Hybrid Systems with Constraints

Control theory is the main subject of this title, in particular analysis and control design for hybrid dynamic systems. The notion of hybrid systems offers a strong theoretical and unified framework to cope with the modeling, analysis and control design of systems where both continuous and discrete dynamics interact. The theory of hybrid systems has been the subject of intensive research over the last decade and a large number of diverse and challenging problems have been investigated. Nevertheless, many important mathematical problems remain open. This book is dedicated mainly to hybrid systems with constraints; taking constraints into account in a dynamic system description has always been a critical issue in control. New tools are provided here for stability analysis and control design for hybrid systems with operating constraints and performance specifications. Contents 1. Positive Systems: Discretization with Positivity and Constraints, Patrizio Colaneri, Marcello Farina, Stephen Kirkland, Riccardo Scattolini and Robert Shorten. 2. Advanced Lyapunov Functions for Lur’e Systems, Carlos A. Gonzaga, Marc Jungers and Jamal Daafouz. 3. Stability of Switched DAEs, Stephan Trenn. 4. Stabilization of Persistently Excited Linear Systems, Yacine Chitour, Guilherme Mazanti and Mario Sigalotti. 5. Hybrid Coordination of Flow Networks, Claudio De Persis, Paolo Frasca. 6. Control of Hybrid Systems: An Overview of Recent Advances, Ricardo G. Sanfelice. 7. Exponential Stability for Hybrid Systems with Saturations, Mirko Fiacchini, Sophie Tarbouriech, Christophe Prieur. 8. Reference Mirroring for Control with Impacts, Fulvio Forni, Andrew R. Teel, Luca Zaccarian. About the Authors 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.

Table of Contents:
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 Padé transformations 4 1.2.1. Preservation of copositive Lyapunov functions 4 1.2.2. Non-negativity of the diagonal Padé 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 ρ(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