Robust Hybrid Control Systems

UNIVERSITY of CALIFORNIA Santa Barbara Robust Hybrid Control Systems A Dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Electrical and Computer Engineering by Ricardo G. Sanfelice Committee in charge: Professor Andrew R. Teel, Chair Professor Petar V. Kokotović Professor João P. Hespanha Professor Bassam Bamieh September 2007 Robust Hybrid Control Systems Copyright c 20071 by Ricardo G. Sanfelice 1Revised on January 2008. To my mother Alicia, and my father Adolfo iii Acknowledgements I am deeply grateful to my advisor, mentor, and friend Andy Teel for having guided me through the challenges of academic life and for teaching me to strive for elegance and mathematical rigor in my research. He has continuously motivated me to think creatively and to always aim for high-quality research. I would like to express my gratitude to my family and friends who have given me constant emotional and spiritual support and encouragement throughout my doctorate program. In particular, I want to thank my wife Christine who has been bold and remained supportive even in the busiest times of this journey. I also want to thank the faculty in the Department of Electrical and Computer Engineering and in the Center of Control, Dynamics, and Computation who directly or indirectly have shaped my career. In particular, I want to thank Petar Kokotovi`cand João Hespanha for their encouragement, advice, and friendship throughout the years. I thank all of the members of my doctoral committee for insightful discussions and useful suggestions made to improve this document. I am grateful to my fellow graduate students, post-docs, and visitors who made graduate school an enjoyable experience and were always available for discussions. I offer special thanks to Rafal Goebel for his helpful comments, advice and friendship; and to Gene, Doca, Ryan, Emre, Dragan, Michael, Max, James, Cai, Sara, Prabir, Payam, Kavyesh, Diane, Chris, and Josh who have been my lab-mates and friends. Finally, I am thankful for the help I have received from the Center for Control, Dynamical Systems’ assistants during the last five years who have aided me very efficiently in the administrative paper work. I am also grateful for the financial support provided by the University of California Santa Barbara and the Office of International Students and Schoolars, and through my advisor by the National Science Foundation, Air Force Office of Scientific Research, and Army Research Office. iv Abstract Robust Hybrid Control Systems by Ricardo G. Sanfelice This thesis deals with systems exhibiting both continuous and discrete dynamics, perhaps due to intrinsic behavior or to the interaction of continuous-time and discrete-time dynamics emerging from its components and/or their interconnection. Such systems are called hybrid systems and permit the modeling of a wide range of engineering systems and scientific processes. In this thesis, hybrid systems are treated as dynamical systems: the interplay between continuous and discrete behavior is captured in a mathematical model given by differential equations/inclusions and difference equations/inclusions, which we simply call hybrid equations. We develop tools for systematic analysis and robust design of hybrid systems, with an emphasis on systems that involve control algorithms, that is, hybrid control systems. To this effect, we identify mild conditions that hybrid equations need to satisfy so that their behavior captures the effect of arbitrarily small perturbations. This leads to novel concepts of generalized solutions that impart a deep understanding not only on the robustness properties of hybrid systems but also on the structural properties of their solutions. In turn, these conditions enable us to generate various tools for hybrid systems that resemble those in the stability theory of classical dynamical systems. These include general versions of Lyapunov and Krasovskii stability theorems, and LaSalle- type invariance principles. Additionally, we establish results on robustness of stability of hybrid control for general nonlinear systems. We also present a novel mathematical framework for numerical simulation of hybrid systems and its asymptotic stability properties. The contributions of this thesis are not limited to the theory of hybrid systems as they have implications in the analysis and design of practically relevant engineering control systems. In this regard, we develop general control strategies for dynamical systems that are applicable, for example, to autonomous vehicles, multi-link pendulums, and juggling systems. v Notation Rn denotes n-dimensional Euclidean space. • R denotes the real numbers. • Z denotes the integers. • R denotes the nonnegative real numbers, i.e., R = [0, ). • ≥0 ≥0 ∞ N denotes the natural numbers including 0, i.e., N = 0, 1,... • { } B denotes the open unit ball in Euclidean space. • Given a set S, S denotes its closure. • Given a set S, co S denotes the convex hull and coS the closure of the convex hull. • Given a set S Rn and a point x Rn, x := inf x y . • ⊂ ∈ | |S y∈S | − | Given sets S ,S Rn, d (S ,S ) denotes the Hausdorff distance between S and S . • 1 2 ⊂ H 1 2 1 2 Given sets S ,S subsets of Rn, S + S := x + x x S , x S . • 1 2 1 2 { 1 2 | 1 ∈ 1 2 ∈ 2 } Given a vector x Rn, x denotes the Euclidean vector norm. • ∈ | | The equivalent notation [xT yT ]T , [x y]T , and (x, y) is used for vectors. • The notation f −1(r) stands for the r-level set of f on dom f, the domain of definition of f, i.e., f −1(r) := • z dom f f(z)= r . { ∈ | } A function α : R≥0 R≥0 is said to belong to the class if it is continuous, zero at zero, and strictly • increasing. → K A function α : R R is said to belong to the class if it belongs to the class and is unbounded. • ≥0 → ≥0 K∞ K A function β : R R R is said to belong to class- if it is continuous, nondecreasing in its • ≥0 × ≥0 → ≥0 KL first argument, nonincreasing in its second argument, and limsց0 β(s,t) = limt→∞ β(s,t)=0. A function β : R≥0 R≥0 R≥0 R≥0 is said to belong to class- if, for each r R≥0, the functions • β( , , r) and β( , r, ×) belong× to class-→ . KLL ∈ · · · · KL vi Contents Acknowledgments iv Abstract v Notation vi List of Figures xi 1 Introduction 1 1.1 Dynamical modeling from a robustness viewpoint . .................... 1 1.2 Toolsforsystematicanalysisanddesign . ................... 3 1.3 Controlstrategiesfor robuststability . ..................... 5 1.4 Robustness of numerical simulations . .................. 6 1.5 Notes ........................................... ........ 7 2 Mathematical Model and Solutions 8 2.1 Preliminaries ................................... ............ 8 2.2 Themodel........................................ ......... 9 2.2.1 Examples ...................................... ....... 10 2.2.2 Hysteresisinfeedbackcontrol . ............... 13 2.3 Timedomainsandarcs .............................. ........... 16 2.4 Solutionstohybridsystems . ............... 19 2.5 Examplesandfurthermodeling . ............... 21 2.5.1 Bouncingballrevisited. ............. 21 2.5.2 Hybridautomaton ............................... ......... 23 2.6 Summary ......................................... ........ 23 2.7 Notesandreferences .............................. ............. 24 vii 3 Generalized Solutions 26 3.1 Hybrid systemswith state perturbations . ................... 26 3.2 Generalizedsolutionsto hybridsystems . ................... 30 3.3 Measurementnoiseinfeedbackcontrol . .................. 34 3.4 Regularhybridsystems .. .. .. .. .. .. .. .. .. .. .. .. ............. 38 3.5 Summary ......................................... ........ 40 3.6 Notesandreferences .............................. ............. 40 4 Stability and Invariance 42 4.1 Stability....................................... ........... 42 4.1.1 Definitions .................................... ........ 42 4.1.2 Lyapunovtheorems .............................. ......... 43 4.2 Invariance...................................... ........... 45 4.2.1 Preliminaries ................................. .......... 45 4.2.2 PropertiesofΩ-limitssets . .............. 46 4.2.3 Convergence via invariance principles . ................. 49 4.2.4 Connections to observability and detectability . ..................... 54 4.3 Examples ........................................ ......... 56 4.4 Summary ......................................... ........ 64 4.5 Notesandreferences .............................. ............. 64 5 Robustness of Hybrid Control 68 5.1 Hybridcontrolofnonlinearsystems . ................. 68 5.2 Robustnesstoperturbations. ................ 71 5.2.1 Robustnessviafilteredmeasurements . ............... 71 5.2.2 Robustness to sensor and actuator dynamics . ................ 73 5.2.3 Robustness to sensor dynamics and smoothing . ................ 74 5.3 Robustness to digital implementation . ................... 76 5.3.1 Sample-and-holdmodel . ........... 76 5.3.2 Closed-loopsystemanalysis . .............. 77 5.4 A benchmark problem: robust global swing-up of a pendulumonacart .............. 79 5.5 Summary ......................................... ........ 82 5.6 Notesandreferences .............................. ............. 83 6 Hybrid Control Applications 85 viii 6.1 Hysteresis-basedcontrol . ................ 85 6.1.1 A robustness motivation to hybrid control . ................. 85 6.1.2 Ageneralrobustnessissue. ............. 87 6.1.3 Controldesignandanalysis . ............. 90 6.1.4 Twonumericalexamples. ........... 94 6.2 Throw-and-catchcontrol. ..............

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