary. Note that ~ E T does not restrict the initial states of the system because the condition im Cz c T guarantees that we have only rate constraints on the input but not on the state which, as argued before, is the only natural case anyway. For this reason, we define here the admissible set of initial conditions based on which the output regulation problems will be defined. Definition 8.1. Given the system (8.1) and the constraint set $, the set ~4(S) := { xo ~ ~P I Czxo ~ $} is said to be the admissible set of initial conditions. Remark 8.2. Note that due to Assumption 8.1 we have that Czxo + Dzu(O) ~ if andonly if Czxo ~ S andDzu(O) ~ $. Henc e the constraint on the initial state is only related to Czxo.

z --

The statements of semiglobal and global regulation problems formulated below dependlargely on the subset of initial conditions inside the admissible set. In the following, the problems using state or measurementfeedbacks are stated separately due to different controller structures. Wewill not discuss the constrained global measurementfeedback output regulation problem. This is because this problem is only solvable under very

198

Saberi et ol.

restrictive conditions. Basically we need to assumethat there exists ~ static feedback u = n(y) and ¢ > 0 such that if ~(0) E A(S), then for all t ~ [0, ¢] where ~(t) = A~(t) + Bn(Cy~(t)). In other words, the system must satisfy the state constraints on the interw~l [0, ¢] with a static output feedback. This is clearly very restrictive and therefore a result for this case is not of muchinterest. From the above we see that we must guarantee that x(t) ~ A($) at all time. Next we consider the constraints induced on the input by z(t) ~ and ~(t) ~ 7". Wedefine: Su:{ueRmlDzue$}

and

Wethen note that the input signal must satisfy u(t) ~ Su and/~(t) E Tu for all t > 0. Consider the constraint operator defined in the previous section with the sets $ and 7" replaced by ,Su and T~ respectively. Then if use u = asu,~-u (uc) and design a controller for a new system with input uc then it is automatically guaranteed that the constraints on the input are satisfied. Therefore, in a sense, our further controller design only needs to concern itself with the constraint on the state x(t) ~ ~4(8). Obviously the nonlinear and dynamic element as.,~-, that we include in our design will play a crucial role in the question whether we achieve stability and/or regulation. Recall that we refer to the state of this dynamicelement as X’~,~. Note that this operator has the additional effect to automatically guarantee a smooth transition if we switch controllers at time 0 without constraint violation. Therefore, in this paper we basically design a controller for the following system ~ : Ax ÷ Bas,,,T,(uc) ~c :

÷ Ew,

y = Cyx + Dyw, z = C~x+ D~as,,7-u (uc), e = C~x + D~w.

(8.7)

However, note that in the end the dynamic element a8.,7-, will be part of the controller. Also, for this new system, we have z(t) ~ ,3 and ~(t) ~ 7" and only if Czx(t) Problem 8.1. Consider the system (8.7), the constraint sets $ ~ and 7" c ~P, and a set l/Y0 c_ ]Rs. The constrained global state feedback output regulation problem is defined as follows. Find, if possible;,

OutputRegulation withConstraints a state feedback law (possibly nonlinear) of the form, i~ : f(z, v, w), v E Rq

uc=g(x,v, w)

199

(8.8)

such that the following conditions hold: 1. The equilibrium point (x, v) -- (0, 0), xs = 0 of the system

= Ax(t)+ S u, u(g(x, ,v0 f(x,v,O)

(8.9)

is locally exponentially stable with A(S) x Nq x 2(s contained in its region of attraction. For any (x(0), v(0)) .4 (8) x Rq, w( O) e and x s~ Xs,0 we have z(t) E and ~(t) E %rforall t >_ For any (x(0),v(0)) A(S) × ~q, w( O) ~ W0, an d x s E0 Xs the solution of the closed-loop system satisfies lim e(t) = If we relax the requirement of global stability to semiglobal stability, obtain the problem:

we

Problem 8.2. Consider the system (8.7), the constraint sets $ and T C RP, and a bounded set W0C ~s. The constrained semiglobal state feedback output regulation problem is defined as follows. For any a priori given (arbitrarily large) boundedset .40 contained in the interior of A(S), find, if possible, a state feedback law (possibly nonlinear) the form (8.8) such that for any a priori given bounded set )20 e Nq, following conditions hold: 1. The equilibrium point (x,v) = (0, 0), Xs = 0 of the system (8.9) locally exponentially stable with Aox ]20 x 2(s contained in its region of attraction. 2. For any (x(0),v(0)) ~ A0 x ]20, w(0) ~ W0, ° ~ 2(s wehave z(t) ~ and ~(t) E T for al l t > 0. 3. For any (x(0),v(0)) ~ .40 x 12o, w(0) ~ 14~o, ° ~ Xs thesolutio n of the closed-loop system satisfies lim e(t) = o whenever 4. For any (x(0), v(0)) ~ Ao x Vo, w(0) E 142o, x s ~Xs, we set w(t) =_for t _>to for someto > 0, wehave l im x( t) =0 and lim v(t) = 0. Moreover,we have z(t) ~ and ~(t) ~ T for al l t > to

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Finally, we consider the measurementfeedback case where, as discussed before, we only consider the case of semiglobal stability: Problem 8.3. Consider the system (8.7), the constraint sets $ C: and T C Rp, and a bounded set 140 C Ns. The constrained semiglobal measurement feedback output regulation problem is defined as follows. For any a priori given (arbitrarily large) boundedset .40 contained the interior of ¢4($), find, if possible, a measurementfeedback law (possibly nonlinear) of the form, i~ =f(y, v, w),

v ¯ Rq

uc = g(y, v, w) such that for any a priori given bounded set ];0 ¯ Rq, the following conditions hold: 1. The equilibrium point (x, v) = (0, 0), xs = 0 of the system, ~ = Ax(t) + Basu,T.(g(C~x, v,

=e(cx, v, o) is locally exponentially stable with .40 x 12o x Xscontained in its region of attraction. o ¯ As, we have 2. For any (x(0),v(0)) ¯ X0 × Vo, w(O) ¯ 14o, x s z(t) ¯ and k(t) ¯ :Yforall t _> 3. For any (x(0), v(0)) ¯ Ao× ]?0, w(0) ¯ 140, ° ¯ A’8 the so luti on of the closed-loop system satisfies lim e(t) = 0 ¯ Xs, whenever 4. For any (x(0), v(0)) ¯ Ao × )20, w(0) ¯ 140, we set w(t) =_ for t > tofor someto > 0, wehave l im x( t) = 0 and lim v(t) = O. Moreover,we have z(t) ¯ and ~(t) ¯ T for al l t _>to.

8.5.

Taxonomy

of

Constraints

In order to formulate the conditions under which the above formulated problems are solvable, it is beneficial to develop a taxonomyof constraints by utilizing the structural properties of the mappingfrom the input to the constrained output, namely the subsystem Ez~ that is characterized by the quadruple (A, B, Cz, Dz). It turns out that the structural properties of this subsystem play dominant roles in the study of constrained semiglobal and

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global stabilization [11]. Specifically, the right invertibility, the location of invariant zeros, and the order of infinite zeros of Ezu determine what can or cannot be achieved. This section is devoted to categorizations of constraints which are to be used in the statements of solvability conditions. The first categorization of constraints is based on whether the subsystem Ezu is right invertible or not. Wefirst recall what is meant by right invertibility. Definition 8.2. The system ~ = Ax + Bu, y = Cx + Du

x E ~

is said to be right invertible if, for any Yref(t) defined on [0, c~), there exists u(t) and x(0) such that y(t) = Yref(t) for all t e [0, oo). Definition 8.3. The constraints

are said to be

¯ right invertible constraints if the subsystem Ezu is right invertible. ¯ non-right invertible invertible.

constraints

if the subsystem Ezu is non-right

The second categorization is based on the location of the invariant zeros of the subsystem Ezu. Werecall next the definition of invariant zeros. Definition 8.4. The invariant zeros of a linear system with a realization (A, B, C, D) are complex numbers A ~ C for which rank(AI~

A -DB)

<normrank(SI~

A -D B)

where by "normrank" we mean the rank of a matrix with entries field of rational functions.

in the

Because of its importance, in the following definition, we specifically label the invariant zeros of the subsystem Ez~ as the constraint invariant zeros of the plant. Definition 8.5. The invariant zeros of the subsystem Ezu are said to be the constraint invariant zeros of the plant associated with the constrained output z. Definition 8.6. The constraints are said to be

202

Saberi et aL minimumphase constraints if all the constraint invariant zeros of the plant associated with the constrained output z are in C-. weakly minimumphase constraints if all the constraint invariant zeros of the plant associated with the constrained output z are in C-t2 Co with the restriction that at least one such constraint invariant zero is in Co and any such constraint invariant zero in CO is simple. weakly non-minimumphase constraints if all the constraint in.variant zeros of the plant associated with the constrained output z ~ are in C- Ucowith at least one non-simple constraint invariant zero O in. C at most weakly non-minimum phase constraints if all the constraint invariant zeros of the plant associated with the constrained °. output z are in C- U C strongly non-minimum phase constraints if any one or more of the constraint invariant zeros of the plant associated with the con+. strained output z are in C

The third categorization is based on the order of the infinite zeros of the subsystem ~zu. See [16] for a definition of infinite zeros of a system. Because of their importance, we label below the infinite zeros of the subsystem Zzu as the constraint infinite zeros of the plant. Definition 8.7. The infinite zeros of the subsystem ~zu are called the constraint infinite zeros of the plant associated with the constrained output z. Wehave the following definition constraints.

regarding the third categorization of

Definition 8.8. The constraints are said to be type one constraints if the order of all constraint infinite zeros is less than or equal to one.

8.6.

Low-gain and Low-high Gain Design for Linear Systems with Actuators Subject to Both Amplitude and Rate Constraints

A numberof powerful analysis and design tools exist to deal with input ~onstraints. Amongthe design methodologies are low-gain~ low-high gain,

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203

scheduled low-gain, scheduled low-high gain, and manyvariations of them. There exist numerous publications and an extensive body of work in the literature on these aspects. This body of work was essentially developed during the last decade by the first two authors and the last author and their students. All this work cannot be surveyed here due to space limitations. Howeverit is worth pointing out a recent book [15] that incorporates some of these aspects regarding output regulation. One point to note here is that the low-gain and low-high gain methods were developed earlier mainly deal with actuators subject to only amplitude constraints. However, recently a new version of low-gain design has been proposed for treating input constraints with both amplitude and rate saturations [20]. This new low-gain design takes into account the dynamics in the rate saturation and extends easily to the low-high gain design for performance improvement. Furthermore, it can be scheduled to achieve global asymptotic stabilization. We review in this section somebasic aspects of low-gain and low-high gain designs as well as somevariations of them as they are needed in certain proofs in a subsequent section. This review will focus on linear systems with both amplitude and rate constraints on input. Weutilize the constraint operator u = crs,T(uc) defined in Section 8.3, and moreover we focus on asymptotically null controllable systems with bounded control*.

8.6.1.

Static

Low-gain State Feedback

There are two approaches to the low-gain design for a linear system with amplitude and rate saturating input. One is the so called direct method and the other one is based on algebraic Riccati equations. Owingto space limitation, we focus here on the Riccati-based method. Consider a linear system, ~c = Ax + Bu, x E Rn, m. u ER

’(8.10)

Wenow recall a fundamental lemmafrom the diverse literature (e.g. see [6,131). Lemma8.1. Suppose (A, B) is asymptotically null controllable with bounded control. Let Q(e) be any parameterized positive definite matrix satisfying: Q(e) > 0 for e > 0, Q(e) -~ 0 as e -~ 0 and ~Q(e) > 0. the algebraic Riccati equation ATp + PA - pBHrp + Q(e) =

(8.11)

* (A, B) is said to be asymptotically null controllable with bounded control if (A, is stabilizable and A has all its eigenvalues in the closed left-half plane.

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Saberiet at’.

has a unique positive definite solution P(e) for any ~ E (0, 1]. Moreover, we have the following properties: 1. A - BBTp(¢) is asymptotically stable for all e > 0. 2. lim~-~0 P(e) = dP(~)> 0 for all e E (0, 3. P(¢) is continuously differentiable with --~-

4. There exists a constant M> 0 such that []p1/2(~)Ap-1/2@)[[ ~_

forany~ ~ (0,1]. 5. Let Fc(e) = BTP(e). There exist positive-valued continuous functions ~(e) and #(e) satisfying lim~-~0 ~(e) = lim~_0 #(e) = 0 such that

~ Lemma8.1 presents a fundamental result which can be used to deriw. all semiglobal stabilization results for linear null controllable systems with both amplitude and rate constraints. The basic idea is that by reducing low-gain parameter ~ we can make the domain of attraction as large as required while amplitude and rate saturations are avoided. Consequently, the closed-loop system always remains in the linear region. The semiglobal stabilization results are compiled in the following theorem. Theorem 8.1. Consider the system (8.10). Let (A,B) be asymptoti-cally null controllable with bounded control. Also, let P(¢) be the unique positive definite solutions of (8.11) in Lemma8.1. Then, given any arbi-trarily large compactset 2d0 C Rn pand any arbitrary constraint sets S C R and T C RP containing the origin as an interior point, there exists an ~* > 0 such that for all ¢ ~ (0, ¢*] the closed loop system with the feedback law u = --BTp(¢)x has the following properties: 1. The equilibrium point x = 0 is asymptotically tained in its domainof attraction.

stable with X0 con-

2. For any x(0) ~ ,-l’o, we haveu(t) ~ for al l t _>0 and/t(t) E 2 r forall t>0. Note that in the above theorem a simple linear control law is used, but the rate constraint on the control can be satisfied only for t > 0. If we want to have the rate constraint satisfied for all t _> 0, including t = 0, then a more complicated control law is needed. This is presented in the next theorem.

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205

Theorem8.2. Consider the system (8.10). Let (A, B) be asymptotically null controllable with boundedcontrol. Also, let P(¢) be the unique positive definite solutions of (8.11) in Lemma8.1. Then, given any arbitrarily large compactset X0C Rn pand any arbitrary constraint sets $ C ]I~ and T C ]~P containing the origin as an interior point, there exists an ¢* > 0 such that for all ~ E (0, ¢*] the closed loop system with the feedback law ue = -BTP(e)x,

u =

has the following properties: 1. The equilibrium point x = 0 and xs = 0 is asymptotically stable with X0 × X’s contained in its domainof attraction. 2. For any x(0) E 2~o and ° e As, we have u(t ) ~ $ for all t > 0 and /~(t) E T for all t _> Note that in order to meet the rate constraint on control at t = 0 a dynamic and nonlinear control law is used in the above theorem. Remark 8.3. The problems we deal with in the preceding two theorems belong to the constrained semiglobal stabilization problem via state feedback, as defined in [11], by choosing the constrained output z = u.

8.6.2.

A New Version

of Low-gain Design

A new low-gain design has been proposed recently in [20] to deal with both amplitude and rate constraints on control input. This new low-gain design results in control laws that are dynamic. For this reason, we sometimes refer to it as the dynamic low-gain design. A key feature of this dynamiclow-gain design, as we shall see soon, is that it allows the development of a low-high gain design for the case when both the amplitude and rate constraints are present. The detailed exposition of what we review below, including the proofs, can be found in [20]. Westart with a lemma which constitutes a building block for this development. Lemma8.2. Suppose (A, B) is asymptotically null controllable with bounded control. Let

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Saber1et al.

where 5 > 0 and Ql(e) is an n × n parameterized positive definite mat:rix satisfying: Ql(e) > 0 for e > 0, QI(S) ~ 0 as e -~ 0 and ~Q~.(e) > 0.

A

(8.12)

Then the following algebraic Riccati equation: A~P + PAc - PBcB~cP+ Q(5, 5) =

(8.13.)

has a unique positive definite solution P(~, 5) with the following properties: 1. Ac - BcBT~p(s,5) is asymptotically stable for all z > 0, 5 > 0. 2. Wehave the following asymptotic behavior: lim P(~, 5) = P(0, 5)

(0° 0) 5I,~

.

Utilizing the above lemma, the following theorem presents a dynamic low-gain feedback law and its properties. Theorem 8.3 [20]. Consider the system (8.10). Let (A, B) be asymptotically null controllable with bounded control. Also, let P(5, 5) be the unique positive definite solution of (8.13). Then, given any arbitrarily large bounded sets X’0 C ~n, 2d~ C ~m, and any arbitrary constraint sets ,S C ]Rp and 7" C ]~P containing the origin as an interior point, there exist ~* > 0 and 5" > 0 such that for all ~ E (0, ~*] and fixed 5* the closed loop system with the dynamic state feedback control law i~c=-(O

I) P(¢,5*)

has the following properties: 1. The equilibrium point (x, uc) = (0, 0), = 0is asymptotically stabl e with X0 × A~I x X’~ contained in its domainof attraction. 0 we have u(t) ~ $ forall 2. For any x(0) E X0, uc(0) ~ X’l, x s ~X’s, t_>0and/t(t)~Tforallt_>0.

Output Regulation with Constraints

8.6.3.

207

A New Low-high Gain Design

Weobserve that during the last decade low-high gain design was developed in the context of dealing with actuators subject to only amplitude saturation (but not rate saturation) in order to enhance the performance that can be achieved with low-gain design. Obviously, one needs to develop a new version of low-high gain design that can improve the performance but can deal with actuators subject to both amplitude and rate saturation. The new low-gain design discussed in the last subsection can be adapted to yield a low-high gain design as presented in the following theorem. Theorem 8.4 [20]. Consider the system (8.10). Let (A, B) be asymptoticMly null controllable with bounded control. Also, let P(e, 6) be the unique positive definite solution of (8.13). Then, given any arbitrarily large bounded sets X0 C Rn, pX1 c ~mand any arbitrary constraint sets S c ]I~ and T C ~P containing the origin as an interior point, there exist e* > 0 and 6" > 0 such that for all e E (0, e*] and fixed 6" the closed loop system with the dynamic state feedback control law,

= where Pl > 0 can be arbitrarily

= large, has the following properties:

1. The equilibrium point (x, uc) = (0, 0), Xs = 0 of the closed-loop system is asymptotically stable with ~0 × W1× 2ds contained in its domainof attraction. 2. For any x(0) E X0, uc(0) C X1, and ° C Xs, wehave u(t) e $ for all t _> 0 and/~(t) G T for all t _> Note that, in contrast to previous low-gain designs, the operator ~rs,T plays an additional role because without it we would immediately have constraint violations. Clearly, the low-highgain design makesfull use of rate capacity by introducing the high gain parameter pl, and hence (as can be shown) it improves the performance of the closed-loop system dramatically. Another approach to avoid the performance deficiency of low-gain design is to use a scheduled low-gain design, which is based on a simple idea, i.e. to increase the gain as the state gets smaller. Remark 8.4. The problems we deal with in Theorems 8.3 and 8.4 also belong to the constrained semiglobal stabilization problem via state feedback, as defined in [11], by choosing the constrained output z = u.

208

8.6.4.

Saberi et al.

Scheduled

Low-gain Design

Wediscussed above low-gain design as well as low-high gain design in the context of semiglobal stability. Onepossibility to achieve global stabilization while still improving local performance is to schedule the low-gain parameter so that the gain is increased as the state trajectory goes to the origin. A simple scheduling mechanism is proposed by Megretski I9] for continuous-time systems with input amplitude saturation. Whenthe input is constrained by both magnitude and rate, the structure in the new version of low-gain design suggests itself a natural scheduling in order to achieve global stabilization while improving the near origin performance. It turns out that we can have two versions of scheduled lowgain design. Both are based on dynamic state feedback. This is presented below. The details of constructing control laws and the proofs are omitted (see the technical report [18]). Theorem8.5. Consider the system (8.10). Let (A, B) be asymptotically null controllable with bounded control. Also, let P(e) be the unique positive definite solutions of (8.11) in Lemma8.1. Then, given any arbitrary constraint sets S C NPand T C NPcontaining the origin as an interior point, there exist sufficiently small constants e* > 0 and ~* > 0 such that the closed loop system with the following nonlinear dynamic control law Uc = -BTP@’(x))x,

u ~--

where e(x) is defined ¢(x(t))

= max{¢E (0,~*] : x(t)TP(e)x(t)tr[BTP(~)B]

~*},

has the following properties: 1. The equilibrium point x = 0 and xs = 0 is globally asymptotically stable. 2. For any x(0) E ’~ and xs° eAs, we have u(t ) ~ 8 for all t >_ 0 and g(t) ~ T for all t _> The following theorem uses a scheduling of the dynamic low-gain design. Theorem 8.6. Consider the system (8.10). Let (A, B) be asymptotically null controllable with boundedcontrol. Also, let P(e, 5) be the unique positive definite solution of (8.13). Then, given any arbitrary constraint sets ~.~ C ]~P and 7- C ]~P containing the origin as an interior point, there exist

Output Regulation with Constraints

209

5* > 0 and a* > 0 such that the closed loop system with the following nonlinear dynamic control law,

-

u=

where

e(x, uc) = max{~e [0, 1]: Uc

()

P(e, d*) x tr[B/P(e,(~*)Bc] ~tc

< a*}, (8.14)

with Bc as given in (8.12), has the following properties: 1. The equilibrium point (a, uc) = (0, 0) and xs = 0 is globally asymptotically stable. 2. For any x(0) E n, uc(0) E TMand xs° ~ Xs, we haveu(t) ~ $ foral l t _> 0 and/t(t) ~ "Yfor all t _> Remark 8.5. The problems we deal with in the preceding two theorems belong to the constrained global stabilization problem via state feedback, as defined in [11], by choosing the constrained output z = u.

8.7.

Main Results for Right-invertible Constraints

The solvability conditions for systems with right invertible constraints are relatively easy to describe, mainlybecause, as will be evident soon, they do not dependon the shape of the constraint set. That is, with right invertible constraints, if the constrained semiglobal or global output regulation problemsare solvable for one specific pair of constraint sets satisfying Assumption 8.1, then these problems are also solvable for all other constraint sets satisfying Assumption 8.1. However, for systems with non-right invertible constraints, the situation is different as we shall see in the next section. It is worth noting here that the right invertible constraints include as a special case the actuator amplitude and rate constraints. Wheneverwe have constraints only on the control variable u, we can set Cz = 0 and Dz = Im so that z = u. In other words, since Ezu can easily be verified to be right invertible, the amplitude and rate constraints on actuators are indeed right invertible constraints. In this sense, the workof [19] is a special case of the present work.

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The main results for right invertible constraints are stated in the ne~:t subsection. The proofs of the theorems are constructive and involve a detailed decomposition of the underlying system. For a concise presentation, all of the proofs are presented in another subsection.

8.7.1.

Results

The following theorems provide solvability conditions for the problems posed in Section 8.4 when the constraints are right invertible. The first theorem concerns Problem 8.1, the global state feedback output regulation problem. Theorem8.7. Consider the system E as given by (8.1), two constraint sets $ and "Y that satisfy Assumption8.1, and a given bounded set ]4;0 C ~s. Assumethat set $ is bounded. Moreover, let Assumptions 8.2, 8.3, and 8.4 hold. Also assume that the constraints are right invertible. Define the system (8.7) based on the data given in (8.1) and the sets $ and 7-. the constrained global state feedback output regulation problem (i.e. Problem8.1) is solvable if the following conditions hold: 1. The constraints

are at most weakly non-minimumphase.

2. The constraints are of type one. 3. There exist Pl, P~ ¯ (0, 1) such that (CzII + DzF)W(t) (1 - p l)

(CzII +Dzr)Sw(t)(1- p~)

(8.15)

for all t _> 0 with w(0) ¯ 1410. Moreover, the first and second conditions are necessary. The next two theorems are concerned with the constrained semiglobal output regulation problems, one utilizing a state feedback controller and the other a measurementfeedback controller. Theorem8.8. Consider the system E as given by (8.1), two constraiut sets $ and q~ that satisfy Assumption8.1, and a given bounded set ]4;0 C ]~8. Moreover, let Assumptions 8.2, 8.3, and 8.4 hold. Also assume that the constraints are right invertible. Define the system (8.7) based on the data given in (8.1) and the sets $ and :r. Then the constrained semiglobal state feedback output regulation problem (i.e. Problem 8.’,2) is solvable if the following conditions hold:

Output Regulation with Constraints 1. The constraints

211

are at most weakly non-minimumphase.

There exist Pl, P2 E (0, 1) such that (8.15) is satisfied for all t with w(0) 3. Hwis a bounded signal. Moreover,the first condition is necessary. Remark 8.6. The third condition in the theorem is not necessary and can be removed. In the absence of Condition 3, uc has to be designed as a nonlinear feedback. However, with Condition 3 uc can be designed as a linear feedback law. Theorem8.9. Consider the system E as given by (8.1), two constraint sets S and 7. that satisfy Assumption 8.1, and a given bounded set ld20 C ms. Moreover, let Assumptions8.2, 8.3, and 8.4 hold. Also assume that the constraints are right invertible. Define the system (8.7) based on the data given in (8.1) and the sets S and 7". Then the constrained semiglobal measurement feedback output regulation problem (i.e. Problem 8.3) is solvable if the following conditions hold: 1. The constraints

are at most weakly non-minimumphase.

2. The constraints are of type one. 3. Hwis a bounded signal. Moreover,the first condition is necessary. Remark8.7. Condition 8.15 is essentially the constraint condition of the reference/disturbance signal. Note that, if from u to e is left-invertible, then the condition (8.15) that all three theorems is close to being necessary in that if there signal w(t) so that

compatibility the subsystem is required is a reference

(CzII + Dzr)W(t) or(CzH + Dzr)S w(t) ¢ 7" for some t > 0, then the constrained semiglobal/global state feedback output regulation problemis not solvable (see also [5, 7]). Remark 8.8. As indicated in Theorems 8.7, 8.8, and 8.9, one of the necessary conditions for the solvability of any of the Problems8.1, 8.2, and 8.3, is that the constraints be at most weakly non-minimumphase. This

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is due to the fact that all the Problems8.1, 8.2, and 8.3 require internal stability either in the semiglobal or global sense. As shownin [11,12], it; is necessary that the constraints be at most weakly non-minimumphase to achieve internal stability either in the semiglobal or global sense. Remark 8.9. Consider the case when we have constraints only on actuator amplitude and rate, i.e. let Cz = 0. In other words, a subset of the input channels is subject to amplitude and rate constraints. Then... it is straightforward to showthat the constraint invariant zeros of P., i.e. the invariant zeros of the subsystem Ez~, coincide with a subset of the eigenvalues of A. This observation implies that the requirement of at most weakly non-minimumphase constraints in Theorems8.7, 8.8, and 8.9 is equivalent to requiring that a particular subset of eigenvaluesof A lie in the closed lefthalf plane. Obviously, such a condition is always satisfied if we are dealing with asymptotically null controllable systems with bounded controls. It; is interesting to consider two cases. One case corresponds to Cz being zero and Dz = Im, that is, all the input channels are subject to amplitude and rate constraints. In this case, the constraint invariant zeros of E coincide with all the eigenvalues of A. Therefore the requirement of at most weakly non-minimumphase constraints in Theorems8.7, 8.8, and 8.9 is equivalent to requiring that the given system be asymptotically null controllable with bounded controls. Another case corresponds to the situation where the subsystem Ezu does not have any invariant zeros, i.e. E does not have any constraint invariant zeros. Hence, for this special case there will not be any constraints on the eigenvalues of A, and the condition of at most weakly non-minimumphase is automatically satisfied. Remark 8.10. Wewould like to point out an important fact, namely, the solvability conditions as given by the previous theorems are independent of any specific shapes of the given constraint sets $ and 2r. In other words, under right invertible constraints, if the constrained semiglobal or global output regulation problems are solvable for some given constraint sets satisfying Assumption8.1, then these problems are also solvable for all other constraint sets satisfying Assumption8.1.

8.7.2.

Proofs

of Theorems

Somepreparations are needed before we get into the proofs. As indicated by the solvability conditions, the proofs of the theorems will largely rely on the structure of the underlying system. For this reason, the original system needs to be rewritten in a special coordinate basis so that the system properties involving invariant zeros and infinite zeros are revealed

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213

naturally. This will greatly facilitate the design of appropriate regulators. A detailed special coordinate basis (scb) is presented in [14,16]. Using the scb coordinates for state, input and output spaces, the subsystem Ezu can be rewritten as (we slightly abuse the notation of scb given in [14, 16] to keep somesystem variables unchanged):

~z = \A21 Z

:

(Cz,1

A2~ Cz,2)

+ Bz

u+

<)

K~ z

(8.16)

x2

where X1 E RTM and x2 E Rn2. This decomposition renders the subsystem characterized by the quadruple (A22, Bz, Ca,z, Dz) strongly controllable and without finite zeros (see [14, 16]). Also, there exists a matrix H such that Azl = B2Hand DzH= 0. Furthermore, if the subsystem is right invertible, then we have Cz,~ = O. After adding the measurementand error equation as well as the exosystem, the system (8.1) with right invertibility constraint can be rewritten as

(

2~1)

:

(’All

Azz

x2

B2

u+

Kz

z+

E2

y -~-(Cy,1

w

(8.17)

z = (0 In view of Assumption8.2, let (H, F) be a solution to the regulator equation (8.3) and decomposeII to be compatible with the above decomposition HT -- (II~ II~). In terms of the new system matrices in (8.17), the regulator equations in (8.3) become II~S = A~IH1 + K~ + E~ [I2S = AzlH~ + Ae~IIz + B2F + Kz~ + E2 0 = Ce,lII~ + Ce,~H~ + De

{

where ~ = Cz,~II2 + DzF.

(8.18)

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Saberi et al.

Also, we introduce the following new variables,

5 = z-~w,

~ = u-Fw.

Then, by virtue of (8.18) it is easy to verify that system (8.17) in terms the new variables becomes ~:1 = AI~:~I + K15

{

~2

A2121 + A~22 + B~fi + K~2 C~,2~ +Dz~ e C~,~1+C~,~2~

(8.20)

This new system has the property that if we design for fi in the above system such that the closed-loop is internally stable then we have e(t) ~ Since z = 2 + ~w, we need to ensure that 5 + ~w satisfies the amplitude and rate constraints, i.e. 5(t) ~w(t) ~ 8,~(t) + ~Sw(t) ~ T for all t ~ 0. In this sense, we have transformed the constrained output regulatkm problem into a constrained stabilization problem [11]. Note t~at in [11] the rate constraint is imposedon z(t) as 2(t) ~ T for all t > 0, excluding t = to avoid sometechnicalities. With the help of the constraint operator as~T we introduced in Section 8.3, this tech~cal point is easily resolved. This will becomeclear in the construction of control laws as given below. Wefirst prove Theorem8.7 with regard to the constrained global state feedback output regulation problem. Because this problem requires that the subsystem Ezu has no infinite zeros of order greater than one, the controller design for a system with this structure is relatively easy. However,in the semiglobal problems, we do not have this condition on the order of infinite zeros. Then the design becomessequential in that we have to go through the order of infinite zeros step by step to get a complete design. Nevertheless, the most basic design concepts of this work are contained in the proof of Theorem 8.7. Proof of Theorem 8.7. The necessity of the first and second conditions follows from the requirementof global stability and the results in [11].. The proof of the s~ciency is based on an explicit construction of the control law. The design follows two steps. Weview the system (8.20) as an interconnection of two subsystems ~ in [11]. First, we focus on the subsystem ~ = A~121 + K150 + KlV.

(8.21)

where ~ = ~0 + v while viewing 50 as an input variable. Note that the conditions of the theorem imply that all the eigenvalues of A~ are in the

215

Output Regulation with Constraints

closed left-half plane. Next, we construct a state feedback law ~0(t) f(21 (t)) such that it satisfies the constraints

¯

¯

and

n

(8.22)

for all t >_ 0, where p~,p~ are ~s given in (8.15) and BNis defined by BN = {¢ e ~P ] ]];~ < N} for some g > 0, while rendering the zero equilibrium point of the closed-loop system of (8.21) and ~0(t) = f(~(t)) globally attractive (i.e. ~(~) ~ 0 ~ t ~ ~) in the presence of signal satisfying: [[v(t)[[

Me-~t, t ~ 0

(8.23)

for given M> 0 and 5 > 0. Moreover, the feedback law 20(t) f( 2~(t)) should render the zero equilibrium point of the closed-loop system with v = 0 locally exponemially stable. Such a nonlinear feedback law 5o(t) f(2~ (t)) can be obtained from [18]. In the second step we design a suitable ~ such that for any initial condition x0 ~ A(S) the output 5 is such that v(t) := 5(t) f( ~l(t)) sa tisfies (8.23). Moreover, the overall closed loop system is asymptotically stable. Finally we have to showthat the constraints are s~tisfied for the original system. Recall that in scb the following matrices have finer structures: Cz,2

(o)

= 622

’ Dz=

0

’

(o) ° o o) B2

B22 B23

where D~ and C~B2~are invertible. Accordingly, we make the following compatible decomposition ~, 2, and f,

£=

525~, 5=

a35~’ f(~)=~A(~,)]’F=

F3F~andfl=

The assumptions on the sets S and T guarantee that we can decompose the sets S and T compatible with the decomposition of 5, S = $1 x Sa, and T = ~ x ~P:

(8.24)

such that z ~ S if and onl~ if ~ + ~w e S~ and ha + ~2w ~ $2. Similarly £ ~ T if and only if ~ + FxSw~ ~. The properties of scb also guarantee that one can choose 53 = F~ such that the system (8.20) with inputs 5~ and 52 and output £ is invertible and the additional invariant zeros introduced

216

Saberi et al.

by the feedback ~3 = F~ are placed at desired locations in the open left half plane. With this choice of ~3 we obtain

}2 = C22(A21~1+ A22~2+ B23F~+ K25) + C22B22~2. We further choosethefeedback laws: fil = D~llfl(~1) and

~ = (C~B~)-~ ( - C~(A~ + A~

-~[~ where fi > 0 can be arbitrary large. Weemphasize ~hat ~he above feedback laws are ~ime invarian~ and nonlinear s~a~e feedback laws. Nex~ we show ~ha~ wi~h the above choice of ~1, ~, and ~ the cons~rain~s on z = ~ + ~w are satisfied, i.e. z(~) ~ ~, ~(~) ~ T. Firs~ from choice of fi~ i~ is obvious

~(~) + ~s~(~) for M1~ ~ 0 using (8.1~) and (8.22). On ~he o~her hand, for ~his particular choice of ~ we find:

where z~ = ~ + ~ and hence:

+e -St

ft J0

[~2~(T)+

We know:

~2w(t)C(1- pl)$2 f2(3~1(t)) ~sw(t) c M~e~& ~(~(t)) c

~f2(~l(T))]

dT

Output Regulation with Constraints

217

for suitable M1, M2, 7 >- 0 where the first set membershipis a consequence of (8.15), the secondfollows from (8.22), the third is guaranteed for suitable M1and 7 because w is generated by an autonomous linear system and the final set membershipagain follows from (8.22) provided M2is large enough. Wefind for any t > 0: z2(t)

:e-5tal

(1 -e -at ) (1 -~)a2+te -St (M~e~t + M2) a3

with hi, a2, a3 ~ ~2. Since 82 is convexand contains 0 in its interior we are guaranteed that z2(t) ~ provided: e-ht+(1-e-ht)(1-~)+te-ht(Mle~t+M2)

(8.25)

for all t k 0. Clearly for t k 1 the inequality is satisfied for 5 large enough. For t ~ [0, 1] we can find the following upper boundfor the left hand side: e-*t + (1 - e -*t) (1 - ~) te -*tM~

(8.26)

%r suitably chosen M3> 0. But it is also easily checked that the deriwtive of (8.26) is unequal to zero for t ~ [0, 1] for 5 l~rge enoughand hence ~ttMns its maximumfor t = 0 or t = 1. For t = 0 and t = 1 we have (8.26) less than or equal to 1 and hence we find that the inequality (8.25) is indeed satisfied and z2(t) ~ ~2 for all t ~ 0 as required. In the following we showthat (8.23) holdsas well. Let vl(t) := 5l(t) fl(~(t)) v2(t ) := ~ 2(t) - A( 2~(t)). Obvi ously, va(t ) = 0 for all t ~ 0. On the other h~nd, it is clear that vz(t) satisfies +2(t) -hvz(t). Since S is bounded,it is easy to verify that v2(0) is bounded. Thus, there exists M > 0 such that ~l~(t)~ ~ -~. Noting v(t) = (v~(t), v~(t)) z, we have shownthat (8.23) holds. . In this way we have obtained a controller which stabilizes the system. Inequality (8.27) guarantees that 2~(t) ~ 0 as t ~ ~, which in guarantees that 5(t) ~ 0. Since the second subsystem is minimumphase and invertible, it implies that k~(t) ~ 0 as t ~ ~. Therefore, we h~ve e(t) ~ ast ~ ~. The above design yields the following input u = ~ + Fw. This feedback will satisfy Ml the constrMnts for t > 0. In order to satisfy the constraints at time t = 0 we apply uc = ~ + Fw to the system (8.7) or, equivalently, the following dynamicfeedback as.,~. (~ + Fw) to the original system (8.1). Wecan show quite easily that there exists ~ T > 0 such that for t > T we have uc(t) = as,,%(uc(t)) ~nd, using this property, it c~n be shown

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Saberiet ,al.

this dynamic feedback solves the constrained global state feedback output regulation problem while satisfying the constraints for all t _> 0. [] Proof of Theorem8.8. The necessity of the first condition in the theorem is a consequenceof the requirement of semiglobal stability (see [11]). The sufficiency of the conditions is based on an explicit construction of a feedback law. By the transformation (8.19), as we did before, the constrained semiglobal output regulation problem can be converted to the corresponding constrained semiglobal stabilization problem. The main ideas have already been presented in the proof of Theorem8.7. The only difference in the semiglobalcase is that the order of infinite zeros of the subsystem Ezu could be greater than one. In this case, the semiglobal stabilization has to follow a sequential design leading to a controller of the form:

u = F(x- l-lw)+

(8.28)

The algorithm developed in [11] can be applied to the system (8.20). For the details we refer to [11]. Obviously, this feedback only satisfies the constraints for t > 0 and in order to resolve this additionM issue at time 0 we apply the dynamic feedback u = crs~,% (F(x Hw ) + Fw

Proof of Theorem 8.9. controller can be designed as

(8.29)

An observer based measurement feedback

(D= (0 u = ~s.,,~, (F~+ (r - Fn)~) where we have used the state feedback law (8.29) with x and w replaced by their measurements. Let Re be the stabilizing solution of the following equation,

0

S+~I

R+R

0 Dy) R + gI = 0.

Wecan then choose the observer gain as (8.30)

Output Regulation with Constraints

219

Choose any small period of time T > 0. Because the constraints are of type one, it is possible to choose an g large enough so that the estimation error (~, @) = (x, w) - (~, @) can be made arbitrarily small after meanwhilethere is no violation of the constraints because of peaking. This guarantees that

satisfies uc = as~,T~(ue) for t _> T, i.e. the saturation ersu,=q is no longer active after time T and the closed loop system operates in the linear region. The analysis after time T follows along similar lines as those,of state feedback regulation design. Wenote that during [0, T], the state resides in (1 + ()X, whereA" is the set of initial states and ( can be arbitrarily small since we can makethe estimation error g decay to zero in any short period of time by choosing a sufficiently large g. Hence, the constrained output still remains in the interior of the constraint set. []

8.8.

Output Regulation with Non-right-invertible Constraints

As we mentionedearlier, for non-right invertible constraints, the solvability conditions intrinsically depend on the shape of the given constraint sets S and 7". Although one could develop certain necessary conditions, finding necessary and sufficient conditions under which Problems 8.1, 8.2, and 8.3 can be solved for non-right-invertible constraints is still an open and challenging problem. However, if we strengthen these problems by requiring that they be solved for all possible sets S and 7" that satisfy Assumption 8.1 (and not merely the given sets), then we can formulate a set of necessary and sufficient conditions that do not depend on the shape of the constraint sets S and T. In what follows we focus on this. Before we state our results, we need to develop certain preliminaries. Consider the decomposition given in (8.16). Wenote that for the nonright invertible constraints, in general Cz,1 ¢ 0 in (8.16). In this case, the transformed system (8.20) for stabilization becomes

wherex~,x2,u areas defined in(8.19). Thistimewe alsoapplied a basis transformation ~’zon theconstraint output2 = Fz~suchthatin scb,the

220

Saberi et al.

5 equation can further be decomposedas 2 ~---

~2 ~ Cz,21

~

Note that an output transformation changes the constraint sets S and T and transforms them in new constraint sets ~ and ~ respectively. This decompositionreveals that somepart of the constraint output is not directly controlled by the input. In fact, this part of the constraints causes the shape dependence of the solvability conditions in the non-right invertible constraints. To proceed, we extract from (8.31) the following subsystem which still has input and output constraints:

{

~ = A~x~ + K~ E~ : ~ Cz,~x~

(S.33)

2 in the state equation contains both 2~ and ~2. But ~2 can be eliminated by using the output equation which expresses ~2 in terms of x~. After this elimination, this subsystem is further decomposedas

]

(8.34)

where x~ represent the zero dynamicsof F~I (see [14, 16]). The following theorem is a consequence of [11]. Theorem 8.10. Consider the system F~ as given by (8.1), two constraint sets S and 7- that satisfy Assumption8.1, and a given bounded set ~/V0 C ]Rs. Moreover, let Assumptions8.2, 8.3, and 8.4 hold. Furthermore, assume that E = 0 (i.e. consider a tracking problem). Define the system (8.7) based on the data given in (8.1) and the sets $ and 7-. Assume there exist pl, P2 @(0, 1) such that (8.15) holds. Then the following statements are equivalent: 1. The constrained semiglobal state feedback output regulation problem (i.e. Problem 8.2) is solvable for all constraint sets $ and 7- satisfying Assumption8.1. 2. A3~ = aI with a <_ 0, the eigenvalues of A~are in the closed left-half plane, and (A~I, K~) stabilizable.

Output Regulation with Constraints

8.9.

221

Tracking Problem with Non-minimum Phase Constraints

As seen earlier, we must have at most weakly non-minimumphase constraints to achieve constrained semiglobal or global output regulation. In fact, we must have at most weakly non-minimumphase constraints even to stabilize the plant in semiglobal or global sense. Supposewe have strongly non-minimumphase constraints. Evidently then neither the constrained semiglobal nor the global output regulation problem is solvable. An interesting question is what can be done in this case. In an attempt to answer this question, let us first recall the following definitions. Definition 8.9. Consider (8.1) with w(0) = 0 and let the constraint sets S C ]~P and ’Y C I~P satisfy Assumption8.1. The set qPs c .4(8) x ]I~ is a constrained stabilizing domain if there exists a state feedback law (possibly nonlinear) of the form,

u = g(z, v) such thkt 1. For any (x(0), v(0)) Pswe have x(t ) -~ andv(t) - -* 0 as t -~oc. 2. For any (x(0),v(0)) ¯ Ps we have the solution of the closed-loop systemsatisfying z ¯ $ and i ¯ 2F for all t ___ 0. Definition 8.10. Consider the system (8.1) with two constraint sets S c ~P and "Y c ~P satisfying Assumption 8.1 and a set W0_C ]~s. The set Pt C A(S) × is a c onstrained tra cking domain if the re exi sts a state feedback law (possibly nonlinear) of the form (8.8) such 1. For any (x(0), v(0)) ¯ Pt and w(0) = 0 x(t ) -* and v(t) -~

2. For any (x(0), v(0)) Pt, and fo r an y w(0) ¯ W0 thesolution of t closed-loop system satisfies (a)

z¯Sandk¯Tforallt_>0.

(b) limt--.~ e(t) = (c) (x(t),v(t)) ¯ T’t for all t >_ 0.

222

Saberi et al.

The main theorem in this section states that, whenever we deal with only tracking problems (i.e. E = 0 in (8.1)) and only magnitude constraints (i.e. 7" = Hn), the constrained tracking domainPt can be as large as the constrained stabilizing domain Ps provided that there exists some e > 0 such that for all 1420satisfying (8.15) and for all w(0) E l/Y0 we have IIw(t) (1 - e)Ps for all t _> 0. This result is in ttme with the result of a recent paper [1] except that our work is suitable for tracking sinusoidal signals and not just for (asymptotically) constant reference signals. Also, our work leads to a method of constructing an appropriate regulator in the case of right invertible constraints. Theorem 8,11. Consider the system (8.1) with E = 0 (i.e. a tracking problem), two constraint sets $ C ][~P and 7" = ]~P (i.e. no rate constraints). satisfying Assumption 8.1, and a given bounded set ~20 C ~s. Define the system (8.7) based on the data given in (8.1) and the set $. Assumptions8.2 - 8.4 hold. Also, assume that there exist Pl ~ (0, 1) such that for all w(0) e ~4~o we have (CzH + DzF)w(t) (1- P l) $ for all > 0. Furthermore, suppose that there exists a state feedback controller that can stabilize the system E with a constrained stabilizing domain Ps. Finally, assumethat there exists an e > 0 such that for all w(0) ~ ]/Y0

II (t) E(1

(8.35)

for all t > 0. Then, any set Pt in the interior of the closure of the set Ps is a constrained tracking domainas defined in Definition 8.10. Proof of Theorem8.11. Note that the result of Theorem 8.11 differs from the results of [1] in that our reference signal is not asymptotically constant as in [1]. The proof therefore requires some modification of the techniques in [1]. To proceed, suppose we already have a controller f which stabilizes the system with domain of attraction T’s. Moreover (8.35) satisfied. Then there exist a set P0, a homogeneousfeedback ](x), and /~ > 0 such that the following hold: 1. Pt C Po C P~. 2. The set P0 is closed. 3. C~x + Dzf(X) ~ for al l x ~ P0. 4. Wehave IIw(t) 6 (1 - ~) P0 for all t _> 0 with w(0) e YP0, and all t _~O.

Output Regulation with Constraints

223

5. The controller u = ](x) guarantees that ¯ po(x(t)) := inf{ (~ > 0: ½xe :P0 } < e-flt~o

(x(O)

(8.36)

for all initial conditions x(0) Note that the condition (8.36) implies in particular that the set :P0 is invariant set of the closed loop system after applying the feedback u = ](x). The construction basically requires finding a homogeneousfeedback and a domainof attraction sufficiently close to the set Ps for the system ic = (A +/5I)x + Bu, (8.37)

z -- Czx + Dzu.

The factor /5 in this modified system will then automatically guarantee (8.36) for the original system. The main technical issue is to prove that for small enough /5 we can find a domain of attraction arbitrarily close to Ps for this modified system while still satisfying our constraints. This property and existence of a suitable feedback can be easily shownbased on, for instance, the ideas in [10]. The main problemis that it is easy to prove the existence of a suitable feedback ] and an associated invariant set P0 but muchmore difficult to actually construct it. Next we define the mapping,

1

v) E:P0},

¯ (x,v) = inf{5 > 0 : v + ~(x-

and the following controller for the system (8.1),

=/(x

-

where @= [1 -~(x, IIw)] w. Then we can use for the rest the same argumentsas in [1] to prove that this controller achieves stability with the desired domain of attraction and output regulation, meanwhilesatisfying our constraints. [] 8.10.

Conclusions

Output regulation problems of linear plants with constraints are addressed in this chapter. Constraints are modeled in terms of what istermed as a constraint output. The constraint output is a function of both the state as well as control variables. The amplitude and the rate of change of the constraint output are restricted to lie in certain sets termed as constraint

224

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sets. Such a model represents the constraints in a broad general framework. Based on such a model, a taxonomyof constraints developed earlier in connection with internal semiglobal and global stabilization of systems with constraints plays a dominating role in output regulation problems as well. Under so called right invertible constraints, we have given here the solvability conditions for both the semiglobal and global output regulation problems by considering both the state as well as the measurement feedback controllers. Wheneverthe solvability conditions are satisfied, the proofs of our results show clearly howto construct the required controllers or regulators that solve the posed output regulation problems. For so called non-right invertible constraints, the posed output regulation problems are challenging and are still open for the general case. However,by strengthening the constrained output regulation problems by requiring that they be solved for all possible constraint sets that satisfy a certain basic assumption (and not merely the given constraint sets), even for non-right invertible constraints, necessary and sufficient condition are provided under which such problems are solvable. One of the basic and important solvability conditions is that the constraints be at most weakly non-minimumphase. For strongly non-minimum phase constraints, it is shownthat wheneverthere exists a controller which achieves stabilization with a certain domainof attraction, it is also possible to achieve tracking (but not disturbance rejection) with the same domain of attraction. References

[1]F.

Blanchini and S. Miani. Any Domain of Attraction for A Linear Constrained System is A Tracking Domain of Attraction, SIAM J. Contr. ~ Opt., 38(3) (2000) 971-994.

[2]J.V.

Breakwell. Time Optimal Pursuit in A Circle. in: T.S. Ba~ar and P. Bernhard, editors, Differential games and applications, pages 72-85. Springer Verlag, 1989.

[3] H.T. Croft. "Lion and Man": A Postscript, Soc., 39 (1964) 385-390.

Journal London Math.

[4] J. Flynn. Lion and Man: The General Case, SIAM J. Contr., (1974) 581-597.

12(4)

[5] J. Han, A. Saberi, A.A. Stoorvogel, and P. Sannuti. Constrained Output Regulation of Linear Plants, in: Proc. 39th CDC,Sydney, Australia (2000) 5053-5058.

Output Regulatfon wfth Constraints

225

[6]P.

Hou, A. Saberi, Z. Lin, and P. Sannuti. Simultaneously External and Internal Stabilization for Continuous and Discrete-time Critically Unstable Systems with Saturating Actuators, Automatica, 34(12) (1998) 1547-1557.

[7] Z. Lin, A.A. Stoorvogel, and A. Saberi. Output Regulation for Linear Systems Subject to Input Saturation, Automatica, 32(1) (1996) 29-47. [8] J.E. Littlewood. (1953).

A Mathematician’s

Miscellany,

Methuen & Co.,

[9]A.

Megretski. L2 BIBOOutput Feedback Stabilization with Saturated Control, in: Proc. 13th IFAC world congress, volume D, San Francisco (1996) 435-440.

[10]C. van Moll. Stabilization of the Null Controllable Region of Linear Systems with Bounded Continuous Feedbacks. Master’s thesis, hoven University of Technology, 1999.

Eind-

[11] A. Saberi, J. Han, and A.A. Stoorvogel. Constrained Stabilization Problems for Linear Plants. To appear in Automatica, 2000.

[12]A. Saberi, J. Han, A.A. Stoorvogel, and G. Shi. Constrained Stabilization Problems for Discrete-time Linear Plants. Submitted for publication, 2000. [13] A. Saberi, Z. Lin, and A. Teel. Control of Linear Systems with Saturating Actuators, IEEE Trans. Ant. Contr., 41(3) (1996) 368-378. [14] A. Saberi and P. Sannuti. Squaring Downof Non-Strictly Proper Systems, Int. J. Contr., 51(3) (1990) 621-629.

[15]A. Saberi, A.A. Stoorvogel, and P. Sannuti. Control of Linear Systems with Regulation and Input Constraints, Communication and Control Engineering Series, Springer Verlag, (2000).

[16]P. Sannuti and A. Saberi. Special Coordinate Basis for Multivariable Linear Systems-Finite and Infinite Zero Structure, Squaring Downand Decoupling, Int. J. Contr., 45(5) (1987) 1655-1704.

[17]G. Shi, A. Saberi, and A.A. Stoorvogel. On Lp (lp) Performance with Global Internal Stability for Linear Systems with Actuators Subject to Amplitude and Rate Saturation, in: American Control Conference, Chicago, IL (2000) 730-734.

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[18] G. Shi, A. Saberi, and A. Stoorvogel. On the Stabilization and Performance of Linear Plants with Actuators Subject to Amplitude an,:l Rate Saturations. In preparation, 2001. [19] G. Shi, A. Saberi, A.A. Stoorvogel, and P. Sannuti. Generalized Output Regulation for Linear Systems with Actuators Subject Amplitude and Rate Saturations, in: Proc. ACC(2000) 1240-1244. [20] A.A. Stoorvogel and A. Saberi. Output Regulation of Linear Plants with Actuators Subject to Amplitude and Rate Constraints, Int. J. Robust gJ Nonlinear Control, 9(10) (1999) 631-657. [21] A.A. Stoorvogel, A. Saberi, and G. Shi. On Achieving Lp (£p) Performancewith Global Internal Stability for Linear Plants with Saturating Actuators, in: Proc. 38th CDC, Phoenix AZ (1999) 2762-2767.

Chapter 9 Optimal Windup and Directionality Compensation in Input-Constrained Nonlinear Systems M. Soroush Drexel University,

Philadelphia,

Pennsylvania

P. Daoutidis University

9.1.

of Minnesota, Minneapolis,

Minnesota

Introduction

Whena plant with actuator saturation nonlinearities is controlled with a dynamic controller, the closed-loop stability and performance maydegrade significantly due to directionality and/or windupeffects. This chapter surveys definitions of these two phenomena,presents an optimal directionality compensator, and develops a dynamic nonlinear controller which compensates for windup and allows for optimal performance in the presence of saturation. Specifically, the notion of directionality in input-constrained systems is defined, and the class of plants that do. not exhibit the directionality are characterized. The performance of the optimal directionality compensator is shownand comparedwith those of clipping and direction preservation, by linear and nonlinear examples. Given a controller output, the directionality

227

228

Soroush and Daoutid’,!s

Directionality Compensator

Figure 1: Directionality

compensation.

compensator calculates an optimal feasible (constrained) plant input that results in a plant response as close as possible to the response of the same plant to the controller output. The compensator can be used for both linear and nonlinear plants, irrespective of the type of controller being used. A nonlinear dynamic controller which includes windup compensation is then presented. In the absence of input constraints, the proposed controller is input-output linearizing, whereas in the presence of input constraints it provides the flexibility of achieving performance optimality and a desired region of closed-loop asymptotic stability. The application and performance of the controller are demonstrated by a chemical reactor example. 9.2.

Directionality

and

Windup

Whena plant with actuator saturation nonlinearities is controlled by an analytical dynamic controller, the closed-loop response maybe considerably poorer than an integral-of-squared-error (ISE) optimal response. Examples of analytical controllers are PID controllers, internal modelcontrollers and input-output linearizing controllers. This poorer closed-loop performance can be due to windup and/or directionality. If the directionality is not compensated for, then the plant actuators naturally render the controller output feasible by clipping (limiting) the controller output components. The problem of directionality compensationis that of calculating a feasible plant input on the basis of a given unconstrained controller output (see Figure 1). It is worth noting that these two problems are not present in modelpredictive control, in which constraints are explicitly accounted for and the controller action is solution to a constrained optimization problem. 9.2.1.

Directionality

The phenomenon of directionality usually occurs in multiple-input; multiple-output (MIMO)plants. In single-input single-output (SISO) plants

Optimal Windup and Directionality

Compensation

229

with actuator constraints, the boundary of the plant-input feasible set (which is naturally closed and convex) consists of only two isolated points, and when the controller output is infeasible, one of the two points that is closest to the unconstrained controller output usually yields an optimal response, i.e. one that is closest to the response of the same plant to the unconstrained controller output. In other words, in SISO plants clipping (limiting) an unconstrained controller output usually leads to an optimal feasible plant input. In MIMO plants with actuator constraints, however, the boundaryof the plant-input feasible set consists of an infinite numberof points, and whenthe controller output is infeasible, the feasible point that is closest (in the plant input space) to the unconstrained controller output may not yield an optimal (in the sense described above) response. other words, in MIMO plants clipping the components of an unconstrained controller output maynot lead to an optimal feasible plant input. Compared to integral windup, the phenomenon of directionality has received less attention. In purely analytical control methods, a feasible plant input, u, has been obtained by one of the following methods: ¯ Clipping [8,28]:

where w is the controller output, and ut~ and Uh~are respectively the lower and upper limits on a plant input ue. Direction preservation [4, 6,12, 17]: ue=wemin(Sa~w) ~

Wl

,...,

satin(w) Wm

, ~=l,.-.,m

In [4], the direction preservation approach has been suggested for directionality compensationin plants with ill-conditioned steady state gain matrix. Optimization formulation of the conditioning technique [7,27]. In this approach, whena controller output is infeasible, a feasible controller output is obtained by calculating (via optimization) a new setpoint value that is closest (in the setpoint space) to the original setpoint value and yields a feasible controller output. Controller detuning [15]. Whencontroller output is infeasible, an optimization problem is solved to obtain the values of controller tunable parameters that result in a feasible controller output.

230

Soroush and Daoutidis ¯ Optimal directionality

compensation of Soroush and Valluri [21, 22].

Unlike SISO plants, in MIMO plants clipping and direction preservation may lead to completely different feasible plant inputs and may steer the plant in wrong directions, leading to very poor closed-loop performance. The directionality compensation via optimization formulation of the conditioning technique mayalso lead to poor performance, because it is based on the controller being used but not on the plant being controlled. For example, whencompletely decentralized control is used, this methodis identical to clipping, irrespective of the nature of the plant under control. 9.2.2.

Windup

Windupis another controller performance degradation phenomenonassociated with actuator saturation, which is typically exhibited by dynamic controllers with slow or unstable modes[6] (a special case being the PI/PID controllers that can exhibit integrator windup). Although this phenomenon has been studied extensively, only a few attempts have been made to define it precisely. Furthermore, while closed-loop-response quality indices such as response time and overshoot have been used to document the presence of windup, at the present time there is no specific measure to quantify windup. Wenote two criteria that have been proposed to cheek whether a dynamic controller does/does not exhibit windup; Criterion 1 [4]: A dynamic controller does not exhibit windup, if the states of the controller are not driven by the error whenthe actuator is in saturation. Criterion 2 [9, 10]: A dynamiccontroller does not exhibit windup, if whenthe actuator is in saturation the closed-loop behavior under the controller is identical to that under a ’reference’ static state feedback law (this characterization is based on the realization that windupis not associated with static feedback controllers). In linear analytical control, the issues of windupand constraint handling as well as closed-loop stability in the presence of input constraints have been studied extensively [1, 2, 5, 7-9, 13, 15, 23, 27, 28]. In nonlinear analytical model-basedcontrol, these issues have also received considerable attention in recent years. More specifically, there have been several approaches to !he problem of windup in input-output linearizing control methods. These include: ¯ Conditional integration (i.e.

turning off integration whena constraint

Optimal Windup and Directionality

Compensation

231

is active). This approach was employedin real-time nonlinear control of pilot-scale polymerizationreactors (e.g. [20]). ¯ Modelpredictive control (MPC)formulation of input-output linearization [19, 26]. Linear anti-windup (or MPC)schemes combined with control laws that enforce input-output linearization, even in the presence of constraints, thus translating the constraints on the manipulated input into state-dependent constraints on the input to the linearizing feedback loop [3,12,14,16]. An observer-based anti-windup approach with a nonlinear gain [10]. This approach allows attenuation of the effect of windup(due to the controller dynamics), at a desired (arbitrarily fast) rate, with asymptotic closed-loop stability being ensured. 9.2.3.

Organization

of this

Chapter

Section 9.3 presents the definition of directionality and the optimal directionality compensator. It also describes the scope of the work and the application and performance of the directionality compensator via numerical simulations. In Section 9.4, a dynamicinput-output linearizing control law that can handle input constraints and constant disturbances and model errors is derived. The performance of the nonlinear control law is shown by a chemical reactor example. 9.3.

Optimal

9.3.1.

Scope

Directionality

Compensation

Weconsider the class of general but affine-in-control, nonlinear multivariable plants ~iescribed by a state-space model of the form 2y ==f(x)h(x) +g(x)u } where

x = Ix 1 ..-

xn] T e Z C ~}~n×l,

l/~

~-

[Ul

(9.1) "*"

Um] T ¯ U C ~rn×l

and y = [Yl "’" Yra]T ¯ ~m×l are the vectors of state variables, plant inputs (manipulated inputs), and controlled outputs respectively. Here U = {u]ul~ <_ ue <_ Uh~, g = 1,...,m}, where ul~, Uh~, g = 1,...,m, are scalar constant quantities, and X is an open connected set. A controller output w is said to be feasible, if and only if w E U. It is assumed that:

232

Soroush and Daoutidis

gl(x), "", gin(x), h(x) and f(x) are smooth vector functions, where gj(x) represents the jth columnof the matrix g(x); the plant is minimumpha~,;e (has asymptotically stable zero dynamics); each controlled output y~ has a finite relative order (degree) ri, which is the smallest integer for which r Lr,-lh " \] locally [LglLrf,-lhi(2c)... ~-gm f i[x)l # O; the characteristic (decoupiing) matrix of plant is locally nonsingular. The characteristic matrix is an m × m matrix whose ij th entry is Lga L}.’- ~ hi(x); it will be denoted C(x). Here L~, and Lg~ are Lie derivative (in the directions of the vectors f and g~ respectively) operators. 9.3.2.

Directionality

Whether a sat(w), yields space) to the u = sat(w) is

plant is SISO or MIMO,clipping a controller output w, the feasible plant input which is closest n th e pl ant in put unconstrained controller output w. In mathematical terms, the solution to the quadratic program U

subject to Ztl~ ~ 72~ ~ Uh~, ~ = 1, ¯ ¯ ¯, m where I1~11 denotes the Euclidean norm of a vector ~. This feasible plant input, which is closest (in the plant input space) to w, may not lead to optimal response (i.e. one that is closest to the response of the same plant to the unconstrained controller output w). Wewill refer to this performance degradation as the directionality, a precise definition of whichis given here.

Definition 9.1. A plant in the form of (9.1) does not exhibit directionality, if and only if for every plant input w E ~mxl the response of the plant to sat(w) is closest (in the output space) to the response of the plant to w. It is worth noting that in several aspects this notion of plant directiom ality is different than the one knownas the dependence of plant gain on the direction of plant input vector. For example, the latter does not exist when the condition number of plant gain matrix is one, while as we will see, the former is not present when plant characteristic matrix is diagonal. 9.3.3.

Optimal Directionality

Compensation

For a plant in the form of (9.1), let ~(r) and ~(~-), r _> t, represent the predicted values of a controlled output y, when the plant is subjected

Optimal Windup and Directionality

Compensation

233

to a given unconstrained controller output w and to a feasible plant input u, respectively. The objective is to calculate a feasible plant input, u, that renders the predicted value of every output y~, ~)~, as close as possible to ~)~. In mathematicMterms, we seek a feasible plant input u that is solution to the constrained minimization problem: (9.2) subject to the input constraints ut~ <_u~(t) <_Uh~,

g = 1, ...,

m

(9.3)

where t represents the present time, II~(T)llp~ is the p~-function norm a scalar function ~(T) over a sufficiently short time interval of the form It, t + The] with Th, > 0: t+7’’~lS(T)lp~d~II~(T)llp~ ~ LJt

, pe >_1,

and ql,"’, qm, are adjustable positive scalar parameters whose values are set according to the relative importance of the controlled outputs: the higher the value of a q~, the smaller the mismatchbetween the constrained and unconstrained plant responses in Ye (the lesser the effect of the constraints on the Ye response). Theorem9.1. [18, 21] For a plant of the form of (9.1), at each time instant t given an unconstrained controller output w, the optimal feasible plant input, denoted by u+, that minimizes the performance index in (9.2) subject to the constraints of (9.3), is the solution to the m-dimensionM quadratic program: m~nI{QC(x)u - QC(x)w[~ (9.4) subject to u~ _< u~ _< u~,

g = 1, ...,

rn

where Q is a constant m × m diagonal matrix given by

(9.5)

234

Soroush and Daoutidis

[ w I Optimal Controller~ Directionality]~-~J [ [Compensator

Figure 2: Optimal directionality

compensation in nonlinear systems.

The quadratic programof (9.4). and (9.5) is trivially solvable. For ample, one can use the computationally efficient, simple method described in [21]. Theorem9.1 indicates that at each time instant, the optimal feasible plant input, u+, is calculated on the basis of C (x) and a given unconstrained controller output w. Let .T[C(x), w] denote the solution to the quadratic program of (9.4) and (9.5). Then, + =.T[C(x), w] represents the opti mal directionality compensator. Thus, the characteristic matrix plays a key role in the optimal directionality compensation: to calculate an optimal feasible plant input in a nonlinear plant, given a controller output, one needs to knowthe characteristic matrix and measurements of the state v£riables of the plant (see also Figure 2). It is the nature of the characteristic matrix, not that of steady state gain matrix, that determines whenit is optimal to use the clipping approach for directionality compensation. It is noteworthy that the characteristic matrix and the steady state gain matrix characterize two different aspects of plant behavior; the former characterizes the sensitivity of plant to input changes over a very short horizon and the latter over an infinite horizon. Structural properties such as singular values and the condition number of the steady state gain matrix and relative gain array also characterize the plant response over an infinite horizon. Using steady state structural properties as a basis for selecting either of the approaches maylead to a very poor closed-loop performance, unless one uses a steady state controller. Remark9.1. For the class of plants with diagonal characteristic m.atrix, the optimal directionality compensatoris identical to m limiters (clippers), i.e. ue--sate(w), ~=l,...,m Thus, for this class of plants the feasible plant input that is closest (in the plant input space) to the unconstrained controller output w, yields an optimal plant response (i.e. one that is closest to the response of the

Optimal Windup and Directionality

~

Compensation

Optimal

235

] u

Controner D~ection~ity ~ / Compensator ]

Figure 3: Optimal directionality

compensation in linear systems.

same plant to the unconstrained controller output w). In other words, this class of plants do not exhibit the directionality, and thus in the presence of input constraints their closed-loop performance is not degraded by the directionality. Remark 9.2. In the case that the weights ql,"’,

qm are chosen such

that

2(r~!) that is, when the controlled outputs are of equal importance irrespective of the values of their relative orders (rl,..., r,~), the quadratic program (9.4) and (9.5) takes the simple form: rain III 2IC(x)u - C(x)w subject to ut~ <_ue <_Uh~, ~ = 1, -..,

(9.6)

m.

Remark9.3. For multivariable time-invariant linear systems described by a state-space model of the form

{

~ = Ax+ Bu y = Cx

where A, B and C are n×n, n×m and m×n matrices characteristic matrix

(9.7) respectively,

the

ClAr~-~ B

: cmAr.~-i B is independent of x. Thus, as shownin Figure 3, in time-invariant linear plants the optimal directionality compens,ator does not require information on the state of plant: u = ~[C, w]. In linear plants, the characteristic

Soroush and Daoutidis

236

matrix will be diagonal, if and only if the diagonal element of every row of the transfer function matrix has the absolutely lowest relative order in that row, where relative order of a rational function is the difference between the orders of the numerator and denominator polynomials of the function. Remark 9.4. A feasible plant input value calculated by the optimal directionality compensator of (9.4) and (9.5) is optimal over a very short time horizon. Because the time horizon was chosen to be very short, the resulting constrained optimization problem is a quadratic program that is easy to solve. However, a calculated optimal plant input value that is optimal over a very short time horizon may not be optimal over a very long (infinite) time horizon, for example, whenthe plant under consideration non-minimum-phase. Remark 9.5. The optimal directionality compensator allows one to minimizethe effect of input constraints on the controlled outputs that are more important in a given plant, by choosing higher values for the weights, q~,"’, qm, corresponding to more important controlled outputs. 9.3.4.

Application

to Two Plants

Decentralized PI Control of a Linear Plant. two-input two-output plant [18]: P(s)

100s + 1 -1 1140-300]

Consider the linear

40

(9.8)

with lull _< 1, i = 1, 2. For this example, rl = 1, r2 = 1, and C = [ -0.01 0.4

-3.0 1 0.4

Twocompletely decentralized PI controllers with kcl = 2.1, kc~ -- 0.36, ~’I1 = 174.0 s and ~-I~ = 22.1 s, and with conditional integration (to prevent integral windup) are used to track asymptotically the set point changes Yspl -- 8 and Ysp~ -- 3. The conditional integration involves turning off the integrator of the ith loop when the input u~ saturates. Figure 4 depicts the closed-loop output response under the same two PI controllers but three different directionality compensators; it shows that while clipping and direction preservation approaches lead to poor responses in Yl, the optimal direction compensator provides a significantly better closed-loop performance.

OptimalWindupand Directionality Compensation

237

I0 7I-5 5

i

i

i

I

I

I

I

I

I

I

I

I

I

I

c~ I-I-3"

-5 1,2 I,I, ~ 0,92 0,7 1,5 10.5-~ 0-0,5 0

60

120

180

240

300

Figure 4: Controlledoutputs and plant inputs of the linear exampleof (9.8): solid = no boundson the inputs; dotted = clipping, dotted-dashed direction preservation,anddashed= optimaldirectionality compensation, whenluyl~
238

Soroush and Daoutidis

I-O Linearizing Control of a Nonlinear Bioreactor. Consider continuous stirred-tank bioreactor described by a mathematical model [18]:

dt dSe dt dX dt

(9.9)

where the specific growth rate

~($1, &)

Se K& + $1 Ks2 + $2’

and $2 denote the outlet concentrations of substrates 1 and 2 respectively, Sfl and Sf2 respectively represent the concentration of substrate 1 in feed 1 and the concentration of substrate 2 in feed 2, and D1 and D2 are respectively the dilution rates of feed streams 1 and 2. The values of the model parameters are the same as in [24]. The controlled outputs and manipulated inputs are as follows: Yl := S1, Y2 ---- $2, Ul ~ D1, and us = D2 with the bounds 0 <_ D1,D2 :~_ 0.4 h-~. The control objective is to operate the reactor at the set points Yspl --- 2.0 kg.m-3 and Ysp~ = 4.9 kg.m-3, by an input-output linearizing controller with an optimal integral windup compensator. For this plant, rl -- 1, r~ = 1, and S1

C =

- S~ S~2 - S~

Figure 5 depicts the startup profiles of the controlled outputs and plant inputs of the bioreactor under the same nonlinear controller but three different directionality compensators. In the presence of the input constraints, clipping (dashed line) cannot operate the plant at the steady state (leads very poor closed-loop response), and direction preservation (dotted-dashed line) results in a relatively better performancecomparedto that of clipping. However,the closed-loop performance under the optimal directionality compensator is of higher quality; it is the closest response to that represented by the solid line (obtained in the absence of the plant input bounds).

Optimal Windup and Directionality

Compensation

239

~3

0 0.8 ,- 0,6;-0,40,20 2

I

],5-

0

~ 2

0

a

Timah Figure 5: Profiles of the controlled outputs and plant inputs of the bioreactor exampleof (9.9): solid = no bounds on the inputs; dotted = clipping, dotted-dashed = direction preservation, and dashed -- optimal directional-1. ity compensation, when 0 < D1, D2 < 0.4 h

240

9.4. 9.4.1.

Soroush and Daoutidis

Windup Compensation Scope

Weconsider the class of multi-input multi-output, continuous-time, nonlinear plants described by a state-space model of the form of (9.1). make the following additional assumptions: (al) every variable of the system of (9.1) is in the form of deviation from its nominalsteady-state value, and thus the origin is an equilibrium point; (a2) the output set-point, de~ noted by Ysp, is achievable at steady state in the sense that there exists a Uo E interior(U), which satisfies f(~) 9( ~)Uo = 0, where ~ E X and h(¢) = ysp; (a3) cleriC(x)]¢ O,Vx~ X; and (a4) the plant is minimum phase (has asymptotically stable zero dynamics) on

9.5.

Nonlinear Controller Design

For a nonlinear plant with a model in the form of (9.1), let us request a linear input-output response of the form: m

/~j

(9.10)

~ = Ysp

i=1 j=i

where/~ij ~ Nmxl, i = 1,...,m, j = 1,...,r~, that are chosen such that all roots of Sj det

I

-~-

~ ~lj

"’"

kj=l

~ j=l

are adjustable

~mjsJ

:

constants

0

J

lie in the left half plane and the matrix

is nonsingular. It is straightforward to show through a direct calculation of the output derivatives that this closed-loop response translates into the following relation: (~(x, u) = (9.11) where

m (I)(x,

~) =~ h(:~)

~zi ~n ~hi(2~) -~ i=1 ~=1

Let the solution for u of (9.11) be denoted by the following state feedback law: u = k~(x, Ysp) (9.12)

Optimal Windup and Directionality

Compensation

241

where

-1 ysp- h(x) + ~(x,y,p)5 [~C(x)]

~ieLhi(x)

"= 4=1

and the time-invariant linear system:

{

~ = A¢~+B¢O(x,u) V = C~

(9.13)

be u minimal-order state-space realization of ~

~

d~w ~(x,

u)

(9.14)

i=1 j=l

where rid ~ ~mxl, i = 1,...,m, j = 1,...,r~ that are chosen such that all roots of det

~ljS j

I +

are adjustable

~mjS j

¯ ..

constants

= 0

j=~ lie in the left half plane and the matrix

[~1~ ... z~] is nonsingular. The above system calculates ~ which is the vector of estimates of the plant outputs. It essentially acts as an observer for the plant outputs and their derivatives up to order (ri - 1), i = 1,..., Theorem9.2. Consider a plant with a model of the form of (9.1) and the dynamic controller:

{

~=Ac~+Bc~(x,u),

w= ~(x, ~ + c~) ~ = y {c(~),

~(0)=0

(9.~5)

where e = ysp - y. Then, (a) the controller has integral action, i.e. in the presence of constant disturbances and model errors, induces an offset-free closed-loop response [24, 25]. (b) whenthe constraints are not active, and fl~e = y~e, g = 1,..., r~, i 1,..., m, and its states are initialized consistently, the controller induces the linear input-output closed-loop response of (9.10) [24, 25].

Soroush .hnd Daoutidis when the constraints are active, flit = 7it, g = 1,..., ri, i :: 1,..., m, and its states are initialized consistently, the controller minimizes a constrained quadratic performance index [24, 25].

(d)whenthe

constraints are active, the origin of the closed-loop systemis locally asymptotically stable, provided that the state feedback law of (9.12) is also locally asymptotically stabilizing and the poles of (9.14:) are placed sufficiently far left in the complexplane [10].

9.5.1.

Application

to a Nonlinear

Chemical Reactor

Weconsider the same chemical reactor described in [20]: 1C CA. = fl(CA,T) + ’~1 T f2(CA, T) + -~O where CAand CA~ are respectively the outlet and inlet concentrations of the reactant, T is the outlet stream temperature, and Q is the rate of heat input to the reactor. The controlled outputs and manipulated inputs are: Yl = CA, Y2 = T, ul = CA~, and u2 = Q with the bounds 5 _< u~ ~ -1. 15 kmol.m-3 and -10 _< u2 _< 10 kJ.s Here rl = r2 = 1 and C = diag{1/~-, 1/(pcV)}. Application of the := control lawofTheorem 9.2,~/~ ~- "~1 = fl111 -~- fl221 = 100 and~/~2~ =,~121 fl~l = fl~ = 0, to this chemical reactor leads to the following mixederrorand state-feedback controller:

(9.17)

Figure 6 depicts the startup profiles of the controlled outputs and manipulated inputs under the nonlinear controller of (9.17). In the absence of the input constraints, the closed-loop plant output responses (solid line) are exactly two completely decoupled, first-order responses.

243

OptimalWindupand Directionality Compensation

-~ 2,0]< 1.002450400v 350,_300250~ :tO

I

I

I

I

I

I

I

I

I

30201000

200

400

I

I

600

800

1000

Time,s Figure6: Profiles of the controlledoutputsandplant inputs of the reactor example of 9.16: solid -- no boundsonthe inputs; dashed-- when5 _

--,u~(i)i = 1,...,m}

By definition the polyhedral set S(uo, c~) C3 S(ul, fie) contains the ellipsoid if(P, ~/-~) used to derive c~e and Remark 10.5. Wecould consider other compact sets to obtain similar results to Lemma 10.1. In this case, definition of c~e(i) and fie(i) given (10.23) and (10.24) would be different. Remark 10.6. In view of our objective, the use of Lemma10.1 necessitates to determine matrices P, ~, scalar 3’ and vectors c~e,/~e.

259

Output Feedback Compensators

10.4.

Control

Strategy

via

Riccati

Equations

In this section we showhowto use two coupled Riccati equations in order to exhibit both the matrices of the dynamic controller and the associated set of stabilizable initial conditions. Thus, we can state the following lemmaregarding the avoidance saturation case, that is the case whenthe behavior of system (10.12) is linear. Remark 10.7. Whensaturations lm) A S(Ul, lm) defined by:

S(uo,

do not occur, that is, when ~(t)

S(~0,1m)={~E~(n+m); --u0(~)_<[ 0 I~ ](~)~
S(u~, lm)= {~ E ~2(,~+m); _ Ul(i) _< K(i)~ _< ul(0,i = 1, ...,

m}

the closed-loop system (10.12) admits the following linear model:

~(t) (A + ~F(t)~ + ~X)~(t)

(10.30)

It is important to point out that, without additional conditions, we cannot conclude that any trajectory initiated in S(Uo, lm) ~ S(Ul, lm) is a trajectory of system (10.30), that is, remains confined in S(uo, lm) A S(u~, lm). The region S(uo, lm) ~ S(u~, lm) is called the linearity region. Lemma10.2. If there exist ~ > 0, X = X~ > 0 and Y -- Y~ > 0 such that Xe = eX = X~’ > 0 and Y~ = eY = Y~ > 0 are solutions following coupled Riccati equations:

to the

(10.31) (,4

- D~D~R~’C)~ + Y~(A - :D1D~R~C)’ -~ - PM VC’R +~(It - ~’-~ ~+ Q~= 0 2~2e ~2)~1 (10.32) Ze = Y~-~ - X, = Z~ > 0

(10.33)

260

Tarbouriech and Garcia

withRiE=~R1 =R~e >0, R2e =eR2÷ 2D2 R~e >0, QIe =eQ1 := Q~ > 0, Q2e = eQ2 = Q~ > 0, then the controller defined in (10.7) with Ac = A+ I3C~ - BcC + (7~ - BcD2)Ig’~Xe Bc = Z:IYe-I(Y~C ’ 1+ 79~D~)R~ Cc = - R~I B’ X~

(10.34)

and the ellipsoid ~(P, -~) with p = [ Xe + Ze_ZE-Ze ]Z~

(10.35)

and with scalar ~/defined as

" max

marx i.

2

’

e ~’~c(i) ,

[0 max

ul(i) i -~ 1, ..., m (10.36>

solve Problem 10.1. Proof. By simplicity denote H = Y, ZeBc. By considering A~, Be, Cc and P given respectively in (10.34) and (10.35) and by computing the timederivative of the quadratic function V(~) = ~P~ along the trajectories linear system (10.30) one obtains ~’(~) = ~’[(A + ~IK DF(t)E)’P + P(A + ~]Kq- DF(t)IE)]~. It follows [21]:

’ Cc. with Q1 = Y~-~(Q~e+eHR2H’)YE-~, Q2 = -Q1, Q3 =¯ Q~+Q~e+C~R~ Therefore, from (10.31), (10.32) and (10.33) one gets V(~) < 0. more, we have to prove that the behavior of system (10.12) remains linear, that is, corresponds to the behavior of system (10.30). For this, suffices to prove that the ellipsoid ~(p,~,-1) is included in the region linearity S(Uo, lm) ~ S(ul, lm). This is verified provided [30] that -~ ~

(

~2

I

i

~0(i)2

- ~(~

min .min Cc(i)(Z:

1 +x[l)ctc(i),~n

[0

Ix: [

t0 )

261

Output Feedback Compensators i = 1,...,m.

[]

In order to obtain a larger domain~(P, ~,--1) of safe initial conditions, we are mainly interested in control saturation. Thus, in the allowance saturation case, the following proposition can be derived. For ease of notation, let us define Aj(at) = diag(In, Fj(at),

In+m)

Proposition 10.1. Given matrix P as defined in (10.31)-(10.35). controller defined in (10.34) and the ellipsoid ~(P, -1) solve Problem 10.1 if there exist vectors at,/~t and a scalar 0~ satisfying: (10.37)

0
(10.38) (10.39)

max(0’(19t),

(10.40)

2

where 7(ft) ft ma(~)Cc(~)~Z~ x i O~t(i)

0’(at) = max

[ 0 Ira(i)

(

--1_~_ Xe -1

, i : 1,...,m and

2 ui(~) ]X~ "1

u~(~)

i~n({) , i = 1, ...,rn.

Proof. The satisfaction of relations (10.38), (10.39), (10.40) that the ellipsoid ~(p,y-1) is included in the polyhedral set S(uo, a.t) S(ul, Bit). Thus from Lemma10.1, for ~ E S(u0, at)N S(ul, ft), then ~ can be computed from the polytopic system (10.25). By computing the timederivative of the quadratic function V(~) = ~P~ along the trajectories polytopic system (10.25), it follows:

262

Tarbouriech and Gm~cia

Hence, if £(j, k) = (A~- ÷ AAj)’P + P(Aj + AAj) PBkK + K’~’kP <: Vj = 1, ..., 2m, Vk = 1, ..., 2m, by convexity one gets ~-(~) < 0. By using the same majoration as in Lemma10.2, the condition £(j, k) < 0 reads condition (10.37). Thus, if conditions (10.37) and (10.40) hold, the topic model (10.25) represents the saturated system (10.12) in ~(P, 3’-1). Therefore in ~(P, -1) system (10.12) i s a symptotically s table. Instead of considering two coupled Riccati equations as expressed in Lemma10.2 we can consider two coupled LMIs. Thus, the following corollary can be stated. Corollary 10.1. Given R2 = R[ > 0. Then the controller (10.7) with

defined in

and the ellipsoid ~(P, ’-1) with -Z

(10.42)

-Zz ]

solve Problem 10.1 if ~ > 0 and R1 -- R~ solutions to 1. there exist W= W WA’ + AW - 13R~13’ + ~:D~ W£’ ] <0

(lO.3)

2. there exists a matrix Q -- Q~ > 0 solution to Q(A - D1D~R2C) + (A - DID~R2C)’Q +~ - C’R~C

QD~

[ < 0

-(I~

(10.4 Set Z = Z’ = Q -

W-1.

The condition Z > 0 reads: I~+m W

> 0

(10.45)

263

Output Feedback Compensators

4. conditions (10.37) and (10.40) are verified with P defined in (10.42). Proof. It readily follows from the application of Lemma10.2 and Proposition 10.1 in which we replace relations (10.31), (10.32), (10.33)

(10.43), (10.44), (10.45). Proposition 10.1 give a sufficient condition for solving Problem10.1 and allows to derive the following simple algorithm. ARE-Algorithm. 1. Given positive definite matrices R1, R:, Q1, Q~ and scalar e solve relations (10.31), (10.32) and (10.33) for X~, Y~ and 2. Derive controller matrices from (10.34) and matrix P from (10.35). = [ Im(~) 3. Compute the scaling parameter ¢(~)

u°~(~)

and

, i = 1,..., mand consider (0 = min(¢(~), ~b(~)). Thus, define ae(i) and /~e(~) as functions of a parameter ( >_ ae(~) = min(1, ~) and fie(~) = rain(l, ~). Build matrices j = 1, ..., 2TM, and Fk(fie), k = 1, ..., m, as described i n Lemma0.1. 1 4. Find the maximalvalue (max such that condition (10.37) is verified. Hence, the ellipsoid in which the stability of the closed-loop system is defined by P and "~ = ~ Components of vectors ae and ~e are defined respectively by min(1, ¢(~) ) and min(1, ~), i = 1, ..., The results derived from Proposition 10.1 or Corollary 10.1 (and therefore the above algorithm) raise some remarks. Note that step 1 can be modified to consider LMIconditions (10.43), (10.44) and (10.45) to derive solution P and controller matrices. computation is done disregarding the way how the ellipsoid ~(P, 3’-;) fits for the polyhedral region S(u0, ae)r~S(u~, ~e). Moreover,note that the fact of obtaining a small scalar 3’ does not guarantee necessarily obtaining a large set if(P, ~/-~). The existence of solutions Xe and Ye to Riccati equations (10.31), (10132)and (10.33)is independent of the choice of R1, R2, (~1 and

264

Tarbouriech and Garcia Q~ [21]. Howeverthe a priori choice of these matrices influence the size of the resulting ellipsoid ~(P, y-l). The way to take into account this fact is not easy to determine. Someglobal nonlinear optimization problems can be set but ~heir solution is not easily attainable [30]. A way to have a supplementary degree of freedom is to consider LMI conditions in step 1.

¯ In order to obtain a controller of order nc < n + m, matrices Xc and Ye solutions to Riccati equations (10.31) and (10.32) must satisfy rank(Zc) = rank(Y¢-1 - Xe) = nc. Such a condition is nonconvex and maybe hard to test [7]. ¯ In the state feedback case, the closed-loop system is that one described in Remark10.1. Then, Proposition 10.1 applies by considering P = Xe solution to relation (10.31) and the feedback gain K [ K1 K2 ] = -Rlfl3’P. In relation (10.37), matrices A, ~ and ]K are replaced by .A, B and the above defined K matrix. The scalar "~ is then ~2 K ~-I r~, defined as ~ = m~ (~(u,), ~(u0)) with ~(u~) m~ 2 ~ u~(~) at(O[

]X7 ~ I~(¢)

0 I~(o

and ~(uo) = max

[32]. , i = 1, ...,

2

u0(~)

10.5.

Control

Strategy

via

Matrix

Inequalities

In this section, we use a formulation via matrix inequalities due to Gahinet and Apkarian [8]. As previously, we distinguish between the avoidance saturation case (next lemma) and the allowance saturation case (next proposition). Let us define the following matrices:

I

O In+,~

0

C

D2

0

0 0 ’

9I= [ B’S N’ B~N T 00

A’S + SA 0

N’Da -I~ 0

0 0

(10.46) 6) = [ Ac ’[ C~

Output Feedback Compensators

265

Lemma10.3. The controller defined in (10.7) and the ellipsoid ~(P, ,-1) with ~ [N S TN]

P=

(10.47)

solve linearly Problem10.1, that is without allowing control amplitude and rate saturation, if there exist matrices S = S’, N, T = T’, (~, and satisfying N’

T > 0

¯ + 03II@’9l+ 9I/(3ff)I <

N’

T

0

S N N’ T 0 Cc(~)

0

(10.48)

(10.49)

(lo.5o) ] k 0, Vi=l,...,m

(10.51)

Proof. Concerning the avoidance of control amplitude and rate saturation avoidance, the proof mimics the one of Lemma10.2. Indeed, the satisfaction of relations (10.50) and (10.51) means that the ellipsoid ~(P, -1) is included in the region of linearity. Furthermore, the proof to showthat the satisfaction of relations (10.48) and (10.49) implies the asymptotic bility of the linear system (10.30) are based on the use of the BoundedReal Lemma and the notion of quadratic stabilizability and in particular its 7t~ control interpretation [8]. [] By considering the occurence of control amplitude and rate saturation, the following proposition can be given. Proposition 10.2. The controller defined in (10.7) and the ellipsoid e(p, .~-1), with P defined in (10.47), solve Problem10.1 if there exist trices S = S~, N, T = T~, O, vectors a~, fl~ and positive scalar ~ satisfying N’

T > 0

(10.52)

266

Tarbouriech and Garcia k~j + ffJl~.O’diag(In+m,F~(/3t))91 9I’diag(In+m,F~03t))Og)Ij < 0, Vj= 1,...,2 m, m k~- l,...,2 (10.53)

N’

T

0

_> 0, Vi = 1, ..., m (10.54)

S N’ 0

N 0 ] T ~3e(i)C’c( _> 0, Vi = 1, ..., m 0 f~(~)Cc(~) 0
(10.55)

(10.56) (10.57)

0 < fli(~) _<1, i = 1, ...,

where ~j and ffJ~j are obtained by replaced in (10.46) ,4, E and C Adiag(In, rj (c~)), gdiag(I,~, Fj (c~e)) and Cdiag(In, Fj (c~e)), respectively. Proof. It readily follows from Lemma10.3 and Proposition 10.1. ~] At this stage, Proposition 10.2 presents sufficient conditions of existence of dynamic controller matrices and associated domains of stability. The major difficulty in applying the above conditions resides in the fact that the decision variables appear as products in the matrix inequalities. Tha~ implies that the computationof a feasible solution (in t3 and P) to relations (10.52)-(10.57) is an NP-hard problem. In practice, to overcomethis difficulty, matrix inequalities maybe recasted into LMIs by transformations, overestimation and/or relaxation techniques. Moreover, note that whenno saturation are allowed (i.e., (~e =/)e =lm., see Lemma10.3) constraints (10.48) and (10.49) can be linearly solved P and O. Whenthere exists a feasible solution to this linear problem, it mayserve as the initialization of an algorithm formed by a sequence of the following relaxation schemes. Relax 1. For given P and O, find ae, ~3e and 7 solutions to (10.53), m

(10.54), (10.55), (10.56) and (10.57) in order to minimizeEae(i) +~3e(i) i=1

Output Feedback Compensators

267

Relax 2. For given P, ae and/~t, find O and 3, solutions to (10.53), (10.54) and (10.55) in order to minimize Relax 3. For given {b, at and ~3e, find P and 3’ solutions to (10.52), (10.53), (10.54) .and (10.55) in order to minimizetrace(P) Relax 4. For given O, at and ~3~, find P and 3’ solutions to (10.52), (10.53), (10.54) and (10.55) in order to minimize Log(det(3’P)). Nevertheless, there are no a priori rules which could help to determine an optimal sequence to use these different relaxation schmes. Furthermore, in the first relaxation, the solution P and O resulting from the previous AREapproach could be used. 10.6.

Illustrative

Examples

Example10.1. Let us first consider the single integrator system studied in [26]. System(10.2) is described by the following data D0=0; A0 = 0; B0 =1; C0=l; E1 =0; E2 =0; u0= 1; ui= 1

DI=0

which gives matrices ,4 and B of (10.8):

In order to compareour results with those described in [26], we consider the state feedback case. From(10.31) the stabilizing state feedback is K -R~IB’X~ and by computing X~ and K from RI~ = 50000 and Qle = I1 0 1 (matrices used in [26])one obtains 0 R

[

P=Xe = 104 0.0224

5.0223

]

; K= [ -0.0045

-1.0045

l

~, By testing the validity of condition (10.37) one gets 3’-1 = 7.894010 c~e = 0.2517 and Be -- 0.2511. In Pigure 2, we compareour stability domain in the space °f [ x ]with the °ne °btained in [26]’u In the above example, even if we obtain a set of admissible conditions with an interesting size, it is clear, due to the implicit dependenceof the choice of R1 and Q~, for the size ~(P, 3’-1) that our result mayconduct conservative results. See [32] for more details about this example.

268

Tarbouriech and Garcia

Figure 2: Stability domain obtained in [26] (dotted line) and obtained from our algorithm (solid line). Example 10.2. Consider system (10.2) described by: A0= Co=

0 0 ;B0= 1 ;DI= 1 1 0 ;Do=O;D2=O;E2=O

’

which gives matrices defined in (10.8):

;C=[1

Z)I=

;£=[

0 0]

0 1 0 ]

Note that the uncertain open-loop system is unstable when F(t) -then neither the global stabilization nor the semi-global stabilization can be studied. For different values of the bounds u0 and u~, Table 1 shows the values of the volume of the ellipsoid ~(P, ,~--1) (computed from x/det(P-~y-~)), the associated values of ae and ~3~, and the profit (denoted GL/N in %) resulting from this ellipsoid (in which the occurrence of the saturation is allowed) with respect to the one included in the linearity region (saturation avoided).

269

Output Feedback Compensators Table 1: Operating conditions for different bounds u0 and Ul. u0 ul

VoI(~(p,v-*)) GL/N(%)

10.7.

1 1 -6 7.1229 10 1.4510105 0.9853 0.9697

5 10 10 5 0.2814 0.0117 5636.88 15185.66 0.9996 0.9803 0.9851 0.9700

10 500 ~ 1.2319 10 63.79 0.9553 0.9745

Concluding Remarks

While actuator nonlinearities and model uncertainty are commonfeature in control systems, to take into account their effects conjointly in the design of controllers is usually difficult and therefore, in general, infrequently addressed in the literature. In this chapter, the problem of stabilization through dynamic output feedback of multiple bounded inputs state space models with uncertainties was addressed. Conditions for the existence of a dynamic output control law of the same order as the system under consideration that guarantees the closedloop asymptotic stability in presence of both control amplitude and rate saturation was proposed. Let us underline that the occurrence of position and rate saturations was effectively considered, which generally allows the stabilization of larger sets of initial conditions, although by increasing the complexity of the solvability conditions. Computationdifficulties have not been hidden, instead uncovered and discussed. The proposed approach, which can be qualified as preliminary results on the subject, suffers from somepotential sources of conservatism because of the use of differential inclusions to represent the saturated closed-loop system or the actuator model considered in the approach. However, our approach can be used to specify~ or design actuator capacity: considering a given control law satisfying some control requirements, this can be carried out by expressing optimization problems in which the bounds u0 and ul are decision variables. The natural continuity of this works should consist in considering other models to represent the dynamics of the actuators. Furthermore, the problem of output rejection of disturbances (generated by exogeneous systems, for example) should be studied.

270

Tarbouriech and Garda

Acknowledgement. The authors would like to thank their colleague and friend Isabelle Queinnec for the interesting discussions concerning the results developed in the chapter.

References [1] D. S. Bernstein and A. N. Michel. Special Issue: Saturating Actuators, Int. J. of Robust and Nonlinear Control, 5 (1995) 375-380. [2] S. Boyd, L. E1 Ghaoui, E. F4ron and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory, SIAMStudies in Applied Mathematics, Philadelphia, Pennsylvania (1994). [3] C. Burgat and S. Tarbouriech. Stability and Control of Saturated Lin-. ear Systems, in A. J, Fossard and D. Normand-Cyrot(Editors) "Non-linear Systems. Volume 2. Stability and Stabilization", Chapmanand Hall, London, UK(1996) 113-197. [4] P. 3. Campoand M. Morari. Robust Control of Processes Subject to Saturation Nonlinearities, Computers and Chemical Engineering, 14 (1990) 343-358. [5] M. Chilali and P. Gahinet. Hoo Design with Pole Placement Constraints: an LMI Approach, IEEE Trans. on Autom. Control, 41, no.3 (1996) 358-367. [6] P. Dorato. Robust Control, IEEE Press Book, (1987). [7] L. E1 Ghaoui, F. Oustry, M. AitRami. A Cone Complementarity Linearization Algorithm for Static Output-feedback and Related Problems, IEEE Trans. Autom. Control, 42, no.8 (1997) 1171-1176. [8] P. Gahinet and P. Apkarian. A Linear Matrix Inequality Approach to ~ Control, Int. J. of Robust and Nonlinear Control, 4 (1994) 421-448. [9] G. Garcia, J. Bernussou, D. Arzelier. Robust Stabilization of Discretetime Linear Systems with Norm-bounded Time Varying Uncertainty, Systems & Control Letters, 22 (1994) 327-339. [10] G. Garcia and S. Tarbouriech. Stabilization with Eigenvalues Placement of a Norm-bounded Uncertain System by Bounded Inputs, Int. J. of Robust and Nonlinear Control, 9 (1999) 599-615. [11] J. M. Gomesda Silva Jr. and S. Tarbouriech. Local Stabilization of Discrete-time Linear Systems with Saturating Controls: an LMI-based Approach, Proc. of American Control Conference, Philadelphia, USA (1998).

Output Feedback Compensators

271

[12] 3. M. Gomesda Silva Jr. and S. Tarbouriech. Local Stabilization of Linear Systems under Amplitude and Rate Saturating Actuators, Proc. of 39th IEEE Conference on Decision and Control, Sydney, Australia (2000). [13] D. Henrion, S. Tarbouriech, G. Garcia. Output Feedback Robust Stabilization of Uncertain Linear Systems with Saturating Controls: an LMI Approach, IEEE Trans. on Autom. Contr., 44, no.ll (1999) 2230-2237. [14] P. Kapasouris and M. Athans. Control systems with Rate and Magnitude Saturation for Neutrally Stable Open-loop Systems, Proc. of the 29th IEEE Conference on Decision and Control, Honolulu, Hawaii (1990) 3404-3409. [15] V. Kapila and W. M. Haddad. Fixed-structure Controller Design for Systems with Actuator Amplitude and Rate Nonlinearities, Proc. of the 37th IEEE Conference on Decision and Control, Tampa, USA (1998) 909-914. H. K. Khalil. Nonlinear Systems, Macmillan Publishing CompanylNew [16] York, NY(1992). [17] Z. Lin. Semi-global Stabilization of Linear Systems with Position and Rate-limited Actuators, Systems & Control Letters, 30 (1997) 1-11. [18] A. P. Molchanovand E. S. Pyatnitskii. Criteria of Asymptotic Stability of Differential and Difference Inclusions Encountered in Control Theory, Systems & Control Letters, 13 (1989) 59-64. [19] T. Nguyen and F. Jabbary. Output Feedback Controllers for Disturbance Attenuation with Actuator Amplitude and Rate Saturation, Proc. of the American Control Conference, San Diego, USA(1999) 1997-2001. [20] T. Nguyen and F. Jabbary. Output Feedback Controllers for Disturbance Attenuation with Actuator Amplitude and Rate Saturation, submitted, communicatedby the authors. [21] I. R. Petersen. A Stabilization Algorithm for a Class of Uncertain Linear Systems, Systems & Control Letters, 8 (1987) 351-357. [22] B. G. Romanchuck. Computing Regions of Attraction with Polytopes: Planar Case, Automatica, 32, no.12 (1996) 1727-1732. [23] L. Rundqwist and R. Hillgren. Phase Compensation of Rate Limiter in JAS 39 Gripen, AIAA, (1996) 69-79. A. Saberi, Z. Lin, A. R. Teel. Control of Linear Systems with Saturat[24] ing Actuators, IEEE Trans. Autora. Control, 41, no.3 (1996) 368-378.

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[25] A. Saberi and A. A. Stoorvogel. Stabilization and Regulation of Linear Systems with Saturated and Rate-limited Actuators, Proc. of American Control Conference, Albuquerque, USA(1997) 3920-3921. [26] J. M. Shewchun and E. Feron. High Performance Bounded Control, Proc. of American Control Conference, Albuquerque, USA(1997) 3250-3254. [27] A. A. Stoorvogel and A. Saberi (Editors). Special Issue: Control Problems with Constraints, Int. J. of Robust and Nonlinear Control, 9 (1999). [28] R. Suarez, J. Alvarez-Ramirez, J. Alvarez. Linear Systems with Single Saturated Input: Stability Analysis, Proc. of 30th IEEE Conference on Decision and Control, Brighton, UK(1991) 223-228. [29] H. J. Sussmann, E. D. Sontag and Y. Yang. A General Result on the Stabilization of Linear Systems using BoundedControls, IEEE Trans. on Autom. Control, 39, no.12 (1994) 2411-2425. [30] S. Tarbouriech arid G. Garcia (Editors). Control of Uncertain Systems with BoundedInputs, Lecture Notes in Control and Information Sci-. ences, 227, Springer- Verlag (1997). [31] S. Tarbouriech, G. Garcia, J. M. Comesda Silva Jr. Stability and Disturbance Tolerance for Linear Systems with Bounded Controls, Proc. of European Control Conference, Porto, Portugal (2001) 3219-3224. [32] S. Tarbouriech, G. Garcia, D. Henrion. Local Stabilization of Linear Systems with Position and Rate BoundedActuators, Proc. of the l~th World IFAC Congress, Beijing, China, F (1999) 459-464. [33] F. Tyan and D. S. Bernstein. Dynamic Output Feedback Compensation for Linear Systems with Independent Amplitude and Rate Saturations, Int. J. of control, 67, no.1 (1997) 89-116. [34] K. Zhou, J. C. Doyle and K. Clover. Robust and Optimal Control, Prentice Hall, EnglewoodCliffs, NewJersey, USA(1996). [35] M. Sznaier. An Exact Solution to General SISO Mixed H2/Ho~ Problems via Convex Optimization, IEEE Trans. on Autom. Control, 39, no.12 (1994) 2511-2417.

Chapter 11 Actuator Saturation Control via Linear Parameter-Varying Control Methods F.

Wu

North Carolina State University,

Raleigh,

North Carolina K. M. Grigoriadis University

11.1.

of Houston, Houston, Texas

Introduction

In all real world controlled engineering systems, actuator capacity is limited by the inherent physical constraints and limitations of the actuator. For example, the output torque of a servomotor in a robotic manipulator, the flow capacity of a fuel valve, or the deflection of a control surface in an aircraft are constrained to operate between certain saturation limits. As a result of saturation, the actual plant input is different from the controller output and the controller states are wrongly updated. This discrepancy between the controller output and the plant input due to actuator saturation is called controller windup [4]. Controller windupcould result in a significant performance degradation, large overshoot or possible instability

273

274

Wuand Grigoriad.is

in spite of the satisfactory performance predicted from a linear design. Any controller with slow or unstable dynamics will exhibit a windu:p in the presence of actuator saturation constraints [15]. Desirable design requirements for linear control synthesis subject to actuator saturation are the closed-loop system stability, recovery of the linear design specifications in the absence of saturation (linear performance recovery), and the smoot].~ degradation of the linear performance in the presence of saturation (graceful performance degradation). While analysis of systems including saturated actuators is relatively easy, the controller synthesis problemin the presence of input nonlinearity is a much more involved ~ask. Attempts to penalize the control output variables so that the actuator limits are never violated for any expected reference commandsoften leads to poor conservative designs in which the control system for the most part operates far from its full capacity. A typical approach to formulate such an actuator saturation problem is to assume a linear time-invariant plant, model the saturation block as a sector-bounded nonlinearity and apply small-gain type condi.tions (Popov, circle, and scaled small-gain theorems) for the stability and performance analysis [24, 34, 35]. For example, the work in [34] examined the actuator saturation controller design by directly accounting for the; nonlinearity and provided a guaranteed domainof attraction for linear and nonlinear controllers. Unfortunately, the synthesis condition was given in the form of coupled Riccati equations which are computationally expensive and they can only be solved locally via homotopy algorithms. Stabilizing nonlinear controllers were developed in [29] for linear systems subject to input nonlinearities utilizing a low and high gain control strategy with respect to the linear and saturated regions. This approach can be considered as an extreme case of a gain-scheduled control with two controllers involved. A popular alternative approach to saturating control is the anti-windup method that employs a two-step design procedure. The main idea of antiwindupcontrol is to design first the linear controller ignoring the saturation nonlinearities and then add anti-windup compensation to minimize the adverse effects of the saturation on the closed-loop performance. One of the earliest attempts to overcomewindupin PID controllers was studied in [i6]. Recently, many anti-windup control schemes were proposed from different perspectives [4,11,15,19]. However,although the two-step anti-windup synthesis appears to be successful for SISOsystems, it is often inadequate for MIMO systems because of the directional nature of the saturation nonlinearities in these systems [15]. In particular, the anti-windup control schemes do not. always guarantee the stability of the closed-loop controlled systems. A general frameworkthat unifies a large class of existing anti-windup control schemes was recently proposed in [12, 23]. This framework is useful

LPV Saturation

Control

275

for understanding different anti-windup control schemes and motivates the development of systematic procedures for designing anti-windup controllers that provide guaranteed stability and performance. A chronological bibliography on saturating actuators can be found in [8]. In this work, we propose a novel anti-windup control design approach for nonlinear systems with input saturation constraints using linear parametervarying (LPV) techniques. The study of LPVsystems has been motivated by the gain-scheduling control design methodology[28, 30]. The classical approach to gain-scheduling involves the design of several LTI controllers for a parameterized family of linearized models of a system and the interpolation of the controller gains. This heuristic approach leads to satisfactory results only if the variations of the operating parameters are sufficiently slow [31, 32]. Recently, a systematic gain-scheduling control design technique has been developed in [27] using a scaled small-gain theorem. In the context of an induced /22 performance index, it was shown that the gain-scheduled control synthesis problem can be solved using LMIoptimization techniques. Based on the machinery from [27], the LPVcontrol frameworkhas been further extended in [3, 6, 7, 36] using single quadratic and parameter-dependent Lyapunov functions. These developments have addressed issues of practical importance, such as, realness of the parameters [7] and boundednessof the parameter variations [6, 36]. Our approach for anti-windup control design is distinct from previous work in several aspects. The systems we considered are in general nonlinear besides the presence of the actuator saturation nonlinearities. The proposed designs directly account for the actuator nonlinearities by representing the status of each saturated actuator as a gain-scheduled varying parameter. Hence, the resulting controller is gain-scheduled with respect to the system operating parameters and the actuator saturation parameters. This systematic gain-scheduling design approach provides a guaranteed stability property, improved performance and dynamic anti-windup. Finally, the controller synthesis conditions for the proposed anti-windup scheme can be formulated in terms of Linear Matrix Inequalities (LMIs), which are solvable using newly developed efficient interior-point optimization algorithms. Therefore, ad-hoc two-step design procedures of designing first the linear controller and subsequently the anti-windup compensation are avoided. The notation in this work is standard. R stands for the set of real numbers, and R+ for the nonnegative real numbers. Rmxn is the set of T. real rn x n matrices. The transpose of a real matrix Mis denoted by M + Ker(M) will be used to denote the orthogonal complement of M. i s the pseudo-inverse of matrix M. A block diagonal matrix with submatrices X1, X2,... , Xp in its diagonal will be denoted by diag {X1, X2 .... , Xp}.

276

Wu and Grigo~qadis

Weuse S n×n to denote the real, symmetric n × n matrices, and n×n S+ to denote positive definite matrices. If M ~ Sn×n, then M> 0 (M _> 0) in.dicates that Mis positive definite (positive semidefinite), and M< 0 (M <_ denotes a negative definite (negative semidefinite) matrix. For x ~ n, i ts T x)7. 1 normis defined as Ilxll := (x The space of square integrable functions is denoted by £2, that is, for any u e £2, Ilull2 := If0 ~ ur(t)u(t)dt] ½ is finite. The notation IIPII~ will be used to denote the induced £2 norm of a system P, that is

for u ~ ~:2 - {0}.

11.2.

LPV System Analysis Synthesis

and Control

A linear parameter varying (LPV) system is a finite-dimensional linear system

[A(p(t),p(t)) 1 [,(t)] e(t)J LC(p(t),/~(t)) D(p(t),~(t))j [d(t)] ’

(11.1)

where x ~ Rn, and the state-space data are continuous functions of a time-varying parameter vector p ~ Rs and its derivative ~. Dependency on the parameter derivative ~ is taken into account to consider the case where equation (11.1) represents a closed-loop system with a controller that depends oI~ ~. It is assumed that the parameter values are not known in advance, but they are measurable in real-time. The vector-valued parameter signal evolves continuously over time and its range is limited to a given compact set P. In addition, its time derivative is bounded and satisfies the constraint Itii[ _< ~i, i = 1,2,... ,s. For notational purpose, denote )~ = {q : ]q~] _< ~, i = 1, 2,..., s}, where )2 is a given convexpolytope Rs that contains the origin. Given the sets P and ~, we define the parameter ~-variation set as follows:

that is, 9~ is the set of all allowable parameter trajectories system.

for the LPV

LPV Saturation 11.2.1.

277

Control

Induced/22

Norm Analysis

Denote the LPVsystem in (11.1) by E~. It is possible to bound the induced /22 norm of this system using a parameter-dependent Lyapunov function and a sequence of LMIconstraints [36]. Theorem 11.1. Given the LPVsystem E~, suppose that there exists a continuously differentiable function WE C1 (Re,S+n×n ) and a scalar 7 > such that AT(p, q)W(p) + W(p)m(p, q)

W(p)B(p,

BT(p,q)W(P)

-71 D(p,q)

c(p,q)

CT(P, q)] DT(p,q)| < -’,/ I .]

(11.2) for all (p, q) E (;o, F). Then the system Evp is exponentially stable

ttr~[[,2<~ forall p(.)~.

Notice that the LMIcondition (11.2) corresponds to an infinite dimensional convex problem due to its parameter dependence. To obtain a finite dimensional optimization problem, the parameter-dependent matrix function W(p) can be approximated using a finite set of basis functions. Also, a finite gridding of the parameter space P can be used to eliminate the dependence on the parameter vector p. A detailed discussion on the gridding technique and the selection of appropriate basis functions for such parameter-dependent LMIscan be found in [2]. 11.2.2.

LPV Controller

Synthesis

Consider an open-loop LPVplant

~(t)] eCt)|= y(t)J

[

A(p(t ) C~(p(t)) C2(p(t))

Bi(p(t)) Z)~(p(t) :D2~(p(t)

B~(p(t)) ]

.Dl~(p(t))|/d(t)/ ¢>~(p(t))J L~(t)j

(11.3)

It is assumedthat: (A1) The triple (.4,~2,C2) is parameter-dependent stabilizable and detectable, i.e. there exist positive-definite matrix functions P(p), Q(p)

278

Wu and Grigoriadis and rectangular matrix functions F(p), L(p) satisfying P(p) [A(p) B2(p)F(p)] + [A(p) + B2(p)F(p)] T P(p) + ~ i=1

~+~ [~(p) + n(p)C2(p)] Q(~)+ Q(~)[A(p) i=1

for all (p, q) (A2) ~1 = 0 and ~e = Suppose a linear, from y to u,

[

~(t)~

parameter-dependent controller

[~(~(t),~(t))

is used in feedback

~(~(t),~(t))~

(~1.4)

where xk ~ Rn. Note that the controller gains ~re ~unctions of the parameters p and ~. In this way, gain-scheduling is achieved emirely by the parameter-dependent controller. T Define Xc~ := Ix T xTk], then the closed-loop LPVsystem of (11.3) and (11.4) can be written

[A~(~(t),~(t))~(~(t),~(t))] [~(t)~ [e~(t)~ ~(t) ] [c~(~(t),~(t)) 9~(z(t),~(t))] d(t) J

(11.~)

wh~

(11.6) The LPV 7-Performance/p-Variation Problem is to find a parameterdependent controller in the form of (11.4) such that the closed-loop system (11.5)-(11.6) satisfies the conditions of Theorem11.1. The following sult provides necessary and sufficient conditions for the solvability of this problem and a parameter-dependent controller that guarantees induced E~ norm performance from d to e less than ~ [6, 36]. Theorem11.2. Given the parameter p-variation set 5~;~, a scalar ~/> 0, and the open-loop LPVsystem (11.3), the LPV~/-Performance p-Variation

LPV Saturation

Control

279

Problemis solvable if and only if there exist continuously differentiable matrix functions R(p), S(p) CI(R8, S~_×n) such th at fo r al l p R(p)CT~(p)

~(p)

-’7I

0 -’7I (11.~)

S(p)l~l (p)

CT(p) 0

o (11.8) (11.9)

s(p where JV’R(R) := Ker

~2(~)],Zs(p):= Ker[C2(p)~2,(p)].

Let M(p)NT(p) := I - R(p)S(p) and F(p) := -[T)T2(p):D~2(p)] + [,TBT2(p)R-~(p)+

then one nth-order, strictly proper controller K~ that solves the feedback problem has state-space matrices T dR dM AK(p, ~) = -~ (p ) -S (p) ~ - Y( p) ~ + AT(p) +S(p) [A(p) B2(p)F(p) + n( p)C2(p)] R( +~-~CT(p) [C~ (p) + ~]2(p)F(p)] R(p)} M-T(p)

(11.10)

BK(p) = N-Z(p)S(p)g(p)

(11.11)

CK(p) = F(p)R(p)M-T(p)

(11.12)

~.(p)=

(~L~3)

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Wu and Grigoriadis

Remark11.1. If the above conditions (11.7)-(11.9) are satisfied, by continuity and compactness of the parameter set P, it is possible to perturb feasible R and S such that the three LMIs(11.7)-(11.9) still and (S - -1) >0 uniformly on P, i.e . M(p) andN(p)can be se lec ted to be nonsingular. Remark11.2. The notation ~ =t= (.) in (11.7)-(11.8)

indicates

every combination of +(.) and -(.) should be included in the inequality. Therefore each inequality actually represents 2s different LMIs that must be checked. The above synthesis condition is presented using a parameter-dependent Lyapunov function. This includes the single quadratic Lyapunov function case as a special case. In many siutations, the inclusion of parameter information into the Lyapunov function is useful to improve controlled performance by exploiting allowable bounds on parameter variation rates. The solvability conditions provided in (1-1.7)-(11.9) are clearly infinite-dimensional, as is the solution space. To approximate this problem with a finite dimensional problem, we restrict the search for a parameterdependent Lyapunovfunction to a span of a finite numberof basis functions. That is, let N.f

Ng

x(0) f (o)xi, Y( O) = i=l

where {fi(0)}~N=~ and {gj( )}~=~ are user-specified scalar basis functions and X~, Yj are new optimization variables to be determined. After such a parameterization, the LPVsynthesis conditions can be solved using a gridding method over the parameter space. Note that the synthesis conditions (11.7)-(11.9) and the controller rameterization ( 11.10 )-(11.13) are slightly different from the ones provided in [36]. This is due to the relaxed assumptions on T~,2 and 792~ terms that are not necessarily full columnand full row rank. However, the proof follows the sameas in [6, 36]. Remark11.3. If the plant is linear time-invariant, the above synthesis conditions recover the necessary and sufficient conditions for 7-/~ control in [17,21].

LPV Saturation

11.3.

281

Control

LPV Anti-Windup

Control

Design

In this section, a new anti-windup controller design framework for parameter-varying plants with input saturation will be presented using an LPVcontrol formulation. The consideration of parameter-varying plants allows the treatment of nonlinear plants via linearization about a family of operating points. Consider the following generalized parameter-varying plant G (with all weighting functions embedded) including an input saturation nonlinearity a(u)

[y()J

~?(t)] F A(p(t)) Bl(p(t)) e(t)| |C~(p(t)) Dl ~(p(t)) D1

(11.14)

TMand where x(t) Rn,u(t) E Rn",y(t) ~ Rn’,d(t) e Rn’~,e(t) e p(.) ~ .~. All state-space matrices have compatible dimensions and are continuous functions of the parameter vector p. The saturation function a(u) specifies the limited actuator capacity on the control input u. We consider decoupled, sector-bounded, static actuator nonlinearities with constant saturation limit u~ax in the ith channel, that is

~(u~) for i = 1,2,... rameters

lud ~

= sign(u~)up~

, n~. Weintroduce the following saturation scheduling pa-

O~(u~) - a(u~), for i = 1,2,...

,nu

(11.15)

Ui

and 0i(0) = 1, that is, O~(u~)~ (0, 1]. Hence, the variable 0~ defines the level of saturation of the ith actuator at each instant of time; see Figure 1. Our objective is to design parameter-varying controllers that are scheduled based on both the operating condition parmeter vector p and the saturation indicator parameter vector 0. Let us define the following saturation matrix operator 0 = diag(0~, 02,...

, On~,)

where 0~ is given by (11.15). Clearly ~ = I represents the situation that all actuators are in their linear regimes. Otherwise, the values on the diagonal elements of O will reflect the status of each saturated actuator. Let ~o denote the set of arbitrary parameter trajectories 0(.) with 0 10~(t)l ~ 1

282

Wu and Crigoriadis

operating point 0

~ = tan(C) 0e (0,1]

Figure 1: Saturation indicator parameter. for all t. Now,the parameter-varying saturation plant G can be rewritten in the standard LPVform

~(~)l ’ A(p(t)) Bl(p(t)) B2(p(t))O(t) e(~)| CI(p(t))Dll(p(t)) D~2(p(t))O(t) a(t)/ ~(~)|= C2(p(t))D2~(p(t))D22(p(t))O(t)uX( t) (t)J o o (o(t)- s)

[

(11.1~)

where (p, 0) e ~;~ x 9me and ~(t) = ~r(u) - u. It is noted that a information on rate limits of the actuators can also be incorporated easily in a similar framework. Let fi -- [pT oT] T, and consider a parameter-varying controller K of the form

[~(t)l rA(fi(t),ts(t)) u(t) ] = [C~(#(t),b(t))

B~,~(fi(t), b(t)) l / y(t) D~,~(~(t), b(t))] L $(t)

(11.17) that takes measurements of y and ~. Weseek to design such a controller to solve the LPV"7-Performance/~-Variation Problem for the parametervarying saturated system (11.14), that is, to guarantee that max lIFt(G, K)II,~ < "7,

(11.18)

LPV Saturation

283

Control

where F~(G, K) is the lower linear fractional transformation (LFT) of the operators G and K, which denotes the closed-loop system of (11.14)- (11.17); see Figure 2.

LPVanti-windup controller

Figure 2: LPVanti-windup controller

structure.

Based on the proposed parameter-varying formulation of this problem, we obtain the following result. Corollary 11.1. Sufficient conditions for the solvability of the LPV anti-windup control synthesis problem (11.18) are provided by Theorem 11.2 using the following substitutions A=A,

BI=B1,

B2=B20,

A parameter-varying controller that solves the LPVanti-windup control synthesis problem is provided by the expressions (11.10)-(11.13) where

Ak=Ak,[B~,y B~,~] = Ck =C~, [Dk,~ Dk,g] = Therefore, the LPVanti-windup control synthesis problem is formulated as an LMIoptimization problemthat can be solved efficiently using interiorpoint optimization algorithms [10, 18, 26]. Note that, since the nonlinear

284

Wu and Grigoriadis

operator er(.) is included in the controller state-space data, the resulting controller is nonlinear. In particular, the output of such a controller will depend on, or be gain-scheduled by the status of saturated actuators. To account for different performance requirements over the linear and the saturated actuator regions, parameter-dependent weighting functions should be used to reshape the saturated LPV system performance. For example, enhanced anti-windup requirement of the LPVcontroller can be achieved using small and large penalties for the control effort over the linear and the saturation operating regions, respectively. Due to the specific structure of our problem, we emphasize that parameter-dependent weighting functions are necessary to achieve a successful LPVanti-windup control design. To observe this, suppose that the saturated system (11.14) does not depend on any parameter p. Then, the converted LPVsystem (11.16) will have its parameter dependency only B2 and D12 terms. Now, it can be easily seen that Ker [B~ z)~T2] = Ker [OB~T OD~T2] = Ker [B~ D~2 ]

and the LMIs specified over the whole parameter space in Theorem 11.2 collapse to parameter-independent LMI constraints. Hence, the solution of ghe LPVsynthesis conditions are constant matrices. In this case, the controller gains of Theorem11.2 take a special form as F(O) = --0 -1 [D~2DI2]+ [TB~R-1 + Df2C1 ]

+ 0] +UlV l] [V=lV;,] Consequently, the resulting LPVcontroller has the following structure

Lj

A~

[ J

where the star.space data A~, B~, C~ coincide with the solution from standard ~ control design when the saturation elements are not present; see [3]. It can be easily verified that due to cancellations of the saturation parameters O, the closed-loop system in this case is parameter-independent. Through this counter argument, we have shown that parameter-dependent weighting functions are needed to provide different closed-loop performance specifications over different parameter ranges ~nd to ensure parameter d~ pendeacy of the closed-loop system on the saturation indicator parameter 0.

LPV Saturation

285

Control

Remark 11.4. The proposed approach for anti-windup controller design based on LPVcontrol theory is different from classical anti-windup schemesin several aspects: 1. Most of the previous anti-windup control schemes are constructed using a ’two-step’ design procedure that consists of designing a controller ignoring the saturation effects and then adding anti-windup compensator to alleviate the performance degradation due to saturation. Such a design procedure often results in conservatism and unnecessarily sacrificed performance. The proposed LPVanti-windup controllers are designed based upon different performance requirements over different operating conditions. 2. The proposed LPV control approach provides a dynamic, gain-scheduled controller for anti-windup design with guaranteed stability and performance properties. However, previous anti-windup schemes often use static compensation mechanismand generally they do not have desired stability and performance properties. 3. The design procedure for anti-windup controller using LPVformulation is simple because the synthesis conditions are formulated as an LMIoptimization problem, for which effective numerical algorithms exist to provide solutions. 4. If there is no actuator nonlinearity (0 = I), this proposed LPVantiwindup design procedure provides the standard optimal 7/~ control synthesis conditions. Hence, linear performance recovery is achieved. 11.4.

Application

to

a Flight

Control

Problem

Despite recent advances, the flight control design problem for modern aircraft continues to be one of the most challenging control problems because of the complexity of the system and the stringent flight performance and reliability specifications. Aircraft control is inherently nonlinear and multivariable and the corresponding models contain significant modeling uncertainty, variable parameters and unknownexternal disturbances (such as wind gusts) that are affecting the system. The complexity of the flight control problem for modernaircraft is increasing, resulting in significant pilot workload. In addition, future aircraft are likely to have manymore control surfaces than current designs and the pilot will have to rely on advances in automatic flight control [1]. The aircraft dynamics are described by the nonlinear coupled rigid body equations of motion and the aircraft body flexible dynamic equations. Linearized aircraft models can be obtained using small disturbance theory but

286

Wu and Grigoriadis

they are only valid for small regions about trim operating conditions. However, the expandedoperational envelope and the wide range of flight operating conditions of modernaircraft yields significant parameter variability in the aircraft model. Hence, the dynamic response characteristics can vary substantially during a typical mission. Single fixed-structure controllers cannot qapture effectively this wide operating range resulting in compromised performance and gain scheduling approaches have been used extensively to provide controllers that are scheduled (interpolated) to each operating condition [20]. Recently, systematic procedures for gain-scheduling have been proposed [13] [9] [5]; however, important control design questions and issues still remain unresolved. Current approaches to flight control are limited by the presence of actuator saturation constraints. Aircraft control surfaces, such as elevators, ailerons, rudder and flaps, and thrust vectoring control have limitations both in their maximumdeflections and in their rate of variations due to geometric or aerodynamic constraints. Several aircraft mishaps have been attributed to the control surface rate limitations such as the F-22 crash in April 1992 [14] and the Gripen crash in August 1992 [33]. Actuator limitations have also been identified as a major contribution to catastrophic pilot induced oscillations (PIOs) [25]. Hence, the proposed approach could result in significant improvementin flight control applications. Wewill apply our proposed LPVanti-windup control synthesis technique to the control of a linearized F-8 aircraft modelconsidered in [22]. An anti-windup control schemewas designed in [22] using a saturation compensator (error governor) that scales downthe control input when saturation occurs, i.e. the error governor plays the role of a static gain-scheduling element. However, the proposed scheme is only applicable to open-loop stable systems. In this example, we will demonstrate the effectiveness of our proposed approach through the design of a dynamic gain-scheduled, anti-windup controller for an unstable plant. The following state equations describe the longitudinal dynamics of the F-8 aircraft

~(t) -!.8

-0.006 -0.014 -0.0001 0

-12 -16.64 -1.5 0

0 -32.2 0 0

x(t)

-19 -0.66 -0.16 0

_3]

’5

(11.19)

=

o -1 =

(11.20)

LPV Saturation

Control

287

The input, output and state variables are as follows J" ~e(t) elevator angle (deg) saturation limit at °) 52(t) flaperon angle (deg) saturation limit at °) ~)(t) pitch angle (rad) y(t) = "~(t) flight path angle (rag)

{

x(t) =

q(t) v(t) a(t) ~(t)

pitch rate (rad/sec) forward velocity (ft/sec) angle of attack (rad) pitch angle (rad)

The open-loop system dynamics has been slightly modified and has an unstable pole at 0.14 to demonstrate the applicability of the proposed LPV design method to an unstable plant. Also, the saturation limit of both control surfaces is reduced to ~15° from ~25°, that is, u~~ = u~~ = 15. Wedefine the following saturation scheduling parameters

~ := { ]~

f ~(~)

~=o

~ = o,

the~ the plant can be converted to the parameter-varying form (11.16). Moreover~we wi]] use someparameter-dependentweighting functions to enhancethe tracking performanceover the linear operating range and to penalize the control magnitudeover the s~turation range (see Figure 3). Specific~l]y~ weselect the following parameter-dependent weightingfunctions

w:(~)=D}+O}(~Z ---~,~-~n~ W~(~) = D[+ O}(~1 A~)-~ where

~ = -o.o~,~= ~i~g{~.~s~o.~s + (~)~o, ~.~s~o.~s + (~)~o}, ~[ =-~oo~, ~[=~i~g{l~.S~L0~l (O~)~0,--~0.0ff~.0~ ~ = -~, D[= ~j~g{~.0e~ - ~.~(~,)~o, ~.0~-~(~)~o}

288

Wuand Grigoriadis

Figure 3: Weighted open-loop interconnection model.

for input saturated

F-8

Note that all the weighting functions above are specified in a balanced realization format, and have continuous parameter dependency. The control system is required to track reference signals with steady state tracking error less than 2%if there is no saturation. Notice that whenone of the actuators saturates (0i < 1), the tracking weights will decrease to emphasizestability over performance. On the other hand, there is a small penalty on the control input to emphasize tracking performance over the linear region. This can be seen clearly from the frequency plot of the weighting function~,; with frozen parameter values, as shownin Figure 4. 11.4.1.

Single

Quadratic

Lyapunov Function

Case

For anti-windup controller synthesis using the proposed LPVframe~. work, we choose a saturation parameter range as follows 0i ~ [0.5, 1], i = 1, 2. Wefirst consider the case of a single quadratic Lyapunovfuncton. Gridding the two-dimensional parameter space uniformly with a 10 x 10 grid, we obtain an induced £2 norm performance level of ~ = 2.159. Such. a gridding method is a standard approach to convert infinite-dimensional LMIconstraints in LPVsynthesis to finite-dimensional ones [2, 36]. Since the optimization only guarantees that the LMI constraints hold at grid points, we further analyzed the resulting solution over a more dense grid of 50 x 50 points, and verified that the synthesis LMIsare indeedsatisfied over the entire parameter space.

LPV Saturation

289

Control

210

°1o

Frequency

(a) Performance weight 210

(b) Control weight Wu Figure 4: Parameter-dependent weighting functions evaluated at frozen parameters: 01 = 02 = 1 (solid line), 01 = 02 = 0.5 (dash line).

Note that implementing the weighting function We° in a controllable canonical realization instead of a balanced realization results in a different closed-loop induced/:2 norm performance of ~, -- 2.233. For an observable canonical realization of We~, we obtain ~ = 2.245. Such a dependencyof the performance level on the weighting function realization is expected due to the choice of a constant Lyapunovfunction. It is easy to see that different realizations of parameter-dependent weighting functions imply parameterdependent (i.e. time-varying) coordinate transformation of the weighted open-loop systems. The LPVcontroller is implemented in the following manner: the con-

290

Wuand Grigoriad~!s

troller takes measurementof the parameters and their rates in real-time, and computes the controller output using equations (11.10)-(11.13). saturation scheduling parameters are measured by comparing the inputs and outputs of the actuator blocks. Their derivatives can be estimated using some standard estimation algorithms. With the synthesized antiwindup LPVcontroller, the closed-loop tracking performance of the aircraft model (11.19)-(11.21) to a reference signal r = [10, T is shown i~ Figure 5.

(a) outputs

(b) saturated control inputs Figure 5: LPVcontrol of F-8 model with input saturation nonlinearity. Figure 6 shows the time history of the saturation indicator parameters; 01(t) (solid line) and 02(t) (dash line). It is observed that whenthe second. input channel (dash line) is saturated, the controller gains change because.. 02 < 1. During the saturation period, the LPVanti-windup controller is.

LPV Saturation

291

Control

Figure 6: Saturation indicator history. seeking to eliminate saturation by adjusting its controller gains accordingly. This is reflected by the degraded tracking performance and the increased penalty on the control effort. After the control signal moves out of the saturation zone, the controller resumesits full tracking capability, whichis characterized by the scheduling parameters 81 --- 8~ -- 1. Also note that 82 varies between 0.6 and 1, which falls into the specified parameter range. For comparison purposes, we have also designed an 7~ controller by ignoring the nonlinear saturation effect. The corresponding performance level is ~, -- 0.647. However,the performanceof such a controller is severely deteriorated in the presence of the input saturation. This is demonstrated by the presence of large oscillations in the response as shownin Figure 7. To better understand the behavior of the LPVand 7-/~ controllers, we applied these same controllers to the F-8 model without actuator saturation. The simulation results are given in Figures 8 and 9. Notice that the control input associated with the 7-/~ controller is significantly higher to provide quick tracking response. On the other hand, the proposed antiwindupLPVdesign penalizes the control action during the saturation range to alleviate the saturation effects. The overall performanceof the LPVcontroller presents desired graceful performance degradation for anti-windup design. 11.4.2.

Parameter-Dependent

Lyapunov Function

Case

For comparison we also examine the parameter-dependent Lyapunov function case. Wechoose saturation parameters 8~ E [0.5, 1], i -- 1,2 and parameter variation bound as [10, 10]. In order to solve the infinite-

292

Wuand Grigoriadis

1

2

3

4

T~nS~e 6

7

8

(a) outputs

0

1

2

3

4

Ti5me

6

7

(b) saturated control inputs Figure 7: 7-/~ control of F-8 tnodel with input saturation nonlinearity. dimensional LMIproblems in (11.7)-(11.9), we parameterize the matrix function space using three basis functions as /1

= 1, f2(~)

~---

~1, f3(~)

=

The synthesized LPVcontroller generally would depend on parameter rate measurement, which is undesirable. This can be avoided by a special choice of matrix functions R, S [6]. Gridding the two-dimensional parame-ter space uniformly using a 5 x 5 grid, we obtain the guaranteed induced/22 norm performance level to be 2.098 and 2.097 for R or S fixed seperately. The achievable performance with various forms of R, S is shown in Table 1 It is observed that the inclusion of parameter dependence into the Lya-

LPV Saturation

293

Control

2

3

4

5 Time

6

7

8

9

10

(a) outputs 25

,

(b) control inputs Figure 8: LPVcontrol of F-8 model without saturation effect. punov function does not provide appreciable performance improvement for this problem. Moreover, in a saturation control problem, no a priori information about parameter variation rates is easily available. Thus, the use of a single quadratic Lyapunovfunction is justified in this case. 11.5.

Conclusions

In this work, we propose a novel anti-windup control synthesis approach for systems with a class of input nonlinearitles, such as, static input saturations, using a linear parameter-varying framework. Saturation indicator parameters are used to schedule the controller depending on the saturation levels. Other input nonlinearities can also be treated using the proposed

294

Wu and Grigori~dis

1:¸

0

1

2

3

4

5 Time

s

(a) outputs

(b) control inputs Figure 9: T/~ control of F-8 model without saturation effect.

approach. Parameter-dependent weighing functions are used to quantify different performance requirements over the linear and the saturation op.erating regions. The dynamic gain-scheduling feature of the proposed LPV anti-windup control provides guaranteed stability, and potential for perfor.mance improvement. The synthesis conditions are in the form of LMIs that can be solved using efficient interior-point algorithms. The method can be applied to nonlinear systems using a standard LPVcontrol approach to develop anti-windup control for a family of linearized models at different operating points. A longitudinal dynamics flight control example with two saturation control inputs is used to demonstratethe effectiveness of the pro-posed approach and the graceful performance degradation in the presence of saturation.

LPV Saturation

295

Control

Table 1: Induced £2 norm of LPV saturation parameter-dependent Lyapunov functions. R fixed S fixed R,S parameter-dependent

control

design using

2.098 2.097 2.096

References [1] J.R. Adams, J.M. ButYington, A.G. Sparks, and S.S. Banda. Robust Multivariable Flight Control, Springer-Verlag, (1994). [2] P. Apk~rian and R. Adams. Advanced Gain-scheduling Techniques for Uncertain System, IEEE Trans. Control Syst. Tech., 6 (1998) 21-32. of Gain[3] P. Apkarian and P. Gahinet. A Convex Characterization scheduled ~ Controllers, IEEE Trans. Automat. Control, 40 (1995) 853-864. [4] K.J./~strSm and B. Wittenmark. Computer Controlled Systems: Theory and Design, Prentice-Hall Inc., (1984). [5] G.J. Balas, I. Fialho, A. Packard, J. Renfrow, and C. Mullaney. On the Design of LPVControllers for the F-14 Aircraft Lateral-directional Axis during Powered Approach, in: Proc. of American Control Conf. (1997) 123-127. [6] G. Becker. Additional Results on Parameter-dependent Controllers for LPVSystems, in: Proc. of 13th IFAC World Congress (1996) 351-356. [7] G. Becker and A. Packard. Robust Performance of Linear Parametrically Varying Systems using Parametrically-dependent Linear Feedback, Systems and Control Letters, 23 (1994) 205-215. [8] D.S. Bernstein and A.N. Michel. A Chronological Bibliography on Saturating Actuators, Int. J. Robust and Nonlinear Control, 5 (1995) 375-380. [9] J.-M. Biannic, P. Apkarian, and W.L. Garrard. Parameter Varying Control of a High-performance Aircraft, AIAA J. Guidance, Control and Dynamics, 20 (1997) 225-233. [10] S.P. Boyd, L. E1 Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in Systems and Control Theory, SIAM,(1994).

[11] P.J. Campoand M. Morari. Robust Control of Processes Subject to Iuput Saturation Nonlinearities, Computers and Chemical Engineering, 14 (1990) 343-358. [12] P.J. Campo, M. Morari, and C.N. Nett. Multivariable Anti-windup and Bumpless Transfer: A General Theory, in: Proc. of American Control Conference (1989) 1706-1711. [13] L.H. Carter and J.S. Shamma. Gain-scheduled Bank-to-turn Autopilot Design using Linear Parameter Varying Transformations, AIAA J. Guidance, Control and Dynamics, 19 (1996) 1067-1063. [14] M.A. Dornheim. Report Pinpoints Factors Leading to YF-22 Crash, Aviation Week ~¢ Space Technology, 137 (1992) 53-54. [15] J. Doyle, R. Smith, and D.F. Enns. Control of Plants with Input Saturation Nonlinearities, in: Proc. of American Control Conference (1987) 1034-1039. [16] H.A. Fertik and C.W. Ross. Direct Digital Control Algorithm with Anti-windup Feature, ISA Transactions, 6 (1967) 317-328. [17] P. Gahinet and P. Apkarian. A Linear Matrix Inequality Approach to 7-t~ Control, Int. J. Robust and Nonlinear Control, 4 (1994) 421-448. [18] P. Gahinet, A. Nemirovskii, A.J. Laub, and M. Chilali. Toolbox. Mathworks, Natick, MA,1995.

LMI Control

[19] R. Hanus, M. Kinnaert, and J.L. Henrotte. Conditioning Technique, A General Anti-windup and Bumpless Transfer Method, Automatica, 23 (1987) 729-739. [20] R.A. Hyde. H~ Aerospace Control Design: A VSTOLFlight Application, Springer-Verlag, (1995). [21] T. Iwasaki and R.E. Skelton. All Controllers for the General ~ Control Problem: LMI Existence Conditions and State Space Formulas, Autoraatica, 30 (1994) 1307-1317. [22] P. Kapasouris, M. Athans, and G. Stein. Design of Feedback Control Systems for Stable Plants with Saturating Actuators, in: Proc. of 27th IEEE Conf. on Dec. and Control (1988) 469-479. [23] M.V. Kothare, P.J. Campo, M. Morari, and C.N. Nett. A United Framework for the Study of Anti-windup Designs, Automatica, 30 (1994) 1869-1883. [24] M.V. Kothare and M. Morari. Multiplier Theory for Stability Analysis of Anti-windup Control Systems, in: Proc. of 3~th IEEE Conf. on Dec. and Control (1995) 3767-3772.

LPV Saturation

297

Control

[25] K. McKay. Summary of an AGARDWorkshop on Pilot Induced Oscillations, in: Proc. of AIAA Guidance, Navigation, Control Conf., Scottsdale, AZ (1994). [26] Y. Nesterov and A. Nemirovski. Interior in Convex Programming, SIAM, (1993).

Point Polynomial Methods

[27] A. Packard. Gain Scheduling via Linear Fractional Transformations, Systems and Control Letters, 22 (1994) 79-92. [28] W.J. Rugh. Analytical Framework for Gain Scheduling, IEEE Control Systems Magazine, 11 (1991) 74-84. [29] A. Saberi, Z. Lin, and A.R. Teel. Control of Linear Systems with Saturating Actuators, IEEE Trans. on Automat. Control, 41 (1996) 368-378. [30] J.S. Shammaand M. Athans. Analysis of Nonlinear Gain-scheduled Control Systems, IEEE Trans. on Automat. Control, 35 (1990) 898907. [31] J.S. Shammaand M. Athans. Guaranteed Properties uled Control for Linear Parameter-varying Plants, (1991) 559-564.

of Gain SchedAutomatica, 27

[32] J.S. Shammaand M. Athans. Gain Scheduling: Potential Hazards and Possible Remedies, IEEE Control Systems Magazine, 12 (1992) 101-107. [33] C.A. Shifrin. SweedenSeeks Cause of Gripen Crash, Aviation Week ~J Space Technology, 139 (1993) 78-79. [34] F. Tyan and D.S. Bernstein. Anti-windup Compensator Synthesis for Systems with Saturation Actuators, Int. J. Robust and Nonlinear Control, 5 (1995) 521-537. [35] M. Vidyasagar. Nonlinear Systems Analysis, Prentice-Hall Inc., (1978). [36] F. Wu, X.H. Yang, A. Packard, and G. Becket. Induced /22 Norm Control for LPVSystems with Bounded Parameter Variation Rates, Int. J. Robust and Nonlinear Control, 6 (1996) 983-998.

Index Absolute stability, 163-187 Anti-windup, 1-31, 77-107, 109-134 compensation, 77-107, 273-297 control, 77-107, 273-297 LPV control, 273-297

[Constraint] type one, 189-226 weakly minimum phase, 189226 weakly non-minimum phase, 189-226 Control bang-bang, 47-76 bounded, 47-76, 163-187, 189226 decentralized, 227-246 extremal, 47-76 fixed-gain, 77-107 global, 1-31 high gain, 109-134 input-output linearizing, 227246 local, 1-31, 163-187 low-and-high gain, 109-134, 163-187 model predictive, 227-246 saturating, 163-187 saturation, 247-272 switching, 77-107 Controller detuning, 227-246 parameter-dependent, 273-297 windup, 273-297 Convex numerical algorithm, 135-161 optimization, 77-107, 163-187

Basin ofattrac~on, 247-272 Bioreactor, 227-246 Chemical reactor, 227-246 Circle analysis, 77-107 criterion, 135-161, 163-187 Clipping, 227-246 Conditional integration, 227-246 Conditioning, 227-246 Congruence transformation, 77-107 Constrained optimization, 227-246 Constraint amplitude, 189-226, 247-272 at most non-minimum phase, 189-226 infinite zero, 189-226 input, 189-226, 227-246 input position and rate, 247272 invariant zero, 189-226 magnitude and rate, 189-226 minimum phase, 189-226 non-right invertible, 189-226 output, 189-226 rate, 189-226 right invertible, 189-226 state, 189-226 strongly non-minimum phase, 189-226 taxonomy of, 189-226

Deadbeatoperator, 189-226 Decomposition Schur, 135-161 singular value, 135-161,163187 299

300

Index

Design dynamic, 189-226 low gain, 189-226 low.high gain, 189-226 scheduled low gain, 189-226 scheduled low-high gain, 189226 Differential inclusion, 247-272 Directionality compensation, 227246 Direction preservation, 227-246 Directly saturating control, 77-107 Disturbance attenuation, 109-134 rejection, 33-45, 163-187 Domainof attraction, 47-76, 77-107 estimate of, 135-161 Domainof performance, 77-107 Energy gain, 163-187 Equilibrium manifold, 1~31 Exosystem, 189-226 Feedback measurement, 189-226 output, 77-107, 135-161, 109134, 163-187, 247-272 smooth, 47-76 state, 1-31, 77-107, 109-134, 135-161, 163-187, 227-246 Flight control, 273-297 Forward invariant region, 1-31 Gain scheduling, 273-297 Gaussian random process, 33-45 H®control, 273-297 Hausdroff distance, 47-76 Index theory, 47-76 Inner approximation, 163-187 Invariant set, 1-31, 77-107 zeros, 189-226 L2

gain, 109-134,163-187

norm, 273-297 Limit cycle, 47-76 Limiter command, 1-31 rate, 109-134, 135-161 Linear fractional transformation (LFT), 273-297 Linear parameter varying (LPV), 273-297 Linear system anti-stable, 47-76 exponentially unstable, 47-76 time-invariant, 77-107 Loop transformation, 163-187 LQR, 47-76 Lure-Postnikov, 163-187 Lyapunovfunction, 47-76, 135-161, 163-187 parameter dependent, 109-134, 273-297 quadratic, 273-297 Lyapunov matrix constant, 109-134 parameter-dependent, 109-134 Magnitude and rate limit, 1-31 Matrix inequality bilinear (BMI), 163-187 bounded real, 109-134 linear (LMI), 77-107, 135-161, 163-187, 247-272, 273-297 nonlinear, 135-161 Measurement governor, 1-31 Model polytopic, 247-272 uncertainty, 247-272 Nonlinearity actuator, 247-272 deadzone, 163-187 input, 273-297 memoryless, 163-187 saturation, 227-246, 273-297 sector-bounded, 77-107, 135161, 163-187, 273-297

301

Index Norm-bounded uncertainty, 272 Null controllable, 189-226 region, 1-31, 47-76 Nyquist plot, 33-45

247-

Optimization interior-point, 273-297 LMI, 273-297 Orthogonal complement, 163-187 Output regulation global, 189-226 semiglobal, 189-226 Performance H2, 77-107 quadratic, 135-161 Popov criterion, 33-45, 77-107, 163-187 Positive invariance, 1-31 Pseudo-invariance, 163-187 Quasi-concave maximization, 77107 Quasi-convex optimization, 77-107 Quasi-linear gain, 33-45 Rate bound, 109-134 limitation, 247-272 Reachable set, 109-134 Reference governor, 1-31 Region of attraction, 109-134, 163-187 stability, 247-272 Riccati equation, 47-76, 189-226, 247-272 Saturated actuator, 273-297 Saturating actuator, 47-76 Saturation amplitude, 135-161 amplitude and rate, 135-161, 247-272 asymmetric, 47-76 degree of, 77-107 input, 247-272

[Saturation] local analysis, 163-187 magnitude, 1-31, 109-134 nonlinearity, 77-107, 247-272 parameter-varying, 273-297 rate, 1-31, 135-161, 247-272 symmetric, 47-76 Scaling matrix, 135-161 Scheduling approach, 109-134 Schur complement, 77-107, 135-161 Semidefinite program, 163-187 Small gain dissipation inequality, 163-187 Spline function, 109-134 S-procedure, 77-107 Stability global, 135-161 local, 135-161, 109-134, 163187 multiplier, 135-161 quadratic, 109-134 region, 163-187 Stabilization global, 163-187 quadratic, 247-272 robust local, 247-272 semi-global, 77-107 Standard deviation, 33-45 Stochastic linearization, 33-45 System discrete-time, 135-161 linear parameter varying (LPV), 109-134 polytopic LPV, 109-134 saturated, 1-31 time-reversed, 47-76 uncertain, 247-272 Tracking problem, 189-226 Trajectory closed, 47-76 extremal, 47-76 Unit saturation

function,

Windup, 1-31 compensation,

163-187

227-246