Control Systems with
Actuator Saturation
Analysis and Design
Tingshu Hu
Zongli Lin
With 67 Figures
Birkhauser
Contents
Preface xiii 1 Introduction 1 1.1 Linear Systems with Actuator Saturation 1
1.2 Notation, Acronyms, and Terminology 3
2 Null Controllability —Continuous-Time Systems 11
2.1 Introduction 11 2.2 Preliminaries and Definitions 12 2.3 General Description of Null Controllable Region 15 2.4 Systems with Only Real Eigenvalues 21 2.5 Systems with Complex Eigenvalues 27 2.6 Some Remarks on the Description of C(T) 33 2.7 Asymptotically Null Controllable Region 34 2.8 Conclusions 35
3 Null Controllability-Discrete-Time Systems 37
3.1 Introduction 37 3.2 Preliminaries and Definitions 38 3.3 General Description of Null Controllable Region 41 3.4 Systems with Only Real Eigenvalues 44 3.5 Systems with Complex Eigenvalues 48 3.6 An Example 50 3.7 Asymptotically Null Controllable Region 51 3.8 Conclusions 53
Contents
4 Stabilization on Null Controllable Region
-Continuous-Time S y s t e m s ' 55
4.1 Introduction \ . 55 4.2 Domain of Attraction-Planar System under
Saturated Linear Feedback 57 4.3 Semi-Global Stabilization-Planar Systems 67 4.4 Semi-Global Stabilization-Higher Order Systems 74 4.5 Conclusions 83
5 Stabilization on Null Controllable Region —
Discrete-Time S y s t e m s 85
5.1 Introduction 85 5.2 Global Stabilization at Set of
Equilibria-Planar Systems 86 5.3 Global Stabilization - Planar Systems 99 5.4 Semi-Global Stabilization - Planar Systems 105 5.5 Semi-Global Stabilization-Higher Order Systems 108 5.6 Conclusions I l l
6 Practical Stabilization on Null Controllable Region 113
6.1 Introduction 113 6.2 Problem Statement and Main Results 114 6.2.1 Problem Statement 114 6.2.2 Main Results: Semi-Global Practical Stabilization . 114 6.3 Proof of Main Results 115
6.3.1 Properties of the Trajectories of Second Order
Linear Systems 115 6.3.2 Properties of the Domain of Attraction 119 6.3.3 Proof of Theorem 6.2.1: Second Order Systems . . . 127 6.3.4 Proof of Theorem 6.2.1: Higher Order Systems . . . 141 6.4 An Example 144 6.5 Conclusions 147 6.A Proof of Lemma 6.3.1 149 6.B Proof of Lemma 6.3.2 153
7 Estimation of the Domain of Attraction under
Saturated Linear Feedback 157
7.2 A Measure of Set Size 159 7.3 Some Facts about Convex Hulls _ 160 7.4 Continuous-Time Systems under State Feedback 163
7.4.1 A Set Invariance Condition Based on
Circle Criterion 164 7.4.2 An Improved Condition for Set Invariance 165 7.4.3 The Necessary and Sufficient
Condition-Single Input Systems 167 7.4.4 Estimation of the Domain of Attraction 169 7.5 Discrete-Time Systems under State Feedback 173 7.5.1 Condition for Set Invariance 173 7.5.2 The Necessary and Sufficient
Condition-Single Input Systems 177 7.5.3 Estimation of the Domain of Attraction 179 7.6 Extension to Output Feedback 180 7.7 Conclusions 181
8 On Enlarging the Domain of Attraction 183
8.1 Introduction 183 8.2 Continuous-Time Systems 183 8.3 Discrete-Time Systems 185 8.4 Conclusions 191
9 Semi-Global Stabilization with Guaranteed
Regional Performance 195
9.1 Introduction 195 9.2 Expansion of the Domain of Attraction 197 9.3 Semi-Globalization-Discrete-Time Systems 199 9.4 Semi-Globalization-Continuous-Time Systems 205 9.5 An Example 207 9.6 Conclusions 208
10 Disturbance Rejection with Stability 211
10.1 Introduction 211 10.2 Continuous-Time Systems 213 10.2.1 Problem Statement 213 10.2.2 Condition for Set Invariance 214
Contents 10.2.3 Disturbance Rejection with Guaranteed Domain
of Attraction 216 10.2.4 An Example 219 10.3 Discrete-Time Systems 221 10.3.1 Problem Statement 221 10.3.2 Condition for Set Invariance 223 10.3.3 Disturbance. Rejection with Guaranteed Domain
of Attraction 225 10.4 Conclusions . 228
11 On Maximizing the Convergence Rate 229
11.1 Introduction 229 11.2 Continuous-Time Systems 233
11.2.1 Maximal Convergence Control and Maximal
Invariant Ellipsoid 233 11.2.2 Saturated High Gain Feedback 242 11.2.3 Overall Convergence Rate 247 11.2.4 Maximal Convergence Control in the Presence
of Disturbances 255 11.3 Discrete-Time Systems 258 11.4 Conclusions 264
12 Output Regulation —Continuous-Time Systems 265
12.1 Introduction 265 12.2 Preliminaries and Problem Statement 267 12.2.1 Review of Linear Output Regulation Theory . . . . 267 12.2.2 Output Regulation in the Presence of
Actuator Saturation 270 12.3 The Regulatable Region 271 12.4 State Feedback Controllers 279 12.5 Error Feedback Controllers 290 12.6 An Example 297 12.7 Conclusions 301
13 Output Regulation —Discrete-Time Systems 305
13.1 Introduction 305 13.2 Preliminaries and Problem Statement 306 13.2.1 Review of Linear Output Regulation Theory . . . . 306
13.2.2 Output Regulation in the Presence of
Actuator Saturation 307 13.3 The Regulatable Region 309 13.4 State Feedback Controllers 315 13.5 Error Feedback Controllers 324 13.6 Conclusions 325
14 Linear Systems with Non-Actuator Saturation 327
14.1 Introduction 327 14.2 Planar Linear Systems under State
Saturation-Continuous-Time Systems 328 14.2.1 System Description and Problem Statement 328 14.2.2 Main Results on Global Asymptotic Stability . . . . 328 14.2.3 Outline of the Proof 330 14.3 Planar Linear Systems under State
Saturation-Discrete-Time Systems 344 14.3.1 System Description and Problem Statement 344 14.3.2 Main Results on Global Asymptotic Stability . . . . 344 14.3.3 Outline of the Proof 347 14.4 Semi-Global Stabilization of Linear Systems
Subject to Sensor Saturation 362 14.4.1 Introduction 362 14.4.2 Main Results 363 14.4.3 An Example 370 14.5 Conclusions 371 Bibliography 375 Index 387
