Stability and Stabilization of Linear Systems with Saturating Actuators (eBook)

Stability and Stabilization of Linear Systems with Saturating Actuators 3
Foreword 6
Preface 8
Contents 11
Notation 17
Part I: Generalities 20
Chapter 1: Linear Systems Subject to Control Saturation-Problems and Modeling 21
1.1 Introduction 21
1.2 The Open-Loop System 22
1.3 The Closed-Loop System 23
1.4 The Region of Linearity 30
1.5 The Region of Attraction 31
1.6 Problems Considered 32
1.6.1 Asymptotic Stability Analysis 33
1.6.1.1 Approaches Considering the RAS Included in the Region of Linearity 34
Polyhedral RAS 34
Ellipsoidal RAS 35
1.6.1.2 Approaches Considering the RAS not Included in the Region of Linearity 35
1.6.2 Asymptotic Stabilization 36
1.6.2.1 Global Stabilization 37
1.6.2.2 Semi-global Stabilization 39
1.6.2.3 Local (Regional) Stabilization 42
Saturation Avoidance 42
Saturation Allowance 44
1.6.3 External Stability Analysis 46
1.6.4 External Stabilization 48
1.6.5 The Anti-windup Problem 49
1.7 Models for the Saturation Nonlinearity 50
1.7.1 Polytopic Models 51
1.7.1.1 Polytopic Model I 51
1.7.1.2 Polytopic Model II 53
1.7.1.3 Polytopic Model III 55
1.7.2 Sector Nonlinearity Models 58
1.7.2.1 Classical Sector Condition 59
1.7.2.2 Generalized Sector Condition 61
1.7.3 Regions of Saturation Models 62
1.8 Equilibrium Points 64
1.9 Conclusion 65
Part II: Stability Analysis and Stabilization 67
Chapter 2: Stability Analysis and Stabilization-Polytopic Representation Approach 68
2.1 Introduction 68
2.2 Asymptotic Stability Analysis 69
2.2.1 Ellipsoidal Sets of Stability 69
2.2.2 Polytopic Approach I 71
2.2.3 Polytopic Approach II 73
2.2.4 Polytopic Approach III 75
2.2.5 Optimization Problems 76
2.2.5.1 Size Criteria 77
2.2.5.2 BMI x LMI Problems 79
2.3 External Stability 82
2.3.1 Amplitude Bounded Exogenous Signals 83
2.3.2 Energy Bounded Exogenous Signals 88
2.4 Stabilization 93
2.4.1 State Feedback Stabilization 93
2.4.1.1 Regional Asymptotic Stabilization 93
A. Regional Guaranteed Stability 95
B. Maximization of an Estimate of the Region of Attraction 95
C. Optimization of the Actuator Size 96
2.4.1.2 Performance Issues 97
2.4.1.3 Trade-off Between Saturation, Size of the Stability Region and Time-Domain Performance 99
2.4.1.4 Piecewise Linear State Feedback 101
2.4.1.5 Regional External Stabilization 103
(A) Amplitude Bounded Exogenous Signals 103
(B) Energy Bounded Exogenous Signals 104
2.4.2 Observer-Based Feedback Stabilization 108
2.4.3 Dynamic Output Feedback Stabilization 112
2.4.4 Global Stabilization 118
2.5 Uncertain Systems 120
2.5.1 Stability Analysis 120
2.5.2 Extensions 125
2.6 Discrete-Time Case 126
2.6.1 Ellipsoidal Regions of Asymptotic Stability 127
2.6.2 Polyhedral Regions of Asymptotic Stability 129
2.6.2.1 Preliminaries 129
2.6.2.2 Maximal Positively Invariant Set in S(|K|,u0alpha) 132
2.6.3 Extensions 136
2.6.3.1 External Stability 136
2.6.3.2 Stabilization 137
2.7 Conclusion 137
Chapter 3: Stability Analysis and Stabilization-Sector Nonlinearity Model Approach 139
3.1 Introduction 139
3.2 Asymptotic Stability Analysis 140
3.2.1 Quadratic Lyapunov Function 141
3.2.2 Lure Lyapunov Function 145
3.2.3 Computational Burden 148
3.2.4 Optimization Problems 149
3.2.4.1 Quadratic Case 149
3.2.4.2 Lure Case 150
3.3 External Stability 152
3.3.1 Amplitude Bounded Exogenous Signals 152
3.3.2 Energy Bounded Exogenous Signals 157
3.3.3 Optimization Issues 161
3.3.3.1 Amplitude Bounded Exogenous Signals 161
3.3.3.2 Energy Bounded Exogenous Signals 162
3.4 Stabilization 162
3.4.1 State Feedback Stabilization 162
3.4.1.1 Asymptotic Stabilization 162
Optimization Issues 164
3.4.1.2 External Stabilization 164
Amplitude Bounded Exogenous Signals 164
Energy Bounded Exogenous Signals 165
Optimization Issues 167
3.4.2 Observer-Based Feedback Stabilization 171
3.4.3 Dynamic Output Feedback Stabilization 177
Amplitude Bounded Exogenous Signals 178
Energy Bounded Exogenous Signals 182
Optimization Issues 185
3.5 Uncertain Systems 187
Norm-Bounded Uncertainty 188
Polytopic Uncertainty 189
Gain Scheduling Approach 190
3.6 Discrete-Time Case 194
3.7 Extensions 196
3.7.1 Nested Saturations 196
3.7.2 Nested Nonlinearities 197
3.7.3 Nonlinear and/or Hybrid Systems 198
3.8 Conclusion 199
Chapter 4: Analysis via the Regions of Saturation Model 200
4.1 Introduction 200
4.2 Polyhedral Regions of Stability 201
4.2.1 Positive Invariance 201
4.2.2 Contractivity-Compact Case 204
4.2.3 Determination of Stability Regions 208
4.3 Ellipsoidal Regions 211
4.3.1 Test Condition 1 212
4.3.2 Test Condition 2 215
4.4 Discrete-Time Case 217
4.4.1 Positive Invariance 217
4.4.2 Contractivity-Compact Case 218
4.5 Unbounded Sets of Stability 219
4.5.1 Unbounded Polyhedra 219
4.5.2 Unbounded Ellipsoidal Sets 221
4.6 Conclusion 222
Chapter 5: Synthesis via a Parameterized ARE Approach or a Parameterized LMI Approach 224
5.1 Introduction 224
5.2 The Parameterized ARE Approach 225
5.2.1 Preliminaries 226
5.2.2 The Single-Input Case 228
5.2.3 The Multi-variable Case 231
5.2.4 Control Computation and Implementation 234
5.3 A Parameterized LMI Approach 239
5.4 Multi-objective Control: Eigenvalues Placement and Guaranteed Cost 245
5.4.1 Preliminaries 245
5.4.2 The Single-Input Case 248
5.4.3 The Multi-variable Case 253
5.5 Disturbance Tolerance 258
5.5.1 Preliminaries 260
5.5.2 tau-Parameterized Control 262
5.5.3 Control Law Computation and Implementation 266
5.6 Nonlinear Bounded Control for Time-Delay Systems 270
5.6.1 Problem Statement 270
5.6.2 Preliminaries 271
5.6.3 Riccati Equation Approach 272
5.6.4 LMI Approach 275
5.7 Conclusion 279
Part III: Anti-windup 280
Chapter 6: An Overview of Anti-windup Techniques 281
6.1 Introduction-Philosophy 281
6.2 General Anti-windup Architecture 282
6.3 A Bit of History 285
Early Academic Study 287
Constrained Input Control 287
Modern Anti-windup 287
6.4 Formulation of Problems 288
6.5 Regional (Local) Versus Global Strategies 292
6.5.1 A Quick Overview in the Regional (Local) Context 292
6.5.2 A Quick Overview in the Global Context 293
6.6 Mismatch-Based Anti-windup Synthesis 294
6.7 Conclusion 295
Chapter 7: Anti-windup Compensator Synthesis 296
7.1 Introduction 296
7.2 Problems Setup 296
7.2.1 Direct Linear Anti-windup 298
7.2.2 Model Recovery Anti-windup 299
7.3 Direct Linear Anti-windup Design 300
7.3.1 Preliminary Elements 301
7.3.2 DLAW Schemes with Global Stability Guarantees 303
7.3.3 DLAW Schemes with Regional Stability Guarantees 307
7.3.4 Some Algorithms 310
7.4 Model Recovery Anti-windup Design 313
7.4.1 Preliminary Elements on the Architecture 313
7.4.2 Some Algorithms 315
7.5 Anti-windup Algorithms Summary 317
7.6 Some Extensions 318
7.6.1 Rate Saturation 319
7.6.2 Sensor Saturations 319
7.6.3 Nested Saturations 320
7.6.4 Time-Delay Systems 321
7.6.5 Anti-windup for Nonlinear Systems 322
7.7 Conclusion 322
Chapter 8: Applications of Anti-windup Techniques 323
8.1 Introduction 323
8.2 Static Anti-windup Examples 323
8.3 Dynamic Anti-windup Examples 327
Enlargement of the Domain of Admissible Initial States 332
Maximization of Disturbance Rejection 334
8.4 Toward More Complex Nonlinear Actuators 336
8.4.1 Actuator with Position and Rate Saturations 337
8.4.2 Dynamics Restricted Actuator 342
8.4.3 Electro-Hydraulic Actuator 347
8.5 Pseudo Anti-windup Strategies when the Dead-Zone Element is not Accessible 351
8.5.1 Anti-windup Scheme Using a Fictitious Linear Element 351
8.5.2 Observer-Based Anti-windup Scheme 355
8.6 Conclusion 363
Appendix A: Some Concepts Related to Stability Theory 365
A.1 Introduction 365
A.2 Stability of Linear Autonomous Systems 366
A.3 Stability for Nonlinear Systems 367
A.3.1 Stability Definition (Lyapunov Stability) 367
A.3.2 Lyapunov's Stability Theorem 369
A.3.3 The Invariance Principle 371
A.3.4 Region of Attraction 372
A.3.5 Set of Equilibrium Points 373
A.4 Application of Lyapunov Stability 373
A.4.1 Absolute Stability 374
A.4.2 Positive Real Function Concept 375
A.4.3 Circle Criterion 376
A.4.4 Popov Criterion 377
A.4.5 Quadratic Stability 378
Appendix B: Quadratic Approach for Robust Control 379
B.1 Introduction 379
B.2 Uncertain Models and Quadratic Stability 380
B.2.1 Uncertain Models 380
B.2.1.1 Norm-Bounded Uncertainty (NB-Uncertainty) 380
B.2.1.2 Bounded-Real Uncertainty (BR-Uncertainty) 381
B.2.1.3 Positive Real Uncertainty (PR-Uncertainty) 381
B.2.1.4 Structured Uncertainty 382
B.2.1.5 Polytopic Uncertainty (P-Uncertainty) 382
B.2.2 Quadratic Stability 382
B.2.2.1 P-Uncertainty 383
B.2.2.2 NB-Uncertainty 384
B.3 Quadratic Stabilizability 385
B.3.1 P-Uncertainty 386
B.3.2 NB-Uncertainty 387
B.4 Guaranteed Cost Control 389
B.4.1 P-Uncertainty 391
B.4.2 NB-Uncertainty 392
B.5 Pole Placement 394
B.5.1 Pole Placement in a Disk 394
B.5.1.1 d-Stabilizability 394
B.5.1.2 P-Uncertainty 395
B.5.1.3 NB-Uncertainty 395
B.5.2 Pole Placement in LMI Regions 397
B.5.2.1 Quadratic R-Stabilizability 398
B.5.2.2 P-Uncertainty 398
B.5.2.3 NB-Uncertainty 399
Appendix C: Linear Matrix Inequalities (LMI) and Riccati Equations 401
C.1 Introduction 401
C.2 Algebraic Lyapunov Equation (ALE) 401
C.2.1 Continuous Algebraic Lyapunov Equation (CALE) 402
C.2.2 Discrete Algebraic Lyapunov Equation (DALE) 403
C.3 Algebraic Riccati Equation (ARE) 404
C.3.1 Continuous Algebraic Riccati Equation (CARE) 404
C.3.1.1 Characterization of Solutions of CARE 405
C.3.1.2 Stabilizing Solution-Riccati Operator 406
C.3.2 Discrete Algebraic Riccati Equation (DARE) 407
C.3.2.1 Stabilizing Solution-Riccati Operator 408
C.4 Linear Matrix Inequalities (LMI) 409
C.5 Schur's Complement 410
C.6 Bilinear Matrix Inequalities (BMI) 412
C.6.1 Eigenvalues and Generalized Eigenvalues Problems 413
C.6.1.1 EigenValue Problems (EVP) 413
C.6.1.2 Generalized EigenValue Problems (GEVP) 414
C.7 The Elimination Lemma and the S-Procedure 415
C.7.1 The Elimination Lemma 415
C.7.2 The S-Procedure 416
C.7.2.1 The S-Procedure for Quadratic Functions and Nonstrict Inequalities 416
C.7.2.2 The S-Procedure for Quadratic Forms and Strict Inequalities 417
C.8 Ellipsoids and Polyhedral Sets 417
C.8.1 Ellipsoid and Invariant Ellipsoid 417
C.8.2 Convex Polyhedron and Invariant Polyhedron 418
C.8.3 Maximum Volume Ellipsoid Contained in a Symmetric Polytope 419
C.8.4 Smallest Volume Ellipsoid Containing a Symmetric Polytope 420
References 422
Index 440