Hybrid Systems with Constraints

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.

• Preface xiChapter 1. Positive Systems: Discretization with Positivity and Constraints 1Patrizio COLANERI, Marcello FARINA, Stephen KIRKLAND, Riccardo SCATTOLINI and Robert SHORTEN1.1. Introduction and statement of the problem 11.2. Discretization of switched positive systems via Padé transformations 41.2.1. Preservation of copositive Lyapunov functions 41.2.2. Non-negativity of the diagonal Padé approximation 71.2.3. An alternative approximation to the exponential matrix 91.3. Discretization of positive switched systems with sparsity constraints 101.3.1. Forward Euler discretization 101.3.2. The mixed Euler-ZOH discretization 111.3.3. The mixed Euler-ZOH discretization for switched systems 141.4. Conclusions 181.5. Bibliography 18Chapter 2. Advanced Lyapunov Functions for Lur’e Systems 21Carlos A. GONZAGA, Marc JUNGERS and Jamal DAAFOUZ2.1. Introduction 212.2. Motivating example 242.3. A new Lyapunov Lur’e-type function for discrete-time Lur’e systems 262.3.1. Definition of discrete-time Lur’e systems 262.3.2. Introduction of a new discrete-time Lyapunov Lur’e-type function 262.3.3. Global stability analysis 292.3.4. Local stability analysis 302.4. Switched discrete-time Lur’e system with arbitrary switching law 372.4.1. Definition of the switched discrete-time Lur’e system 372.4.2. Switched discrete-time Lyapunov Lur’e-type function 382.4.3. Global stability analysis 382.4.4. Local stability analysis 402.5. Switched discrete-time Lur’e system controlled by the switching law 462.5.1. Global stabilization 462.5.2. Local stabilization 482.6. Conclusion 512.7. Bibliography 52Chapter 3. Stability of Switched DAEs 57Stephan TRENN3.1. Introduction 573.1.1. Systems class: definition and motivation 573.1.2. Examples 593.2. Preliminaries 623.2.1. Non-switched DAEs: solutions and consistency projector 623.2.2. Lyapunov functions for non-switched DAEs 663.2.3. Classical distribution theory 673.2.4. Piecewise-smooth distributions and solvability of [3.1] 693.3. Stability results 713.3.1. Stability under arbitrary switching 723.3.2. Slow switching 743.3.3. Commutativity and stability 753.3.4. Lyapunov exponent and converse Lyapunov theorem 773.4. Conclusion 813.5. Acknowledgments 813.6. Bibliography 81Chapter 4. Stabilization of Persistently Excited Linear Systems 85Yacine CHITOUR, Guilherme MAZANTI and Mario SIGALOTTI4.1. Introduction 864.2. Finite-dimensional systems 894.2.1. The neutrally stable case 904.2.2. Spectra with non-positive real part 914.2.3. Arbitrary rate of convergence 974.3. Infinite-dimensional systems 1014.3.1. Exponential stability under persistent excitation 1034.3.2. Weak stability under persistent excitation 1054.3.3. Other conditions of excitation 1064.4. Further discussion and open problems 1104.4.1. Lyapunov-based arguments for the existing results 1114.4.2. Generalization of theorem 4.5 to higher dimensions 1114.4.3. Generalizations of theorem 4.8 1124.4.4. Properties of ρ(A, T ) 1164.4.5. Stabilizability at an arbitrary rate for systems with several inputs 1174.4.6. Infinite-dimensional systems 1184.5. Bibliography 118Chapter 5. Hybrid Coordination of Flow Networks  121Claudio De PERSIS, Paolo FRASCA5.1. Introduction 1215.2. Flow network model and problem statement 1235.2.1. Load balancing 1245.3. Self-triggered gossiping control of flow networks 1255.4. Practical load balancing 1275.5. Load balancing with delayed actuation and skewed clocks 1325.6. Asymptotical load balancing 1365.7. Conclusions 1415.8. Acknowledgments 1415.9. Bibliography 141Chapter 6. Control of Hybrid Systems: An Overview of Recent Advances 145Ricardo G. SANFELICE6.1. Introduction 1456.2. Preliminaries 1496.2.1. Notation 1496.2.2. Notion of solution for hybrid systems 1506.3. Stabilization of hybrid systems 1516.4. Static state feedback stabilizers 1556.4.1. Existence of continuous static stabilizers 1576.5. Passivity-based control 1596.5.1. Passivity 1606.5.2. Linking passivity to asymptotic stability 1646.5.3. A construction of passivity-based controllers 1676.6. Tracking control 1696.7. Conclusions 1766.8. Acknowledgments 1766.9. Bibliography 177Chapter 7. Exponential Stability for Hybrid Systems with Saturations  179Mirko FIACCHINI, Sophie TARBOURIECH, Christophe PRIEUR7.1. Introduction 1797.2. Problem statement 1817.2.1. Saturated reset systems 1827.3. Set theory and invariance for nonlinear systems: brief overview 1857.3.1. Invariance for convex difference inclusions 1867.4. Quadratic stability for saturated hybrid systems 1907.4.1. Set-valued extensions of saturated functions 1907.4.2. Continuous-time quadratic stability 1927.4.3. Discrete-time quadratic stability 1947.4.4. Exponential stability for saturated hybrid systems 1957.4.5. Exponential Lyapunov functions for saturated hybrid systems 1987.5. Computational issues 2037.6. Numerical examples 2057.7. Conclusions 2077.8. Bibliography 208Chapter 8. Reference Mirroring for Control with Impacts  213Fulvio FORNI, Andrew R. TEEL, Luca ZACCARIAN8.1. Introduction 2138.2. Hammering a surface 2168.2.1. The reference hammer dynamics 2168.2.2. Using dwell-time logic to avoid Zeno solutions 2188.2.3. The controlled hammer dynamics 2198.2.4. Instability with standard feedback tracking 2208.2.5. Using a mirrored reference to design a hybrid stabilizer 2218.3. Global tracking of a Newton’s cradle 2248.3.1. The reference cradle 2248.3.2. The controlled cradle 2258.3.3. Using a mirrored reference to design a hybrid stabilizer 2268.3.4. Simulations 2298.4. Global tracking in planar triangles 2308.4.1. The reference mass 2318.4.2. The controlled mass 2338.4.3. Using a family of mirrored references to design a hybrid stabilizer 2338.4.4. Simulations 2398.5. Global state estimation on n-dimensional convex polyhedra 2408.5.1. The reference dynamics 2418.5.2. The observer dynamics 2438.5.3. Estimation by hybrid reformulation of the observer dynamics 2448.5.4. Simulations 2468.6. Proof of the main theorems 2478.6.1. A useful Lyapunov result 2478.6.2. Proofs of theorems 8.1–8.4 2488.7. Conclusions 2518.8. Acknowledgments 2528.9. Bibliography 252List of Authors 257Index 261