Franco Blanchini Stefano Miani

Set-Theoretic Methods in Control

Birkhäuser Boston • Basel • Berlin

Contents

Preface vii

1 Introduction 1 1.1 Notation 1

1.1.1 Acronyms 3 1.2 Basic ideas and motivations 3

1.2.1 The spirit of the book 3 1.2.2 Solving a problem 5 1.2.3 Conservative or intractable? 6 1.2.4 How to avoid reading this book 8 1.2.5 How to benefit from reading this book 9 1.2.6 Past work referencing 9

1.3 Outline of the book 9 1.3.1 The link with Lyapunov's theory 10 1.3.2 Uncertain Systems 13 1.3.3 Constrained control 18 1.3.4 Required background 25

1.4 Related topics and reading 26

2 Lyapunov and Lyapunov-like functions 27 2.1 State space modeis 27

2.1.1 Differential inclusions 29 2.1.2 Model absorbing 30 2.1.3 The pitfall of equilibrium drift 32

2.2 Lyapunov derivative 34 2.2.1 Solution of a System of differential equations 34 2.2.2 The upper-right Dini derivative 35 2.2.3 Derivative along the Solution of a differential equation.. 36 2.2.4 Special cases of directional derivatives 37

2.3 Lyapunov functions and stability 39 2.3.1 Global stability 40

xii Contents

2.3.2 Local stability and ultimate boundedness 43 2.4 Control Lyapunov Functions 45

2.4.1 Associating a control law with a Control Lyapunov Function: state feedback 46

2.4.2 Associating a control law with a Control Lyapunov Function: Output feedback 53

2.4.3 Finding a Control Lyapunov Function 54 2.4.4 Polytopic Systems 54 2.4.5 The convexity issue 57 2.4.6 Fake Control Lyapunov Functions 57

2.5 Lyapunov-like functions 60 2.6 Discrete-time Systems 62

2.6.1 Converse Lyapunov theorems 68 2.6.2 Literature review 69

2.7 Exercises 70

3 Convex sets and their representation 73 3.1 Convex functions and sets 73

3.1.1 Operations between sets 76 3.1.2 Minkowski function 79 3.1.3 The normal and the tangent cones 81

3.2 Ellipsoidal sets 83 3.3 Polyhedral sets 86 3.4 Other families of convex sets 94 3.5 Exercises 96

4 Invariant sets 99 4.1 Basic defmitions 99 4.2 Nagumo's theorem 101

4.2.1 Proof of Nagumo's theorem for practical sets and regulär / 104

4.2.2 Generalizations of Nagumo's theorem 106 4.2.3 An example of application of Nagumo's theorem 108 4.2.4 Discrete-time Systems 110 4.2.5 Positive invariance and fixed point theorem 112

4.3 Convex invariant sets and linear Systems 114 4.4 Ellipsoidal invariant sets 120

4.4.1 Ellipsoidal invariant sets for continuous-time Systems . . 120 4.4.2 Ellipsoidal invariant sets for discrete-time Systems 124

4.5 Polyhedral invariant sets 125 4.5.1 Contractive polyhedral sets for continuous-time

Systems 126 4.5.2 Contractive sets for discrete-time Systems 135 4.5.3 Associating a control with a polyhedral control

Lyapunov function and smoothing 138 4.5.4 Existence of positively invariant polyhedral C-sets 142

Contents xiii

4.5.5 The positive description 143 4.6 Other classes of invariant sets and historical notes 144 4.7 Exercises 146

5 Dynamic programming 149 5.1 Infinite-time reachability set 149

5.1.1 Linear Systems with linear constraints 156 5.1.2 State in a tube: time-varying and periodic case 164 5.1.3 Historical notes and comments 167

5.2 Backward computation of Lyapunov functions 168 5.3 The largest controlled invariant set 171 5.4 The uncontrolled case: the largest invariant set 179

5.4.1 Comments on the results 184 5.5 Exercises 188

6 Set-theoretic analysis of dynamic Systems 191 6.1 Set propagation 191

6.1.1 Reachable and controllable sets 191 6.1.2 Computation of set propagation under polytopic

uncertainty 194 6.1.3 Propagation of uncertainties via ellipsoids 197

6.2 O-Reachable sets with bounded inputs 198 6.2.1 Reachable sets with pointwise-bounded noise 198 6.2.2 Infinite-time reachability and l\ norm 207 6.2.3 Reachable sets with energy-bounded noise 209 6.2.4 Historical notes and comments 212

6.3 Stability and convergence analysis of polytopic Systems 212 6.3.1 Quadratic stability 213 6.3.2 Joint spectral radius 213 6.3.3 Polyhedral stability 215 6.3.4 The robust stability radius 217 6.3.5 Best transient estimate 218

6.4 Performance analysis of dynamical Systems 220 6.4.1 Peak-to-peak norm evaluation 221 6.4.2 Step response evaluation 226 6.4.3 Impulse and frequency response evaluation 228 6.4.4 Norm evaluation via LMIs 229

6.5 Periodic System analysis 231 6.6 Exercises 233

7 Control of parameter-varying Systems 235 7.1 Robust and gain-scheduling control 237 7.2 Stabilization of LPV Systems via quadratic

Lyapunov functions 241 7.2.1 Quadratic stability 242

Contents

7.2.2 Quadratic stabilizability 242 7.2.3 Quadratic Lyapunov functions: the discrete-time case . . 244 7.2.4 Quadratic stability and Hoc norm 245 7.2.5 Limits of quadratic functions and linear Controllers . . . . 246 7.2.6 Notes about quadratic stabilizability 251

7.3 Polyhedral Lyapunov functions 251 7.3.1 Polyhedral stabilizability 251 7.3.2 Universality of polyhedral Lyapunov functions

(and their drawbacks) 256 7.3.3 Smoothed Lyapunov functions 261

7.4 Gain-scheduling linear Controllers and duality 263 7.4.1 Duality in a quadratic framework 267

7.5 Exercises 268

Control with time-domain constraints 271 8.1 Input constraints 274

8.1.1 Construction of a constrained control law and its associated domain of attraction 278

8.1.2 The stable-unstable decomposition 283 8.1.3 Systems with one or two unstable eigenvalues 284 8.1.4 Region with bounded complexity for constrained

input control 291 8.2 Domain of attraction for input-saturated Systems 295 8.3 State constraints 299

8.3.1 A case study 301 8.3.2 Assigning an invariant (and admissible) set 306

8.4 Control with rate constraints 312 8.4.1 The rate-bounding Operator 314

8.5 Output feedback with constraints 315 8.6 The tracking problem 317

8.6.1 Reference management device 319 8.6.2 The tracking domain of attraction 324 8.6.3 Examples of tracking problems 330

8.7 Exercises 333

(Sub-)Optimal control 337 9.1 Minimum-time control 337

9.1.1 Worst-case controllability 337 9.1.2 Time optimal Controllers for linear discrete-time

Systems 341 9.1.3 Time optimal Controllers for uncertain Systems . . . . . . . . 342

9.2 Optimal peak-to-peak disturbance rejection 347 9.3 Constrained receding-horizon control 352

9.3.1 Receding-horizon: the main idea 352 9.3.2 Recursive feasibility and stability 355

Contents xv

9.3.3 Receding horizon control in the presence of disturbances 360

9.4 Relatively optimal control 365 9.4.1 The linear dynamic Solution 369 9.4.2 The nonlinear static Solution 377

9.5 Exercises 386

10 Set-theoretic estimation 389 10.1 Worst-case estimation 390

10.1.1 Set membership estimation for linear Systems with linear constraints 396

10.1.2 Approximate Solutions 403 10.1.3 Bounding ellipsoids 408 10.1.4 Energy-bounded disturbances 408

10.2 Including observer errors in the control design 410 10.3 Literature review 412 10.4 Exercises 412

11 Related topics 415 11.1 Adaptive control 415

11.1.1 A surge control problem 420 11.2 The domain of attraction 425

11.2.1 Systems with constraints 426 11.3 Hybrid and switching Systems 430

11.3.1 Switching and switched Systems 432 11.3.2 Switching among Controllers 436 11.3.3 Relay Systems 441

11.4 Planar Systems 447 11.5 Exercises 449

12 Appendix 451 12.1 Remarkable properties of the Euler auxiliary System 451 12.2 MAXIS-G: a Software for the computation of invariant sets

for constrained LPV Systems 456 12.2.1 Software availability 458 12.2.2 Web addresses 458

References 459

Index 477

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