Feedback Control Theory John Doyle,Bruce Francis,Allen Tannenbaum CMacmillan Publishing Co.,1990
Feedback Control Theory John Doyle Bruce Francis Allen Tannenbaum c Macmillan Publishing Co
Contents Preface iⅱ 1 Introduction 1 1.1 Issues in Control System Design·····.··. 1 1.2 What Is in This Book 7 2 Norms for Signals and Systems 11 2.1 Norms for Signals.··············· 11 2.2 Norms for☒Systems·.············ 13 2.3 Input-Output Relationships.··..·.. 。 15 2.4 Power Analysis (Optional)......... 16 2.5 Proofs for Tables 2.1 and 2.2 (Optional).... 18 2.6 Computing by State-Space Met hods (Optional).. 21 3 Basic Concepts 27 3.1 Basic Feedback Loop.....··. 27 3.2 Internal Stability ... 30 3.3 Asy mptotic Track ing ... 33 3.4 Performance....·.·· 35 4 Uncertainty and Robustness 39 4.1 Plant Uncertainty 39 4.2 Robust Stability........... 43 4.3 Robust Performance.....·· 47 4.4 Robust Performance More Generally 51 4.5 Conclusion·...··········· 52 5 Stabilization 57 5.1 Controller Parametrization:Stable Plant......................... 57 5.2 Coprime Factorization.·.···.·········, 59 5.3 Coprime Factorization by State-Space Met hods (Optional)............... 63 5.4 Controller Parametrization:General Plant........................ 64 5.5 Asy mptotic Properties................................... 66 5.6 Strong and Simultaneous Stabilization ......................... 68 5.7Cart-Pendulum Example...,,,······················ 73
Contents Preface iii Introduction Issues in Control System Design What Is in This Book Norms for Signals and Systems Norms for Signals Norms for Systems InputOutput Relationships Power Analysis Optional Proofs for Tables and Optional Computing by StateSpace Methods Optional Basic Concepts Basic Feedback Loop Internal Stability Asymptotic Tracking Performance Uncertainty and Robustness Plant Uncertainty Robust Stability Robust Performance Robust Performance More Generally Conclusion Stabilization Controller Parametrization Stable Plant Coprime Factorization Coprime Factorization by StateSpace Methods Optional Controller Parametrization General Plant Asymptotic Properties Strong and Simultaneous Stabilization CartPendulum Example i
6 Design Constraints 79 6.1 Algebraic Constraints 79 6.2 Analytic Constraints ...... 80 7 Loopshaping 93 7.1 The Basic Technique of Loopshaping 93 7.2 The Phase Formula (Optional) 96 7.3 Examples.,..······ 100 8 Advanced Loopshaping 107 8.1 Optimal Controllers..... 107 8.2 Loopshaping with C..·· 108 8.3 Plants with RHP Poles and Zeros. 113 8.4 Shaping S,T,orQ...·..··. 125 8.5 Furt her Notions of Optimality.,,·· 128 9 Model Matching 139 9.1 The Model-Mat ching Problem.... ..139 9.2 The Nevanlinna-Pick Problem.... 140 9.3 Nevanlinna's Algorithm ... 143 9.4 Solution of the Model-Mat ching Problem 147 9.5 State-Space Solution (Optional)..... 149 10 Design for Performance 153 10.1P-I Stable.....····· 153 10.2P-1 Unstable..·....... 。 158 10.3 Design Example:Flexible Beam 159 10.4 2-Norm Minimization....... 164 11 Stability Margin Optimization 169 11.1 Optimal Robust Stability 169 ll.2 Conformal Mapping..·..· 173 11.3 Gain Margin Optimization... 174 11.4 Phase Margin Optimization . 179 12 Design for Robust Performance 183 12.1 The Modified Problem....... ·..183 12.2 Spectral Factorization 184 12.3 Solution of the Modified Problem...... 185 12.4 Design Example:Flexible Beam Continued 191 References 197
Design Constraints Algebraic Constraints Analytic Constraints Loopshaping The Basic Technique of Loopshaping The Phase Formula Optional Examples Advanced Loopshaping Optimal Controllers Loopshaping with C Plants with RHP Poles and Zeros Shaping S T or Q Further Notions of Optimality Model Matching The ModelMatching Problem The NevanlinnaPick Problem Nevanlinnas Algorithm Solution of the ModelMatching Problem StateSpace Solution Optional Design for Performance P Stable P Unstable Design Example Flexible Beam Norm Minimization Stability Margin Optimization Optimal Robust Stability Conformal Mapping Gain Margin Optimization Phase Margin Optimization Design for Robust Performance The Modied Problem Spectral Factorization Solution of the Modied Problem Design Example Flexible Beam Continued References
Preface Striking developments have taken place since 1980 in feedback control theory.The subject has become both more rigorous and more applicable.The rigor is not for its own sake,but rat her that even in an engineering discipline rigor can lead to clarity and to methodical solutions to problems. The applicability is a consequence both of new problem formulations and new mathematical so- lutions to these problems.Moreover,computers and software have changed the way engineering design is done.These developments suggest a fresh presentat ion of the subject,one that exploits these new developments while emphasizing their connection with classical control. Control systems are designed so that certain designated signals,such as tracking errors and act uator inputs,do not exceed pre-specified levels.Hindering the achievement of this goal are uncertainty about the plant to be controlled (the mat hematical models that we use in represent ing real physical systems are idealizat ions)and errors in measuring signals(sensors can measure signals only to a certain accuracy).Despite the seemingly obv ious requirement of bringing plant uncertainty explicitly into control problems,it was only inthe early 1980s that control researchers re-established the link to the classical work of Bode and ot hers by formulating a tractable mat hematical notion of uncertainty in an input-output framework and developing rigorous mat hematical techniques to cope with it.This book formulates a precise problem,called the robust performance problem,with the goal of achieving specified signal levels in the face of plant uncertainty. The book is addressed to students in engineering who have had an undergraduate course in signals and systems,including an introduct ion to frequency-domain met hods of analyzing feedback control systems,namely,Bode plots and the Ny quist criterion.A prior course on state-space theory would be advantageous for some optional sections,but is not necessary.To keep the development elementary,the systems are single-input/single-output and linear,operating in continuous time. Chapters 1 to 7 are intended as the core for a one-semester senior course;they would need supplementing with additional examples.These chapters constit ute a basic treat ment of feedback design,containing a detailed formulation of the control design problem,the fundamental issue of performance/stability robust ness tradeoff,and the graphical design technique of loopshaping, suitable for benign plants (stable,minimum phase).Chapters 8 to 12 are more advanced and are intended for a first graduate course.Chapter 8 is a bridge to the latter half of the book, extending the loopshaping technique and connecting it with notions of optimality.Chapters 9 to 12 treat controller design via optimizat ion.The approach in these latter chapters is mat hematical rather than graphical,using elementary tools involving interpolat ion by analytic functions.This mat hematical approach is most useful for mult ivariable systems,where graphical techniques usually break down.Nevert heless,we believe the setting of single-input/single-output systems is where this new approach should be learned. There are many people to whom we are grateful for their help in this book:Dale Enns for sharing his expertise in loopshaping;Raymond Kwong and Boyd Pearson for class testing the book;and Munther Dahleh,Ciprian Foias,and Karen Rudie for reading earlier drafts.Numerous 逝
Preface Striking developments have taken place since in feedback control theory The sub ject has become both more rigorous and more applicable The rigor is not for its own sake but rather that even in an engineering discipline rigor can lead to clarity and to methodical solutions to problems The applicability is a consequence both of new problem formulations and new mathematical so lutions to these problems Moreover computers and software have changed the way engineering design is done These developments suggest a fresh presentation of the sub ject one that exploits these new developments while emphasizing their connection with classical control Control systems are designed so that certain designated signals such as tracking errors and actuator inputs do not exceed prespecied levels Hindering the achievement of this goal are uncertainty about the plant to be controlled the mathematical models that we use in representing real physical systems are idealizations and errors in measuring signals sensors can measure signals only to a certain accuracy Despite the seemingly obvious requirement of bringing plant uncertainty explicitly into control problems it was only in the early s that control researchers reestablished the link to the classical work of Bode and others by formulating a tractable mathematical notion of uncertainty in an inputoutput framework and developing rigorous mathematical techniques to cope with it This book formulates a precise problem called the robust performance problem with the goal of achieving specied signal levels in the face of plant uncertainty The book is addressed to students in engineering who have had an undergraduate course in signals and systems including an introduction to frequencydomain methods of analyzing feedback control systems namely Bode plots and the Nyquist criterion A prior course on statespace theory would be advantageous for some optional sections but is not necessary To keep the development elementary the systems are singleinputsingleoutput and linear operating in continuous time Chapters to are intended as the core for a onesemester senior course they would need supplementing with additional examples These chapters constitute a basic treatment of feedback design containing a detailed formulation of the control design problem the fundamental issue of performancestability robustness tradeo and the graphical design technique of loopshaping suitable for benign plants stable minimum phase Chapters to are more advanced and are intended for a rst graduate course Chapter is a bridge to the latter half of the book extending the loopshaping technique and connecting it with notions of optimality Chapters to treat controller design via optimization The approach in these latter chapters is mathematical rather than graphical using elementary tools involving interpolation by analytic functions This mathematical approach is most useful for multivariable systems where graphical techniques usually break down Nevertheless we believe the setting of singleinputsingleoutput systems is where this new approach should be learned There are many people to whom we are grateful for their help in this book Dale Enns for sharing his expertise in loopshaping Raymond Kwong and Boyd Pearson for class testing the book and Munther Dahleh Ciprian Foias and Karen Rudie for reading earlier drafts Numerous iii
Caltech students also struggled with various versions of this material:Gary Balas Carolyn Beck, Bobby Bodenheimer,and Roy Smith had particularly helpful suggest ions.Finally,we would like to thank the AFOSR,ARO,NSERC,NSF,and ONR for partial financial support during the writing of this book. iⅳ
Caltech students also struggled with various versions of this material Gary Balas Carolyn Beck Bobby Bodenheimer and Roy Smith had particularly helpful suggestions Finally we would like to thank the AFOSR ARO NSERC NSF and ONR for partial nancial support during the writing of this book iv