Preface xi
Chapter 1. Positive Systems: Discretization with Positivity
and Constraints 1
Patrizio COLANERI, Marcello FARINA, Stephen KIRKLAND, Riccardo
SCATTOLINI and Robert SHORTEN
1.1. Introduction and statement of the problem 1
1.2. Discretization of switched positive systems via Pade
transformations 4
1.2.1. Preservation of copositive Lyapunov functions 4
1.2.2. Non-negativity of the diagonal Pade
approximation 7
1.2.3. An alternative approximation to the exponential
matrix 9
1.3. Discretization of positive switched systems with sparsity
constraints 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 switched
systems 14
1.4. Conclusions 18
1.5. Bibliography 18
Chapter 2. Advanced Lyapunov Functions for Lur?e
Systems 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-time
Lur?e systems 26
2.3.1. Definition of discrete-time Lur?e
systems 26
2.3.2. Introduction of a new discrete-time Lyapunov
Lur?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 arbitrary
switching law 37
2.4.1. Definition of the switched discrete-time Lur?e
system 37
2.4.2. Switched discrete-time Lyapunov Lur?e-type function
38
2.4.3. Global stability analysis 38
2.4.4. Local stability analysis 40
2.5. Switched discrete-time Lur?e system controlled by the
switching 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 projector
62
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 Linear
Systems 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 persistent
excitation 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 higher
dimensions 111
4.4.3. Generalizations of theorem 4.8 112
4.4.4. Properties of ?(A, T ) 116
4.4.5. Stabilizability at an arbitrary rate for systems with
several 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 flow
networks 125
5.4. Practical load balancing 127
5.5. Load balancing with delayed actuation and skewed
clocks 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 Recent
Advances 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 with
Saturations 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: brief
overview 185
7.3.1. Invariance for convex difference inclusions 186
7.4. Quadratic stability for saturated hybrid
systems 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 systems
195
7.4.5. Exponential Lyapunov functions for saturated hybrid
systems 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 with
Impacts 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 Zeno
solutions 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 hybrid
stabilizer 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 hybrid
stabilizer 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 hybrid
stabilizer 233
8.4.4. Simulations 239
8.5. Global state estimation on n-dimensional convex polyhedra
240
8.5.1. The reference dynamics 241
8.5.2. The observer dynamics 243
8.5.3. Estimation by hybrid reformulation of the observer
dynamics 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
