ISBN-10:
1848215274
ISBN-13:
9781848215276
Pub. Date:
05/13/2013
Publisher:
Wiley
Hybrid Systems with Constraints / Edition 1

Hybrid Systems with Constraints / Edition 1

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Product Details

ISBN-13: 9781848215276
Publisher: Wiley
Publication date: 05/13/2013
Series: ISTE Series
Pages: 288
Product dimensions: 9.60(w) x 6.30(h) x 0.90(d)

About the Author

Jamal Daafouz is an expert in the area of switched and polytopic systems and has published several major results in leading journals (IEEE TAC, Automatica, Systems and Control Letters, etc.). He serves as an Associate Editor for the key journal IEEE TAC and is a member of the Editorial Board of the IEEE CSS society.

Sophie Tarbouriech is an expert in the area of nonlinear systems with constraints and has published several major results in leading journals (IEEE TAC, Automatica, Systems and Control Letters, etc.) and books. She is a member of the Editorial Board of the IEEE CSS society and has also served as an Associate Editor for the key journal IEEE TAC.

Mario Sigalotti is an expert in applied mathematics and switched systems and has published several results in leading journals (IEEE TAC, Automatica, Systems and Control Letters, etc.). He heads the INRIA team GECO and is a member of the IFAC Technical Committee on Distributed Parameter Systems.

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Table of Contents

Preface xi

Chapter 1. Positive Systems: Discretization with Positivityand Constraints 1
Patrizio COLANERI, Marcello FARINA, Stephen KIRKLAND, RiccardoSCATTOLINI and Robert SHORTEN

1.1. Introduction and statement of the problem 1

1.2. Discretization of switched positive systems via Padétransformations 4

1.2.1. Preservation of copositive Lyapunov functions 4

1.2.2. Non-negativity of the diagonal Padéapproximation 7

1.2.3. An alternative approximation to the exponentialmatrix 9

1.3. Discretization of positive switched systems with sparsityconstraints 10

1.3.1. Forward Euler discretization 10

1.3.2. The mixed Euler-ZOH discretization 11

1.3.3. The mixed Euler-ZOH discretization for switchedsystems 14

1.4. Conclusions 18

1.5. Bibliography 18

Chapter 2. Advanced Lyapunov Functions for Lur’eSystems 21
Carlos A. GONZAGA, Marc JUNGERS and Jamal DAAFOUZ

2.1. Introduction 21

2.2. Motivating example 24

2.3. A new Lyapunov Lur’e-type function for discrete-timeLur’e systems 26

2.3.1. Definition of discrete-time Lur’esystems 26

2.3.2. Introduction of a new discrete-time LyapunovLur’e-type function 26

2.3.3. Global stability analysis 29

2.3.4. Local stability analysis 30

2.4. Switched discrete-time Lur’e system with arbitraryswitching law 37

2.4.1. Definition of the switched discrete-time Lur’esystem 37

2.4.2. Switched discrete-time Lyapunov Lur’e-type function38

2.4.3. Global stability analysis 38

2.4.4. Local stability analysis 40

2.5. Switched discrete-time Lur’e system controlled by theswitching law 46

2.5.1. Global stabilization 46

2.5.2. Local stabilization 48

2.6. Conclusion 51

2.7. Bibliography 52

Chapter 3. Stability of Switched DAEs 57
Stephan TRENN

3.1. Introduction 57

3.1.1. Systems class: definition and motivation 57

3.1.2. Examples 59

3.2. Preliminaries 62

3.2.1. Non-switched DAEs: solutions and consistency projector62

3.2.2. Lyapunov functions for non-switched DAEs 66

3.2.3. Classical distribution theory 67

3.2.4. Piecewise-smooth distributions and solvability of[3.1] 69

3.3. Stability results 71

3.3.1. Stability under arbitrary switching 72

3.3.2. Slow switching 74

3.3.3. Commutativity and stability 75

3.3.4. Lyapunov exponent and converse Lyapunov theorem 77

3.4. Conclusion 81

3.5. Acknowledgments 81

3.6. Bibliography 81

Chapter 4. Stabilization of Persistently Excited LinearSystems 85
Yacine CHITOUR, Guilherme MAZANTI and Mario SIGALOTTI

4.1. Introduction 86

4.2. Finite-dimensional systems 89

4.2.1. The neutrally stable case 90

4.2.2. Spectra with non-positive real part 91

4.2.3. Arbitrary rate of convergence 97

4.3. Infinite-dimensional systems 101

4.3.1. Exponential stability under persistentexcitation 103

4.3.2. Weak stability under persistent excitation 105

4.3.3. Other conditions of excitation 106

4.4. Further discussion and open problems 110

4.4.1. Lyapunov-based arguments for the existing results 111

4.4.2. Generalization of theorem 4.5 to higherdimensions 111

4.4.3. Generalizations of theorem 4.8 112

4.4.4. Properties of ρ(A, T ) 116

4.4.5. Stabilizability at an arbitrary rate for systems withseveral inputs 117

4.4.6. Infinite-dimensional systems 118

4.5. Bibliography 118

Chapter 5. Hybrid Coordination of Flow Networks 121
Claudio De PERSIS, Paolo FRASCA

5.1. Introduction 121

5.2. Flow network model and problem statement 123

5.2.1. Load balancing 124

5.3. Self-triggered gossiping control of flownetworks 125

5.4. Practical load balancing 127

5.5. Load balancing with delayed actuation and skewedclocks 132

5.6. Asymptotical load balancing 136

5.7. Conclusions 141

5.8. Acknowledgments 141

5.9. Bibliography 141

Chapter 6. Control of Hybrid Systems: An Overview of RecentAdvances 145
Ricardo G. SANFELICE

6.1. Introduction 145

6.2. Preliminaries 149

6.2.1. Notation 149

6.2.2. Notion of solution for hybrid systems 150

6.3. Stabilization of hybrid systems 151

6.4. Static state feedback stabilizers 155

6.4.1. Existence of continuous static stabilizers 157

6.5. Passivity-based control 159

6.5.1. Passivity 160

6.5.2. Linking passivity to asymptotic stability 164

6.5.3. A construction of passivity-based controllers 167

6.6. Tracking control 169

6.7. Conclusions 176

6.8. Acknowledgments 176

6.9. Bibliography 177

Chapter 7. Exponential Stability for Hybrid Systems withSaturations  179
Mirko FIACCHINI, Sophie TARBOURIECH, Christophe PRIEUR

7.1. Introduction 179

7.2. Problem statement 181

7.2.1. Saturated reset systems 182

7.3. Set theory and invariance for nonlinear systems: briefoverview 185

7.3.1. Invariance for convex difference inclusions 186

7.4. Quadratic stability for saturated hybridsystems 190

7.4.1. Set-valued extensions of saturated functions 190

7.4.2. Continuous-time quadratic stability 192

7.4.3. Discrete-time quadratic stability 194

7.4.4. Exponential stability for saturated hybrid systems195

7.4.5. Exponential Lyapunov functions for saturated hybridsystems 198

7.5. Computational issues 203

7.6. Numerical examples 205

7.7. Conclusions 207

7.8. Bibliography 208

Chapter 8. Reference Mirroring for Control withImpacts  213
Fulvio FORNI, Andrew R. TEEL, Luca ZACCARIAN

8.1. Introduction 213

8.2. Hammering a surface 216

8.2.1. The reference hammer dynamics 216

8.2.2. Using dwell-time logic to avoid Zenosolutions 218

8.2.3. The controlled hammer dynamics 219

8.2.4. Instability with standard feedback tracking 220

8.2.5. Using a mirrored reference to design a hybridstabilizer 221

8.3. Global tracking of a Newton’s cradle 224

8.3.1. The reference cradle 224

8.3.2. The controlled cradle 225

8.3.3. Using a mirrored reference to design a hybridstabilizer 226

8.3.4. Simulations 229

8.4. Global tracking in planar triangles 230

8.4.1. The reference mass 231

8.4.2. The controlled mass 233

8.4.3. Using a family of mirrored references to design a hybridstabilizer 233

8.4.4. Simulations 239

8.5. Global state estimation on n-dimensional convex polyhedra240

8.5.1. The reference dynamics 241

8.5.2. The observer dynamics 243

8.5.3. Estimation by hybrid reformulation of the observerdynamics 244

8.5.4. Simulations 246

8.6. Proof of the main theorems 247

8.6.1. A useful Lyapunov result 247

8.6.2. Proofs of theorems 8.1–8.4 248

8.7. Conclusions 251

8.8. Acknowledgments 252

8.9. Bibliography 252

List of Authors 257

Index 261

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