Robust Systems Theory and Applications (Adaptive and Learning Systems for Signal Processing, Communications and Control Series)

ISBN 0-471-17627-3

llllllllllllllllllllllllllllllii11MIIijl11

ROBUST SYSTEMS THEORY AND APPLICATIONS

Adapti ve and Learning Systems for Signal Processing, Comm unicati ons, and Control Editor: Simon Haykin

Werbos I THE ROOTS OF BACKPROPAGATION: From Ordered Derivativ es to Neural Networks and Political Forecas ting Krstic, Kanellak opoulos , and Kokotov icl NONLINEAR AND ADAPTIVE CONTROL DESIGN Nikias and Shoo I SIGNAL PROCESSING WITH ALPHA-STABLE DISTRIBUTIONS AND APPLICATIONS Diamant aras and Kung I PRINCIPAL COMPONENT NEURAL NETWORKS: THEORY AND APPLICATIONS Tao and Kokotov icl ADAPTIVE CONTROL OF SYSTEMS WITH ACTUATOR AND SENSOR NONLINEARITIES Tsoukalas I FUZZV AND NEURAL APPROACHES IN ENGINEERING

ROBUST SYSTEMS THEORY AND APPLICATIONS Ricardo

s. Sanchez-Pena Mario Sznaier

Hrycej I NEUROCONTROL: TOWARDS AN INDUSTRIAL CONTROL METHODOLOGY Beckerm an I ADAPTIVE COOPERATIVE SYSTEMS Cherkas sky and Mulier I LEARNING FROM DATA: CONCEPTS, THEORY. AND METHODS Passino and Burgess I STABILITY OF DISCRETE EVENT SYSTEMS S6nchez -Pena and Sznaier I ROBUST SYSTEMS THEORY AND APPLICATIONS Vapnik I STATISTICAL LEARNING THEORY

A Wiley-Interscience Publicati on

JOHN WILEY & SONS, INC. New York I Chiches ter I Weinhei m I Brisbane I Singapo re I Toronto

To our parents Lolita and Migu el, Reb eca and Mar cos

This book is printe d on acid-f ree paper . Copy right

©

@

1998 by John Wiley & Sons. Inc. All rights reserv ed.

Published simul taneo usly in Canad a. No part of this public ation may he reprod uced, stored in a retrie val system or transm itted in any form or by any mean s, electr onic, mechanical. photo copyi ng, recording, scann ing or otherw ise, excep t as permi tted under Sectio n 107 or 108 of the 1976 Unite d States Copyr ight Act. without either the prior writte n permi ssion of the Publis her, or autho rizatio n through paym ent of the appro priate per-co py fee to the Copy right Clear ance Cente r, 222 Rosew ood Drive , Danv ers, MA 01923, (978) 750-8 400, fax (978) 750-4744. Requ ests to the Publisher for permi ssion shoul d be addre ssed to the Permissions Depa rtmen t, John Wiley & Sons, Inc., 605 Third Aven ue, New York, NY 10158 -0012, (212) 850-6011, fax (212) 850-6008, E-Mail: PERM REQ@ W!LE Y.CO M.

Library of Congress Cataloging-in-Publica tion Data:

Sanch ez-Pe na, Ricar do Robu st system s theor y and applic ations / Ricar do Sanch ez-Pe na, Mario Sznaier. p. em. Includ es biblio graph ical refere nces and index. ISBN 0-471-17627-3 l. Autom atic contro l. 2. Contr ol theory . 3. Syste m identification. I. Sznaier, Mario . II. Title. TJ213.S11544 1998 629 .S--dc21 98-6420 CJP Printe d in the Unite d States of Amer ica

10 9 8 7 6 5 4 3 2 1

i] f

i'

CONTENTS

1 Introduction

1

General Control Problem / 1 1.1.1 Experimental Phase / 2 1.1.2 Simulation Phase / 3 1.1.3 Theoretical Phase / 5 1.2 Why Feedback? / 11 1.3 Feedback Loop Trade-off / 14 1.4 Objectives of an Applied Theory / 17 1.5 Objectives of This Book / 19 1.5.1 General Remarks / 19 1.5 .2 Scope / 19 1.5.3 How to Use This Book / 21

1.1

2 SISO Systems 2.1 2.2 2.3 2.4

2.5

2.6 2.7 2.8

23

Introduction / 23 Well Posedness / 24 Nominal Internal Stability / 26 Robust Stability / 29 2.4.1 Phase and Gain Margins / 29 2.4.2 Global Dynamic Uncertainty / 32 Nominal Performance / 42 2.5.1 Known Disturbance/Noise/Reference / 42 2.5.2 Bounded Disturbances at the Output / 44 2.5.3 Other Performance Criteria / 47 Robust Performance / 50 Extension to MIMO Systems / 55 Problems / 57 vii

viii

3

Stabilization 3.1 3.2

3.3 3.4

3.5 3.6 3.7 3.8 3.9

4

61

Introduction I 6! Well Posedness and Internal Stability 1 63 3.2.1 Well Posedness 1 63 3.2.2 Internal Stability 1 64 Open-Loop Stable Plants 1 65 The General Case I 68 3.4.1 Special Problems 1 69 3.4.2 The Output Feedback Case I 75 Controller Structure and Separation Principle Closed-Loop Mappings 1 79 A Coprime Factorization Approach 1 80 3.7.1 Coprime Factorizations I 80 LFTs and Stability I 88 Problems I 90

5.6 5.7

4.7

4.8 4.9

4.7.1 Permanent Magnet Stepper Motor 4.7.2 Loop Shaping Q(s) 1 119 Related Design Procedures 1 123 Problems I 125

4.5

4.6

1 78

5.4 5.5

Introduction

I

Introduction I 207 7.1.1 Stability Margin I 210 7.2 Structured Dynamic Uncertainty I 215 7.2.1 Computation I 215 7.2.2 Analysis and Design I 219 7.3 Parametric Uncertainty I 223 7.3.1 Introduction I 223 7.3.2 Research Directions I 225 7.3.3 Kharitonov's Theorem I 230 7.3.4 Mapping Theorem I 233 7.4 Mixed Type Uncertainty I 237 7.4.1 Introduction I 237 7.4.2 Mixed fL I 239 7.5 Problems I 242

1 109

127

127

The LQR Case: Kalman's Inequality

207

7.1

The Classical Linear Quadratic Regulator Problem The Standard 1{2 Problem 1 133 Relaxing Some of the Assumptions 1 141 Closed-Loop Properties 1 142 5.5.1

157

7 Structured Uncertainty

1 99

1l2 Optimal Control 5.1 5.2 5.3

145 146

Introduction I 157 The Standard 'H 00 Problem I 158 6.2.1 Background: Hankel and Mixed Hankel-Toeplitz Operators I 161 6.2.2 Proof of Theorem 6.1 I 176 6.3 Relaxing Some of the Assumptions I 179 6.4 LMI Approach to 'H= Control I 180 6.4.1 Characterization of All Output Feedback 'Hcxc Controllers I 183 6.4.2 Connections with the DGKF Results I 188 6.5 Limiting Behavior I 192 6.6 The Youla Parametrization Approach I 193 6.7 Problems I 203

1 97

Nominal Performance and Robust Stability Robust Performance I 100 4.5.1 Sensor Uncertainty 1 101 4.5.2 Actuator Uncertainty 1 102 Design Procedure I 105 Examples I 109

I I

ix

6.1 6.2

93

Introduction I 93 Nominal Performance Robust Stability I 98

5.5.2 Some Consequences of Kalman's Inequality 5.5.3 Stability Margins of Optimal 1i 2 Controllers Related Problems I 149 Problems I 152

6 1loo Control

Loop Shaping 4.1 4.2 4.3 4.4

5

CONTENTS

CONTENTS

1 143

1 128 8

245

£ 1 Control 8.1

Introduction

I

245

X

CONTENTS

CONTENTS

Robust Stability Revisited / 247 8.2.1 Robust Stability Under LTV Perturbations j 250 8.2.2 Stability Under Time Invariant Perturbations j 253 8.3 A Solution to the SISO £1 Control Problem / 254 8.3.1 Properties of the Solution / 262 8.3.2 The MIMO Case / 266 8.4 Approximate Solutions / 267 8.4.1 An Upper Bound of the £1 Norm / 268 8.4.2 The * Norm / 269 8.4.3 Full State Feedback / 273 8.4.4 All Output Feedback Controllers for Optimal *-Norm Problems / 276 8.5 The Continuous Time Case / 282 8.5.1 Solution Via Duality / 282 8.5.2 Rational Approximations to the Optimal £ 1 Controller / 285 8.6 Problems / 288 8.2

9

Model Order Reduction

10.2.1 Input Data / 325 10.2.2 Consistency / 326 10.2.3 Identification Error / 328 10.2.4 Convergence / 334 10.2.5 Validation / 338 10.3 Frequency-Domain Identification / 339 10.3.1 Preliminaries / 340 10.3.2 Sampling Procedure / 342 10.3.3 Consistency / 344 10.3.4 Identification Procedures / 346 10.4 Time-Domain Identification / 361 10.4.1 Preliminaries / 362 10.4.2 Identification Procedures / 366 10.5 Further Research Topics / 372 10.5.1 Unstable Systems / 372 10.5.2 Nonuniforrnly Spaced Experimental Points / 372 10.5.3 Model Reduction / 373 10.5.4 Continuous Time Plants / 373 10.5.5 Sample Complexity / 373 10.5.6 Mixed Time/Frequency Experiments / 374 10.5.7 Mixed Parametric/Nonpara metric Models / 374

293

Introduction / 293 Geometry of State-Space Realizations / 295 9.2.1 Controllable/Unobse rvable Spaces / 295 9.2.2 Principal Components / 297 9.3 Hankel Singular Values / 304 9.3.1 Continuous Systems / 304 9.3.2 Discrete Systems / 306 9.4 Model Reduction / 310 9.4.1 Introduction / 310 9.4.2 Hankel Operator Reduction / 310 9.4.3 Balanced Realizations / 312 9.4.4 Balanced Truncation / 314 9.5 Algorithms / 318 9.5.1 Approximation Error / 319 9.6 Problems / 321 9.1

9.2

10 Robust Identification 10.1 10.2

Introduction / 323 General Setup / 324

11

Application Examples SAC-C Attitude Control Analysis / 377 11.1.1 Introduction / 377 11.1.2 Linear Model / 378 11.1.3 Design Constraints / 381 11.1.4 Robustness Analysis / 383 11.1.5 Simulations / 385 11.2 Controller Design for a D 2 0 Plant / 389 11.2.1 Model of the Plant / 389 11.2.2 Robustness Analysis / 391 11.2.3 Controller Design / 397 11.3 X-29 Parametric Analysis / 400 11.3.1 Linear Model / 400 11.3.2 Results / 405 11.4 Control of a DC-to-DC Resonant Converter / 407 11.4.1 Introduction / 407

11.1

323

xi

377

xii

CONTENTS

11.4.2 11.4.3 11.4.4 11.4.5 11.4.6 11.4.7 11.4.8

CONTENTS

The Conventi onal Parallel Resonant Converte r Small Signal Model I 409 Control Objectives I 410 Analysis of the Plant I 411 Control Design I 414 Controlle r Synthesis I 420 Simulatio n Results I 421

I

407

B.9 Inverse I 469 B.10 Linear Fractiona l Transform ations I 470 B.ll Norm Computa tions I 475 B.11.1 1t 2 Norm Computa tion I 475 B.11.2 Hoo Norm Computa tion I 476 B.11.3 £1 Norm Computa tion I 476 B.12 Problems I 479

Bibliography

427

C

A

445

Index

Mathem atical Background Algebraic Structure s I 445 A.l.l Field I 445 A.l.2 Linear Vector Space I 446 A.1.3 Metric, Norm, and Inner Products I 447 A.2 Function Spaces I 449 A.2.1 Introduct ion I 449 A.2.2 Banach and Hilbert Spaces I 449 A.2.3 Operator and Signal Spaces I 450 A.2.4 Isomorph ism I 452 A.2.5 Induced Norms I 453 A.2.6 Some Importan t Induced System Norms I 455 A.3 Duality and Dual Spaces I 457 A.3.1 The Dual Space I 457 A.3.2 Minimum Norm Problems I 458 A.4 Singular Values I 460 A.4.1 Definitio n I 460 A.4.2 Propertie s and Applicati ons I 461 A.l

B System Comput ations B.l Series I 465 B.2 Change of Variables I 466 B.3 State Feedback I 467 B.4 State Estimatio n I 467 B.5 Transpos e System I 467 B.6 Conjugat e System I 467 B.7 Addition I 468 B.8 Output Feedback I 468

465

xiii

Riccati Equations

481 487

PREFACE

Robustness against disturbances and model uncertainty is at the heart of control practice. Indeed, in the (completely unrealistic) case where both all external disturbances and a model of the system to be controlled are exactly known, there is no need for feedback: Optimal performance can be achieved with an open loop controller. The main ingredients of present day robust control theory were already present in the classical work of Bode and in many popular frequency domainbased design techniques. With the advent of state-space methods in the mid 1960s the issue of robustness took a backseat to other topics, but was never completely abandoned by the control community. especially control practitioners. Interest in robust control rose again in the late 1970s where it was shown that many popular control methods (including optimal LQR control and controller design based on the cascade of an observer and state feedback) led to closed-loop systems very sensitive to model perturbations. Moreover, widely accepted "ad-hoc" recipes for "improving robustness," such as using artificially large noise levels in the design, had precisely the opposite effect. Robust control has undergone extensive developments in the past two decade.s, leading to powerful formalisms, such as 7-i 00 , 1-L-synthesis/analysis and, more recently, £1 optimal control, that, coupled with newly developed control-oriented identification techniques, have been successfully applied to challenging practical problems. A salient feature of the framework is that it is oriented towards applications and thus is based on "practical," realistic assumptions. There are many excellent books that cover specialized topics (7-i 2 , 1-ioo, £1 , parametric uncertainty, linear matrix inequalities) with others scattered in the technical journals. Our intention in writing this book is to provide a selfcontained overview of robust control that illustrates all the issues involved, ranging from the transformation of experimental signals from the physical plant to a set of models (robust identification), to the synthesis of a controller for that set of models (robust control). The purpose of the book is twofold: to serve as a textbook for courses at the Master's/beginning Ph.D. level and as a reference for control practitioners. It assumes that the reader has a background in classical and state-space control methods. In order to keep the XV

xvi

PREFACE

PREFACE

text size at a manageable level, in some cases only basic results are covered, and the reader is referred to more specialized literature for further coverage. In all cases we have strived to link the theory with practical applications. To this end, in addition to the examples covered throughout the book, the last chapter contains several worked out application problems that stress specific practical issues: nonlinearities, unknown time delays, infinite dimensional plants, actuator and sensor limitations. They are all extracted from our practical experience in different engineering fields. Furthermore, due to the fact that most of the problems to be solved by the theory presented here are computer-intensive, we stress the algorithmic and computational aspect of the solution along with the mathematical theory. We (and our graduate students) have tried to eliminate obvious mistakes. However, as anyone who has tried knows, it is virtually impossible to make a "perfect" book. We encourage readers to send corrections, comments and general feedback to either one of the authors. Finally, as attested by the list of more than 300 references, we have tried to give credit where it is due. However, owing to the sheer volume of literature published on robust control, we may have inadvertently failed to do so on occasion. We apologize in advance to readers or authors who may feel that this is the case, and we encourage them to send us comments.

ACKNOWLEDGMENTS

We owe a special debt to Professor Thanasis Sideris from whom both of us learned a great deal (not only about robust control) and who was very influential in shaping our subsequent research. We would also like to acknowledge the important contribution that Professor Manfred Morari had in the development of the concepts presented in this book, and in advising Ricardo S. Sanchez Peiia to make the right decisions during hard times in his career. Mario Sznaier is specially indebted to Professor Mark Damborg for his mentoring. Among Professor Damborg's many contributions, the two that stand out are attracting him to the field of control and making sure that he got a Ph.D. degree when he showed signs of enjoying life too inuch as a graduate student. The reviews by Professors Peter Dorato and Roy Smith at initial stag~s • of this book have been instrumental in encouraging the authors to carry out the project and in shaping its final form. Preliminary versions of the book were tested in courses offered by the authors at Penn State University and the University of Buenos Aires. Feedback from students enrolled in these courses was fundamental to its fine tuning. We are also indebted to our graduate and post-doctoral students, Takeshi Amishima, Pablo Anigstein, Juanyu Bu, Tamer Inane, Cecilia Mazzaro, Pablo Parrilo and Zi-Qin Wang, for reviewing the book in great detail and providing some of the material. In particular, Cecilia developed very efficiently the

xvii

examples in Chapter 10 and Takeshi, Juanyu, and Zi-Qin contributed some of the material in Chapters 8 and 11. The book was shaped to a large extent by interactions with our colleagues. In particular, many of the ideas and developments in Chapter 8 arose from research carried out jointly with Professor Franco Bianchini. Dr. Hector Rotstein influenced the presentation of the Hoo material in Chapter 6. Professors S. P. Bhattacharyya, D. Bernstein, N. K. Bose, 0. Crisalle, M. Dahleh, M.A. Dahleh, J. C. Doyle, M. Fan, C. V Hollot, M. Khammash, A. Megretski, A. Packard, J. Boyd Pearson, A. Ray, A. Saberi, A. Stoorvogel, R. Suarez, R. Tempo and J. Vagners contributed (directly or indirectly) to this book through many discussions and comments during the years. The first author would like to thank his colleagues at LAE, School of Engineering of the University of Buenos Aires and at the Argentine Space Agency (CONAE), especially Professors C. Godfrid and A. Perez, and Ing. Beto Alonso for important support. The financial support of CONAE, the University of Buenos Aires and Fundaci6n Antorchas during his stays at Penn State in 1994 and 1996 are also gratefully acknowledged. Mario Sznaier would like to thank the National Science Foundation, and in particular Dr. Kishan Baheti, Director of the Engineering Systems Program, for supporting his research program. This book would have not been possible without this support. We are also indebted to Ms. Lisa Van Horn and Mr. George Telecki at John Wiley and Sons, Inc. for their assistance throughout this project. Finally we would like to thank our children, Gadiel, Lucila and Pablo, and spouses, Monica and Octavia, for their infinite patience, support, and understanding for the time this book has stolen from them. RICARDO

s. SANCHEZ-PENA MARIO SZNAIER

Buenos Aires. Argentina University Park. Pennsylvania

ROBUST SYSTEMS THEORY AND APPLICATIONS

1 INTRODUCTION

1.1

GENERAL CONTROL PROBLEM

The purpose of this book is to introduce the reader to the theory of control systems, with particular emphasis on the applicability of the results to practical problems. Therefore it considers not only the control viewpoint but also the identification of systems as part of the problem. Also, as for any theory of systems oriented toward practical applications, robustness is essential and will be the underlying concept throughout the book. For these reasons the 'title of the book includes the following words: Theory, Applications, Robust, and Systems. It is important to start with a wide angle view of the general control picture, so that we can locate ourselves in the particular approach focused by this book. For this reason, the general problem of implementing a controller, which actuates over a physical process in a prescribed way, will be divided into three well defined phases. These phases are all equally important but involve very different technical tools, which go from pure mathematics to microprocessor technology. In today's practical control problems, all phases should be considered and it is very difficult to have a clear knowledge of all the technical fields involved. Although in this book we concentrate on the theoretical and computational aspects of the whole problem, we first make a general overview of the disciplines involved at each stage and the interrelation among them. This is the basic objective of robust identification and control theory: to produce theoretical results that are closely related to the computational and experimental aspects of the control problem. In the next subsection we will describe each phase in detail.

2

1.1.1

GENERAL CONTROL PROBLEM

INTRODUCTION

Experimental Phase

This phase is where the control problem is generated and where the last test on the controller is performed. This means that the actual control problem to be solved comes always from a physical setup: an aircraft, a distillation plant, a robotic manipulator. The performance specifications are also formulated in terms of variables involved in this physical experience: stable dynamics at high angle of attack, constant concentration of the distilled product, fast response to commanded trajectories. Also, in many cases, once the practical problem has been formulated, an experiment should be performed to·obtain a mathematical model as "close" as possible to the real physical system to be controlled. Finally, the last test to prove the effectiveness of the controller is performed on the physical system. The experimental phase is basically technological. It involves engineering disciplines (electrical, mechanical, chemical, aeronautical, etc.) or even biology or economics, depending on what we define as the system to be controlled. It also involves knowledge on the sensors, actuators, and technical aspects of microprocessors and 1/0 devices, for computer control. We may as well include in this phase the low- and high-level programming, as in the case of communication programs in Assembler and control algorithm coding, respectively. The interaction of the physical plant with its environment is performed by "real world" signals provided by the plant to the sensors and application of physical stimulations to the plant through the actuators, which modify its behavior. These input and output signals, as well as the physical laws that describe the system behavior, provide the only information we have of the physical plant. In many cases there are no physical laws that explain the plant behavior or they are extremely complex. A trial and error control, or direct experimentation using the input and output signals (without any analysis), is unrealizable, at least for the class of problems in which we are interested. A more elaborate and "scientific" procedure should (and can) be performed to control a given plant. Furthermore, in many cases trial and error would not even be practical because of costs (chemical plants) or simply to prevent accidents (aircraft, nuclear power plant). The information on the system-input-o utput signals and physical lawsis never complete (to complicate things). The reason is that the physical laws, which may be applied to the (sensor, actuator, plant) behavior describe only-.. a certain aspect of the problem, but not all. To make this point, consider the following example. To describe the output voltage on a given resistor through which a given current is circulating, we may use Ohm's law. Nevertheless, the temperature, gravitational field, and magnetic field could also influence the measurement, as well as the physics of the voltage sensor and current supply. A description of a particular physical process using all the many aspects involved could, to say the least, be impractica1. 1 Sometimes, also for practical 1 It is an epistemological question if a complete mathematical picture of physical reality is possible.

3

reasons, a simplification of the physical laws should be applied to de~cri_be the system. Actually, the laws are not modified, but instead new simphfymg assumptions are added. The above facts determine that all practical control problems have a certain amount of uncertainty in its more general meaning. It should be noted that this uncertainty is a consequence of the observation of the system and its environment, not due to the system itself. In particular, in the control jargon, this lack of knowledge is called noise when applied to the sensing procedure, disturbance for the external stimulations to the plant, and uncertainty when it applies to the description of the dynamical behavior of the plant by a mathematical model. These three elements-in particular, the latter-play a fundamental role in the orientation of control theory since the 1980s and in the motivation for Robust Identification and Control.

1. 1.2 Simulation Phase For many different reasons it is highly convenient to have a computational model of the plant. By this we mean a representation, in a computer, of the relevant inputs and outputs of the physical system and their relationship. This representation can be very elaborate as in a hardware in the loop scheme, where parts of _the system are represented co~putation~lly_ and others are connected physically to the computer I/0 ports. Otherwise It can be a complete computational model of the plant obtained from a mathematical model that includes all relevant physical phenomena involved. To illustrate some of the many reasons justifying a simulation of the plant, we cite the following "real world" examples, which will be described in greater detail in Chapter 11. 1. In many cases direct experimentation with the hardware is imposs_ible, as in the case of satellites. In these situations, it is usual to start with a high-fidelity simulation, which includes models of the system (satellite), actuators (inertial wheels, gas jets, magnetic torque coils), sensors (s~­ lar and Earth magnetic field sensors), and environment (Sun Ephemens and Earth magnetic field model along the orbit). These models include the nonlinear dynamics, flexible phenomena, and all the necessary elements to represent the system. According to required tests to be performed along the project, pieces of real hardware may be connected to this hi-fi simulator replacing their simulations, that is, sensors or actuators. 2. The cost of experimentation could be a strong reason to simulate a plant, like the case of many chemical processes, where the startup of 2ft could be argued that in this case we have a comhination of experimental and simulation phases. What we mean by simulation in this book is in general an object that is not exactly the real plant or environment. In any case. the important point is to realize that there arc d1ffcrent phases in the control problem.

4

INTRODUCTION

the process could involve many millions of dollars. The high costs could be a consequence of very expensive materials or long experimentation. Testing over a computer model provides a very fast and inexpensive way of verifying a controller (see D 2 0 plant in Chapter 11 and [228]). 3. A very important reason for simulation is when human lives_ are involved, as in the case of an experimental aircraft. Therefore a computer simulation gives a safe means of testing high-risk maneuvers (see NASA's X-29 experimental aircraft in Chapter 11). The disciplines involved are related to the areas of computer science, optimization, and numerical methods. The mathematical models to be implemented by a computer code could be obtained from physical laws or from direct experimentation on the plant. In this last case, the classical Parameter Identification procedures ([184, 293]) or the very recent Robust Identification ([146]) techniques (see Chapter 10) could be used. The main goal of the simulation phase is to have a copy as "close" as possible to the experimental plant. The word experimental has been emphasized to distinguish it from the true plant, which is not the same concept, as mentioned in the previous subsection and detailed next. For simulation, we are interested only in certain aspects of the physical system but not in all of them. Take, for example, the dynamics of the longitudinal axis of a rigid aircraft, which is a particular aspect to be considered not at all related to the thermal gradients of the wings of the same aircraft. Therefore, for simulation purposes, we only consider certain input and output signals that are relevant to our problem. Furthermore, even for a particular aspect of the problem, we may consider different levels of detail. Take the same example, where in one case we may consider the aircraft as a rigid body and in a more detailed analysis we may add the flexible modes. Therefore, for simulation purposes, we only consider the experimental plant, which is a particular aspect of the true plant, that arises in the experimental setup. The trade-off in this phase of the problem is defined in terms of how "close" we want the computational model to represent the plant versus how "big" a computer will be needed to perform this task. In the latter, the main limitations are computation time and available memory. As the computational model describes the plant better and better, the computation time will grow, which could pose a serious limitation in many real-time applications. For example, even for reasonable size minicomputers, a complete moder of the orbit, Earth magnetic field, Sun Ephemeris, gravitational and aerodynamic disturbance models, and satellite dynamics could take much more time for simulation than for real time. This is the case of low-orbit satellites with short orbital periods (SAC-C in Chapter 11 has a 90 minute period). Also, in many cases, a "good" computational model could involve a great amount of memory, like in many finite element representations of physical systems. In any case, this trade-off determines that certain simplifying assumptions should he made when representing a physical system by a computer code.

GENERAL CONTROL PROBLEM

5

Typical assumptions for the nonlinear model of an aircraft are rigid body dynamics and symmetry. The simplifying assumptions mentioned above, in the simulation phase, are only related to the computational constraints. Instead, for the theoretical analysis and synthesis phase, new hypothesis should be added to simplify the problem even more. This is related to the available mathematical tools to design and analyze a feedback loop, as will be seen next.

1 .1 .3 Theoretical Phase Introduction In history and, in particular, in the history of science, humans have tried to solve first the simpler problems and continue with the more complicated ones. This, which seems almost a natural law of human behavior, also applies to the theory of control systems. For this reason in classical control theory, systems were first described as single-input single-output (SISO), while later on, the more general multiple-input multiple-output (MIMO) description was introduced by modern control theory through the state-space description. In the same line of thought, the optimization of a quadratic functional (LQG optimal control) appeared earlier than the 'Hoo optimal control problem. In this latter case, the linear 'Hoo control problem was stated ([329]) almost ten years before the first solutions ([22, 157, 309, 310]) to the 'Hou nonlinear control problem were produced. At this point, there are not yet fully developed analysis and design tools to deal with control systems described by coupled nonlinear partial differential equation descriptions with (possibly) uncertain parameters. Therefore the theoretical phase of a control problem, which will be the main concern ,of this book, deals with a much more simplified mathematical model of the plant. Historically, the most usual system description for design and analysis of feedback systems has been finite-dimensional (lumped), linear, time invariant (FDLTI) mathematical models. For this class of models there is a well established theory, which also covers many of the most usual applications. 3 A visual idea of all the simplifying assumptions the designer needs to make to obtain a FDLTI mathematical model from the physical system may be seen in Figure 1.1. The usual steps that should be taken to compute a FDLTI model, from a very complex mathematical representation of a particular plant, are presented there. Some applications, which will be detailed in Chapter 11, illustrate a few of the steps. System Model At this point we will distinguish two different approaches that compute a FDLTI model from a physical system. The first one obtains a simplified version of a very complex mathematical model. This could be a set of partial differential equations or a time delayed or a high order set of ordinary differential equations. This complex mathematical model is usually 1 Not

hccausc physical systems are themselves FDLfl. hut hccausc for most practical purposes they can he considered to he so.

6

GENERAL CONTROL PROBLEM

INTRODUCTION

Physical System (Law! of Ph1J!ic!)/(Identification)

I

i

I

i

I

_j

i

I

I

I Nonlinear O.D.E. ;.:-----1: Nonlinear P.D.E. ;-:----...;

L_____j

Linear P.D.E.

(Local

Linear O.D.E. {FDLTI Model)

z(t) = Az(t) + Bu(t) } y(t)

= Cz(t) + Du(t)

y(•)

= G(•). u(•)

v Physical System

FDLTIModel

Figure 1.1. ''Translation" of a physical system to a FDLTI model.

obtained from physical laws describing the system. This model simplification procedure is called approximation and is purely mathematical, once the model of the system, obtained from physical laws, is known.

7

There are a variety of methods that approximate systems described by partial differential equations (infinite-dimensional systems) by ordinary differential equations. Among others, we can mention finite elements, finite differences, averaging techniques, splines, and FIT methods ([24, 25, 139, 169]). More recently, the balanced realization and truncation of infinite-dimensional models provides simplified finite-dimensional models ([79, 134]). In the case of systems with time delays, which are also infinite dimensional, although the above methods can be used, the most usual and practical approach is to use Pade approximants. The second approach is based on direct experimentation over the physical system and is called in general identification. This approach consists in computing a mathematical model, usually FDLTI, from the data measured at the system's output when a particular input signal is applied. The classical parameter identification procedures ([184, 293]) assume a particular mathematical structure for the model and, through an optimization method, obtain a set of parameters that minimize the "difference" between the model and the system's output, for the same applied input. More recent techniques called robust identification obtain directly a FDLTI model, without assuming any parameter structure, from the output data of the system when a particular set of signals are applied to the input. These latter techniques are related directly to robust control and will be described in greater detail in Chapter 10. From Figure 1.1 we observe that sometimes we should linearize a nonlinear model of the system. The classical linearization technique consists in computing a linear version of the nonlinear model for a particular set of parameters (operating point) of the latter. This is called a local model ([189]), which is approximately valid only in a small neighborhood of the selected values of these parameters. In some applications, the parameters have fixed values and, ' for that reason, there is only one working point and one linear model ([268]). Otherwise, if the parameters can take different values, the designer should synthesize a linear controller for each linear model. Next, a test should be made (analytical, by simulation or experimentation) to guarantee a smooth transition from one controller to another (see Example below). This is called gain scheduling and is a well known procedure to control nonlinear plants. Recent results in this area attack this problem through the use of linear parameter varying (LPV) controllers ([222]). Also, for certain classes of nonlinear models (linear analytic models), a state transformation and a nonlinear state feedback can be computed, which produce a (global) linear model ([89, 153, 156]). Although this is not a robust result,4 it is very useful in many practical applications ([1, 2, 44, 106, 107]), for example, robotic manipulators, aircraft/spacecraft control, and motor control. 4 The

state-feedback law, which linearizes the model, depends on exact knowledge of the nonlinear model; therefore feedback linearization is not robust against uncertainty in the nonlinear model.

8

INTRODUCTION

GENERAL CONTROL PROBLEM

Disturbances The designer not only assumes certain properties of the system (e.g., linearity, time invariance) to obtain a simpler model but also needs to make certain hypotheses on the disturbances applied to the system by its environment. In classical and modern control, the disturbances were usually assumed to be certain classes of signals: steps, ramps, impulses, sinusoids. In a stochastic approach, the covariance matrix of the stochastic proce-ss representing the disturbance was assumed to be known exactly. Very seldom does the designer have a clear knowledge of the disturbing signals acting on the system. Therefore the less one assumes about their nature, the higher the possibility of producing an effective control, despite these disturbances. As a matter of fact, a control design procedure that optimally rejects a step disturbing signal will no longer be optimal if the actual disturbance is a square wave. For this reason, in robust control, the only assumption on the disturbances is the fact that they are bounded (usually in energy), which is a very realistic hypothesis. If the designer has some extra information on the frequency distribution of the disturbances, that information could be incorporated into the design procedure and produce a less conservative result. Note that even in this case, there is no assumption on the form of the disturbance. Uncertainty Due to the assumptions on the system and the disturbances, a fair amount of uncertainty between the mathematical model and the experimental results will appear. In the simulation phase, the simplifying hypotheses are made due to computational restrictions (time, memory). Instead, in the theoretical phase, the assumptions that generate the uncertainty are a consequence of the mathematical tools that the analyst/designe r has decided to use. We can evaluate separately the effect on the feedback loop of the uncertainty due to the modeling of the system and due to the external disturbances. From basic control courses we know that the latter has no effect on the sta~op, a~ lea~i? _l~~r sysie_§s~5 -altho-ugh)t~~~-d~grade the Rerformance. 6il1Ile other hand, tile uncertamty in the model can have a critical effect on the stability of the closed loop. Specifically, we know that if the design of a stabilizing controller has been based on an erroneous plant model, the loop can be unstable. This is clear from the basics of control systems, as in the case of models with uncertain gains or phases. Take, for example, a controller designed for a model of the plant with a gain that e~­ actly doubles the gain of the actual plant. If this model has a gain margin smaller than 2 (and the designer doesn't know it), the controller could make the loop unstable. From the above arguments, we can conclude that