This book develops a general analysis and synthesis framework for impulsive and hybrid dynamical systems. Such a framewo
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English Pages 496 [521] Year 2014
Table of contents :
Contents
Preface
Chapter One. Introduction
Chapter Two. Stability Theory for Nonlinear Impulsive Dynamical Systems
Chapter Three. Dissipativity Theory for Nonlinear Impulsive Dynamical Systems
Chapter Four. Impulsive Nonnegative and Compartmental Dynamical Systems
Chapter Five. Vector Dissipativity Theory for LargeScale Impulsive Dynamical Systems
Chapter Six. Stability and Feedback Interconnections of Dissipative Impulsive Dynamical Systems
Chapter Seven. EnergyBased Control for Impulsive PortControlled Hamiltonian Systems
Chapter Nine. Optimal Control for Impulsive Dynamical Systems
Chapter Ten. Disturbance Rejection Control for Nonlinear Impulsive Dynamical Systems
Chapter Eleven. Robust Control for Nonlinear Uncertain Impulsive Dynamical Systems
Chapter Twelve. Hybrid Dynamical Systems
Chapter Thirteen. Poincare Maps and Stability of Periodic Orbits for Hybrid Dynamical Systems
Appendix A. System Functions for the Clock Escapement Mechanism
Bibliography
Index
Impulsive and Hybrid Dynamical Systems
PRINCETON SERIES IN APPLIED MATHEMATICS
Edited by Ingrid Daubechies, Princeton University Weinan E, Princeton University Jan Karel Lenstra, Eindhoven University Endre Stili, University of Oxford
The Princeton Series in Applied Mathematics publishes high quality advanced texts and monographs in all areas of applied mathematics. Books include those of a theoretical and general nature as well as those dealing with the mathematics of specific applications areas and realworld situations.
Impulsive and Hybrid Dynamical Systems
Stability, Dissipativity, and Control
Wassim M. Haddad VijaySekhar Chellaboina Sergey G. Nersesov
PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD
Copyright © 2006 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire OX20 1SY All Rights Reserved
Library of Congress CataloginginPublication Data Haddad, Wassim M., 1961Impulsive and hybrid dynamical systems : stability, dissipativity, and control. / Wassim M. Haddad, VijaySekhar Chellaboina, and Sergey G. Nersesov. p. cm. (Princeton series in applied mathematics) Includes bibliographical references and index. ISBN13: 978691127156 (cl : alk. paper) ISBNlO: 0691127158 (cl : alk. paper) 1. Automatic control. 2. Control theory. 3. Dynamics. 4. Discretetime systems. I. Chellaboina, VijaySekhar, 1970 II. Nersesov, Sergey G., 1976 III. Title. IV. Series. TJ213.H232006 003' .85dc22
2005056496
British Library CataloginginPublication Data is available This book has been composed in Times Roman in
H\'JEX
The publisher would like to acknowledge the authors of this volume for providing the cameraready copy from which this book was printed. Printed on acidfree paper.
00
pup. princeton.edu Printed in the United States of America 10987654321
To my wife Lydia, the true grace, sapience, ectropy, and balance in my life
W.M.H. To my wife Padma, who like a beautiful lotus that adorns a lake ever enriches my life
v.c. To my parents Garry and Ekatherina and my brother A rtyom, with gratitude and appreciation
s.
G. N.
The highest form of pure thought is in mathematics. Plato
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Mathematics, rightly viewed, possesses not only truth, but supreme beautya beauty ... sublimely pure, capable of a stern perfection such as only the greatest art can show. Bertrand Russell
There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world. Nikolai Lobachevsky
From the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician. Sir James Jeans
This, therefore, is mathematics: she reminds you of the invisible forms of the soul; she gives life to her own discoveries; she awakens the mind and purifies the intellect; she brings light to our intrinsic ideas; and she abolishes oblivion and ignorance which are ours by birth. Proclus
Contents
Preface
xiii
Chapter 1. Introduction
1.1 1.2
Impulsive and Hybrid Dynamical Systems A Brief Outline of the Monograph
Chapter 2. Stability Theory for Nonlinear Impulsive Dynamical Systems
2.1 2.2 2.3 2.4
Introduction Nonlinear Impulsive Dynamical Systems Stability Theory of Impulsive Dynamical Systems An Invariance Principle for StateDependent Impulsive Dynamical Systems 2.5 Necessary and Sufficient Conditions for QuasiContinuous Dependence 2.6 Invariant Set Theorems for StateDependent Impulsive Dynamical Systems 2.7 Partial Stability of StateDependent Impulsive Dynamical Systems 2.8 Stability of TimeDependent Impulsive Dynamical Systems 2.9 Lagrange Stability, Boundedness, and Ultimate Boundedness 2.10 Stability Theory via Vector Lyapunov Functions
Chapter 3. Dissipativity Theory for Nonlinear ImpUlsive Dynamical Systems
3.1 3.2 3.3 3.4
Introduction Dissipative Impulsive Dynamical Systems: InputOutput and State Properties Extended KalmanYakubovichPopov Conditions for Impulsive Dynamical Systems Specialization to Linear Impulsive Dynamical Systems
Chapter 4. Impulsive Nonnegative and Compartmental Dynamical Systems 4.1 4.2
4.3 4.4 4.5
Introduction Stability Theory for Nonlinear Impulsive Nonnegative Dynamical Systems Impulsive Compartmental Dynamical Systems Dissipativity Theory for Impulsive Nonnegative Dynamical Systems Specialization to Linear Impulsive Dynamical Systems
1 1
4
9 9 11 20 27 32
38 44 56 63 71 81
81 84 103 119 125 125 126 131 135 143
x
CONTENTS
Chapter 5. Vector Dissipativity Theory for LargeScale Impulsive Dynamical Systems
5.1 5.2 5.3 5.4
Introduction Vector Dissipativity Theory for LargeScale Impulsive Dynamical Systems Extended KalmanYakubovichPopov Conditions for LargeScale Impulsive Dynamical Systems Specialization to LargeScale Linear Impulsive Dynamical Systems
Chapter 6. Stability and Feedback Interconnections of Dissipative Impulsive Dynamical Systems
6.1 6.2 6.3 6.4 6.5
Introduction Stability of Feedback Interconnections of Dissipative Impulsive Dynamical Systems Hybrid Controllers for Combustion Systems Feedback Interconnections of Nonlinear Impulsive Nonnegative Dynamical Systems Stability of Feedback Interconnections of LargeScale Impulsive Dynamical Systems
Chapter 7. EnergyBased Control for Impulsive PortControlled Hamiltonian Systems
7.1 7.2 7.3 7.4 7.5
Introduction Impulsive PortControlled Hamiltonian Systems EnergyBased Hybrid Feedback Control EnergyBased Hybrid Dynamic Compensation via the EnergyCasimir Method EnergyBased Hybrid Control Design
Chapter 8. Energy and EntropyBased Hybrid Stabilization for Nonlinear Dynamical Systems
8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8
Introduction Hybrid Control and Impulsive Dynamical Systems Hybrid Control Design for Dissipative Dynamical Systems Lagrangian and Hamiltonian Dynamical Systems Hybrid Control Design for EulerLagrange Systems Thermodynamic Stabilization EnergyDissipating Hybrid Control Design EnergyDissipating Hybrid Control for Impulsive Dynamical Systems 8.9 Hybrid Control Design for Nonsmooth EulerLagrange Systems 8.10 Hybrid Control Design for Impact Mechanics Chapter 9. Optimal Control for Impulsive Dynamical Systems 9.1 9.2 9.3
Introduction Impulsive Optimal Control Inverse Optimal Control for Nonlinear Affine Impulsive Systems
147 147 150 175 186
191 191 191 199 208 214
221 221 222 227 233 242
249 249 251 258 265 267 271 277 300 308 313 319 319 319 330
CONTENTS
904
9.5 9.6
Nonlinear Hybrid Control with Polynomial and Multilinear Performance Functionals Gain, Sector, and Disk Margins for Optimal Hybrid Regulators Inverse Optimal Control for Impulsive PortControlled Hamiltonian Systems
XI
333 337 345
Chapter 10. Disturbance Rejection Control for Nonlinear Impulsive Dynamical Sy~ems
10.1 Introduction 10.2 Nonlinear Impulsive Dynamical Systems with Bounded Disturbances 10.3 Specialization to Dissipative Impulsive Dynamical Systems with Quadratic Supply Rates IDA Optimal Controllers for Nonlinear Impulsive Dynamical Systems with Bounded Disturbances 10.5 Optimal and Inverse Optimal NonlinearNonquadratic Control for Affine Systems with £2 Disturbances Chapter 11. Robust Control for Nonlinear Uncertain Impulsive Dynamical Systems
11.1 Introduction 11.2 Robust Stability Analysis of Nonlinear Uncertain Impulsive Dynamical Systems 11.3 Optimal Robust Control for Nonlinear Uncertain Impulsive Dynamical Systems 1104 Inverse Optimal Robust Control for Nonlinear Affine Uncertain Impulsive Dynamical Systems 11.5 Robust Nonlinear Hybrid Control with Polynomial Performance Functionals Chapter 12. Hybrid Dynamical Systems
Introduction LeftContinuous Dynamical Systems Specialization to Hybrid and Impulsive Dynamical Systems Stability Analysis of LeftContinuous Dynamical Systems Dissipative LeftContinuous Dynamical Systems: InputOutput and State Properties 12.6 Interconnections of Dissipative LeftContinuous Dynamical Systems
12.1 12.2 12.3 1204 12.5
Chapter 13. Poincare Maps and Stability of Periodic Orbits for Hybrid Dynamical Systems
13.1 13.2 13.3 1304 13.5 13.6 13.7
Introduction LeftContinuous Dynamical Systems with Periodic Solutions Specialization to Impulsive Dynamical Systems Limit Cycle Analysis of a Verge and Foliot Clock Escapement Modeling Impulsive Differential Equation Model Characterization of Periodic Orbits
351 351 352 358 366 375
385 385 386 395 402 406 411 411 412 418 422 427 435
443 443 444 451 458 459 462 464
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CONTIN~
13.8 Limit Cycle Analysis of the Clock Escapement Mechanism 13.9 Numerical Simulation of an Escapement Mechanism
468 472
Appendix A. System Functions for the Clock Escapement Mechanism
477
Bibliography
485
Index
501
Preface
Dynamical systems theory holds the supreme position among all mathematical disciplines as it provides the foundation for unlocking many of the mysteries in nature and the universe which involve the evolution of time. Dynamical systems theory is used to study ecological systems, geological systems, biological systems, economic systems, pharmacological systems, physiological systems, neural systems, cognitive systems, and physical systems (e.g., mechanics, quantum mechanics, thermodynamics, fluids, magnetic fields, galaxies, etc.), to cite but a few examples. Many of these systems involve an interacting mixture of continuous and discrete dynamics exhibiting discontinuous flows on appropriate manifolds, and hence, give rise to hybrid dynamics. The increasingly complex nature of engineering systems involving controller architectures with realtime embedded software also gives rise to hybrid systems, wherein the continuous mathematics of the system dynamics and control interact with the discrete mathematics of logic and computer science. Modern complex engineering systems additionally involve multiple modes of operation placing stringent demands on controller design and implementation of increasing complexity. Such systems typically possess a multiechelon hierarchical hybrid control architecture characterized by continuoustime dynamics at the lower levels of the hierarchy and discretetime dynamics at the higher levels of the hierarchy. The lowerlevel units directly interact with the dynamical system to be controlled while the higherlevel units receive information from the lowerlevel units as inputs and provide (possibly discrete) output commands which serve to coordinate and reconcile the (sometimes competing) actions of the lowerlevel units. The hierarchical controller organization reduces processor cost and controller complexity by breaking up the processing task into relatively small pieces and decomposing the fast and slow control functions. Typically, the higherlevel units perform logical checks that determine system mode operation, while the lowerlevel units execute continuousvariable commands for a given system mode of operation. The ability of developing an analysis and control design framework for hybrid
xiv
PREFACE
dynamical systems is imperative in light of the increasingly complex nature of dynamical systems which have interacting continuoustime dynamics as well as discreteevent dynamics, such as advanced high performance tactical fighter aircraft, variablecycle gas turbine engines, air and ground transportation systems, and swarms of air and space vehicles. Hybrid dynamical systems is an emerging discipline within dynamical systems theory and control, and hence, the term hybrid system has many meanings to different researchers and practitioners. We define a hybrid dynamical system as an interacting countable collection of dynamical systems involving a mixture of continuoustime dynamics and discrete events that includes impulsive dynamical systems, hierarchical systems, and switching systems as special cases. In this monograph we develop a unified analysis and control design framework for impulsive and hybrid dynamical systems using a Lyapunov and dissipative systems approach. The monograph is written from a systemtheoretic point of view and can be viewed as a contribution to mathematical system theory and control system theory. The material in this book is thus intended to complement the monographs on qualitative analysis, asymptotic analysis, and stability analysis of impulsive dynamical systems [1214,93,148]. After a brief introduction on impulsive and hybrid dynamical systems in Chapter 1, fundamental stability theory for nonlinear impulsive dynamical systems is developed in Chapter 2. In Chapter 3, we extend classical dissipativity theory to impulsive dynamical systems. Chapter 4 provides a treatment of nonnegative and compartmental impulsive dynamical systems. A detailed treatment of vector dissipativity theory for largescale impulsive dynamical systems is given in Chapter 5, while stability results for feedback interconnections of impulsive dynamical systems are given in Chapter 6. In Chapters 7 and 8 we develop energybased hybrid controllers for EulerLagrange, portcontrolled Hamiltonian, and dissipative dynamical systems. A detailed treatment of optimal hybrid control is given in Chapter 9, while Chapters 10 and 11 provide extensions to hybrid disturbance rejection control and hybrid robust control, respectively. Next, Chapter 12 develops a unified dynamical systems framework for a general class of systems possessing leftcontinuous flows which include hybrid, impulsive, and switching systems as special cases. Finally, in Chapter 13 we generalize Poincare's theorem to leftcontinuous dynamical systems for analyzing the stability of periodic orbits of impulsive and hybrid dynamical systems. The first author would like to thank V. Lakshmikantham for his
PREFACE
xv
valuable discussions on impulsive differential equations over the recent years. In addition, the authors thank V. Lakshmikantham, Anthony N. Michel, and Tomohisa Hayakawa for their constructive comments and feedback. In some parts of the monograph we have relied on work we have done jointly with Dennis S. Bernstein, Sanjay P. Bhat, Qing Hui, NataSa A. Kablar, and Alexander V. Roup; it is a pleasure to acknowledge their contributions. The results reported in this monograph were obtained at the School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, the Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, and the Department of Mechanical, Aerospace, and Biomedical Engineering of the University of Tennessee, Knoxville, between June 1999 and July 2005. The research support provided by the Air Force Office of Scientific Research and the National Science Foundation over the years has been instrumental in allowing us to explore basic research topics that have led to some of the material in this monograph. We are indebted to them for their support.
Atlanta, Georgia, USA, March 2006, Wassim M. Haddad Knoxville, Tennessee, USA, March 2006, VijaySekhar Chellaboina Villanova, Pennsylvania, USA, March 2006, Sergey G. Nersesov
Chapter One Introd uction
1.1 Impulsive and Hybrid Dynamical Systems
Modern complex engin~ering systems are highly interconnected and mutually interdependent, both physically and through a multitude of information and communication network constraints. The complexity of modern controlled dynamical systems is further exacerbated by the use of hierarchical embedded control subsystems within the feedback control system, that is, abstract decisionmaking units performing logical checks that identify system mode operation and specify the continuousvariable sub controller to be activated. These multiechelon systems (see Figure 1.1) are classified as hybrid systems (see [6,126,161] and the numerous references therein) and involve an interacting countable collection of dynamical systems possessing a hierarchical structure characterized by continuoustime dynamics at the lowerlevel units and logical decisionmaking units at the higher level of the hierarchy. The lowerlevel units directly interact with the dynamical system to be controlled, while the logical decisionmaking, higherlevel units receive information from the lowerlevel units as inputs and provide (possibly discrete) output commands, which serve to coordinate and reconcile the (sometimes competing) actions of the lowerlevel units. The hierarchical controller organization reduces processor cost and controller complexity by breaking up the processing task into relatively small pieces and decomposing the fast and slow control functions. Typically, the higherlevel units perform logical checks that determine system mode operation, while the lowerlevel units execute continuousvariable commands for a given system mode of operation. Due to their multiechelon hierarchical structure, hybrid dynamical systems are capable of simultaneously exhibiting continuoustime dynamics, discretetime dynamics, logic commands, discrete events, and resetting events. Such systems include dynamical switching systems [29,101,140]' nonsmooth impact systems [28,32], biological systems [93], sampleddata systems [71], discreteevent systems [139], intelligent vehicle/highway systems [113], constrained mechanical sys
2
CHAPTER 1 1\
I I
SUPREMAL COORDINATOR
I I
I
HIERARCHICAL LEVEL
I
I I
I
\ \
\
n \\ ~
~ __ ~
K
~
\ \
\
DECISIONMAKING UNIT \
h __ ~ __ ~ \\ .... K ~ ~ \
COORDINATION I I I I
"'"
I
\
PERFORMANCE RESULTS
\ \ _ _ .I.
I
OUTPUT MEASUREMENTS DYNAMICAL SYSTEM
Figure L1 Multiechelon dynamical system [87].
terns [28], and flight control systems [158], to cite but a few examples. The mathematical descriptions of many hybrid dynamical systems can be characterized by impulsive differential equations [12,14,79,93, 148]. Impulsive dynamical systems can be viewed as a subclass of hybrid systems and consist of three elementsnamely, a continuoustime differential equation, which governs the motion of the dynamical system between impulsive or resetting events; a difference equation, which governs the way the system states are instantaneously changed when a resetting event occurs; and a criterion for determining when the states of the system are to be reset. Since impulsive systems can involve impulses at variable times, they are in general timevarying systems, wherein the resetting events are both a function of time and the system's state. In the case where the resetting events are defined by a prescribed sequence of times which are independent of the system state, the equations are known as timedependent differential equations [12,14,35,61,62,93]. Alternatively, in the case where the resetting events are defined by a manifold in the state space that is independent of time, the equations are autonomous and are known as statedependent differential equations [12,14,35,61,62,93]. Hybrid and impulsive dynamical systems exhibit a very rich dynamical behavior. In particular, the trajectories of hybrid and impulsive dynamical systems can exhibit multiple complex phenomena such as Zeno solutions, noncontinuability of solutions or deadlock, beating or livelock, and confluence or merging of solutions. A Zeno solution involves a system trajectory with infinitely many resettings in finite
INTRODUCTION
3
time. Deadlock corresponds to a dynamical system state from which no continuation, continuous or discrete, is possible. A hybrid dynamical system experiences beating when the system trajectory encounters the same resetting surface a finite or infinite number of times in zero time. Finally, confluence involves system solutions that coincide after a certain point in time. These phenomena, along with the breakdown of many of the fundamental properties of classical dynamical system theory, such as continuity of solutions and continuous dependence of solutions on the system's initial conditions, make the analysis of hybrid and impulsive dynamical systems extremely challenging. The range of applications of hybrid and impulsive dynamical systems is not limited to controlled dynamical systems. Their usage arises in several different fields of science, including computer science, mathematical programming, and modeling and simulation. In computer science, discrete program verification and logic is interwoven with a continuous environment giving rise to hybrid dynamical systems. Specifically, computer software systems interact with the physical system to admit feedback algorithms that improve system performance and system robustness. Alternatively, in mathematical linear and nonlinear optimization with inequality constraints, changes in continuous and discrete states can be computed by a switching dynamic framework. Modeling and simulating complex dynamical systems with multiple modes of operation involving multiple system transitions also give rise to hybrid dynamical systems. Among the earliest investigations of dynamical systems involving continuous dynamics and discrete switchings can be traced back to relay control systems and bangbang optimal control. Dynamical systems involving an interacting mixture of continuous and discrete dynamics abound in nature and are not limited to engineering systems with programmable logic controllers. Hybrid systems arise naturally in biology, physiology, pharmacology, economics, biocenology, demography, chemistry, neuroscience, impact mechanics, quantum mechanics, systems with shock effects, and cosmology, among numerous other fields of science. For example, mechanical systems subject to unilateral constraints on system positions give rise to hybrid dynamical systems. These systems involve discontinuous solutions, wherein discontinuities arise primarily from impacts (or collisions) when the system trajectories encounter the unilateral constraints. In physiological systems the blood pressure and blood flow to different tissues of the human body are controlled to provide sufficient oxygen to the cells of each organ. Certain organs such as the kidneys normally require higher blood flows than is necessary to satisfy ba
4
CHAPTER 1
sic oxygen needs. However, during stress (such as hemorrhage) when perfusion pressure falls, perfusion of certain regions (e.g., brain and heart) takes precedence over perfu~ion of other regions, and hierarchical controls (overriding controls) shut down flow to these other regions. This shutting down process can be modeled as a resetting event giving rise to a hybrid system. As another example, biomolecular genetic systems also combine discrete events, wherein a gene is turned on or off for transcription, with continuous dynamics involving concentrations of chemicals in a given cell. Even though many scientists and engineers recognize that a large number of life science and engineering systems are hybrid in nature, these systems have been traditionally modeled, analyzed, and designed as purely discrete or purely continuous systems. The reason for this is that only recently has the theory of impulsive and hybrid dynamical systems been sufficiently developed to fully capture the interaction between the continuous and discrete dynamics of these systems. Even though impulsive dynamical systems were first formulated by Mil'man and Myshkis [123,124]' 1 the fundamental theory of impulsive differential equations is developed in the monographs by Bainov, Lakshmikantham, Perestyuk, Samoilenko, and Simeonov [1214,93,148]. These monographs develop qualitative solution properties, existence of solutions, asymptotic properties of solutions, and stability theory of impulsive dynamical systems. In this monograph we build on the results of [1214,93,148] to develop invariant set stability theorems, partial stability, Lagrange stability, boundedness and ultimate boundedness, dissipativity theory, vector dissipativity theory, energybased hybrid control, optimal control, disturbance rejection control, and robust control for nonlinear impulsive and hybrid dynamical systems. 1.2 A Brief Outline of the Monograph
The main objective of this monograph is to develop a general analysis and control design framework for nonlinear impulsive and hybrid IMil'man and Myshkis were the first to develop qualitative analysis results for impulsive dynamical systems. However, work on impact and hybrid systems can be traced back to ancient Greek scientists and mathematicians such as Aristotle, Archimedes, and Heron. Problems related to Heron's work on hybrid automata (Ilcei QVTOjLQT07rOL"ITLK,TJr;) as well as problems on impact dynamics attracted the interest of numerous physicists and mathematicians who followed with relevant contributions made in the last three centuries. Notable contributions include the work of Leibniz, Newton, (Jacob) Bernoulli, d'Alembert, Poisson, Huygens, Coriolis, Darboux, Routh, Appell, and Lyapunov.
INTRODUCTION
5
dynamical systems. The main contents of the monograph are as follows. In Chapter 2, we establish notation and definitions, and develop stability theory for nonlinear impulsive dynamical systems. Specifically, Lyapunov stability theorems are developed for timedependent and statedependent impulsive dynamical systems. Furthermore, we state and prove a fundamental result on positive limit sets for statedependent impulsive dynamical systems. Using this result, we generalize the KrasovskiiLaSalle invariant set theorem to impulsive dynamical systems. In addition, partial stability, Lagrange stability, boundedness, ultimate boundedness, and stability theorems via vector Lyapunov functions are also established. In Chapter 3, we extend the notion of dissipative dynamical systems [165,166] to develop the concept of dissipativity for impulsive dynamical systems. Specifically, the classical concepts of system storage functions and system supply rates are extended to impulsive dynamical systems. In addition, we develop extended KalmanYakubovitchPopov conditions in terms of the hybrid system dynamics for characterizing dissipativeness via system storage functions and hybrid supply rates for impulsive dynamical systems. Furthermore, a generalized hybrid energy balance interpretation involving the system's stored or accumulated energy, dissipated energy over the continuoustime dynamics, and dissipated energy at the resetting instants is given. Specialization of these results to passive and nonexpansive impulsive systems is also provided. In Chapter 4, we extend the results of Chapters 2 and 3 to develop stability and dissipativity results for impulsive nonnegative and compartmental dynamical systems. In Chapter 5, we develop vector dissipativity notions for largescale nonlinear impulsive dynamical systems. In particular, we introduce a generalized definition of dissipativity for largescale nonlinear impulsive dynamical systems in terms of a hybrid vector inequality, a vector hybrid supply rate, and a vector storage function. Dissipativity properties of the largescale impulsive system are shown to be determined from the dissipativity properties of the individual impulsive subsystems making up the largescale system and the nature of the system interconnections. Using the concepts of dissipativity and vector dissipativity, in Chapter 6 we develop feedback interconnection stability results for impulsive nonlinear dynamical systems. General stability criteria are given for Lyapunov, asymptotic, and exponential stability of feedback impulsive dynamical systems. In the case of quadratic hybrid supply rates corresponding to net system power and weighted inputoutput energy, these results generalize the positivity and small gain theorems to the case of nonlinear impulsive dynamical systems.
6
CHAPTER 1
In Chapter 7, we develop a hybrid control framework for impulsive portcontrolled Hamiltonian systems. In particular, we obtain constructive sufficient conditions for hybrid feedback stabilization that provide a shaped energy function for the closedloop system while preserving a hybrid Hamiltonian structure at the closedloop level. A novel class of energybased hybrid controllers is proposed in Chapter 8 as a means for achieving enhanced energy dissipation in EulerLagrange, portcontrolled Hamiltonian, and dissipative dynamical systems. These controllers combine a logical switching architecture with continuous dynamics to guarantee that the system plant energy is strictly decreasing across resetting events. The general framework leads to closedloop systems described by impulsive differential equations. In addition, we construct hybrid controllers that guarantee that the closedloop system is consistent with basic thermodynamic principles. In particular, the existence of an entropy function for the closedloop system is established that satisfies a hybrid Clausiustype inequality. Extensions to hybrid EulerLagrange systems and impulsive dynamical systems are also developed. In Chapter 9, a unified framework for hybrid feedback optimal and inverse optimal control involving a hybrid nonlinear nonquadratic performance functional is developed. It is shown that the hybrid cost functional can be evaluated in closed form as long as the cost functional considered is related in a specific way to an underlying Lyapunov function that guarantees asymptotic stability of the nonlinear closedloop impulsive system. Furthermore, the Lyapunov function is shown to be a solution of a steadystate, hybrid HamiltonJacobiBellman equation. Extensions of the hybrid feedback optimal control framework to disturbance rejection control and robust control are addressed in Chapters 10 and 11, respectively. In Chapter 12, we develop a unified dynamical systems framework for a general class of systems possessing leftcontinuous flows, that is, leftcontinuous dynamical systems. These systems are shown to generalize virtually all existing notions of dynamical systems and include hybrid, impulsive, and switching dynamical systems as special cases. Furthermore, we generalize dissipativity, passivity, and nonexpansivity theory to leftcontinuous dynamical systems. Specifically, the classical concepts of system storage functions and supply rates are extended to leftcontinuous dynamical systems providing a generalized hybrid system energy interpretation in terms of stored energy, dissipated energy over the continuoustime dynamics, and dissipated energy over the resetting events. Finally, the generalized dissipativity notions are used to develop general stability criteria for feedback
INTRODUCTION
7
interconnections of leftcontinuous dynamical systems. These results generalize the positivity and small gain theorems to the case of leftcontinuous and hybrid dynamical systems. Finally, in Chapter 13 we generalize Poincare's theorem to dynamical systems possessing leftcontinuous flows to address the stability of limit cycles and periodic orbits of leftcontinuous, hybrid, and impulsive dynamical systems. It is shown that the resetting manifold provides a natural hyperplane for defining a Poincare return map. In the special case of impulsive dynamical systems, we show that the Poincare map replaces an nthorder impulsive dynamical system by an (n 1)thorder discretetime system for analyzing the stability of periodic orbits.
Chapter Two Stability Theory for Nonlinear Impulsive Dynamical Systems
2.1 Introduction
One of the most basic issues in system theory is stability of dynamical systems. System stability is characterized by analyzing the response of a dynamical system to small perturbations in the system states. Specifically, an equilibrium point of a dynamical system is said to be stable if, for small values of initial disturbances, the perturbed motion remains in an arbitrarily prescribed small region of the state space. More precisely, stability is equivalent to continuity of solutions as a function of the system initial conditions over a neighborhood of the equilibrium point uniformly in time. If, in addition, all solutions of the dynamical system approach the equilibrium point for large values of time, then the equilibrium point is said to be asymptotically stable. The most complete contribution to the stability analysis of nonlinear dynamical systems was introduced in the late nineteenth century by the Russian mathematician Alexandr Mikhailovich Lyapunov in his seminal work entitled The General Problem of the Stability of Motion [110112]. Lyapunov's results, which include the direct and indirect methods, along with the KrasovskiiLaSalle invariance principle [15,91,98,99]' provide a powerful framework for analyzing the stability of nonlinear dynamical systems as well as designing feedback controllers which guarantee closedloop system stability. Lyapunov's direct method states that if a positivedefinite function of the states of a given dynamical system can be constructed for which its time rate of change due to perturbations in a neighborhood of the system's equilibrium is always negative or zero, then the system's equilibrium point is stable or, equivalently, Lyapunov stable. Alternatively, if the time rate of change of the positive definite function is strictly negative, then the system's equilibrium point is asymptotically stable. In light of the increasingly complex nature of dynamical system analysis and design, such as nonsmooth impact systems [28,32], biological systems [93], demographic systems [106], hybrid systems [30,
10
CHAPTER 2
169], sampleddata systems [71], discreteevent systems [139], systems with shock effects, and feedback systems with impulsive or resetting controls [35,61,62]' dynamical systems exhibiting discontinuous flows on appropriate manifolds arise naturally. The mathematical descriptions of such systems can be characterized by impulsive differential equations [12,14,79,93,148]. To analyze the stability of dynamical systems with impulsive effects, Lyapunov stability results have been presented in the literature [12,9295,105,148,153,154,170]. In particular, local and global asymptotic stability conclusions of an equilibrium point of a given impulsive dynamical system are provided if a smooth (at least continuously differentiable) positivedefinite function of the nonlinear system states (Lyapunov function) can be constructed for which its time rate of change over the continuoustime dynamics is strictly negative and its difference across the resetting times is negative. However, unlike dynamical systems possessing continuous flows, KrasovskiiLaSalletype invariant set stability theorems [15,91,98,99] have not been addressed for impulsive dynamical systems. This is in spite of the fact that systems theory with impulsive effects has dominated the Russian and Eastern European literature [12,14,79,9295,148,153,154]. There appear to be (at least) two reasons for this state of affairs, namely, solutions of impulsive dynamical systems are not continuous in time and are not continuous functions of the system's initial conditions, which are two key properties needed to establish invariance of positive limit sets, and hence an invariance principle. In this chapter, we develop Lyapunov and invariant set stability theorems for nonlinear impulsive dynamical systems. In particular, invariant set theorems are derived, wherein system trajectories converge to the largest invariant set of Lyapunov level surfaces of the impulsive dynamical system. For statedependent impulsive dynamical systems with continuously differentiable Lyapunov functions defined on a compact positively invariant set (with respect to the nonlinear impulsive system), the largest invariant set is contained in a hybrid level surface composed of a union involving vanishing Lyapunov derivatives and differences of the system dynamics over the continuoustime trajectories and the resetting instants, respectively. In addition, if the Lyapunov derivative along the continuoustime system trajectories is negative semidefinite and no system trajectories can stay indefinitely at points where the function's derivative or difference identically vanishes, then the system's equilibrium is asymptotically stable. These results provide less conservative conditions for examining the stability of statedependent impulsive dynamical systems as compared to the
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
11
classical results presented in [12,93,148,153,170]. In addition, partial stability, Lagrange stability, boundedness, ultimate boundedness, stability of timedependent impulsive dynamical systems, and stability theorems via vector Lyapunov functions are also established. 2.2 Nonlinear ImpUlsive Dynamical Systems
In this section, we develop notation and introduce some basic properties of impulsive dynamical systems [12,14,79,9295,105,148,153,154]. The notation used in this monograph is fairly standard. Specifically, ~ denotes the set of real numbers, Z+ denotes the set of nonnegative integers, Z+ denotes the set of positive integers, ~n denotes the set of n x 1 column vectors, ~nxm denotes the set of n x m real matrices, §n denotes the set of n x n symmetric matrices, Nn denotes the set of n x n nonnegativedefinite matrices, 1pm denotes the set of n x n positivedefinite matrices, (.) T denotes transpose, (.)# denotes group generalized inverse, and In or I denotes the n x n identity matrix. Furthermore, £2 denotes the space of squareintegrable Lebesgue measurable functions on [0,(0) and £2 denotes the space of squaresummable sequences on Z+. In addition, we denote the boundary, the interior, o
and the closure of the set S by 8S, S, and S, respectively. We write I . I for the Euclidean vector norm, R(M) and N(M) for the range space and the null space of a matrix M, spec(M) for the spectrum of the square matrix M, ind(M) for the index of M (that is, the size of the largest Jordan block of M associated with ). = 0, where). E spec(M)), ® for the Kronecker product, and EEl for the Kronecker sum. Furthermore, we write V' (x) for the Frechet derivative of V at x, B£(o), 0 E ~n, c > 0, for the open ball centered at 0 with radius c, M ~ 0 (respectively, M > 0) to denote the fact that the Hermitian matrix M is nonnegative (respectively, positive) definite, inf to denote infimum (that is, the greatest lower bound), sup to denote supremum (that is, the least upper bound), and x(t) 7 M as t 7 00 to denote that x(t) approaches the set M (that is, for each c > 0 there exists T > 0 such that dist(x(t), M) < c for all t > T, where dist(p, M) ~ infxEM lip  xii). Finally, the notions of openness, convergence, continuity, and compactness that we use throughout the monograph refer to the topology generated on ~n by the norm II . II. As discussed in Chapter 1, an impulsive dynamical system consists of three elements:
i) a continuoustime dynamical equation, which governs the mo
12
CHAPTER 2
tion of the system between resetting events; ii) a difference equation, which governs the way the states are instantaneously changed when a resetting event occurs; and iii) a criterion for determining when the states of the system are to be reset.
Thus, an impulsive dynamical system has the form
x(t) = fe(x(t)), ~x(t) = fd(X(t)),
x(o) = Xo, (t, x(t)) ¢ S, (t,x(t)) E S,
(2.1) (2.2)
where t ~ 0, x(t) E V ~ ~n, V is an open set with 0 E V, ~x(t) ~ x(t+)x(t), where x(t+) ~ x(t)+ fd(X(t)) = lime:+o x(t+c), x(t) E Z, fe : V t ~n is continuous, fd : S t ~n is continuous, and S c [0, 00) x V is the resetting set. A function x : Ixo t V is a solution to the impulsive dynamical system (2.1) and (2.2) on the interval Ixo ~ ~ with initial condition x(O) = Xo, if x(·) is leftcontinuous and x(t) satisfies (2.1) and (2.2) for all t E Ixo. We assume that the continuoustime dynamics feO are such that the solution to (2.1) is jointly continuous in t and Xo between resetting events. A sufficient condition ensuring this is Lipschitz continuity of Ie (.). Alternatively, uniqueness of solutions in forward time along with the continuity of feO ensure that solutions to (2.1) between resetting events are continuous functions of the initial conditions Xo E V even when feO is not Lipschitz continuous on V (see [41, Theorem 4.3, p. 59]). More generally, feO need not be continuous. In particular, if feO is discontinuous but bounded and x(·) is the unique solution to (2.1) between resetting events in the sense of Filippov [44], then continuous dependence of solutions between resetting events with respect to the initial conditions hold [44]. We refer to the differential equation (2.1) as the continuoustime dynamics, and we refer to the difference equation (2.2) as the resetting law. In addition, we use the notation s(t, T, xo) to denote the solution x(t) of (2.1) and (2.2) at time t ~ T with initial condition X(T) = Xo. Finally, a point Xe E V is an equilibrium point of (2.1) and (2.2) if and only if s(t, T, x e) = Xe for all T ~ 0 and t ~ T. Note that Xe E V is an equilibrium point of (2.1) and (2.2) if and only if fe(x e) = 0 and fd(X e) = O. For a particular trajectory x(t), we let tk denote the kth instant of time at which (t, x(t)) intersects S, and we call the times tk the resetting times. Thus, the trajectory of the system (2.1) and (2.2) from the initial condition x(O) = Xo is given by 'IjJ(t, 0, xo) for 0 < t ::; it,
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
13
where 1jJ(t, 0, xo) denotes the solution to continuoustime dynamics (2.1). If and when the trajectory reaches a state Xl ~ X(tl) satisfying (tl,XI) E S, then the state is instantaneously transferred to xt ~ Xl + fd(XI), according to the resetting law (2.2). The trajectory x(t), tl < t ~ t2, is then given by 1jJ(t, tl, xt), and so on. Note that the solution x(t) of (2.1) and (2.2) is leftcontinuous, that is, it is continuous everywhere except at the resetting times tk, and
Xk ~ X(tk) = lim X(tk  6),
(2.3)
E>O+
xt ~ X(tk)
+ fd(X(tk)) = E>O+ lim X(tk + 6),
(2.4)
for k = 1,2, .... To ensure the wellposedness of the resetting times we make the following additional assumptions: AI. If (t, x(t)) E
0< 6
0 such that, for all
6
6,
1jJ(t + 6, t,x(t))
~
S.
A2. If (tk' X(tk)) E 8S n S, then there exists o ~ 6 < 6,
6
> 0 such that, for all
Assumption Al ensures that if a trajectory reaches the closure of S at a point that does not belong to S, then the trajectory must be directed away from S, that is, a trajectory cannot enter S through a point that belongs to the closure of S but not to S. Furthermore, A2 ensures that when a trajectory intersects the resetting set S, it instantaneously exits S. Finally, we note that if Xo E S then the system initially resets to xt = Xo + fd(XO) ~ S, which serves as the initial condition for continuoustime dynamics (2.1). It follows from A2 that resetting removes the pair (tk' Xk) from the resetting set S. Thus, immediately after resetting occurs, the continuoustime dynamics (2.1), and not the resetting law (2.2), becomes the active element of the impulsive dynamical system. Furthermore, it follows from Al and A2 that no trajectory can intersect the interior of S. Specifically, it follows from Al that a trajectory can only reach S through a point belonging to both S and its boundary. And, from A2, it follows that if a trajectory reaches a point in S that is on the boundary of S, then the trajectory is instantaneously
14
CHAPTER 2
removed from S. Since a continuous trajectory starting outside of S and intersecting the interior of S must first intersect the boundary of S, it follows that no trajectory can reach the interior of S. To show that the resetting times tk are well defined and distinct, assume T = inf{t E lR+ : 1jJ(t, 0, xo) E S} < 00. Now, ad absurdum, suppose h is not well defined, that is, min{ t E lR+ : 1jJ(t, 0, xo) E S} does not exist. Since 1jJ(., 0, xo) is continuous, it follows that 1jJ(T, 0, xo) E as and since, by assumption, min{t E lR+ : 1jJ(t,O,xo) E S} does not exist it follows that 1jJ(T, 0, xo) E 8\S. Note that 1jJ(t, 0, xo) = s(t,O,xo), for every t such that 1jJ(T, 0, x) rf S for all ~ T ~ t. Now, it follows from Al that there exists E > such that s(T + 0,0, xo) = 1jJ(T+8,0,xo), 0 E (O,E), which implies that inf{t E lR+: 1jJ(t,O,xo) E S} > T, which is a contradiction. Hence, 1jJ(T, 0, xo) E as n Sand inf{t E lR+ : 1jJ(t,O,xo) E S} = min{t E lR+ : 1jJ(t,O,xo) ED}, which implies that the first resetting time tl is well defined for all initial conditions Xo E D. Next, it follows from A2 that t2 is also well defined and t2 i tl. Repeating the above arguments it follows that the resetting times tk are well defined and distinct. Since the resetting times are well defined and distinct, and since the solution to (2.1) exists and is unique, it follows that the solution of the impulsive dynamical system (2.1) and (2.2) also exists and is unique over a forward time interval. However, it is important to note that the analysis of impulsive dynamical systems can be quite involved. In particular, such systems can exhibit Zenoness and beating, as well as confluence, wherein solutions exhibit infinitely many resettings in a finite time, encounter the same resetting surface a finite or infinite number of times in zero time, and coincide after a certain point in time. In this monograph we allow for the possibility of confluence and Zeno solutions; however, A2 precludes the possibility of beating. Furthermore, since not every bounded solution of an impulsive dynamical system over a forward time interval can be extended to infinity due to Zeno solutions, we assume that existence and uniqueness of solutions are satisfied in forward time. For details see [12,14,93]. The following two examples demonstrate some of this rich behavior inherent in impulsive dynamical systems.
°
°
Example 2.1 Consider the scalar impulsive dynamical system originally studied in [12] given by
±(t) =0, x(O) = xo, (t, x(t)) rf S, ~x(t) = x 2 (t) sgnx(t)  x(t), (t, x(t)) E S, where sgnx ~ x/lxi, x
i 0,
sgn(O) ~ 0, and S
= {(t, x)
(2.5) (2.6) E lR+ x lR :
15
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
·1
·2
·3~~~4~
·4~~~~~~~~~~~
o
10
Time
Figure 2.1 System trajectories for Xo
=
12
3 and Xo
14
= 4.
Ixl < 3 and t = x + 6k, k E Z+}. The trajectory set, 0, xo) of (2.5) and (2.6) with Ixol ~ 3 does not intersect S, and hence, is continuous (see Figure 2.1). Alternatively, if 1 < Ixol < 3, then the trajectory set, 0, xo) reaches S a finite number of times. In particular, the trajectory set, 0, 21/4) reaches S three times with t3 = 2 (see Figure 2.2). If 0 < Xo < 1, then set, 0, xo) reaches S infinitely many times and limk>oo tk = 00 and limk>oo X(tk) = 0 (see Figure 2.3). If, alternatively, 1 < Xo < 0, then the trajectory s(t,O,xo) reaches S infinitely many times in a finite time. In this case, limk>oo tk = 6 and limk>oo X(tk) = 0 (see Figure 2.4). Finally, (2.5) and (2.6) exhibits confluence. In particular, the trajectories set, 0, 21/4) and set, 0, 4) coincide after t > 2 (see Figures 2.1 and 2.2). l:,. Example 2.2 In this example we consider an impulsive system with a nonconvergent Zeno solution. Specifically, consider the impulsive dynamical system with continuoustime dynamics given by
x(t) = [
Ir(x(t)) 
0
11 sgn(x3(t))
1 ,
x(O) = Xo,
x(t) ¢ Z, (2.7)
where x = [Xl, X2, x3F and rex) ~ y'xi
+ x~,
and discretetime dy
16
CHAPTER 2
·1
.2 L_L._L.,L'''',':_':_.I o 4 Time
Figure 2.2 System trajectory for Xo = 21/4.
namics given by
~x(t) =
er;(;m) [ er;(;m)
1) 1)
+ In r(x(t))] sin[O(x(t)) + In r(x(t))]
cos[O(x(t))
1  x(t),
(r(x(t))1)3 r(x(t))
x(t) where O(x) ]R3:
E
Z, (2.8)
~ tan 1 (~), 'Tk(XO) ~ tk, and resetting set Z ~ {x
E
X3 = O}. Note that if r(xo) > 1 then r(x(tk)) = (k + 1)r(xo)  k , kr(xo)  (k 1)
which implies that Iimktoo r(x(tk))
O(X(tk)) = O(xo)
=
(2.9)
1. Furthermore,
k
1)k + r(xo)] + t:t In r(x(td) = O(xo) + In [(r(xo) r(xo) , "
(2.10) and hence, O(X(tk)) t 00 as k t 00. Finally, X3(tk) = (r(~(~01)3 t 0 as k t 00. Hence, the sequence of impact points is bounded but does not converge (see Figure 2.5).
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
17
·1
Time
Figure 2.3 System trajectory for Xo = 1/2.
Finally, to show that the solution of this system is Zeno note that
and
Tk+1(XO)  Tk(XO)
=
IX3(T:(Xo))1 Ir (+()) I= Tk Xo  1
2 (r(Tk(XO))  1) . (2.13)
Now, using the fact that r(Tk(xo))  1 = kr[x~f2(Ll)' it follows that
Since the series L~l (k~a)2 converges for all a E ffi., limk+oo Tk(XO) = Tl(XO) + L~l[Tk+1(XO) Tk(XO)] exists, and hence, the trajectory of the system (2.7) and (2.8) is Zeno. 6. Clearly, Example 2.2 shows that not every bounded Zeno solution is extendable. In fact, the impulsive dynamical system (2.7) and (2.8) is not even leftcontinuous at t = T(XO) for if it were leftcontinuous, then necessarily limk+oo S(Tk(XO), 0, xo) = S(T(XO), 0, xo), where T(XO)
18
CHAPTER 2
1~4~4
2
3
4 Time
Figure 2.4 System trajectories for Xo
=
1/2, Xo
= 1,
Xo
= 1, and Xo = O.
denotes the Zeno time (accumulation time). However, if a Zeno solution is convergent and the continuous and discrete parts of the states converge to a unique value at the Zeno time r(xo), then by reinitializing the impulsive dynamical system at r(xo) using s(r(xo), 0, xo) as the system initial condition, a Zeno solution can be extended. In [12,79,9295,105,153,154]' the resetting set S is defined in terms of a countable number of functions rk : D > (0, (0), and is given by S
= U{(rk(x), x): XED}. k
(2.15)
The analysis of impulsive dynamical systems with a resetting set of the form (2.15) can be quite involved. Furthermore, since impulsive dynamical systems of the form (2.1) and (2.2) involve impulses at variable times they are timevarying systems. In this monograph, we will consider impulsive dynamical systems involving two distinct forms of the resetting set S. In the first case, the resetting set is defined by a prescribed sequence of times which are independent of the state x. These equations are thus called timedependent impulsive dynamical systems. In the second case, the resetting set is defined by a region in the state space that is independent of time. These equations are called statedependent impulsive dynamical systems. Timedependent impulsive dynamical systems can be written as (2.1) and (2.2) with S defined as b.
S=TxD,
(2.16)
19
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS 1.2
0.8 0.6
X3
0.4 0.2 0 0.2 1.5 0.5
o
1.5
0.5 1
Figure 2.5 Phase portrait of Zeno system with nonconverging trajectory.
where (2.17)
°
and ~ tl < t2 < ... are prescribed resetting times. Now, (2.1) and (2.2) can be rewritten in the form of the timedependent impulsive
dynamical system
°rt
x(t) = fc(x(t)), ~x(t) = fd(X(t)),
t =J tk,
x(O) = xo, t = tk.
(2.18) (2.19)
T and tk < tk+1, it follows that Assumptions Al and A2 are Since satisfied. Since timedependent impulsive dynamical systems involve impulses at a fixed sequence of times, they are timevarying systems. Example 2.3 To show that timedependent impulsive dynamical systems are timevarying systems consider the scalar timedependent impulsive dynamical system ±(t) = 0, ~x(t) =
1,
x(to) = 0, t
t
=J 2,
= 2.
(2.20) (2.21)
Since
s(t, 1,0) = {
~:
1 2,
2,
(2.22)
20
CHAPTER 2
and s
°
(t _ 1 0) "
= {O, 1 < t ~ 3, 1, t>3 ,
(2.23)
it follows that s(t, 1,0) =1= s(t  1,0,0), 2 < t ~ 3, and hence, (2.20) and (2.21) is time varying. 6. Statedependent impulsive dynamical systems can be written as (2.1) and (2.2) with S defined as
s ~ [0,(0)
x
z,
(2.24)
where Z c V. Therefore, (2.1) and (2.2) can be rewritten in the form of the statedependent impulsive dynamical system
x(t) = fc(x(t)), ~x(t) = fd(X(t)),
x(O) = Xo, x(t) E Z.
x(t) f/.
z,
(2.25) (2.26)
We assume that if x E Z, then x + f d (x) f/. Z. In addition, we assume that if at time t the trajectory x(t) E Z\Z, then there exists c > such that for all < 8 < c, x(t + 8) f/. Z. These assumptions represent the specialization of A1 and A2 for the particular resetting set (2.24). It follows from these assumptions that for a particular initial condition, the resetting times 7k(XO) ~ tk are distinct and well defined. Since the resetting set Z is a subset of the state space and is independent of time, statedependent impulsive dynamical systems are timeinvariant systems. Finally, note that if x* E V satisfies fd(X*) = 0, then x* f/. Z. To see this, suppose x* E Z. Then x* + fd(X*) = x* E Z, contradicting A2. Thus, if x = Xe is an equilibrium point of (2.25) and (2.26), then Xe f/. Z, and hence, Xe E V is an equilibrium point of (2.25) and (2.26) if and only if fc(x e ) = 0. In addition, note that it follows from the definition of 7k () that 71 (x) > 0, x f/. Z, and 71 (x) = 0, x E Z. Finally, since for every x E Z, x + fd(X) f/. Z, it follows that 72(X) = 71 (x) + 71 (x + fd(X)) > 0.
°
°
2.3 Stability Theory of Impulsive Dynamical Systems
In this section, we present Lyapunov, asymptotic, and exponential stability theorems for nonlinear statedependent impulsive dynamical systems. The following definition introduces several types of stability corresponding to the zero solution x(t) == of (2.25) and (2.26).
°
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
21
Definition 2.1 i} The zero solution x(t) == 0 to {2.25} and {2.26} is Lyapunov stable if, for all E > 0, there exists 8 = 8(E) > 0 such that if IIx(O) II < 8, then Ilx(t)11 < E, t 2:: O. ii} The zero solution x(t) == 0 to {2.25} and {2.26} is asymptotically stable if it is Lyapunov stable and there exists 8 > 0 such that if Ilx(O)11 < 8, then limt+oo x(t) = O. iii} The zero solution x(t) == 0 to {2.25} and {2.26} is exponentially stable if there exist positive constants a, (3, and 8 such that if Ilx(O) II < 8, then Ilx(t)11 :s; allx(O)lle/3t, t 2:: O. iv} The zero solution x(t) == 0 to {2.25} and {2.26} is globally asymptotically stable if it is Lyapunov stable and for all x(O) E jRn, limt+oo x(t) = O. v} The zero solution x(t) == 0 to {2.25} and {2.26} is globally exponentially stable if there exist positive constants a and (3 such that Ilx(t) II :s; allx(O) Ile/3t, t 2:: 0, for all x(O) E jRn. vi} Finally, the zero solution x(t) == 0 to {2.25} and {2.26} is unstable if it is not Lyapunov stable. Theorem 2.1 Consider the nonlinear impulsive dynamical system 9 given by {2.25} and {2.26}. Suppose there exists a continuously differentiable function V : D  t [0,(0) satisfying YeO) = 0, V(x) > 0, xED, x # 0, and
V'(x)fc(x):s; 0, x vex + fd(x)):S; Vex),
rf Z, x
E
(2.27)
Z.
(2.28)
Then the zero solution x(t) == 0 to {2.25} and {2.26} is Lyapunov stable. Furthermore, if the inequality {2.27} is strict for all x # 0, then the zero solution x(t) == 0 to {2.25} and {2.26} is asymptotically stable. Alternatively, if there exist scalars a, (3, E > 0, and p 2:: 1 such that
allxll P :s; Vex) :s; (3llxII P , xED, V'(x)fc(x) :s; EV(X), x rf Z,
(2.29) (2.30)
and {2.28} holds, then the zero solution x(t) == 0 to {2.25} and {2.26} is exponentially stable. Finally, if, in addition, D = jRn and Vex)
t
00
as Ilxll
t
00,
(2.31)
then the above asymptotic and exponential stability results are global.
22
CHAPTER 2
Proof. Let c > 0 be such that Be(O) 0 since 0 rt oBe(O) and V(x) > 0, x E V, x # o. Next, let (3 E (0,0:) and define Vf3 ~ {x E Be(O): V(x) S {3}. Now, let Xo E Vf3 and note that for S = [0,00) x Z it follows from Assumptions Al and A2 that the resetting times rk(xo) are well defined and distinct for every trajectory of (2.25) and (2.26). Prior to the first resetting time, we can determine the value of V(x(t)) as
V(x(t))=V(x(O))
+ fat V'(x(r))fc(x(r))dr,
t E [O,rl(xO)]. (2.33)
Between consecutive resetting times rk (xo) and rk+l (xo), we can determine the value of V (x( t)) as its initial value plus the integral of its rate of change along the trajectory x(t), that is,
V(x(t)) = V(x(rk(xo))
+ fd(x(rk(xo)))) +
It
V'(x(r))fc(x(r))dr,
Tk(XO)
t E (rk(xo),rk+l(xo)], (2.34) for k = 1,2, .... Adding and subtracting V(x(rk(xO))) to and from the right hand side of (2.34) yields
V(x(t)) = V(x(rk(xo)))
+ [V(x(rk(xo)) + fd(x(rk(xo))))
 V(x(rk(xo)))]
+
it
V'(x(r))fc(x(r))dr,
Tk(XO)
t and in particular, at time rk+l (xo),
E
(rk(XO), rk+l (xo)],
(2.35)
23
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
By recursively substituting (2.36) into (2.35) and ultimately into (2.33), we obtain
V(x(t)) = V(x(O))
+ fat V'(x(T))fc(x(T))dT
k
+ 2)V(X(Ti(XO)) + fd(X(Ti(XO)))) i=l
 V(xh(xo)))]'
If we allow to ~ 0 and L:?=l ~ 0, then (2.37) is valid for k E From (2.37) and (2.28) we obtain
V(x(t))
~ V(x(O)) + fat V'(x(T))fc(x(T))dT,
t
~ O.
Z+.
(2.38)
Furthermore, it follows from (2.27) that
V(x(t))
~
V(x(O))
~
(3,
x(O) E'Df3,
t
~ O.
(2.39)
Next, suppose, ad absurdum, there exists T > 0 such that Ilx(T)11 ~ Hence, since Ilxoll < E, there exists tl E (0, T] such that either Ilx(tl)11 = E or X(tl) E Z, Ilx(tl)11 < E, and Ilx(tt)11 ~ E. If Ilx(tl)11 = E then V(X(tl)) ~ a > (3, which is a contradiction. Alternatively, if X(tl) E Z, Ilx(tl)11 < E, and Ilx(tt)11 ~ E, then Ilx(tt)11 = Ilx(tl) + fd(X(tl))11 ~ 'r/, which implies that V(x(tt)) ~ a > (3 contradicting (2.39). Hence, V(x(t)) :::; (3 and Ilx(t)11 < c, t ~ 0, which implies that 'Df3 is a positive invariant set (see Definition 2.3) with respect to (2.25) and (2.26). Next, since V(·) is continuous and V(O) = 0, there exists 8 = 8(E) E (0, c) such that V(x) < (3, x E B8(0). Since B8(0) C'Df3 C BE:(O) ~ 'D and 'Df3 is a positive invariant set with respect to (2.25) and (2.26), it follows that for all Xo E B8(0), x(t) E BE: (0) , t ~ 0, which establishes Lyapunov stability. To prove asymptotic stability, suppose (2.28) and (2.37) hold, and let Xo E B8(0). Then it follows that x(t) E BE: (0) , t ~ 0. However, V(x(t)), t ~ 0, is monotonically decreasing and bounded from below by zero. Next, it follows from (2.28) and (2.37) that E.
V(x(t))  V(x(s))
~
it
V'(X(T))fc(x(T))dT,
t> s,
(2.40)
and, assuming strict inequality in (2.27), we obtain
V(x(t)) < V(x(s)),
t> s,
(2.41 )
24
CHAPTER 2
provided x(s) i= O. Now, suppose, ad absurdum, x(t), t ~ 0, does not converge to zero. This implies that V(x(t)), t ~ 0, is lower bounded by a positive number, that is, there exists L> 0 such that V(x(t)) ~ L > 0, t ~ o. Hence, by continuity of V(·) there exists ~' > 0 such that V(x) < L, x E 8(dO), which further implies that x(t) rt 8(dO), t ~ O. Next, let Ll ~ mino'~lIxll~e V'(x)fc(x) which implies that  V'(x)fc(x) ~ L 1, ~' ::; Ilxll ::; c, and hence, it follows from (2.37) that
V(x(t))  V(xo) ::; lot V'(x(T))fc(x(T))dT ::; LIt,
(2.42)
and hence, for all Xo E 8 0 (0),
V(x(t)) ::; V(xo)  LIt. Letting t > V(X2~L, it follows that V(x(t)) < L, which is a contradiction. Hence, x(t)  t 0 as t  t 00, establishing asymptotic stability. To show exponential stability, note that it follows from (2.29), (2.30), and (2.28) that the zero solution x(t) == 0 to (2.25) and (2.26) is asymptotically stable. Hence, there exists ~ > 0 such that for all Xo E 8 0(0), x(t)  t 0 as t  t 00. Next, let Xo E 8 0(0) and note that it follows from (2.30) that prior to the first resetting time
V(x(t)) ::; cV(x(t)),
0::; t::; Tl(XO),
Xo E 8 0(0),
(2.43)
which implies that
Similarly, between the first and second resetting times
V(x(t)) ::; cV(x(t)),
Tl(XO) < t ::; T2(XO),
Xo
E
8 0(0),
(2.45)
8 0(0).
(2.46)
which, using (2.28) and (2.44), yields
V(x(t))::; V(X(Tl(XO)) + fd(X(Tl(xo))))ee(t Tl(XO)) ::; V(X(Tl(xo)))ee(t Tl(XO)) ::; V(xo)e eTl(XO)ee(t Tl(XO))
= V(xo)eet,
Tl(XO) < t ::; T2(XO),
Xo
E
Recursively repeating the above arguments for t E (Tk (xo), Tk+ 1 (xo) 1, k = 3,4, ... , it follows that
V(x(t)) ::; V(xo)eet,
t ~ 0,
Xo
E
8 0(0).
(2.47)
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
~ ,
q,(t)
()
k,
m~ 0
hl
25
",(t)
Figure 2.6 Twomass system with constraint buffers. Now, it follows from (2.29) and (2.47) that for all t 2': 0,
allx(t)II P ~ V(x(t))
~ V(xo)ec:t ~
,8llxoIIPec:t,
xo E 88(0), (2.48)
and hence,
t 2': 0,
xo
E 88(0),
(2.49)
establishing exponential stability of the zero solution x(t) == 0 to (2.25) and (2.26). Finally, global asymptotic and exponential stability follow from standard arguments. Specifically, let x(O) ERn, and let ,8 ~ V(x(O)). It follows from (2.31) that there exists c > 0 such that V(x) > ,8 for all x E R n such that Ilxll > Co Hence, Ilx(O)1I ~ c, and, since V(x(t)) is strictly decreasing, it follows that Ilx(t)1I < c, t > O. The remainder of the proof is identical to the proof of asymptotic (respectively, exponential) stability. 0 In the proof of Theorem 2.1, we note that assuming strict inequality in (2.27), the inequality (2.41) is obtained provided x(s) =I O. This proviso is necessary since it may be possible to reset the states to the origin, in which case x(s) = 0 for a finite value of s. In this case, for t > s, we have V(x(t)) = V(x(s)) = V(O) = O. This situation does not present a problem, however, since reaching the origin in finite time is a stronger condition than reaching the origin as t t 00. Example 2.4 Consider the twomass, twospring system with buffer constraints of length ~ shown in Figure 2.6. Between collisions the system dynamics, with state variables defined in Figure 2.6, are given by
ml(h (t) + (k1 + k 2)ql (t)  k2q2(t) = 0, m2ii2(t)  k2ql(t)
+ k2q2(t) =0,
ql (0)
= q01, (it (0) = Q01,
t 2': 0, (2.50) Q2(0) = Q02, Q2(0) = Q02. (2.51)
CHAPTER 2
26
At the instant of a collision, the velocities of the masses change according to the law of conservation of linear momentum and the loss of kinetic energy due to a collision so that
ml(i1(tt) + m2(i2(tt) = ml(iI (tk) + m2q2(tk),
(2.52) (2.53)
ql(tt)  q2(tt) = e(ql(tk)  q2(tk)),
where e E [0,1) is the coefficient of restitution. Solving (2.52) and (2.53) for ql(tt) and q2(tt), the resetting dynamics are given by
~ql (tk) = ql (tt) 
ql (tk) =  (1 + e)m2 (ql (tk)  q2(tk)), (2.54)
~q2(tk) =q2(tt) 
q2(tk) = (1 + e)ml (ql(tk)  q2(tk)). ml +m2
ml+m2
(2.55)
b. ql, X2 = b..ql, X3 = b. q2, an dX4 b= . ·q2, we can rewn·t e . D efi mng Xl = (2.50), (2.51), (2.54), and (2.55) in state space form (2.25) and (2.26) with X £. [XI, X2, X3, x4jT,
fc(x)
=
(kl+k2)X2 [ ~::
+ .kJ.. ml X3
1,fd(X) [ (1+e)m2~ X2 =
 m 1+m 20
.!s2 x _.kJ.. x m2 I m2 3
(1+e)ml ml +m2

X4
)
1 ,
(x  x ) 2
4
(2.56)
D = ]R4, and Z = {x
E ]R4 : Xl  X3 = L, X2 > X4}. Note that Xe = 0 is an equilibrium point of the system. To analyze the stability of the zero solution x(t) == 0 consider the Lyapunov function candidate
V(x) =
~ [mlx~ + m2x~ +
Now, it follows that V(x)
klxI + k2(X3  XI)2] ,
= 0, xED, x ¢ Z,
(2.57)
Z,
(2.58)
and
~V(x) = (e 2 1)mlm2(x2  X4)2 < 0 2(ml+m2)
xED.

and hence, by Theorem 2.1 the zero solution x(t) (2.54), and (2.55) is Lyapunov stable.
,
x
E
== 0 to (2.50), (2.51), ~
To examine the stability of linear l statedependent impulsive systems set fc(x) = Acx and fd(X) = (Ad  In)x in Theorem 2.1. Considering the quadratic Lyapunov function candidate V (x) = X T Px, lImpulsive dynamical systems with fc(x) = Ax and fd(X) = (Ad  I)x are not linear. However, this minor abuse in terminology provides a natural way of
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
27
where P > 0, it follows from Theorem 2.1 that the conditions
xT(A; P + PAc)x < 0, xT(AJPAd P)x:::;O,
x x
rf. Z,
(2.59)
Z,
(2.60)
E
establish asymptotic stability for linear statedependent impulsive systems. These conditions are implied by P > 0, A~ P + PAc < 0, and AJ PAd  P :::; 0, which can be solved using a Linear Matrix Inequality (LMI) feasibility problem [27]. 2.4 An Invariance Principle for StateDependent Impulsive Dynamical Systems
In this section, we develop an invariance principle for statedependent impulsive dynamical systems. The following key assumption and several definitions are needed for the statement of the next fundamental result on positive limit sets for impulsive dynamical systems. Recall that a statedependent impulsive dynamical system is time invariant, and hence, s(t + r, r, xo) = s(t, 0, xo) for all Xo E V, t, r E [0,(0). For simplicity of exposition, in the remainder of this section we denote the trajectory s(t, 0, xo) by s(t, xo). Furthermore, let
Teo ~ [0, (0)\{rl(xo),r2(xo), . .. }.
Assumption 2.1 Consider the impulsive dynamical system 9 given by {2.25} and {2.26}, and let s(t, xo), t 2:: 0, denote the solution to {2.25} and {2.26} with initial condition Xo. Then, for every Xo E V and for every c > and t E Teo, there exists 8(c, xo, t) > such that if Ilxo  yll < 8(c, xo, t), Y E V, then Ils(t, xo)  s(t, y)11 < c.
°
°
Assumption 2.1 is a generalization of the standard continuous dependence property for dynamical systems with continuous flows to dynamical systems with leftcontinuous flows. Specifically, by letting Teo = Txo = [0,(0), where Txo denotes the closure of the set Teo, the quasicontinuous dependence property (i.e., Assumption 2.1) specializes to the classical continuous dependence of solutions of a given dynamical system with respect to the system's initial conditions Xo E V [162]. If, in addition, Xo = 0, s(t,O) = 0, t 2:: 0, and 8(c, 0, t) can be chosen independent of t, then continuous dependence implies the classical Lyapunov stability of the zero trajectory differentiating between impulsive dynamical systems with nonlinear vector fields versus impulsive dynamical systems with linear vector fields, and considerably simplifies the presentation.
28
CHAPTER 2
s(t,O) = 0, t 2: O. Hence, Lyapunov stability of motion can be interpreted as continuous dependence of solutions uniformly in t for all t 2: O. Conversely, continuous dependence of solutions can be interpreted as Lyapunov stability of motion for every fixed time t [162]. Analogously, Lyapunov stability of motion of an impulsive dynamical system as defined in [93] can be interpreted as quasicontinuous dependence of solutions (i.e., Assumption 2.1) uniformly in t for all
t
E
Tx o'
Recall that for x E V, the map s (', x) : ~ ~ V defines the solution curve or trajectory of (2.25) and (2.26) through the point x in V. Identifying s(·,x) with its graph, the trajectory or orbit of a point Xo E V is defined as the motion along the curve
Oxo ~ {x E V: x = s(t, xo), t E ~}.
(2.61)
For t 2: 0, we define the positive orbit through the point Xo E V as the motion along the curve
0; ~ {x
E V: x
= s(t, xo),
t
2: O}.
(2.62)
Similarly, the negative orbit through the point Xo E V is defined as
0;0 ~ {x
E V: x
= s(t, xo),
t ~
O}.
(2.63)
Hence, the orbit Ox of a point x E V is given by 0i U 0;; = {s(t, x) : t2:0}U{s(t,x): t~O}.
Definition 2.2 A point p E V is a positive limit point of the trajectory s(·, x) of {2.25} and {2.26} if there exists a monotonic sequence {tn}~=o of positive numbers, with tn ~ 00 as n ~ 00, such that s(tn,x) ~ p as n ~ 00. A point q E V is a negative limit point of the trajectory s(·,x) of {2.25} and {2.26} if there exists a monotonic sequence {tn}~=o of negative numbers, with tn ~ 00 as n ~ 00, such that s(tn, x) ~ q as n ~ 00. The set of all positive limit points of s(t, x), t 2: 0, is the positive limit set w(x) of s(·, x) of {2.25} and {2.26}. The set of all negative limit points of s(t,x), t ~ 0, is the negative limit set a(x) of s(·, x) of {2.25} and {2.26}. In the literature, the positive limit set is often referred to as the wlimit set while the negative limit set is referred to as the alimit set. Note that p E V is a positive limit point of the trajectory s(t, xo), t 2: 0, if and only if there exists a monotonic sequence {tn}~=o C Tx o' with tn ~ 00 as n ~ 00, such that s(tn, xo) ~ p as n ~ 00. To see this, let p E w(xo) and recall that Txo is a dense subset of the
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
29
semiinfinite interval [0,(0). In this case, it follows that there exists an unbounded sequence {tn}~=o, with tn  00 as n  00, such that limn~oo s(tn, xo) = p. Hence, for every E > 0, there exists n > such that Ils(tn, xo)  pil < E/2. Furthermore, since s(·, xo) is leftcontinuous and 'La is a dense subset of [0, (0), there exists in E'La, in ~ tn, such that IIs(in,xo)s(tn,xo)II < E/2, and hence, IIs(in'xo)pil ~ IIs(tn,xo)pll+IIs(tn,xO)s(tn,xo)II < E. Using this procedure, with E = 1,1/2,1/3, ... , we can construct an unbounded sequence {tk}k=l c 'La such that limk~oo S(tk' xo) = p. Hence, p E w(xo) if and only if there exists a monotonic sequence {tn}~=o C 'La' with tn  00 as n  00, such that s(tn, xo)  pas n  00. For the next definition, St(x) denotes the flow s(t,·) : V _JRn of (2.25) and (2.26) for a given t E R
°
Definition 2.3 A set MeV ~ JRn is a positively invariant set with respect to the nonlinear dynamical system (2.25) and (2.26) if st(M) ~ M for all t 2: 0, where st(M) ~ {St(x): x EM}. A set MeV ~ JRn is a negatively invariant set with respect to the nonlinear dynamical system (2.25) and (2.26) if st(M) ~ M for all t ~ 0. A set M ~ V is an invariant set with respect to the dynamical system (2.25) and (2.26) if st(M) = M for all t E JR.
°
In the case where t 2: in (2.25) and (2.26), note that a set M ~ V is a negatively invariant set with respect to the nonlinear dynamical system (2.25) and (2.26) if, for every y E M and every t 2: 0, there exists x E M such that s(t, x) = y and S(7, x) EM for all 7 E [0, tJ. Hence, if M is negatively invariant, then M ~ st(M) for all t 2: 0; the converse, however, is not generally true. Furthermore, a set MeV is an invariant set with respect to (2.25) and (2.26) (defined over t 2: 0) if st(M) = M for all t 2: 0. Note that a set M is invariant if and only if M is positively and negatively invariant.
°
Definition 2.4 The trajectory s(·, x) of (2.25) and (2.26) is bounded if there exists 'Y > such that II s (t, x)" < 'Y, t E R Next, we state and prove a fundamental result on positive limit sets for impulsive dynamical systems. This result generalizes the classical results on positive limit sets to systems with leftcontinuous flows. An analogous result holds for negative limit sets and is left as an exercise for the reader. Furthermore, we use the notation x(t)  M ~ V as t  00 to denote that x(t) approaches M, that is, for each E > 0, there exists T > such that dist(x(t), M) < E for all t > T.
°
30
CHAPTER 2
Theorem 2.2 Consider the impulsive dynamical system 9 given by (2.25) and (2.26), assume Assumption 2.1 holds, and suppose that for Xo E 'D the trajectory s(t, xo) of 9 is bounded for all t ~ 0. Then the positive limit set w(xo) of s(t, xo), t ~ 0, is a nonempty, compact
invariant set. Furthermore, s(t, xo)
+
w(xo) as t
+ 00.
Proof. Let s(t, xo), t ~ 0, denote the solution to 9 with initial condition Xo E 'D. Since s(t, xo) is bounded for all t ~ 0, it follows from the BolzanoWeierstrass theorem [146] that every sequence in the positive orbit 0;;0 ~ {s(t,xo): t E [O,oo)} has at least one accumulation point Y E 'D as t + 00, and hence, w(xo) is nonempty. Furthermore, since s(t, xo), t ~ 0, is bounded it follows that w(xo) is bounded. To show that w(xo) is closed let {Yi}~O be a sequence contained in w(xo) such that limi+oo Yi = Y E 'D. Now, since Yi + Y as i + 00 it follows that for every c > 0, there exists i such that Ily  Yill < c/2. Next, since Yi E w(xo) it follows that for every T > 0, there exists t ~ T such that IIs(t, xo)  Yill < c/2. Hence, it follows that for every c > and T > 0, there exists t ~ T such that IIs(t, xo)  yll :S IIs(t, xo)  Yill + IIYi  yll < c, which implies that Y E w(xo), and hence, w(xo) is closed. Thus, since w(xo) is closed and bounded, w(xo) is compact. Next, to show positive invariance of w(xo) let Y E w(xo) so that there exists an increasing unbounded sequence {t n } ~=o C 'Lo such that s(tn, xo) + Y as n + 00. Now, it follows from Assumption 2.1 that for every c > and t E Ty, there exists 8(c, y, t) > such that IIyzll < 8(c, y, t), z E 'D, implies IIs(t, y)s(t, z)II < c or, equivalently, for every sequence {yd~l converging to Y and t E Ty, limi+oo s(t, Yi) = s(t, y). Now, since by assumption there exists a unique solution to g, it follows that the semigroup property S(T,S(t,Xo)) = S(t+T,XO) for all Xo E 'D and t, T E [0,(0) holds. Furthermore, since s(tn, xo) + Y as n + 00, it follows from the semigroup property that s(t, y) = s(t, limn+oo s(tn, xo)) = lim n+ oo s(t + tn, xo) E w(xo) for all t E Ty. Hence, s(t, y) E w(xo) for all t E Ty. Next, let t E [0,00 )\Ty and note that, since Ty is dense in [0,(0), there exists a sequence {Tn}~=o such that Tn :S t, Tn E Ty, and lim n+ oo Tn = t. Now, since s(·, y) is leftcontinuous it follows that lim n+ oo S(Tn, y) = s(t, y). Finally, since w(xo) is closed and S(Tn, y) E w(xo), n = 1,2, ... , it follows that s(t,y) = limn+oos(Tn,Y) E w(xo). Hence, St(w(xo)) ~ w(xo), t ~ 0, establishing positive invariance of w(xo). Now, to show invariance of w(xo) let Y E w(xo) so that there exists an increasing unbounded sequence {tn}~o such that s(tn, xo) + Y as n + 00. Next, let t E 'Lo and note that there exists N E Z+
°
°
°
31
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
such that tn > t, n ~ N. Hence, it follows from the semigroup property that s(t, s(tn  t, xo)) = s(tn, xo)  t y as n  t 00. Now, it follows from the BolzanoWeierstass theorem [146] that there exists a subsequence znk of the sequence Zn = s(tn  t, xo), n = N, N + 1, ... , such that znk  t Z E [) and, by definition, Z E w(xo). Next, it follows from Assumption 2.1 that limk>oo s(t, znk) = s(t, limk>oo znk)' and hence, y = s(t, z), which implies that w(xo) ~ St(w(xo)), t E Tx o. Next, let t E [0, oo)\Txo' let i E Txo be such that i> t, and consider y E w(xo). Now, there exists Z E w(xo) such that y = s(i, z), and it follows from the positive invariance of w(xo) that Z = s(i  t, z) E w(xo). Furthermore, it follows from the semigroup property of 9 (Le., S(T,S(t,Xo)) = s(t + T,XO) for all Xo E [) and t,T E [0,00)) that s(t, z) = s(t, s(i  t, z)) = s(i, z) = y, which implies that for all t E [0,00 )\Txo and for every y E w(xo), there exists Z E w(xo) such that y = s(t, z). Hence, w(xo) ~ St(w(xo)), t ~ 0. Now, using positive invariance of w(xo) it follows that St(w(xo)) = w(xo), t ~ 0, establishing invariance of the positive limit set w(xo). Finally, to show s(t, xo)  t w(xo) as t  t 00, suppose, ad absurdum, s(t, xo) f+ w(xo) as t  t 00. In this case, there exists an E > and a sequence {tn}~=o, with tn  t 00 as n  t 00, such that
°
inf
PEW(xo)
Ils(tn' xo)  pil
~
E,
n ~ 0.
However, since s(t, xo), t ~ 0, is bounded, the bounded sequence {s(t n , xo)}~o contains a convergent subsequence {s(t~, xo)}~o such that s(t~, xo)  t p* E w(xo) as n  t 00, which contradicts the original supposition. Hence, s(t, xo)  t w(xo) as t  t 00. 0 Note that the compactness of the positive limit set w(xo) depends only on the boundedness of the trajectory s(t, xo), t ~ 0, whereas leftcontinuity and Assumption 2.1 are key in proving invariance of the positive limit set w(xo). In classical dynamical systems, where the trajectory s(·,·) is assumed to be continuous in both its arguments, both the leftcontinuity and the quasicontinuous dependence properties are trivially satisfied. Finally, we note that unlike dynamical systems with continuous flows, the positive limit set of an impulsive dynamical system may not be connected.
Example 2.5 To demonstrate the importance of the quasicontinuous dependence property for the invariance of positive limit sets, consider the statedependent impulsive dynamical system
x(t) = x(t),
x(t) # 0,
(2.64)
32
CHAPTER 2
~x(t)
= 1, x(t) = 0,
(2.65)
where t 2': 0, x(t) E JR, and x(o) = Xo. In this case, the trajectory s(·,·) is given by s(O, xo) = Xo, Xo E JR, and for all t > 0,
(t
) _ {etxo, Xo
s ,Xo 
t, e
°
# 0,
Xo 
,
(2.66)
which shows that for every Xo E JR, the trajectory set, xo) is leftcontinuous in t and approaches the positive limit set containing only the origin. However, note that the dynamical system does not satisfy the quasicontinuous dependence property and the origin is not an invariant set. L
2.5 Necessary and Sufficient Conditions for QuasiContinuous Dependence
In this section, we develop necessary and sufficient conditions for quasicontinuous dependence of solutions for statedependent impulsive dynamical systems. The following result provides sufficient conditions that guarantee that the impulsive dynamical system 9 given by (2.25) and (2.26) satisfies Assumption 2.1. For this result, the following definition of stability with respect to a compact positively invariant set is needed.
Definition 2.5 Let MeV be a compact positively invariant set for the nonlinear impulsive dynamical system {2.25} and {2.26}. M is Lyapunov stable if for every open neighborhood 01 ~ V of M, there exists an open neighborhood 02 ~ Ch of M such that x(t) E 0 1 for all Xo E 02 and t 2': 0. Proposition 2.1 Consider the nonlinear impulsive dynamical system 9 given by {2.25} and {2.26}. Assume Ai and A2 hold, and assume that either of the following statements holds: i} For all Xo E V, limk>oo Tk(XO) t
rt
° :S °:S
Tl (xo)
oo Tk(XO) t 00, and Z\Z is a Lyapunov stable, compact positively invariant set with respect to the dynamical system g.
Then 9 satisfies Assumption 2.1.
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
33
Proof. To show that i) implies Assumption 2.1, assume that for all z, 0 < T1(XO) < 00, T10 is continuous, and limk>oo Tk(XO) t 00. In this case, it follows from the definition of Tk(XO) that for every Xo E V and k E {1, 2, ... , },
Xo rt.
Tk(XO) = Tkj(XO)
+ Tj[S( Tkj(XO), xo) + fd(S( Tkj(XO), xo))], j = 1, ... , k,
(2.67)
where TO(Xo) ~ O. Since feO is such that the solutions to (2.25) are continuous with respect to the initial conditions between resetting events, it follows that for every k = 0,1, ... , and t E (Tk(XO), Tk+1(XO)], ¢(".) is continuous in both its arguments. Specifically, note that since T1 (xo) is continuous it follows that "71 (xo) ~ S(T1 (xo), xo) = ¢(T1(XO),Xo) is continuous on V. Hence, it follows from (2.67) and the continuity of fdO that T2(XO) = T1(XO)+Tds(T1(XO), xo)+ fd(S(T1 (xo), xo))] is also continuous which implies that "72(XO) ~ S(T2(XO), xo) = ¢(T2(XO)  T1(XO),"71(XO) + fd("71(XO)) is continuous on V. By recursively repeating this procedure for k = 3,4, ... , it follows that Tk(XO) and "7k(XO) ~ S(Tk(XO), xo) are continuous on V. Next, let t E 'Ix o be such that Tk(XO) < t < Tk+1 (xo). Now, noting that s(t, xo) = ¢(tTk(XO), S(Tk(XO), xo)+ fd(S(Tk(XO), xo))), it follows from the continuity of fdO and TkO that s(t, xo) is a continuous function of Xo for all t E 'Ix o such that Tk(XO) < t < Tk+1(XO) for some k. Hence, since limk>oo Tk(XO) t 00, g satisfies Assumption 2.1. Next, consider the case in which Xo E Z. Note that in this case T1(XO) = 0 and T2(XO) = T1(XO + fd(XO)). Since Xo E Z, it follows that Xo + fd(XO) rt. z, and since T1(') is continuous on V and fd(') is continuous on Z, it follows that T2() is continuous on Z. Now, Assumption 2.1 for all Xo E Z can be shown as above. Alternatively, if ii) is satisfied then as in the proof of i) it can be shown that for all Xo E V, Xo rt. Z\Z, s(t, xo) is a continuous function of Xo for all t E 'Ix o ' Next, if Z\Z is a Lyapunov stable, compact positively invariant set with respect to g, then for all Xo E Z\Z, 'Ix o = [0,00). Now, the continuity of s(t,xo) for all t E [0,00) follows from the Lyapunov stability of Z\Z, and hence, g satisfies Assumption 2.1. 0 If, for every Xo E V, the solution s(t, xo) to (2.25) and (2.26) is a Zeno solution, that is, limk>oo Tk(XO) t T(XO) < 00, and the resetting sequence {Tk (xo)} ~o is uniformly convergent in Xo, then condition ii) of Proposition 2.1 implies that g satisfies Assumption 2.1. To see this, note that since {Tk ()} ~1 is a uniformly convergent sequence of continuous functions, it follows that T(') is a continuous
34
CHAPTER 2
function. Now, noting that for all t > T(Xo), t E Tx o ' s(t, xo) = 'lj;(tT(XO), S(T+(XO), xo)), it follows that s(t, xo) is a continuous function of Xo for all t E Tx o' which proves that 9 satisfies Assumption 2.l. Proposition 2.1 requires that the first resetting time T10 be continuous at Xo ED. The following result provides sufficient conditions for establishing the continuity of T1 (.) at Xo ED.
Proposition 2.2 Consider the nonlinear impulsive dynamical system
9 given by {2.25} and {2.26}. Assume there exists a continuously differentiable function X : D  t lR such that the resetting set of 9 is given by Z = {x ED: X(x) = o} and X'(x)fc(x) =I 0, x E Z. Then T10 is continuous at Xo ED, where < T1 (xo) < 00.
° °< T1(XO) < s(t,xo) = 'lj;(t,xo), t
Proof. Let Xo rJ. Z be such that 00. It follows E [0,T1(XO)], from the definition of T10 that X(s(t,xo)) =I 0, t E (0,T1(XO)), and X(S(T1(XO),Xo)) = 0. Without loss of generality, let X(s(t,xo)) > 0, t E (0,T1(XO)). Since x £: 'lj;(T1(XO),Xo) E Z, it follows by assumption that X'(x)fc(x) =I 0, and hence, there exists 0 > such that X('lj;(t,x)) > 0, t E [0,0), and X('lj;(t,x)) < 0, t E (0,0]. (This fact can be easily shown by expanding X ('lj;( t, x)) via a Taylor series expansion about x and using the fact that X'(x)Jc(i:) =I 0.) Hence, X('lj;(t,xo)) > 0, t E [t1,T1(XO)), and X('lj;(t,xo)) < 0, t E (T1(XO),t2], where t1 £: T1(XO)  0 and t2 £: T1(XO) + O. Next, let c £: min{X('lj;(t1' xo)), X ('lj;(t2 , xo))}. Now, since XO and 'lj;(.,,) are jointly continuous, it follows that there exists 0 > Osuch that
°
sup IX('lj;(t, x))  X ('lj;(t, xo))1 < c, O~t~i2
x
E
B8(XO),
(2.68)
°
which implies that X('lj;(tb x)) > and X ('lj;(t2 , x)) < 0, x E B8(XO)' Hence, it follows that t1 < T1(X) < t2, X E B8(XO)' The continuity of T1 (.) at Xo now follows immediately by noting that 0 can be chosen arbitrarily small. 0 The first assumption in Proposition 2.2 implies that the resetting set Z is an embedded submanifold [80], while the second assumption assures that the solution of 9 is not tangent to the resetting set Z. The next result provides a partial converse to Proposition 2.1. For this result, we introduce the following assumption in place of Al and
A2.
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
A3.
35
feO is locally Lipschitz continuous on fd(X) fe(x)
=1=
=
V, Z is closed, and 0, x E Z\8Z. If x E 8Z such that fd(X) = 0, then 0. If x E Z such that fd(X) =1= 0, then x + fd(X) (j. Z.
The following definitions are needed for the statement of the next result.
Definition 2.6 Let V ~ JRn, f : V t JR, and x E V. f is lowersemicontinuous at x E V if for every sequence {xn} ~=o C V such that limn>oo Xn = x, f(x) ~ lim infn>oo f(xn). Note that a function f : V t JR is lowersemicontinuous at x E V if and only if for each a E JR the set {x E V : f(x) > a} is open. Equivalently, a bounded function f : V t JR is lowersemicontinuous at x E V if and only if for each c > 0, there exists 0 > such that Ilx  yll < 0, y E V, implies f(x)  f(y) ~ c.
°
Definition 2.7 Let V ~ JR n , f : V t JR, and x E V. f is uppersemicontinuous at x E V if for every sequence {x n }~=o C V such that limn>oo Xn = x, f(x) ~ lim sUPn>oo f(x n ), or, equivalently, for each a E JR the set {x E V: f(x) < a} is open. As in the case of continuous functions, a function f is said to be lower (respectively, upper) semicontinuous on V if f is lower (respectively, upper) semicontinuous at every point x E V. Clearly, if f is both lower and uppersemicontinuous, then f is continuous.
Proposition 2.3 Consider the nonlinear impulsive dynamical system
9 given by (2.25) and (2.26), and assume A3 holds. If 9 satisfies Assumption 2.1, then 71 (.) is lowersemicontinuous at every x (j. Z. Furthermore, for every x (j. Z such that 71(X) < 00, 710 is continuous at x. Finally, for every x (j. Z such that 71(X) = 00, 71(X n ) t 00 for every sequence {xn}~=1 such that Xn t X.
Proof. Assume 9 satisfies Assumption 2.1. Let x (j. Z and let b. {xn}~=l (j. Z be such that Xn t x and 71(Xl) ~ 71(X2) ~ ... ~ L = limn>oo71(Xn). First, assume 71(Xl) < 00 so that L , 71(X2), ... < 00. Since feO is locally Lipschitz continuous on V it follows that 'lj;(., .) is jointly continuous, and hence, it follows that 'lj;( 71 (x n ), Xn) t 'lj;(L, x) as n t 00. Next, since Z is closed and 'lj;(71 (x n ), xn) E Z for every n = 1, 2, ... , it follows that 'lj;( L , x) E Z which implies that L ~ 71(X) = inf{t E JR+: 'lj;(t,x) E Z}, establishing the lower semicontinuity of 710 at x. Alternatively, if L = 00 so that
36
CHAPTER 2
7t(X1) = T1(X2) = ... = 00, lower semicontinuity follows trivially since T1(X) ~ L = 00. Next, note that since feO is locally Lipschitz continuous on V it follows that 'ljJ(t, x), t ~ 0, cannot converge to any equilibrium in a finite time, and hence, fe('ljJ(T1(X),X)) =1= 0, which implies that fd('ljJ(T1(X),X)) =1= 0. Let {Xn}~1 ¢ Z be such that Xn t x as n t 00 and T1(X1) ~ T1(X2) ~ ... ~ T+ ~ lim n ...... oo T1(X n ). Suppose, ad absurdum, T+ > T1(X), let c > be such that T1(X) < T+  c < T2(X), and let M E Z+ be such that T+  c < T1(X n ), n> M. Now, since g satisfies Assumption 2.1, it follows that 8(T+  c,xn ) t S(T+  c,x), and for every n ~ M, 8(T+  c,xn ) = 'ljJ(T+  c,x n ). Furthermore, limn ...... oo S(T+  c, xn) = limn ...... oo 'ljJ(T+  c, xn) = 'ljJ(T+  c, x). Hence,
°
'ljJ (T+  c  T1 (x), 'ljJ (T1 (x), x) + f d ( 'ljJ (T1 (x), x) ) =8(T+c,X) = lim 8(T+  c,xn ) n ...... oo
= 'ljJ(T+  c,x) = 'ljJ(T+  c  T1(X), 'ljJ( T1 (x), x)).
(2.69)
Now, since feO is locally Lipschitz continuous on V it follows that the solution 'ljJ(t,x), t E R, is unique both forward and backward in time, and hence, it follows that 'ljJ(T1(X),X) = 'ljJ(T1(X),X) + fd('ljJ(T1(X),X)), or, equivalently, fd('ljJ(T1(X),X)) = 0, which is a contradiction. Hence, T+ ~ T1(X), and thus, T10 is uppersemicontinuous at x. Hence, T10 is continuous at x. Finally, let x ¢ Z be such that T1(X) = 00 and let {xn}~=1 E Z be such that Xn t x. Suppose, ad absurdum, that {T1 (xn)}~=1 has a bounded subsequence {T1 (x nj )}~1. Let T ~ limj ...... oo T1 (xnj) < 00. Now, since 'ljJ(., .) is jointly continuous it follows that limj ...... oo 'ljJ( T1 (x nj ), x nj ) = 'ljJ(T, x). Next, since Z is closed and 'ljJ(T1(X nj ),Xnj ) E Z, j = 1,2, ... , it follows that 'ljJ(T, x) E Z which implies that T1(X) = inf{t E R+: 'ljJ(t,x) ¢ Z} ~ T < 00, which is a contradiction. Hence, limn ...... oo T1(X n ) = 00. 0 The following result shows that all convergent Zeno solutions to (2.25) and (2.26) converge to Z\Z if Al and A2 hold, while all convergent Zeno solutions converge to an equilibrium point if A3 holds. Proposition 2.4 Consider the nonlinear impulsive dynamical sys
tem g given by {2.25} and {2.26}.
If the trajectory 8(t, xo), t
~
0, to {2.25} and {2.26} is convergent, bounded, and Zeno, that is, there exists T(XO) < 00 such that Tk(XO) t T(XO) as k t 00 and
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
limk+oo s(7k(XO), Xo)
hold:
37
= s(7(XO), xo), then the following statements
i) If At and A2 hold, and 72(') is continuous on Z, then S(7(XO), xo) EZ\Z. ii) If A3 holds, then S(7(Xo),Xo) is an equilibrium point.
Proof. If the trajectory s(t, xo), t ~ 0, is Zeno, then there exists 7(Xo) < 00 such that 7k(XO) + 7(XO) as k + 00 and, since 71 (xo) < 72(XO) < ... < 7(XO), it follows that 71(XO) < 00. Next, note that there exists Y1 E Z such that s (71 (xo), xo) = Y1, and hence, it follows that 72(XO) = 71 (Xo) +71 (Y1 + fd(Y1)) = 71(XO)+72(Y1). By recursively repeating this procedure for k = 3,4, ... , it follows that k1 7k(XO) = 71(XO) + 72(Yi),
L
i=l
where Yi ~ S(7i(XO),Xo), i = 1,2, .... Now, since 7(XO) = limk+oo7k (xo) it follows that 7(XO) = 71 (Xo) + L:~1 72(Yk). Hence, it follows that 72(Yk) + as k + 00. Now, if the trajectory s(t, xo), t ~ 0, is bounded, then the sequence {Yk}k=O is also bounded and it follows from the BolzanoWeierstrass theorem [146J that there exists a convergent subsequence {YkJ~l such that limi+oo Yk i = Y E Z. Hence, since s(·, xo) is leftcontinuous, it follows that Y = limi+oo Yk i = limi+oo s( tki (xo), xo) = s(limi+oo tki (xo), xo) = s(7(XO), xo). i) Assume Al and A2 hold, and assume 72() is continuous on Z. Next, ad absurdum, suppose Y E Z. Since 720 is continuous on Z it follows that 72(Y) = 72 (limi+oo Yk;) = limi+oo 72 (YkJ = 0, which contradicts the fact that 72(X) > 0, x E Z. Thus, Y E Z\Z or, equivalently, S(7(XO), xo) E Z\Z. ii) Finally, assume A3 holds. Furthermore, note that Yk = 'lj;( 72 (Yk1), Yk1 + fd(Yk1)), k = 2,3, ... , and since 'lj;(.,,) is jointly continuous and 72(Yk) + as k + 00, it follows that
°
°
y= lim Yk k+oo = 'lj;( lim 72(Yk), lim (Yk + fd(Yk))) k+oo k+oo = 'lj;(0, Y + fd{Y)) =Y + fd{Y), which implies that fd{Y) = 0. Now, since Z is closed it follows that Y E Z, and since fd(Y) = 0, it follows from A3 that fc(Y) = 0, which proves the result.
0
38
CHAPTER 2
2.6 Invariant Set Theorems for StateDependent Impulsive Dynamical Systems
In this section, we generalize the KrasovskiiLaSalle invariance principle to statedependent impulsive dynamical systems. This result characterizes impulsive dynamical system limit sets in terms of continuously differentiable functions. In particular, we show that the system trajectories converge to an invariant set contained in a union of level surfaces characterized by the continuoustime dynamics and the resetting system dynamics. Henceforth, we assume that feO, fdO, and Z are such that the dynamical system g given by (2.25) and (2.26) satisfies Assumption 2.1. For the next result V1(y) denotes the ,level set of V(·), that is, V1(y) ~ {x E Ve : V(x) = ,}, where, E JR, Ve ~ V, and V : Ve t JR is a continuously differentiable function, and let M')' denote the largest invariant set (with respect to g) contained in V1(y). Theorem 2.3 Consider the impulsive dynamical system g given by {2.25} and {2.26}, assume Ve C V is a compact positively invariant set with respect to {2.25} and {2.26}, and assume that there exists a continuously differentiable function V : Ve t JR such that V'(x)fe(x):s 0, x EVe, x V(x + fd(X)):S V(x), x EVe,
tt z, x E Z.
(2.70) (2.71)
Let R ~ {x E Dc : x tt z, V'(x)fe(x) = O} U {x EVe: x E Z, V(x + fd(X)) = V(x)} and let M denote the largest invariant set contained in R. If Xo EVe, then x(t) t M as t t 00.
Proof. Using identical arguments as in the proof of Theorem 2.1 it follows that for all t E (rk (xo), rk+1 (xo)], V(x(t))  V(x(O))
=
lot V'(x(r))fe(x(r))dr
k
+ 2)V(x(ri(xo)) + fd(x(ri(xo))))
 V(x(ri(xo)))].
i=l
Hence, it follows from (2.70) and (2.71) that V(x(t)) :s V(x(O)), t ~ O. Using a similar argument it follows that V(x(t)) :s V(x(r)), t ~ r, which implies that V(x(t)) is a nonincreasing function of time. Since V(·) is continuous on a compact set Ve there exists (3 E JR such that V(x) ~ (3, x EVe. Furthermore, since V(x(t)), t ~ 0, is
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
39
nonincreasing, 'Yxo ~ limt>oo V(x(t)), Xo EVe, exists. Now, for all y E w(xo) there exists an increasing unbounded sequence {tn}~=o such that x(tn )  t y as n  t 00, and, since V(·) is continuous, it follows that V(y) = V(limn>oo x(tn )) = limn>oo V(x(t n )) = 'Yxo' Hence, y E Vl(,,(xo) for all y E w(xo), or, equivalently, w(xo) ~ Vl(,,(xo)' Now, since Ve is compact and positively invariant, it follows that x(t), t ~ 0, is bounded for all Xo EVe, and hence, it follows from Theorem 2.2 that w(xo) is a nonempty, compact invariant set. Thus, w(xo) is a subset of the largest invariant set contained in Vl(,,(xo)' that is, w(xo) ~ M 1xo ' Hence, for every Xo EVe, there exists 'Yxo E JR such that w(xo) ~ M 1xo ' where M1xo is the largest invariant set contained in Vl(,,(xo)' which implies that V(x) = 'Yxo' x E w(xo). Now, since M"Yxo is a invariant set, it follows that for all x(o) E M 1xo ' x(t) E
M 1xo ' t ~ 0, and hence, V(x(t)) ~ dV~(t)) = V'(x(t))Je(x(t)) = 0, for all x(t) rf Z, and V(x(t) + Jd(X(t))) = V(x(t)), for all x(t) E Z. Thus, M1xo is contained in M which is the largest invariant set contained in R. Hence, x(t)  t M as t  t 00. 0 Example 2.6 Consider the nonlinear statedependent impulsive dynamical system
X3(t) X4(t) XI(t)  2X4(t) 
Jxr~:)~x~(t) Jxr~:)~x~(t)
[~:~~ll ~: l'
X2(t)
+ 2X3(t) 
X4(0)
[~:m 1 ~~~1~; 1' =[
X4(t)
[~~:m 1
x(t)
X40
x(t) rf Z,
(2.72)
(2.73)
E Z\Z,
X3(t)
e)(x~(t) + l' +
= [ (1 +
~X4(t)
= [
(1
X3(t)) e)(xI(t)  X4(t))
x(t)
E Z, (2.74)
where t ~ 0, XI(t),X2(t),X3(t),X4(t) E JR, e E (0,1), Z = {x E Ve: + x~ = 1, XIX3 + X2X4 < O}, x ~ [Xl X2 x3 X4J T , and
xi
40
CHAPTER 2
xi
xi
D = Dc = {x E ]R4: + x~ 2: 1, XlX4  X2X3 = + xn. First, note that Z = {x E Dc: + x~ = 1, XlX3 + X2X4 ~ O}, and hence, Z\Z = {x E Dc: + x~ = 1, XlX3 + X2X4 = O} which can be shown to be a compact invariant set with respect to the dynamical system 9 given by (2.72)(2.74). Furthermore, note that Dc is an invariant set with respect to the impulsive dynamical system (2.72)(2.74). To x~ and see this, consider the function ¢(x) ~ XlX4  X2X3 note that (p(x) is identically zero along the solutions of (2.72), and ¢(x + ~x)  ¢(x) = 0 for all x E Dc. Next, we use Proposition 2.1 to show that the dynamical system (2.72)(2.74) satisfies Assumption 2.1. To see this, note that it can be shown that
xi
xi
xi 
2(xi+x~)(Vxr+xIl)+(XlX3+X2X4)2 ~ yx 1+x 2
2(XlX3
2 2 1 ,xl +x 2 > ,
+ X2X4), XlX3
xi + x~ = 1, + X2X4 > 0,
(2.75) which shows that 71 (x) is continuous for all x ¢ Z. Furthermore, it follows from (2.75) that 7l(X) t 0 as x t 8Z which implies that 71 (.) is continuous on D. Finally, it can be shown that for all xED, the sequence {7k(Xnr=1 is a uniformly convergent sequence. Now, it follows from ii) of Proposition 2.1 that the dynamical system 9 given by (2.72)(2.74) satisfies Assumption 2.1. Next, (2.72)(2.74) can be written in the form of (2.25) and (2.26) with
and
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
Now, consider the function V: 'Dc
V( X) = ( Xl2 + X22)1/2
t
41
lR given by
+ 12 (XIX32 + X22X4)2 ' Xl +X 2
(2.76)
and note that V'(x)fc(x) = 0 for all X f/. Z, which implies that Z\Z is Lyapunov stable. Furthermore, since e E (0,1) note that V(x + fd(x)) = V(x) if and only if XIX3 + X2X4 = O. Hence, the set {x E Z: V(x + fd(x)) = V(x)} = 0 and the set R = 'Dc\Z. Now, note that the largest invariant set M contained in R = 'Dc \Z is {x E 'Dc: xi + x~ = 1, XIX3 + X2X4 = O}, and hence, it follows from Theorem 2.3 that the solution x(t), t 2': 0, to (2.72)(2.74) approaches the invariant set {x E 'Dc: xi + x~ = 1, XIX3 + X2X4 = O} as t t 00 for all initial conditions contained in 'Dc. Finally, Figure 2.7 shows the phase portrait of the states Xl versus X2 for the initial condition [Xl (0) X2(0) X3(0) X4(0)]T = [200 2jT E 'Dc. Alternatively, this can also be shown using Proposition 2.3. Specifically, it follows from Proposition 2.3 that (Xl (t), x2 (t)) t Z\Z as t t r(xo) and since Z\Z is an invariant set, it follows from Theorem 2.3 that (XI(t),X2(t)) t {x E 'Dc: xi + x~ = 1, XIX3 + X2X4 = O} as t t 00. 6 The following corollaries to Theorem 2.3 present sufficient conditions that guarantee local asymptotic stability of the nonlinear impulsive dynamical system (2.25) and (2.26). For these results, recall that if the zero solution x( t) == 0 to (2.25) and (2.26) is asymptotically stable, then the domain of attraction 'DA ~ 'D of (2.25) and (2.26) is given by 'DA ~ {xo E 'D: if x(to)
= xo, then t>oo lim x(t) = O}.
(2.77)
Corollary 2.1 Consider the nonlinear impulsive dynamical system {2.25} and {2.26}, assume 'Dc C 'D is a compact positively invariant o
set with respect to {2.25} and {2.26} such that 0 E 'Dc, and assume there exists a continuously differentiable function V : 'Dc t lR such that V(O) = 0, V(x) > 0, x =1= 0, and {2.70} and {2.71} are satisfied. Furthermore, assume that the set R ~ {x E 'Dc: x f/. Z, V' (x) fc (x) = O} U {x E'Dc: x E Z, V(x + fd(x)) = V(x)} contains no invariant set other than the set {O}. Then the zero solution x(t) == 0 to {2.25} and {2.26} is asymptotically stable and 'Dc is a subset of the domain of attraction of {2.25} and {2.26}. Proof. Lyapunov stability of the zero solution x(t) == 0 to (2.25) and (2.26) follows from Theorem 2.1. Next, it follows from Theorem
42
CHAPTER 2
1.5
0.5
0.5
1
1.5
_2L~~~~~~~~
2
1.5
1
0.5
0
0.5
xlI}
Figure 2.7 Phase portrait of
Xl
1.5
versus
2
X2.
2.3 that if Xo E 'Dc, then w(xo) ~ M, where M denotes the largest invariant set contained in R, which implies that M = {a}. Hence, x(t) ~ M = {a} as t ~ 00, establishing asymptotic stability of the zero solution x(t) == 0 to (2.25) and (2.26). 0 Setting'D = ]Rn and requiring V(x) ~ 00 as Ilxll ~ 00 in Corollary 2.1, it follows that the zero solution x(t) == 0 to (2.25) and (2.26) is globally asymptotically stable. Similar remarks hold for Corollaries 2.2 and 2.3 below.
Example 2.7 Consider a bouncing ball, with coefficient of restitution e E (0,1), on a horizontal surface under a normalized gravitational field. Modeling the surface collisions as instantaneous, it follows from Newton's equations of motion that the bouncing ball dynamics are characterized by the statedependent impulsive differential equations
[
~~~~~ ] = [ ~~~ ] , (Xl(t),X2(t))
rf. Z,
(2.78)
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
43
where t 2: 0, XI(t),X2(t) E ]R, XI(t) 2: 0, sgn(xI) ~ xI/lxII, Xl i= 0, sgn(O) ~ 0, Z = {(XI,X2) E V: Xl = 0, X2 < O}, and V = {(XI,X2) E ]R2: Xl
2: o}.
First, we use Proposition 2.1 to show that the impulsive dynamical system (2.78) and (2.79) satisfies Assumption 2.1. Note that Z = {(XI,X2) E]R2: Xl = 0, X2 SO}, and hence, Z\Z = {(O,O)} is a compact invariant set with respect to the dynamical system 9 given by (2.78) and (2.79). Next, it can be shown that
X2 TI (XI,X2 ) = {
+ J x§ + 2XI, 2X2,
Xl
Xl
> 0,
= 0, X2 > 0,
(2.80)
°
which shows that TI(XI,X2) is continuous for all (XI,X2) ¢ Z. furthermore, it follows from (2.80) that TI(X) > as X > az which implies that TI() is continuous on V. Finally, it can be shown that for all (XI,X2) E V, the sequence {Tk(Xl,X2)}~1 is a uniformly convergent sequence. Now, it follows from ii) of Proposition 2.1 that the dynamical system 9 given by (2.78) and (2.79) satisfies Assumption 2.1. Next, (2.78) and (2.79) can be written in the form of (2.25) and (2.26) with X ~ [Xl, x2jT, fc(x) = [X2' sgn(xI)]T, and fd(X) = [0, (1 + e)x2jT. Now, consider the function V : ]R2 > ]R given by V(x) = Xl + ~x§ and note that V'(x)fc(x) = for all X ¢ Z, which implies that Z\Z = {(O,O)} is Lyapunov stable. Furthermore, since e E (0,1), note that V(x + fd(X)) = V(x) if and only if X2 = 0. Hence, the set {(Xl, X2) E Z: V(x + fd(X)) = V(x)} = 0 and the set 'R = {(Xl, X2) E ]R2: Xl 2: O} \Z. Now, note that the largest invariant set M contained in 'R = {(Xl, X2) E ]R2: Xl 2: O} \Z is {(O,O)}, and hence, since V(x) is radially unbounded, it follows from Theorem 2.3 that (Xl(t), X2(t)) > (0,0) as t > 00. Alternatively, this can also be shown using Proposition 2.3. Specifically, it follows from Proposition 2.3 that (XI(t),X2(t)) > Z\Z = {(O,O)} as t > T(Xo) and since Z\Z = {(O, O)} is an invariant set it follows from Corollary 2.1 that (Xl(t),X2(t)) > {(O,O)} as t > 00. 6.
°
Corollary 2.2 Consider the nonlinear impulsive dynamical system (2.25) and (2.26), assume Vc c V is a compact positively invariant
set with respect to (2.25) and (2.26) such that
°EVe, and assume o
44
CHAPTER 2
there exists a continuously differentiable function V : 'Dc that V(O) = 0, V(x) > 0, x # 0, V'(x)fc(x) < 0,
x E'Dc ,
x
rt z,
x
# 0,
t
lR. such (2.81)
and {2.71} is satisfied. Then the zero solution x(t) == 0 to {2.25} and {2.26} is asymptotically stable and 'Dc is a subset of the domain of attraction of {2.25} and {2.26}. Proof. It follows from (2.81) that V'(x)fc(x) = 0 for all x E 'Dc \Z if and only if x = O. Hence, R = {O} U {x E 'Dc: x E Z, V(x + fd(x)) = V(x)}, which contains no invariant set other than {O}. Now, the result follows as a direct consequence of Corollary 2.1. 0 Corollary 2.3 Consider the nonlinear impulsive dynamical system {2.25} and {2.26}, assume 'Dc C 'D is a compact positively invariant o
set with respect to {2.25} and {2.26} such that 0 E 'Dc, and assume that for all Xo E 'Dc, Xo # 0, there exists r ~ 0 such that x(r) E Z, where x(t), t ~ 0, denotes the solution to {2.25} and {2.26} with the initial condition Xo. Furthermore, assume there exists a continuously differentiable function V : 'Dc t lR. such that V(O) = 0, V(x) > 0, x #0, V(x
+ fd(x))
 V(x) < 0,
x E 'Dc,
x E Z,
(2.82)
and {2.70} is satisfied. Then the zero solution x(t) == 0 to {2.25} and {2.26} is asymptotically stable and 'Dc is a subset of the domain of attraction of {2.25} and {2.26}.
rt
Proof. It follows from (2.82) that R = {x E'Dc : x z, V'(x)fc(x) = O}. Since, for all Xo E 'Dc, Xo # 0, there exists r ~ 0 such that x( r) E Z, it follows that the largest invariant set contained in R is {O}. Now, the result is as a direct consequence of Corollary 2.1. 0
2.7 Partial Stability of StateDependent Impulsive Dynamical Systems
In many engineering applications, partial stability, that is, stability with respect to part of the system's states, is often necessary. In particular, partial stability arises in the study of electromagnetics [173], inertial navigation systems [155], spacecraft stabilization via
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
45
gimballed gyroscopes and/or flywheels [163], combustion systems [9], vibrations in rotating machinery [108], and biocenology [144], to cite but a few examples. For example, in the field of biocenology involving LotkaVolterra predatorprey models of population dynamics with age structure, if some of the species preyed upon are left alone, then the corresponding population increases without bound while a subset of the prey species remains stable [144, pp. 260269]. The need to consider partial stability in the aforementioned systems arises from the fact that stability notions involve equilibrium coordinates as well as a hyperplane of coordinates that is closed but not compact. Hence, partial stability involves motion lying in a subspace instead of an equilibrium point. Additionally, partial stabilization, that is, closedloop stability with respect to part of the closedloop system's state, also arises in many engineering applications [108,163]. Specifically, in spacecraft stabilization via gimballed gyroscopes asymptotic stability of an equilibrium position of the spacecraft is sought, while requiring Lyapunov stability of the axis of the gyroscope relative to the spacecraft [163]. Alternatively, in the control of rotating machinery with mass imbalance, spin stabilization about a nonprincipal axis of inertia requires motion stabilization with respect to a subspace instead of the origin [108]. Perhaps the most common application where partial stabilization is necessary is adaptive control, wherein asymptotic stability of the closedloop plant states is guaranteed without necessarily achieving parameter error convergence. In this section, we introduce the notion of partial stability for nonlinear statedependent impulsive dynamical systems. Specifically, consider the nonlinear statedependent impulsive dynamical system
XI(t) = fIe(XI(t), X2(t)), X2(t) = he(XI(t), X2(t)), L\xI(t) = fId(XI(t), X2(t)), L\x2(t) = hd(XI(t), X2(t)), where t 2:: 0,
XI(O) = XlO, (XI(t), X2(t)) X2(0) = X20, (XI(t), X2(t)) (XI(t), X2(t)) E Z, (XI(t), X2(t)) E Z,
rt Z, (2.83) rt Z, (2.84) (2.85) (2.86)
'D ~ lRn1 , 'D is an open set such that 0 E 'D, X2 E lRn2 , L\xI(t) = XI(t+)  XI(t), L\x2(t) = X2(t+)  X2(t), fIe : 'D X lRn2 t lRn1 is such that for every X2 E lR n2 , fIe(O, X2) = 0 and fIe (, X2) is locally Lipschitz in XI, he : 'D X lRn2 t lRn2 is such that for every Xl E 'D, he(XI,') is locally Lipschitz continuous on 'D in X2, fId : 'D X lRn2 t lRn1 is continuous and fId (0, X2) = 0 for all X2 E lRn2 , hd : 'D x lR n2 t lRn2 is continuous, and Z c 'D x lRn2. For a particular trajectory x(t) = (XI(t),X2(t)), t 2:: 0, we let tk (= Xl E
46
CHAPTER 2
Tk(XlO, X20)) denote the kth instant of time at which x(t) intersects Z. Furthermore, we make the following assumptions:
Al'. If x(t) E Z\Z, then there exists c > 0 such that, for all 0 c, x (t + 8) 0, fe : V t IRn is locally Lipschitz continuous on V, fe(O) = 0, and fd : V t IRn is such that fdO is continuous and fd(O) = O. In this case, since the vector fields of (2.139) and (2.140) are timeindependent, the ndimensional phase
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
61
portrait of (2.139) and (2.140) is not affected by the periodic resettings of the time variable. That is, when t = T, then t is reset to zero. The timedependent impulsive dynamical system (2.139) and (2.140) with periodic resettings can hence be equivalently characterized as a statedependent impulsive dynamical system with an additional state representing time, that is, i(t) = ic(x(t)), ~x(t) = id(X(t)),
x(to) = xo, x(t) E Z,
x(t)
f Z,
(2.141) (2.142)
where (t) _ [ x(t) ] x T(t) ,
(2.143)
(2.144)
Z=
{x = [xT, TF E V x [0, TJ : T = T}, Xo E V, and TO = to E [0, T). Note that the solution x(t), t ~ to, to (2.139) and (2.140) is equivalently characterized by the partial solution x(t), t ~ to, to (2.141) and (2.142).
Theorem 2.7 Consider the timedependent impulsive dynamical system (2.139) and (2.140) with tk = kT, k = 1,2, ... , and T > 0. Assume that Vc C V is a compact positively invariant set with respect to (2.139) and (2.140). Furthermore, assume there exists a continuously differentiable function V : Vc + lR such that V(O) = 0, V(x) > 0, x # 0, x EVe, and (2.145) (2.146)
V'(x)fc(x) ~ 0, V(x + fd(x))  V(x) ~ 0,
Let R'Y ~ {x E Vc : V(x) = ')'}, where ')' > 0, and let M'Y denote the largest invariant set contained in R'Y' If for each')' > 0, M'Y contains no system trajectory, then the zero solution x(t) == to (2.139) and (2.140) is uniformly asymptotically stable.
°
Proof. Uniform Lyapunov stability follows from ii) of Theorem 2.6 since V(x), x EVe, is a positivedefinite function on a compact set Vc. To show asymptotic stability, consider the statedependent representation (2.141) and (2.142) of the timedependent impulsive dynamical system (2.139) and (2.140). Note that for the impulsive dynamical system given by (2.141) and (2.142) the state variable T(t), t ~ to,
62
CHAPTER 2
is defined over the interval [0, T], and hence, the set Dc x [0, T] is a compact positively invariant set with respect to (2.141) and (2.142). Furthermore, defining X(x) ~ T  T, x E Dc x [0, T], it follows that i = E Dc x [0, T] : X(x) = O}. Note that X'(x)ic(x) = 1 =1= 0, x E Z, and hence, it follows from Proposition 2.2 that Tl() is con< Tl(XO) < 00. (Specifically, tinuous at Xo E Dc x [0, T], where Tl(XO) = T  to which is continuous at Xo E Dc x [0, T].) Next, since (2.141) and (2.142) possesses an infinite number ofresettings, it follows from Proposition 2.1 that (2.141) and (2.142) satisfy Assumption 2.1. Now, it follows from Theorem 2.2 that the positive limit set w(xo) of (2.141) and (2.142) with Xo E Dc x [0, T] is a nonempty, compact invariant set, which further implies that the positive limit set w(xo) of (2.139) and (2.140) with Xo E Dc is a nonempty, compact invariant set. Next, it follows from (2.145) and (2.146) that V(x(t)) is nonincreasing for all t ~ 0, which implies that, since V(·) is continuous, V(x(t)) t 'Y ~ as t t 00. Hence, the positive limit set w(xo) is contained in M'Y. Now, since by assumption M'Y contains no system trajectory for each 'Y > 0, it follows that 'Y = 0, establishing uniform asymptotic stability of the zero solution x(t) == to (2.139) and (2.140). 0
fx
°
°
°
Finally, we analyze a timevarying periodic dynamical system as a special case of a statedependent impulsive dynamical system. Consider the nonlinear periodic dynamical system given by
x(t) = f(t,x(t)),
x(to) = Xo,
t ~ to,
(2.147)
°
where x(t) E lR n , t ~ to, f : [to, 00) x]Rn t ]Rn is such that f(t + T, x) = f(t, x), t ~ 0, x E lR n , where T > is given. Furthermore, we assume f (., .) is such that for every to E lR+ and Xo E lRn , there exists a unique solution x(t), t ~ to, to (2.147). Note that with T = tto, the solution x(t), t ~ to, to the nonlinear timevarying dynamical system (2.147) is equivalently characterized by the solution Xl(T), T ~ 0, of the nonlinear autonomous dynamical system
Xl(T) =f(X2(T),Xl(T)), X2(T) = 1, X2(0) = to,
Xl(O) = Xo,
T ~ 0,
(2.148) (2.149)
where Xl() and X2() denote differentiation with respect to T. Next, using the fact that f(t+T,x) = f(t,x), where t ~ to and x E lRn , the solution x(t), t ~ to, of the nonlinear timevarying dynamical system (2.147) can be also characterized as a solution Xl(T), T ~ 0,
63
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
of nonlinear statedependent impulsive dynamical system given by
[ XI(O) ] = [ Xo ] [ :h(r) ] = [ !(X2(r),XI(r)) ] x2(r) 1 'X2(0) to' (xI(r),x2(r)) ¢ Z, (2.150) (xI(r),x2(r))
E
Z,
(2.151)
where Z = {(Xl, X2) E ]Rn x ]R: X2 = T}. Note that the solution to (2.150) and (2.151) is bounded if the solution x(t), t :2 to, to (2.147) is bounded. Now, it follows from Propositions 2.2 and 2.1 that the nonlinear statedependent dynamical system given by (2.150) and (2.151) satisfies Assumption 2.1. Hence, it follows from Theorem 2.2 that for every (to,xo) E [0,00) x ]Rn, the positive limit set w(to,xo) of the solution (xI(r),x2(r)), r:2 0, to (2.150) and (2.151) is nonempty, bounded, and invariant or, equivalently, for every (to, xo) E [0,00) x]Rn the positive limit set w(to, xo) of the solution x(t), t :2 to, to the nonlinear periodic dynamical system (2.147) is nonempty, bounded, and invariant. Although both solutions xI(r), r :2 0, to (2.148) and (2.149) and (2.150) and (2.151), are equivalent to the solution x(t), t :2 to, to (2.147), note that the solution to (2.148) and (2.149) is always unbounded, whereas the solution of the impulsive dynamical system (2.150) and (2.151) is always bounded if the solution x(t), t :2 to, to (2.147), is bounded. The assumption that !(.,.) is periodic is critical in casting (2.147) as a statedependent impulsive dynamical system (2.150) and (2.151). In light of the above, it follows that the positive limit set of (2.147) is nonempty, bounded, and invariant. This of course is a classical result for timevarying periodic dynamical systems [162, p. 153]. 2.9 Lagrange Stability, Boundedness, and Ultimate Boundedness
In the previous sections we introduced the concepts of stability and partial stability for nonlinear impulsive dynamical systems. In certain engineering applications, however, it is more natural to ascertain whether for every system initial condition in a ball of radius 8 the solution of the nonlinear impulsive dynamical system is bounded. This leads to the notions of Lagrange stability, boundedness, and ultimate boundedness. These notions are closely related to what is known in the literature as practical stability. In this section, we present Lyapunov
64
CHAPTER 2
like theorems for boundedness and ultimate bounded ness of nonlinear impulsive dynamical systems.
Definition 2.11 i} The nonlinear statedependent impulsive dynamical system given by (2.83}(2.86) is Lagrange stable with respect to Xl if, for every XlO E 1) and X20 E ]Rn2, there exists E = E(XlO, X20) > such that IlxI(t)11 < E, t ~ 0. ii} The nonlinear statedependent impulsive dynamical system given by (2.83}(2.86) is bounded with respect to Xl uniformly in X2 if, for every X20 E ]Rn2, there exists 'Y > such that, for every 8 E (0, 'Y), there exists E = E(8) > such that IIXlOll < 8 implies IIXI(t)1I < E, t ~ 0. The nonlinear statedependent impulsive dynamical system (2.83}(2.86) is globally bounded with respect to Xl uniformly in X2 if, for every X20 E ]Rn2 and 8 E (0,00), there exists E = E(8) > Osuch that IIXlOll < 8 implies IIXI(t)1I < E, t ~ 0. iii} The nonlinear statedependent impulsive dynamical system given by (2.83}(2.86) is ultimately bounded with respect to Xl uniformly in X2 with bound E if, for every X20 E ]Rn2, there exists 'Y > such that, for every 8 E (O,'Y), there exists T = T(8,E) > such that IIXlOll < 8 implies IIXI(t)1I < E, t ~ T. The nonlinear statedependent impulsive dynamical system (2.83}(2.86) is globally ultimately bounded with respect to Xl uniformly in X2 with bound c if, for every X20 E ]Rn2 and 8 E (0,00), there exists T = T( 8, E) > such that IIxlO II < 8 implies IIXI(t)1I < E, t ~ T.
°
°
°
°
°
°
Note that if a nonlinear statedependent impulsive dynamical system is globally bounded with respect to Xl uniformly in X2, then it is Lagrange stable with respect to Xl. Alternatively, if a nonlinear statedependent impulsive dynamical system is (globally) bounded with respect to Xl uniformly in X2, then there exists E > such that it is (globally) ultimately bounded with respect to Xl uniformly in X2 with a bound c. Conversely, if a nonlinear statedependent impulsive dynamical system is (globally) ultimately bounded with respect to Xl uniformly in X2 with a bound E, then it is (globally) bounded with respect to Xl uniformly in X2. The following results present Lyapunovlike theorems for boundedness and ultimate boundedness. For these results recall that V(XI,X2) = V'(XI,X2)fc(XI, X2), where fc(xI,x2) = If!c(xI,x2) fic(xI,x2)jT, and L~.v(XbX2) = V(XI + hd(XI, X2), X2 + hd(XI, X2))  V(XI, X2), for a given continuously differentiable function V : 1) X ]Rn2 ~ R
°
Theorem 2.8 Consider the nonlinear statedependent impulsive dynamical system (2. 83}(2. 86}. Assume there exist a continuously dif
65
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
ferentiable function V : V x lR n2 (3(.) such that
7
lR and class K functions oc() and
(Xl, X2) E V x lR n2 , (2.152) X2 E lR n2 , (XI,X2) rf Z,
a(llxIiI) ::; V(XI' X2) ::; (3(ll xIII), V(XI,X2)::; 0, Xl E V, IlxI11 2: fL,
(2.153) ~V(XI,X2)::; 0,
Xl E V,
X2 E lR n2 ,
IlxI11 2: fL,
(XI,X2) E Z, (2.154)
where fL
o= 6
> 0 is such that B a l((3(/l))(O)
sUP(Xl,X2)EB/L(O)xlRn2nZ V(XI
C V.
Furthermore, assume
+ hd(XI, X2), X2 + hd(XI, X2))
ex
ists. Then the nonlinear statedependent impulsive dynamical system {2.83}{2.86} is bounded with respect to Xl uniformly in X2. Furthermore, for every 8 E (0, ,), XlO E 138 (0) implies that IlxI(t)11 ::; c, t 2: 0, where
(2.155) where TJ 2: max{(3(fL),O} and, ~ sup{r > 0: Ba1((3(r))(0) C V}. If, in addition, V = lRn1 and a(·) is a class Koo function, then the nonlinear statedependent impulsive dynamical system {2.83}{2.86} is globally bounded with respect to Xl uniformly in X2 and for every XlO E lRn1 , Ilxl(t)11 ::; c, t 2: 0, where c is given by {2.155} with
8 = IlxlOll. Proof. First, let 8 E (O,fL] and assume IlxlOll ::; 8. If Il xl(t)11 ::; /L,
2: 0, then it follows from (2.152) that IlxI(t)11 ::; fL ::; a 1 ((3(fL)) ::; a 1 (TJ), t 2: O. Alternatively, if there exists T > 0 such that IlxI(T)11 > fL, then it follows that there exists T < T such that either Ilxl (T) II = fL, (Xl(T),X2(T)) rf Z, and Ilxl(t)11 > fL, t E (T,T], or (XI(T),X2(T)) E Z, Ilxl(T)11 ::; fL, and IlxI(t)11 > fL, t E (T, T]. Hence, it follows from
t
(2.152)(2.154) that
a(llxI(T)II) ::; V(xI(T),X2(T)) ::; V(Xl(T),X2(T))::; (3(fL) :s; TJ, if Ilxl(T)11 = fL and (Xl(T),X2(T)) rf Z, or a(llxl(T)ID:S; V(xl(T),X2(T)):S; V(XI(T+),X2(T+)) = V(Xl(T) + hd(Xl(T),X2(T)),X2(T) + hd(Xl(T),X2(T))) :S;O :s; TJ, if Ilxl(T)11 :s; fL and (Xl(T),X2(T)) E Z. In either case, it follows that Ilxl(T)11 :s; a 1 (TJ)·
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Next, let fl E (Jl, 'Y) and assume XlO E 88 (0) and IlxlOll > Jl. Now, for every i > 0 such that Ilxl(t)1I ~ Jl, t E [O,~, it follows from (2.152) and (2.153) that
a(llxl(t)ll)
~ V(Xl(t),X2(t)) ~ V(XlO,X20) ~ {3(fl),
t ~ 0,
which implies that Ilxl(t)11 ~ aI ({3(c5)) , t E [O,~. Next, if there exists T > 0 such that Ilxl (T) II ~ Jl, then it follows as in the proof of the first case that Ilxl(t)11 ~ a l (7]), t ~ T. Hence, if XlO E B8(0)\Bp,(0), then Ilxl(t)11 ~ a l (max{7],{3(fl)}), t ~ O. Finally, if V = lRnl and a(·) is a class Koo function it follows that {3(.) is a class Koo function, and hence 'Y = 00. Hence, the nonlinear statedependent impulsive dynamical system (2.83)(2.86) is globally bounded with respect to Xl uniformly in X2· 0 Theorem 2.9 Consider the nonlinear statedependent impulsive dynamical system (2. 83}(2. 86}. Assume there exist a continuously differentiable function V : V x lRn2 ~ lR and class K functions a(·) and {3() such that (2.152) and (2.154) hold. Furthermore, assume that there exists a continuous function W : V ~ lR such that W(Xl) > 0, Ilxlll > Jl, and V(Xl' X2) ~  W(Xl),
Xl E V,
Ilxlll > Jl,
X2 E lR n2 , (Xl,X2) r;j Z, (2.156)
where Jl > 0 is such that Ba1(,B(p,))(0) C V. Finally, assume () ~ V(XI +!Id(XI,X2),X2+hd(XI,X2)) exists. Then the nonlinear statedependent impulsive dynamical system given by (2.83}(2.86) is ultimately bounded with respect to Xl uniformly in X2 with bound E ~ a l (7]), where 7] > max{{3(Jl), ()}. Furthermore, lim SUPt.oo Ilxl(t)11 ~ a l ({3(Jl)). If, in addition, V = lR nl and a(·) is a class Koo function, then the nonlinear statedependent impulsive dynamical system (2.83}(2.86) is globally ultimately bounded with respect to Xl uniformly in X2 with bound E.
sUP(Xl,X2)EB!L(O)xlR.n2nz
Proof. First, let fl E (0, Jll and assume IlxlOll ~ fl. As in the proof of Theorem 2.8, it follows that Ilxl(t)11 ~ a l (7]) = E, t ~ O. Next, let fJ' E (Jl, 'Y), where 'Y ~ sup{r > 0: Ba1(,B(r))(0) C V}, and assume XlO E B8(0) and IlxlO II > Jl. In this case, it follows from Theorem 2.8 that Ilxl(t)11 ~ a l (max{7],{3(c5)}), t ~ O. Suppose, ad absurdum, Ilxl(t)11 ~ {31(7]), t ~ 0, or, equivalently, Xl(t) E 0 ~
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
67
t 2: O. Since 0 is compact, W(·) is continuous, and W(XI) > 0, IlxIil 2: (3I('f}) > {t, it follows from Weierstrass' Bo:1(,8(8))(0)\B,81(1))(0),
theorem [146] that k ~ minx1Eo W(XI) from (2.154) and (2.156) that
> 0 exists. Hence, it follows (2.157)
which implies that
a(llxl (t)ll) :::; (3(ll xlOll)  kt :::; (3( (3( 0 such that Ilx(t)11 < c, t 2: to· ii} The nonlinear timedependent impulsive dynamical system given by (2.115) and (2. 116} is uniformly bounded if there exists'Y > 0 such that, for every 0 such that, for every
(2.167) (2.168) (2.169)
°V(x+ is such that B c V. Furthermore, assume fd(x)) exists. Then the nonlinear statedependent a l(,B(I1,»)(O)
sUPxEBI'(O)nz
impulsive dynamical system {2.25} and {2.26} is bounded. If, in addition, V =]R.n and V(x)  00 as IIxll  00, then the nonlinear statedependent impulsive dynamical system {2.25} and {2.26} is globally bounded. Proof. The result is a direct consequence of Corollary 2.4.
0
Corollary 2.7 Consider the nonlinear statedependent impulsive dynamical system {2.25} and {2.26}. Assume there exists a continuously differentiable function V : V  ]R. and class J( functions a(·) and (30 such that {2.167} and {2.169} hold, and
V'(x)fc(x)
< 0,
x E V,
x
rf Z,
IIxll > fL,
(2.170)
71
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
where I"
>0
is such that Bee 1 (,8(jL)) (0) C 1) with'r]
> 13(1").
Further
more, assume 0 ~ sUPXEB/L(O)nZ V(x+ fd(x)) exists. Then the nonlinear statedependent impulsive dynamical system (2.25) and (2.26) is ultimately bounded with bound E ~ 0:1('r]), where 'r] ~ max{J3(I") , O}. Furthermore, lim sUPt>oo Ilx(t)11 :::; 0: 1 (13(1"))' If, in addition, 1) = IRn and V(x) + 00 as IIxll + 00, then the nonlinear statedependent impulsive dynamical system (2.25) and (2.26) is globally ultimately bounded with bound E. Proof. The result is a direct consequence of Corollary 2.5.
0
2.10 Stability Theory via Vector Lyapunov Functions
In this section, we introduce the notion of vector Lyapunov functions for stability analysis of nonlinear impulsive dynamical systems. The use of vector Lyapunov functions in dynamical system theory offers a very flexible framework since each component of the vector Lyapunov function can satisfy less rigid requirements as compared to a single scalar Lyapunov function. Specifically, since for many nonlinear dynamical systems constructing a system Lyapunov function can be a difficult task, weakening the hypothesis on the Lyapunov function enlarges the class of Lyapunov functions that can be used for analyzing system stability. Moreover, in certain applications, such as the analysis of largescale nonlinear dynamical systems, several Lyapunov functions arise naturally from the stability properties of each individual subsystem. To develop the theory of vector Lyapunov functions for nonlinear impulsive dynamical systems, we first introduce some results on vector differential inequalities and the vector comparison principle. The following definitions introduce the notions of class W and class Wd functions involving quasimonotone increasing and nondecreasing functions, respectively.
Definition 2.14 ([151]) A function We = [WeI"'" Weq]T : IRq + IRq is of class W if Wei (Z') :::; Wei(Z"), i = 1, ... , q, for all z', z" E IRq such that zj :::; z'j, z~ = z~', j = 1, ... , q, i =1= j, where Zi denotes the ith component of z.
Definition 2.15 ([96]) A function Wd = [Wdb"" WdqjT : IRq + IRq is of class Wd if Wdi(Z') :::; Wdi(Z"), i = 1, ... , q, for all z', z" E IRq such that z~ :::; z~', i = 1, ... , q.
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CHAPTER 2
Note that if WdO E Wd, then WdO E W. If weO E W we say that We satisfies the Kamke condition [88,164J. Note that if we(z) = Wez, where We E IRqx q , then the function weO is of class W if and only if We is essentially nonnegative, that is, all the offdiagonal entries of the matrix We are nonnegative. Alternatively, if Wd(Z) = WdZ, where Wd E IRqx q , then the function Wd(') is of class Wd if and only if Wd is nonnegative, that is, all the entries of the matrix Wd are nonnegative. Furthermore, note that it follows from Definition 2.14 that every scalar (q = 1) function we(z) is of class W. Next, we consider the nonlinear comparison system given by
= we(z(t)),
z(t)
z(to)
= zo,
t EI
zo '
(2.171)
where z(t) E Q ~ IRq, t E I zo ' is the comparison system state vector, Izo ~ T ~ [0, 00) is the maximal interval of existence of a solution z(t) of (2.171), Q is an open set, 0 E Q, and We : Q t IRq is Lipschitz
continuous on Q. For the results of this section we write x ~ ~ 0 (respectively, x » 0), x E IRn, to indicate that every component of x is nonnegative (respectively, positive). Furthermore, we denote the nonnegative and positive orthants of IRn by oc;:. and IR+., respectively. That is, if x E IR n , then x E oc;:. and x E IR+. are equivalent, respectively, to x ~~ 0 and x > > O. Proposition 2.5 Consider the nonlinear comparison system {2.171}. Assume that the function We : Q t IRq is continuous and weO is of class W. If there exists a continuously differentiable vector function V = [VI, ... ,VqjT : Izo t Q such that
V(t) «we(V(t)), then V(to)
«
t EI
zo '
(2.172)
zo, Zo E Q, implies V(t) «z(t),
where z(t), t E I
zo '
t
E I zo '
(2.173)
is the solution to {2.171}.
Proof. Since V(t), t E I zo ' is continuous it follows that for sufficiently small r > 0,
V(t) «z(t),
t E
[to, to + rJ.
(2.174)
Now, suppose, ad absurdum, inequality (2.173) does not hold on the entire interval Izo' Then there exists i E Izo such that V(t) « z(t), t E [to, i), and for at least one i E {1, ... ,q}, (2.175)
73
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
and
Vj(i) ~ zj(i), Since
weO
E
I i,
j
j
= 1, ... ,q.
(2.176)
W, it follows from (2.172), (2.175), and (2.176) that (2.177)
which, along with (2.175), implies that for sufficiently small f > 0,
Vi(t) > Zi(t), t E [i  f, i). This contradicts the fact that V(t) z(t), t E [to, i), and establishes (2.173).
«
0
Next, we present a stronger version of Proposition 2.5 where the strict inequalities are replaced by soft inequalities. Proposition 2.6 Consider the nonlinear comparison system {2.171}. Assume that the function We : Q + ~q is continuous and weO is of class W. Let z(t), t E L zo ' be the solution to {2.171} and [to, to +r] ~
Lzo be a compact interval. If there exists a continuously differentiable vector function V : [to, to + r] + Q such that
V(t) ~~ we(V(t)),
t E [to, to
+ r],
(2.178)
then V(to)
~~
Zo,
Zo E Q,
(2.179)
implies V(t)
~~
z(t),
t E [to, to
+ r].
(2.180)
Proof. Consider the family of comparison systems given by
i(t) = we(z(t)) 
+ ~e,
z(to) = Zo
+ ~e,
T
b.
(2.181)
where E > 0, n E Z+, e = [1, ... ,1] , and t E Lzo+~e, and let the solution to (2.181) be denoted by S(n)(t, Zo + ~e), E Lzo+~e. Now, it follows from Theorem 3 of [42, p. 17] that there exists a compact interval [to, to+r] ~ Lzo such that S(n)(t, zo+~e), t E [to, to+ r], is defined for all sufficiently large n. Moreover, it follows from Proposition 2.5 that
V(t)«
S(n)(t, Zo
+ ~e) «
for all sufficiently large m E ~e), t E [to, to
+ r],
n
E
S(m)(t, Zo
t
+ ;;e),
n> m, t E [to, to + r], (2.182)
Z+. Since the functions S(n)(t, Zo +
Z+, are continuous in t, decreasing in n,
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and bounded from below, it follows that the sequence of functions S(n)(·, Zo + ~e) converges uniformly on the compact interval [to, to +T] as n t 00, that is, there exists a continuous function z : [to, to + T] t Q such that
S(n)(t, Zo uniformly on [to, to that
+ T].
+ ~e)
z(t),
t
n
t
00,
(2.183)
Hence, it follows from (2.182) and (2.183)
V(t) :S:S z(t),
t
E
[to, to
+ T].
(2.184)
Next, note that it follows from (2.181) that
S(n)(t, Zo
+ ~e) = Zo + ~e +
rt wc(S(n)((T, Zo + ~e))d(T,
Jto
t
[to, to + TJ,
E
(2.185)
= Zo and, since wcO is a continuous function, wc(z(t)) as n t 00 uniformly on [to, to + T]. Hence, taking the limit as n t 00 on both sides of (2.185) yields
which implies that z(to)
wc(S(n)(t, Zo
+~e))
t
z(t) = Zo +
r wc(.i((T))d(T,
Jto
t
E
[to, to + TJ,
(2.186)
which implies that z(t) is the solution to (2.171) on the interval [to, to + T]. Hence, by uniqueness of solutions of (2.171) we obtain that z(t) = z(t), [to, to + T]. This along with (2.184) proves the result. 0 Next, consider the nonlinear dynamical system given by
x(t) = f(x(t)),
x(to) = Xo,
t
E L xo '
(2.187)
where x(t) E D ~ jRn, t E L xo ' is the system state vector, Lxo is the maximal interval of existence of a solution x(t) of (2.187), D is an open set, 0 ED, and fO is Lipschitz continuous on D. The following result is a direct consequence of Proposition 2.6. Corollary 2.8 Consider the nonlinear dynamical system {2.187}. Assume there exists a continuously differentiable vector function V : D t Q ~ jRq such that
V'(x)f(x) :S:S wc(V(x)),
xED,
(2.188)
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
where We : Q + ~q is a continuous function, We (.) z(t) = we(z(t)),
z(to) = zo,
t
E
75
W, and
E I zo '
(2.189)
has a unique solution z(t), t E I zo ' where x(t), t E I xo ' is a solution to {2.187}. If [to, to + r] ~ Ixo nIzo is a compact interval, then V(xo) ::;::; Zo,
Zo
(2.190)
E Q,
implies V(x(t)) ::;::; z(t), Proof. Define 'T}(t) ~ V(x(t)), t plies
t
E
[to, to + r].
(2.191)
E I xo ' and note that (2.188) im
(2.192) Moreover, if [to, to + r] ~ Ixo n Izo is a compact interval, then it follows from Proposition 2.6, with V(xo) = 'T}(to) ::;::; Zo, that
V(x(t)) = 'T}(t) ::;::; z(t),
t
E
[to, to + r],
(2.193)
o
which establishes the result.
If in (2.187) f : ~n + ~n is globally Lipschitz continuous, then (2.187) has a unique solution x(t) for all t 2 to. An alternative sufficient condition for global existence and uniqueness of solutions to (2.187) is continuous differentiability of f : ~n + ~n and uniform boundedness of f'(x) on ~n. Note that if the solutions to (2.187) and (2.189) are globally defined for all Xo E V and Zo E Q, then the result of Corollary 2.8 holds for any arbitrarily large but compact interval [to, to + r] C ~+. For the remainder of this section we assume that the solutions to the systems (2.187) and (2.189) are defined for all t 2 to· Continuous differentiability of fO and weO provides a sufficient condition for the existence and uniqueness of solutions to (2.187) and (2.189) for all t 2 to. Next, we consider the statedependent impulsive dynamical system given by
x(t) = fe(x(t)), L1x(t) = fd(X(t)),
x(to) = Xo, x(t) E Z,
x(t) ¢ Z,
t
E I xo ' (2.194)
(2.195)
where x(t) E V ~ ~n, t E I xo ' is the system state vector, Ixo is the maximal interval of existence of a solution x(t) to (2.194) and (2.195), V is an open set, 0 E V, fe : V + ~n is Lipschitz continuous and
76
CHAPTER 2
satisfies fe(O) = 0, fd : V t ffin is continuous, D.x(t) ~ x(t+)  x(t), and Z C V ~ ffi.n is the resetting set. We assume that Al and A2 hold so that the required properties for the existence and uniqueness of solutions to (2.194) and (2.195) are" satisfied. Theorem 2.10 Consider the impulsive dynamical system (2. 194} and
(2. 195}. Assume there exists a continuously differentiable vector function V : V t Q ~ ffi.q such that V'(x)fe(x) ~~ we(V(x)), x rt z, V(x + fd(X)) ~~ V(x) + Wd(V(X)), x
E
(2.196) (2.197)
Z,
where We : Q t ffi.q and Wd : Q t ffi.q are continuous functions, weC) E W, Wd() E Wd, and the comparison impulsive dynamical system z(t) = we(z(t)), D.z(t) =Wd(Z(t)),
z(to) = Zo, x(t) E Z,
has a unique solution z(t), t compact interval, then V(xo)
rt z,
Zo,
t E I zo ' (2.198)
(2.199)
If [to, to
E Izo.
~~
x(t)
Zo
+ T]
~ Ixo
n Izo is a
E Q,
(2.200)
implies V(x(t))
~~
z(t),
t
E
[to, to + T],
(2.201)
where x(t), t E I xo ' is the solution to (2.194) and (2.195) with initial condition Xo E V.
Proof. Without loss of generality, let Xo rt z, Xo E V. If Xo E Z, then by Assumption A2, xo+ fd(XO) rt Z serves as the initial condition for the continuoustime dynamics. If for Xo rt Z the solution x(t) rt Z for all t E [to, to + 7], then the result follows from Corollary 2.8. Next, suppose the interval [to, to + 7] contains the resetting times Tk(XO) < Tk+1(XO), k E {1,2, ... ,m}. Consider the compact interval [to, 71(XO)] and let V(xo) ~~ Zo0 Then it follows from (2.196) and Corollary 2.8 that
V(x(t))
~~
z(t),
t
E
[to, 71 (xo)],
(2.202)
where z(t), t E Izo, is the solution to (2.198). Now, since WdC) E Wd it follows from (2.197) and (2.202) that
V(X(Tt(XO))) ~~ V(X(Tl(XO))) + Wd(V(X(Tl(XO)))) ~~ Z(71(XO)) + Wd(Z(71 (xo))) = Z(Tt(XO)).
(2.203)
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
77
Consider the compact interval [Tt(XO), T2(XO)]. Since V(X(Tt(XO)))
SS Z(Tl(XO)), it follows from (2.196) that V(x(t)) SS z(t),
t E [Tt(XO) , T2(XO)].
(2.204)
Repeating the above arguments for t E [T: (xo), Tk+1 (xo)], k = 3, ... , m, yields (2.201). Finally, in the case of infinitely many resettings over the time interval [to, to + T], let limk>oo Tk(XO) = Too(xo) E (to, to + T].
In this case, [to, Too (xo)] = [to, Tl(XO)] u [ U~l[Tk(XO), r k+1(x o)]]. Repeating the above arguments, the result can be shown for the interval [to, roo (xo)]. 0
If the solutions to (2.194), (2.195), and (2.198), (2.199) are globally defined for all Xo E V and Zo E Q, then the result of Theorem 2.10 holds for any arbitrarily large but compact interval [to, to + r] C IR+. For the remainder of this section we assume that the solutions to the systems (2.194), (2.195), and (2.198), (2.199) are defined for all t 2:: to. Theorem 2.11 Consider the impulsive dynamical system (2.194) and (2.195). Assume that there exist a continuously differentiable vector function V : V t Q n IR~ and a positive vector p E IR+ such that V(O) = 0, the scalar function v : V t IR+ defined by v(x) ~ pTV(x), X E V, is such that v(x) > 0, x i= 0, and
V'(x)fe(x) SS we(V(x)), x ¢ Z, V(x + fd(x)) SS V(x) + Wd(V(X)), x E Z,
(2.205) (2.206)
where We : Q t IRq and Wd : Q t IRq are continuous, w e() E W, and we(O) = o. Then the following statements hold:
WdO E Wd,
i) If the zero solution z(t) == 0 to i(t) = we(z(t)), ~z(t) =Wd(Z(t)),
z(to) = Zo, x(t) E Z,
x(t) ¢ Z,
is Lyapunov stable, then the zero solution x(t) and (2.195) is Lyapunov stable.
t 2:: to, (2.207) (2.208)
== 0 to (2.194)
ii) If the zero solution z(t) == 0 to (2.207) and (2.208) is asymptotically stable, then the zero solution x(t) == 0 to (2.194) and (2.195) is asymptotically stable. iii) If V = IR n , Q = IRq, v : IRn t IR+ is radially unbounded, and the zero solution z(t) == 0 to (2.207) and (2.208) is globally asymptotically stable, then the zero solution x(t) == 0 to (2.194) and (2.195) is globally asymptotically stable.
78
CHAPTER 2
iv} If there exist constants v : D t ~+ satisfies
1/
2: 1, a > 0, and (3 > 0 such that (2.209)
and the zero solution z(t) == 0 to (2.207) and (2.208) is exponentially stable, then the zero solution x(t) == 0 to (2.194) and (2.195) is exponentially stable. v} If D = ~n, Q = ~q, there exist constants 1/ 2: 1, a > 0, and (3 > 0 such that v : ~n t ~+ satisfies (2.209), and the zero solution z(t) == 0 to (2.207) and (2.208) is globally exponentially stable, then the zero solution x(t) == 0 to (2.194) and (2.195) is globally exponentially stable.
Proof.
Assume there exist a continuously differentiable vector function V : D t Q n ~~ and a positive vector p E ~~ such that v(x) = pTV(x), XED, is positive definite, that is, v(O) = 0 and v(x) > 0, x I O. Since v(x) = pTV(x) ::; maxi=l, ... ,q{Pi}eTV(x), x E D, where e ~ [1, ... , I]T, the function eTV(x), xED, is also positive definite. Thus, there exist r > 0 and class K functions a, (3 : [0, r] t ~+ such that Br(O) cD and
a(llxll) :::; eTV(x)
:::;
(3(llxll),
(2.210)
x E Br(O).
i) Let c > 0 and choose 0 < t < min{ c, r}. It follows from Lyapunov stability of (2.207) and (2.208) that there exists IL = IL(t) = IL(c) > 0 such that if Ilzolh < IL, where Ilzlll ~ I:i=llzil and Zi is the ith component of z, then Ilz(t)lll < a(t), t 2: to. Now, choose zo = V(xo) 2:2: 0, Xo E D. Since V(x), xED, is continuous, the function eTV(x), XED, is also continuous. Hence, for IL = IL(t) > 0 there exists 6 = 6(IL(t)) = 6(c) > 0 such that 6 < t and if Ilxoll < 6, then eTV(xo) = e T zo = Ilzolh < IL, which implies that Ilz(t)lh < a(t), t 2: to. In addition, it follows from (2.205) and (2.206), and Theorem 2.10 that 0 ::;::; V(x(t)) ::;::; z(t) on any compact interval [to, to+T], and hence, eTz(t) = IIz(t)lh, [to, to+T]. Let T > to be such that x(t) E Br(O), t E [to, to + T]. Thus, using (2.210), it follows that for Ilxo II < 6,
a(llx(t)ID ::; eTV(x(t)) ::; eT z(t) < a(t),
t E [to, to
+ T],
(2.211)
which implies Ilx(t)11 < t < c, t E [to, to + T]. Now, suppose, ad absurdum, that for some Xo E B8(0) there exists f> to + T such that Ilx(f)11 2: t. Then, for Zo = V(xo) and the compact interval [to, ~
79
STABILITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
it follows from Theorem 2.10 that V(x(i)) :s:s z(i), which implies that a(t) :S a(llx(i)ll) :S eTV(x(i)) :S eT z(i) < a(t). This is a contradiction, and hence, for a given c > 0 there exists 5 = 5(c) > o such that for all Xo E B8(0), Ilx(t)11 < c, t ~ to, which implies Lyapunov stability of the zero solution x(t) == 0 to (2.194) and (2.195). ii) It follows from i) and the asymptotic stability of (2.207) and (2.208) that the zero solution x(t) == 0 to (2.194) and (2.195) is Lyapunov stable, and there exists IL > 0 such that if Ilzolll < IL, then limhoo z(t) = O. As in i), choose Zo = V(xo) ~~ 0, Xo E 1). It follows from Lyapunov stability of the zero solution x(V == 0 to (2.194) and (2.195), and the continuity of V : 1) + Q n ~+ that there exists 5 = 5(1L) > 0 such that if Ilxoll < 5, then Ilx(t)11 < r, t ~ to, and eTV(xo) = e T Zo = Ilzolll < IL. Thus, by asymptotic stability of (2.207) and (2.208), for any arbitrary c > 0 there exists T = T(c) > to such that Ilz(t)lh < a(c), t ~ T. Thus, it follows from (2.205) and (2.206) and Theorem 2.10 that 0 :S:S V(x(t)) :S:S z(t) on any compact interval [T, T + TJ, and hence, eTz(t) = Ilz(t)lll, t E [T, T + TJ, and, by (2.210),
a(llx(t)ll) :S eTV(x(t)) :S eT z(t)
< a(c),
t E [T, T
+ TJ.
(2.212)
Now, suppose, ad absurdum, that for some Xo E B8(0), limhoo x(t)
# 0, that is, there exists a sequence {tn}~=l' with tn + 00 as n
+ 00,
such that Ilx(tn)11 ~ t, n E Z+, for some 0 < t < r. Choose c = t and i > T + T such that at least one tn E [T, ~. Then it follows from (2.212) that a(c) :S a(llx(tn)ll) < a(c), which is a contradiction. Hence, there exists 5 > 0 such that for all Xo E B8(0), limhoo x(t) = 0 which along with Lyapunov stability implies asymptotic stability of the zero solution x(t) == 0 to (2.194) and (2.195). iii) Suppose 1) = ~n, Q = ~q, v : ~n + ~+ is radially unbounded, and the zero solution z(t) == 0 to (2.207) and (2.208) is globally asymptotically stable. In this case, V : ~n + ~~ satisfies (2.210) for all x E ~n, where the functions a, (3 : ~+ + ~+ are of class /(00' Furthermore, Lyapunov stability of the zero solution x(t) == 0 to (2.194) and_~2.195) follows from i). Next, for any Xo E ~n and Zo = V(xo) E ~+, identical arguments as in ii) can be used to show that limt>oo x(t) = 0, which proves global asymptotic stability of the zero solution x(t) == 0 to (2.194) and (2.195). iv) Suppose (2.209) holds. Since p E ~~, then (2.213)
80
CHAPTER 2
where & ~ a/ maxi=l, ... ,q{Pi} and !:J ~ f3/ mini=l, ... ,q{Pi}. It follows from the exponential stability of (2.207) and (2.208) that there exist positive constants 'Y, J.I" and 'rJ such that if Ilzolll < J.I" then (2.214)
Choose Zo = V(xo) 2:2: 0, Xo E 1). By continuity of V : 1)  t Q n ~~, there exists 8 = 8(J.I,) > 0 such that for all Xo E B8(0), eTV(xo) = e T Zo = IIzolll < J.I,. Furthermore, it follows from (2.213), (2.214), and Theorem 2.10 that for all Xo E B8(0) the inequality &llx(t)IIV ::; eTV(x(t)) ::; e T z(t)::; 'Yllzollle7J(tto) ::; 'Y!:Jll xoll v e 7J (tt o ) (2.215) holds on any compact interval [to, to for any Xo E B8(0),
(1) "
+ r].
This in turn implies that
1
Ilx(t)1I :5
Ilxolle;('"'),
t
E
[to,to + rl·
(2.216)
Now, suppose, ad absurdum, that for some Xo E B8(0) there exists i > to + r such that (2.217)
Then for the compact interval [to,~, it follows from (2.216) that
(t)
1
Ilx(i)11 ::; v Ilxolle;(ito), which is a contradiction. Thus, inequality (2.216) holds for all t 2: to establishing exponential stability of the zero solution x(t) == 0 to (2.194) and (2.195). v) The proof is identical to the proof of iv). 0 Note that for stability analysis each component of a vector Lyapunov function need not be positive definite, nor does it need to have a negative definite time derivative along the trajectories of (2.194) and (2.195). This provides more flexibility in searching for a vector Lyapunov function as compared to a scalar Lyapunov function for addressing the stability of impulsive dynamical systems. Finally, note that in the case where Wd(Z) == 0, (2.207) and (2.208) specialize to a continuoustime dynamical system, and hence, standard stability methods can be used to examine the stability of (2.207).
Chapter Three Dissipativity Theory for Nonlinear Impulsive Dynamical Systems
3.1 Introduction
In control engineering, dissipativity theory provides a fundamental framework for the analysis and control design of dynamical systems using an inputoutput system description based on systemenergyrelated considerations. The notion of energy here refers to abstract energy notions for which a physical system energy interpretation is not necessary. The dissipation hypothesis on dynamical systems results in a fundamental constraint on their dynamic behavior, wherein a dissipative dynamical system can deliver only a fraction of its energy to its surroundings and can store only a fraction of the work done to it. Many of the great landmarks of feedback control theory are associated with dissipativity theory. In particular, dissipativity theory provides the foundation for absolute stability theory; which in turn forms the basis of the Lure problem, as well as the circle and Popov criteria, which are extensively developed in the classical monographs by Aizerman and Gantmacher [IJ, Lefschetz [100], and Popov [142J. Since absolute stability theory concerns the stability of a dynamical system for classes of feedback nonlinearities which, as noted in [53,54J, can readily be interpreted as an uncertainty model, it is not surprising that absolute stability theory (and hence dissipativity theory) also forms the basis of modernday robust stability analysis and synthesis [53, 55, 66J. The key foundation in developing dissipativity theory for general nonlinear dynamical systems was presented by J. C. Willems [165, 166J in his seminal twopart paper on dissipative dynamical systems. In particular, Willems [165J introduced the definition of dissipativity for general dynamical systems in terms of an inequality involving a generalized system power input, or supply rate, and a generalized energy function, or storage function. The storage function is bounded from below by the available system storage and bounded from above by the required supply. The available storage is the amount of internal
82
CHAPTER 3
generalized stored energy which can be extracted from the dynamical system, and the required supply is the amount of generalized energy that can be delivered to the dynamical system to transfer it from a state of minimum potential to a given state. Hence, as noted above, a dissipative dynamical system can deliver only a fraction of its stored generalized energy to its surroundings and can store only a fraction of generalized work done to it. Dissipativity theory exploits the notion that numerous physical dynamical systems have certain inputoutput system properties related to conservation, dissipation, and transport of mass and energy. Such conservation laws are prevalent in dynamical systems such as mechanical, fluid, electromechanical, electrical, combustion, structural, biological, physiological, biomedical, ecological, and economic systems, as well as feedback control systems. To see this, consider the singledegreeoffreedom springmassdamper mechanical system given by
Mx(t)
+ Cx(t) + Kx(t) = u(t),
x(O) = xo,
x(O) = xo,
t 2: 0,
(3.1) where M > 0 is the system mass, C 2: 0 is the system damping constant, K 2: 0 is the system stiffness, x(t), t 2: 0, is the position of the mass M, and u(t), t 2: 0, is an external force acting on the mass M. The energy of this system is given by
v.;s (x,x.)
= '2IM·x 2 + '2lK x.2
(3.2)
Now, assuming that the measured output of this system is the system velocity, that is, y(t) = x(t), it follows that the time rate of change of the system energy along the system trajectories is given by
Vs(x, x) =Mxx + Kxx = uy  Cx 2 •
(3.3)
Integrating (3.3) over the time interval [0, TJ, it follows that
Vs(x(T), x(T)) = Vs(x(O), x(O))+
lT U(t)y(t)dt1TCx (t)dt, (3.4) 2
which shows that the system energy at time t = T is equal to the initial energy stored in the system plus the energy supplied to the system via the external force u minus the energy dissipated by the system damper. Equivalently, it follows from (3.3) that the rate of change in the system energy, or system power, is equal to the external supplied system power through the input port u minus the internal system power dissipated by the viscous damper. Note that in the
DISSIPATIVITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
83
case where the external input force u is zero and C = 0, that is, no system supply or dissipation is present, (3.3), or, equivalently, (3.4), shows that the system energy is constant. Furthermore, note that since C;::: and V(x(T), x(T)) ;::: 0, T;::: 0, it follows from (3.4) that
°
loT u(t)y(t)dt;::: Vs(xo,xo),
(3.5)
loT u(t)y(t)dt :::; Vs(xo, xo).
(3.6)
or, equivalently,
Equation (3.6) shows that the energy that can be extracted from the system through its inputoutput ports is less than or equal to the initial energy stored in the system. This is precisely the notion of dissipativity. Since Lyapunov functions can be viewed as generalizations of energy functions for nonlinear dynamical systems, the notion of dissipativity, with appropriate storage functions and supply rates, can be used to construct Lyapunov functions for nonlinear feedback systems by appropriately combining storage functions for each subsystem. Even though the original work on dissipative dynamical systems was formulated in the state space setting, describing the system dynamics in terms of continuous flows on appropriate manifolds, an inputoutput formulation for dissipative dynamical systems extending the notions of passivity [171], nonexpansivity [172], and conicity [147,171] was presented in [73,75,127]. In this chapter we develop dissipativity theory for nonlinear impulsive dynamical systems. Specifically, we extend the notions of classical dissipativity theory using generalized storage functions and hybrid supply rates for impulsive dynamical systems. The overall approach provides an interpretation of a generalized hybrid energy balance for an impulsive dynamical system in terms of the stored or accumulated generalized energy, dissipated energy over the continuoustime dynamics, and dissipated energy at the resetting instants. Furthermore, as in the case of dynamical systems possessing continuous flows, we show that the set of all possible storage functions of an impulsive dynamical system forms a convex set, and is bounded from below by the system's available stored generalized energy which can be recovered from the system, and bounded from above by the system's required generalized energy supply needed to transfer the system from an initial state of minimum generalized energy to a given state. In addition,
84
CHAPTER 3
for timedependent and statedependent impulsive dynamical systems, we develop extended KalmanYakubovichPopov algebraic conditions in terms of the system dynamics for characterizing dissipativeness via system storage functions for impulsive dynamical systems.
3.2 Dissipative Impulsive Dynamical Systems: InputOutput and State Properties In this section, we extend dissipativity theory to nonlinear impulsive dynamical systems. Specifically, we consider controlled impulsive dynamical systems having the form
x(t) = fe(x(t))
+ Ge(x(t))ue(t),
x(O) = Xo,
(t, x(t), ue(t)) ¢ S, (3.7) ~x(t) = fd(X(t)) + Gd(X(t))Ud(t), (t, x(t), ue(t)) E S, (3.8) Ye(t) = he(x(t)) + Je(x(t))ue(t), (t,x(t),ue(t)) ¢ S, (3.9) Yd(t) = hd(x(t)) + Jd(X(t))Ud(t), (t, x(t), ue(t)) E S, (3.10) ~ 0, x(t) E 'D ~ jRn, 'D is an open set with 0 E 'D, ~x(t) = x(t+)  x(t), ue(t) E Ue ~ jRmc, Ud(tk) E Ud ~ jRmd, tk denotes the kth instant of time at which (t,x(t), ue(t)) intersects S for a particular trajectory x(t) and input ue(t), Ye(t) E Yc ~ jRlc, Yd(tk) E Yd ~ jRld, fe : 'D ~ jRn is Lipschitz continuous on 'D and satisfies fe(O) = 0, Ge : 'D ~ jRnxmc, fd : 'D ~ 'D is continuous on 'D and satisfies fd(O) = 0, Gd : 'D ~ jRnxmd, he : 'D ~ jRlc and satisfies he(O) = 0, Je : 'D ~ jRlcxmc, hd : 'D ~ jRld and satisfies hd(O) = 0, Jd : 'D ~
where t
jRld xmd , and S C [0, 00) x 'D x Ue is the resetting set. Here, we assume that u e () and Ud(') are restricted to the class of admissible inputs consisting of measurable functions such that (ue(t),Ud(tk)) E Ue X Ud for all t ~ 0 and k E Z[O,t) ~ {k : 0 ~ tk < t}, where the constraint set Ue X Ud is given with (0,0) E Ue X Ud. More precisely, for the impulsive dynamical system 9 given by (3.7)(3.10) defined on the state space 'D ~ jRn, U ~ Ue X Ud and Y ~ Ye X Yd define an input and output space, respectively, consisting of leftcontinuous bounded Uvalued and Yvalued functions on the semiinfinite interval [0,00). The set U ~ Ue X Ud, where Ue ~ jRmc and Ud ~ jRmd, contains the set of input values, that is, for every U = (Ue,Ud) E U and t E [0,00), u(t) E U, ue(t) E Ue, and Ud(tk) E Ud. The set Y ~ Yc X Yd, where Yc ~ jRlc and Yd ~ jRld, contains the set of output values, that is, for every Y = (Ye, Yd) E Y
DISSIPATIVITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
85
and t E [0,00), y(t) E Y, yc(t) E Yc, and Yd(tk) E Yd. The spaces U and Y are assumed to be closed under the shift operator, that is, if u(·) E U (respectively, y(.) E Y), then the function UT (respectively, YT) defined by UT ~ u(t+T) (respectively, YT ~ y(t+T)) is contained in U (respectively, Y) for all T 2 o. For convenience, we use the notation s(t, T, Xo, u) to denote the solution x(t) of (3.7) and (3.8) at time t 2 T with initial condition X(T) = Xo, where U = (Uc,Ud) : ~x T  t Uc X Ud and T ~ {tI, t2, .. .}. Thus, the trajectory of the system (3.7) and (3.8) from the initial condition x(O) = Xo is given by 'ljJ(t, 0, Xo, u) for 0 < t :::; tl. If and when the trajectory reaches a state Xl ~ X(tl) satisfying (tl' XI, UI) E S, where UI ~ Uc(tl), then the state is instantaneously transferred to xt ~ Xl + fd(XI) + Gd(XI)Ud, where Ud E Ud is a given input, according to the resetting law (3.8). The trajectory x(t), tl < t :::; t2, is then given by 'ljJ(t, tI, xt, u), and so on. As in the uncontrolled case, the solution x(t) of (3.7) and (3.8) is leftcontinuous, that is, it is continuous everywhere except at the resetting times tk, and
Xk ~ X(tk) = lim X(tk  c), 0:>0+
+ fd(X(tk)) + Gd(X(tk))Ud(tk) = lim X(tk + c), Ud(tk) E Ud, 0:>0+
(3.11)
xt ~ X(tk)
(3.12)
for k = 1,2, .... Furthermore, the analogs to Assumptions Al and A2 become: AI. If (t, x(t), uc(t)) E all 0 < & < c,
8\S, then there exists c > 0 such that, for
'ljJ(t + &, t, x(t), uc(t + &)) ¢ S. A2. If (tk' X(tk), Uc(tk)) E as n S, then there exists c for all 0 :::; & < c and Ud(tk) E Ud,
> 0 such that,
Timedependent impulsive dynamical systems can be written as
(3.7)(3.10) with S defined as e:,
S=Tx V x Uc .
(3.13)
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CHAPTER 3
Now (3.7)(3.10) can be rewritten in the form of the timedependent impulsive dynamical system
x(t) = fc{x{t)) + Gc{x{t))uc{t), x{o) = XO, ~x{t) = fd(X(t)) + Gd{X{t))Ud{t), t = tk, Yc{t) = hc{x{t)) + Jc{x{t))uc(t), t =1= tk, Yd(t) = hd{X{t)) + Jd(X(t))Ud(t), t = tk.
t =1= tk, (3.14)
(3.15) (3.16) (3.17)
°
Since ¢ T and tk < tk+1, it follows that Assumptions Al and A2 are satisfied. Standard continuoustime and discretetime dynamical systems as well as sampleddata systems can be treated as special cases of impulsive dynamical systems. In particular, setting fd{X) = 0, Gd{x) = 0, hd{x) = 0, and Jd(X) = 0, it follows that (3.14){3.17) has an identical state trajectory as the nonlinear continuoustime system
x(t) = fc(x(t)) + Gc(x(t))uc(t), Yc{t) = hc(x{t)) + Jc{x{t))uc{t).
x(o)
=
XO,
t ~ 0,
(3.18)
(3.19)
Alternatively, setting fc(x) = 0, Gc{x) = 0, hc(x) = 0, Jc{x) = 0, tk = kT, and T = 1, it follows that (3.14){3.17) has an identical state trajectory as the nonlinear discretetime system
x(k + 1) = fd(X(k)) + Gd(x(k))Ud(k), Yd(k) = hd(x(k)) + Jd(x(k))Ud(k).
x(O) = XO,
k E Z+, (3.20) (3.21)
Finally, to show that (3.14)(3.17) can be used to represent sampleddata systems, consider the continuoustime nonlinear system (3.18) and (3.19) with piecewise constant input uc(t) = Ud(tk), t E (tk' tk+1], and sampled measurements Yd(tk) = hd(X(tk)) + Jd(X(tk))Ud(tk). Defining x = [xT, uJF, it follows that the sampleddata system can be represented as
!i: = j{x{t)),
(3.22)
t =1= tk,
~x(t) = [~ ~I] x(t) + [ ~ ] Ud(t), y{t) = h{x{t)),
t = tk,
t =1= tk,
(3.24)
Yd(t) = hd{x(t)) + Jd(X(t))Ud(t),
t
= tk,
where
j(x) = [ fc{x) hd(X) = hd{x),
+OGc(x)Uc ], Jd(x) = Jd(X).
(3.23)
h{x) = hc(x) + Jc(x)u c,
(3.25)
DISSIPATIVITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
87
Statedependent impulsive dynamical systems can be written as (3.7)(3.10) with S defined as
S ~ [0,(0) x
z,
(3.26)
where Z ~ Zx x Ue and Zx c 1). Therefore, (3.7)(3.10) can be rewritten in the form of the statedependent impulsive dynamical system
x(t) = fe(x(t»
+ Ce(x(t))ue(t),
x(O) = Xo,
(x(t), ue(t)) ¢ Z,
Llx(t) = fd(X(t» + Cd(X(t»Ud(t), (x(t), ue(t» E Z, Ye(t) = he(x(t») + Je(x(t)ue(t), (x(t), ue(t» ¢ Z, Yd(t) = hd(X(t)) + Jd(X(t»)Ud(t), (x(t),ue(t) E Z.
(3.27) (3.28) (3.29) (3.30)
We assume that if (x, ue) E Z, then (x + fd(X) + Cd (X)Ud, Ue) ¢ Z, Ud E Ud. In addition, we assume that if at time t the trajectory (x(t),ue(t» E Z\Z, then there exists c > 0 such that for 0 < 6 < c, (x(t + 6), ue(t + 6» ¢ Z. These assumptions represent the specialization of Al and A2 for the particular resetting set (3.26). Finally, in the case where S ~ [0, (0) x 1) x ZUc' where ZUc CUe, we refer to (3.27)(3.30) as an inputdependent impulsive dynamical system, while in the case where S ~ ([0,00) x Zx x Ue) U ([0, (0) x 1) x ZuJ we refer to (3.27)(3.30) as an input/statedependent impulsive dynamical system. Next, we develop dissipativity theory for nonlinear impulsive dynamical systems. Specifically, we consider nonlinear impulsive dynamical systems 9 of the form given by (3.7)(3.10) with t E JR, (t,x(t),ue(t» ¢ S, and (t,x(t),ue(t» E S replaced by X(t,x(t),ue(t» # 0 and X(t, x(t), ue(t» = 0, respectively, where X : JR x 1) x Ue + R Note that setting X(t, x(t), ue(t» = (t  tl)(t  t2)···, where tk + 00 as k + 00, (3.7)(3.10) reduce to (3.14)(3.17), while setting X(t,x(t),ue(t» = X(x(t),ue(t», where X : 1) x Ue + JRn is a support function characterizing the manifold Z, (3.7)(3.10) reduce to (3.27)(3.30). Furthermore, we assume that the system functions feO, fd(·), CeO, Cd(·), heO, hd(·), Je(), and Jd() are continuous mappings. In addition, for the nonlinear dynamical system (3.7) we assume that the required properties for the existence and uniqueness of solutions are satisfied such that (3.7) has a unique solution for all t E JR [14, 93J. For the impulsive dynamical system 9 given by (3.7)(3.10) a function (se(u e, Ye), Sd(Ud, Yd», where Se : Ue x Yc + JR and Sd : Ud x Yd + JR are such that se(O,O) = and Sd(O, 0) = 0, is called a hybrid supply
°
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CHAPTER 3
rate if Se (Ue, Ye) is locally integrable for all inputoutput pairs satisfying (3.7)(3.10), that is, for all inputoutput pairs ue(t) E Ue and
J/
Ye(t) E Yc satisfying (3.7)(3.10), sJ, .) satisfies ISe( ue(s), Ye(s))1 ds < 00 for all t, i 2: o. Note that since all inputoutput pairs Ud(tk) E Ud and Yd (tk) E Yd are defined for discrete instants, Sd (, .) satisfies L:kEZ . ISd(Ud(tk), Yd(tk))1 < 00, where k E Z[t i" ~ {k : t ::; tk < i}. [t,t) ,'J
Definition 3.1 An impulsive dynamical system g of the form (3.7)(3.10) is dissipative with respect to the hybrid supply rate (se, Sd) if the dissipation inequality 0::;
iT
L
Se(ue(t), Ye(t))dt +
to
Sd(Ud(tk), Yd(tk)),
T 2: to,
kEZ[to,T)
(3.31) is satisfied for all T 2: to and all (ueO, Ud (.)) E Ue XUd with x(to) = O. An impulsive dynamical system g of the form (3.7}(3.10) is exponentially dissipative with respect to the hybrid supply rate (se, Sd) if there exists a constant £ > 0, such that the dissipation inequality (3.31) is satisfied with se(ue(t), Ye(t)) replaced by eetse(ue(t), Ye(t)) and Sd(Ud(tk), Yd(tk)) replaced by eetksd(Ud(tk), Yd(tk)), for all T 2: to and all (UeO,Ud(')) E Ue X Ud with x(to) = O. An impulsive dynamical system is lossless with respect to the hybrid supply rate (se, Sd) if g is dissipative with respect to the supply rate (se, Sd) and the dissipation inequality (3.31) is satisfied as an equality for all T 2: to and all (UeO,Ud(')) E Ue X Ud with x(to) = x(T) = O. Next, define the available storage Va(to, xo) of the impulsive dynamical system g by
Va(to,xo)
~
inf
(uc(o),Ud(o)), T~to
[
r
T
lto
+
se(ue(t),Ye(t))dt
L
Sd(Ud(tk), Yd(tk))] ' (3.32)
kEZ[to,T)
where x(t), t 2: to, is the solution to (3.7)(3.10) with admissible inputs (ueO, Ud(')) E Ue XUd and x(to) = Xo. Note that Va(to, xo) 2: 0 for all (t, x) E ~ x V since Va (to , xo) is the supremum over a set of numbers containing the zero element (T = to). It follows from (3.32) that the available storage of a nonlinear impulsive dynamical system g is the maximum amount of generalized stored energy which can be
DISSIPATIVITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
89
g at any time T. Furthermore, define the available exponential storage of the impulsive dynamical system g by
extracted from
~
Va(to,xo)
inf
(ucO,UdO), T'2:to
[iT to
+
ectsc(uc(t),Yc(t))dt
L
kEZ[to,T)
ectkSd(Ud(tk), Yd(t k ))] , (3.33)
where E > 0 and x(t), t ~ to, is the solution of (3.7)(3.10) with admissible inputs (U c() , Ud()) E Uc x Ud and x(to) = Xo· Note that in the case of (timeinvariant) statedependent impulsive dynamical systems, the available storage is time invariant, that is, Va (to, xo) = Va (xo). Furthermore, the available exponential storage satisfies
Va(to, xo) = 
+ =
inf
(Uc(,),Ud('))' T'2:to
L
kEZ[to,T)
_ecto
+
[iT to
ectkSd(Ud(tk), Yd(t k ))] inf
(UC('),UdO), T'2:0
L
ectsc(uc(t) , Yc(t))dt
[
(T ectsc(uc(t), Yc(t))dt
Jo
ectkSd(Ud(tk), Yd(t k ))]
kEZ[O,T) = ectoVa(xo),
(3.34)
where
Va(xo)
~
inf
(uc(,),Ud('))' T'2:0
[
Jo(T ectsc(uc(t), Yc(t))dt +
L
ectkSd(Ud(tk), Yd(tk))]. (3.35)
kEZ[O,T)
Next, we show that the available storage (respectively, available exponential storage) is finite if and only if g is dissipative (respectively, exponentially dissipative). In order to state this result we require two additional definitions.
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CHAPTER 3
Definition 3.2 Consider the impulsive dynamical system 9 given by (3.7)(3.10) with hybrid supply rate (se, Sd)' A continuous nonnegative definite function Vs : ~ x 1) ~ ~ satisfying Vs(t,O) = 0, t E ~, and
Vs(T,x(T))
~ Vs(to,x(to)) + iT Se(ue(t),Ye(t))dt +
to
L
Sd(Ud(tk), Yd(tk)),
(3.36)
kEZ[to,T)
where x(t), t 2 to, is a solution to (3.7)(3.10) with (ue(t), Ud(tk)) E Ue X Ud and x(to) = xo, is called a storage function for g. A continuous nonnegativedefinite function Vs : ~ x 1) ~ ~ satisfying Vs(t,O) = 0, t E ~, and eETVs(T, x(T))
~ eEto Vs (to, x(to)) + iT eEt Se( ue(t), Ye(t) )dt +
L
to
eEtksd(Ud(tk),Yd(tk)),
(3.37)
kEZ[to,T)
where
E
> 0, is called an exponential storage function for g.
Note that Vs (t, x (t)) is leftcontinuous on [to, (0) and is continuous everywhere on [to, (0) except on an unbounded closed discrete set T = {tl' t2, ... }, where T is the set of times when the jumps occur for x(t), t 2 to.
Definition 3.3 An impulsive dynamical system 9 is completely reachable if for all (to, xo) E ~ x 1), there exist a finite time ti ~ to, square integrable input ue(t) defined on [ti' to], and input Ud(tk) defined on k E lZ[tj,to)' such that the state x(t), t 2 ti, can be driven from X(ti) = to x(to) = Xo.
°
Theorem 3.1 Consider the impulsive dynamical system 9 given by (3.7)(3.10) and assume that 9 is completely reachable. Then 9 is dissipative (respectively, exponentially dissipative) with respect to the hybrid supply rate (se, Sd) if and only if the available system storage Va(to, xo) given by (3.32) (respectively, the available exponential system storage Va (to , xo) given by (3.33)) is finite for all to E ~ and Xo E 1), and Va(t,O) = 0, t E R Moreover, if Va(t, 0) = 0, t E ~, and Va (to , xo) is finite for all to E ~ and Xo E 1), then Va(t, x), (t,x) E ~ x 1), is a storage function (respectively, exponential storage
91
DISSIPATIVITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
function) for g. Finally, all storage functions (respectively, exponential storage functions) Vs (t, x), (t, x) E lR xV, for 9 satisfy 0::; Va(t,x) ::; Vs(t,x),
(t,x)
E
lR x V.
(3.38)
Proof. Suppose Va(t,O) = 0, t E lR, and Va(t, x), (t, x) E lR x V, is finite. Now, it follows from (3.32) (with T = to) that Va(t, x) 2: 0, (t, x) E lR x V. Next, let x(t), t 2: to, satisfy (3.7)(3.10) with admissible inputs (uc(t), Ud(tk)), t 2: to, k E Z[to,t), and x(to) = Xo. Since Va(t,x), (t,x) E lR x V, is given by the infimum over all admissible inputs (u c () , Ud(')) E Uc x Ud and T 2: to in (3.32), it follows that for all admissible inputs (U c() , Ud(')) and t E [to, Tj,
L
 Va(to, xo)::; iT sc( uc(t), yc(t) )dt + Sd(Ud(tk), Yd(tk)) to kEZ[to,T) = it Sc(Uc(S),Yc(s))ds
to
+
iT
L
+
Sd(Ud(tk),Yd(tk))
kEZ[to,t)
Sc(Uc(S),Yc(s))ds +
L
Sd(Ud(tk),Yd(tk)),
kE~~
t
which implies
L
Va(to,xo)  i t sc(uc(t),Yc(t))dt to
: ; iT t
Hence,
Va(to, Xo) +
it to
2: 
+
kEZ[to,t)
sc( uc(s), yc(s ))ds +
sc(uc(t), Yc(t))dt + inf
(uc(·),Ud(·)), T?t
[iT t
Sd(Ud(tk),Yd(tk))
L
kEZ[t,T)
L
Sd(Ud(tk), Yd(tk)).
Sd(Ud(tk), Yd(tk))
kEZ[to,t)
sc(uc(s), Yc(s))ds
L
Sd(Ud(tk), Yd(tk))] kEZ[t,T) = Va(t, x(t)) 2: 0,
(3.39)
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which implies that
it
L
Se(ue(t),Ye(t))dt+
to
Sd(Ud(tk),Yd(tk)) 2 Va(to,XO). (3.40)
kEiE[to,t)
Hence, since by assumption Va(to,O) = 0, to E JR, g is dissipative with respect to the hybrid supply rate (se, Sd). Furthermore, Va(t, x), (t, x) E JR x D, is a storage function for g. Conversely, suppose g is dissipative with respect to the hybrid supply rate (se, Sd) and let to E JR and Xo E D. Since g is completely reachable it follows that there exists i :::; t < to, ue(t), t 2 i, and Ud(tk), k E Z[i,to)' such that x(i) = 0 and x(to) = Xo. Hence, since g is dissipative with respect to the hybrid supply rate (se, Sd) it follows that, for all T 2 to,
0:::;
iT
L
Se(ue(t), Ye(t))dt +
t
L
(to
= it Se(ue(t), Ye(t))dt + +
iT
Sd(Ud(tk), Yd(tk))
kEiE[i,to)
L
Se(ue(t), Ye(t))dt +
to
< W(to, xo) :::;
Sd(Ud(tk), Yd(tk)),
kEiE[to,T)
and hence, there exists W : JR x D 00
Sd(Ud(tk), Yd(tk))
kEiE[i,T)
iT
~
JR such that
Se(ue(t), Ye(t))dt +
to
L
Sd(Ud(tk), Yd(tk)).
kEiE[to,T)
(3.41) Now, it follows from (3.41) that, for all (to, xo) E JR x D,
Va(to, xo) = +
inf
(uc(·),Ud(o)), T;::to
L
[ (T se(ue(t), Ye(t))dt lto
Sd(Ud(tk), Yd(t k ))]
kEiE[to,T)
:::;  W(to, xo),
(3.42)
and hence, the available storage Va(t, x), (t, x) E JR x D, is finite. Furthermore, with x(to) = 0, it follows that for all admissible ue(t),
DISSIPATIVITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
t
~
to, and Ud(tk), k
iT
93
E Z[to,oo)'
Sc(Uc(t), yc(t))dt +
to
L
Sd(Ud(tk), Yd(tk))
~ 0,
T
~ to,
kEZ[to,T)
(3.43)
which implies that
:S 0,
(3.44)
or, equivalently, Va(to, x(to)) = Va (to , 0) :S 0. However, since Va(t, x) ~ 0, (t, x) E lR x V, it follows that Va(to,O) = 0, to E R Moreover, if Vs(t,x), (t,x) E lR x V, is a storage function then it follows that, for all T ~ to and Xo E V,
Vs(to, xo)
~ Vs(T, x(T))
 L
iT to
sc(uc(t), yc(t))dt
Sd(Ud(tk), Yd(tk))
kEZ[to,T)
~
 [loT
L
sc(uc(t), Yc(t))dt +
Sd(Ud(tk), Yd(tk))] '
kEZ[to,T)
which implies
Vs(to,xo)
~+
inf
(uc(·),Ud(·)), T?,to
L
[iT to
sc(uc(t),Yc(t))dt
Sd(Ud(tk), Yd(t k ))]
kEZ[to,T)
= Va(to, xo). Finally, the proof for the exponentially dissipative case follows a 0 similar construction and, hence, is omitted. The following corollary is immediate from Theorem 3.1. Corollary 3.1 Consider the impulsive dynamical system g given by (3.7){3.10) and assume that g is completely reachable. Then g is dissipative (respectively, exponentially dissipative) with respect to the
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CHAPTER 3
hybrid supply rate (se, Sd) if and only if there exists a continuous storage function {respectively, exponential storage function} Vs(t, x), (t, x) E lR x V, satisfying {3.36} (respectively, {3.37}}. Proof. The result follows from Theorem 3.1 with Vs(t, x) (t,x) ElRxV.
= Va(t, x), 0
The next result gives necessary and sufficient conditions for dissipativity and exponential dissipativity over an interval t E (tk, tk+lJ involving the consecutive resetting times tk and tk+1. Theorem 3.2 Assume 9 is completely reachable. 9 is dissipative with respect to the hybrid supply rate (se, Sd) if and only if there exists a continuous, nonnegativedefinite function Vs : lR x V t lR such that, for all k E Z+,
Vs(i, x(i))  Vs(t, x(t))
~ 1£ Se(Ue(S), Ye(s))ds,
tk < t
~ i ~ tk+1, (3.45)
Vs(tk, X(tk)
+ fd(x(tk)) + Gd(X(tk))Ud(tk)) 
Vs(tk, X(tk))
~ Sd(Ud(tk), Yd(tk)).
(3.46)
Furthermore, 9 is exponentially dissipative with respect to the hybrid supply rate (se, Sd) if and only if there exist a continuous, nonnegativedefinite function Vs : lR x V t lR and a scalar c > 0 such that eeiVs(i,x(i))  ectVs(t,x(t))
~
1£
ecSSe(Ue(S),Ye(s))ds,
tk < t ~ i ~ tk+b (3.47) Vs(tk, X(tk) + fd(x(tk)) + Gd(X(tk))Ud(tk))  Vs(tk, X(tk)) ~ Sd(Ud(tk), Yd(tk)). (3.48) Finally, 9 is loss less with respect to the hybrid supply rate (se, Sd) if and only if there exists a continuous, nonnegativedefinite function Vs : lR x V t lR such that {3.45} and {3.46} are satisfied as equalities. Proof. Let k E Z+ and suppose 9 is dissipative with respect to the hybrid supply rate (se, Sd). Then, there exists a continuous nonnegativedefinite function Vs : lR x V t lR such that (3.36) holds. Now, since for tk < t ~ i ~ tk+1, lE[t,£) = 0, (3.45) is immediate.
DISSIPATIVITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
95
Next, note that
which, since Z[tk, t+) = {k}, implies (3.46). k
Conversely, suppose (3.45) and (3.46) hold, let t ;: : t ;:::: 0, and let Z[t,i) = {i, i + 1, ... ,j}. (Note that if Z[t,i) = 0 the converse is a direct consequence of (3.36).) In this case, it follows from (3.45) and (3.46) that ~ ~ Vs(t,~ x(t))  Vs(t, x(t))  Vs(t,~ x(t))  Vs(tj+ ,x(tj+))
+ Vs(tj+ ,x(tj+))
 Vs(tj_l' X(tj_l)) +Vs(tj_l,X(tj_l))  ...  Vs(tt,x(ti)) +Vs(tt, x(ti))  Vs(t, x(t)) = Vs(t, x(t))  Vs(tj+ ,x(tj+ )) +Vs(tj, x(tj) + Jd(X(tj)) + Gd(X(tj))Ud(tj))  Vs(tj, x(tj)) + Vs(tj, x(tj))  Vs(tj_l' X(tj_l)) + ... +Vs(ti,x(td + Jd(X(ti)) + Gd(X(ti))Ud(ti))  Vs(ti, x( ti)) + Vs(ti, X(ti))  Vs( t, x( t)) ~
~
~ It+[t se(ue(s), Ye(s))ds + Sd(Ud(tj), Yd(tj)) ]
+ [tj se(ue(s),Ye(s))ds + ... ltj_l +Sd(Ud(ti), Yd(ti))
lti it
+ =
se(ue(s), Ye(s))ds
Se( ue(s), Ye(S) )ds
+
L
Sd(Ud(tk), Yd(tk)),
kEZ[t,t)
which implies that g is dissipative with respect to the hybrid supply rate (se, Sd).
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Finally, similar constructions show that g is exponentially dissipative (respectively, lossless) with respect to the hybrid supply rate (se, Sd) if and only if (3.47) and (3.48) are satisfied (respectively, (3.45) and (3.46) are satisfied as equalities). 0 If in Theorem 3.2 Vs (., x(·)) is continuously differentiable almost everywhere on [to, 00) except on an unbounded closed discrete set T = {tl, t2, ... }, where T is the set of times when jumps occur for x(t), then an equivalent statement for dissipativeness of the impulsive dynamical system g with respect to the hybrid supply rate (se, Sd) is
Vs(t, x(t)) ~ se(ue(t), Ye(t)), tk < t ~ tk+l, ~Vs(tk' X(tk)) ~ Sd(Ud(tk), Yd(tk)), k E Z+,
(3.50) (3.51)
where Vs(·,·) denotes the total derivative of Vs(t,x(t)) along the state trajectories x(t), t E (tk' tk+l], of the impulsive dynamical system (3.7)(3.10) and ~ Vs(tk, X(tk)) ~ Vs(tt, x(tt))  Vs(tk, X(tk)) = Vs(tk, X(tk)+ fd(x(tk))+Gd(x(tk))ud(tk)) Vs(tk, X(tk)), k E Z+, denotes the difference of the storage function Vs (t, x) at the resetting times tk, k E Z+, of the impulsive dynamical system (3.7)(3.10). Furthermore, an equivalent statement for exponential dissipativeness of the impulsive dynamical system g with respect to the hybrid supply rate (se, Sd) is given by
Vs(t, x(t))
+ cVs(t, x(t)) ~ se(ue(t), Ye(t)), tk < t
~ tk+1,
(3.52)
and (3.51). The following theorem provides sufficient conditions for guaranteeing that all storage functions (respectively, exponential storage functions) of a given dissipative (respectively, exponentially dissipative) impulsive dynamical system are positive definite. For this result we need the following definition.
Definition 3.4 An impulsive dynamical system g given by (3.7)(3.10) is zerostate observable if (ue(t), Ud(tk)) == (0,0) and (Ye(t), Yd(tk)) == (0,0) implies x(t) == 0. An impulsive dynamical system g given by (3.7)(3.10) is strongly zerostate observable if ue(t) == and Ye(t) == implies x(t) == 0. Finally, an impulsive system g is minimal if it is zerostate observable and completely reachable.
°
°
Note that strong zerostate observability is a stronger condition than zerostate observability. In particular, strong zerostate observability implies zerostate observability; however, the converse is not necessarily true.
DISSIPATIVITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
97
Theorem 3.3 Consider the nonlinear impulsive dynamical system 9 given by (3.7)(3.10) and assume that 9 is completely reachable and zerostate observable. Furthermore, assume that 9 is dissipative (respectively, exponentially dissipative) with respect to the hybrid supply rate (se, Sd) and there exist functions /'i,e : Yc  t Ue and /'i,d : Yd  t Ud such that /'i,e(O) = 0, /'i,d(O) = 0, se(/'i,e(Ye), Ye) < 0, Ye =1= 0, and Sd(/'i,d(Yd),Yd) < 0, Yd =1= 0. Then all the storage functions (respectively, exponential storage functions) v;, (t, x), (t, x) E R x V, for 9 are positive definite, that is, v;,(., 0) = and v;,(t,x) > 0, (t,x) E R x V, x =1= 0.
°
Proof. It follows from Theorem 3.1 that the available storage Va(t, x), (t, x) E R x V, is a storage function for g. Next, suppose, ad absurdum, there exists (to, xo) E R x V such that Va(to, xo) = 0, Xo =1= 0, or, equivalently,
Furthermore, suppose there exists [ts, tr) C R such that Ye(t) =1= 0, t E [ts, tr), or Yd(tk) =1= 0, for some k E Z+. Now, since there exists /'i,e : Ye  t Ue and /'i,d : Yd  t Ud such that /'i,e(O) = 0, /'i,d(O) = 0, se(/'i,e(Ye), Ye) < 0, Ye =I= 0, and Sd(/'i,d(Yd), Yd) < 0, Yd =1= 0, the infimum in (3.53) occurs at a negative value, which is a contradiction. Hence, Ye(t) = for almost all t E R, and Yd(tk) = for all k E Z+. Next, since 9 is zerostate observable it follows that x = 0, and hence, Va(t,x) = if and only if x = 0. The result now follows from (3.38). Finally, the proof for the exponentially dissipative case is similar and, hence, is omitted. 0
°
°
°
Next, we introduce the concept of a required supply for a nonlinear impulsive dynamical system given by (3.7)(3.10). Specifically, define the required supply Vr (to, xo) of the nonlinear impulsive dynamical system 9 by
Vr(to, xo)
~ (uc(·),Ud(·)), inf [ (to se(ue(t), Ye(t))dt T:Sto iT +
L kEZ[T,tol
Sd(Ud(tk), Yd(tk))] '
(3.54)
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where x(t), t ~ T, is the solution of (3.7)(3.10) with x(T) = 0 and x(to) = Xo. It follows from (3.54) that the required supply of a nonlinear impulsive dynamical system is the minimum amount of generalized energy which can be delivered to the impulsive dynamical system in order to transfer it from an initial state x(T) = 0 to a given state x(to) = Xo. Similarly, define the required exponential supply of the nonlinear impulsive dynamical system 9 by Vr(to, xo)
~ (Uc(),Ud(·)),T:::;to inf [ (to ect se( ue(t), Ye(t) )dt iT +
L
eciksd(Ud(tk), Yd(tk))] ' (3.55)
kEZ[T,tO)
where E > 0 and x(t), t ~ T, is the solution of (3.7)(3.10) with x(T) = 0 and x(to) = Xo. Note that since, with x(to) = 0, the infimum in (3.54) is zero it follows that Vr(to, O) = 0, to E R Next, using the notion of the required supply, we show that all storage functions are bounded from above by the required supply and bounded from below by the available storage. Hence, as in the case of dynamical systems with continuous flows [166], a dissipative impulsive dynamical system can deliver to its surroundings only a fraction of its stored generalized energy and can store only a fraction of the generalized work done to it. Theorem 3.4 Consider the nonlinear impulsive dynamical system 9 given by (3.7)(3.10) and assume that 9 is completely reachable. Then 9 is dissipative (respectively, exponentially dissipative) with respect to the hybrid supply rate (se, Sd) if and only if 0 :S Vr(t, x) < 00, t E JR, x E V. Moreover, if Vr(t, x) is finite and nonnegative for all (to,xo) E JR x V, then Vr(t, x), (t,x) E JR x V, is a storage function (respectively, exponential storage function) for g. Finally, all storage functions (respectively, exponential storage functions) l/;, (t, x), (t, x) E JR xV, for 9 satisfy
o ~ Va(t, x)
~ l/;,(t, x)
:S Vr(t, x) < 00,
(t, x) E JR x V.
(3.56)
Proof. Suppose 0 ~ Vr(t,x) < 00, (t,x) E JR x V. Next, let x(t), t E JR, satisfy (3.7)(3.10) with admissible inputs (ue(t), Ud(tk)), t E JR, k E Z[to,t), and x(to) = Xo. Since Vr(t, x), (t, x) E JR x V, is given by the infimum over all admissible inputs (UeO,Ud(')) E Ue X Ud and T ~ to in (3.54), it follows that for all admissible inputs (UeO,Ud('))
DISSIPATIVITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
and T
~
t
~
99
to,
and hence,
Vr(to, xo)
~ (uc(·),Ud(·», inf [ (t Se(Ue(S), Ye(s))ds TS,t iT +
L
+
ito
Sd(Ud(tk), Yd(t k ))]
kEZ[T,t)
= Vr(t, x(t)) +
+
L
Se(Ue(S), Ye(s))ds +
L
ito
Sd(Ud(tk), Yd(tk))
kEZ[t,to)
Se(Ue(S), Ye(s))ds
Sd(Ud(tk), Yd(tk)),
(3.57)
kEZ[t,to)
which shows that Vr(t,x), (t,x) E ~ x V, is a storage function for g, and hence, g is dissipative. Conversely, suppose g is dissipative with respect to the hybrid supply rate (se, Sd) and let to E ~ and Xo E 'O. Since g is completely reachable it follows that there exist T < to, ue(t), T ~ t < to, and Ud(tk), k E Z[T,oo), such that x(T) = 0 and x(to) = Xo. Hence, since g is dissipative with respect to the hybrid supply rate (se, Sd) it follows that, for all T ~ to,
o~ and hence,
to
se(ue(t), Ye(t))dt +
L kEZ[T,tO)
Sd(Ud(tk), Yd(tk)),
(3.58)
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CHAPTER 3
L
+
Sd(Ud(tk), Yd(tk))] ,
(3.59)
kEZ[T,to)
which implies that
o ~ Vr(to, xo) < 00,
(to, xo) E lR. x 1).
(3.60)
Next, if Vs(',·) is a storage function for g, then it follows from Theorem 3.1 that
o ~ Va(t, x) ~ Vs(t, x),
(t, x) E lR. x 1).
(3.61)
Furthermore, for all T E lR. such that x(T) = 0 it follows that Vs(to, xo)
~ Vs(T, 0)+
[0
se(ue(t), Ye(t))dt+
L
Sd(Ud(tk), Yd(tk)),
kEZ[T,tO)
(3.62) and hence, Vs(to, xo)
~ (uc(·),Ud(·)), inf [ (to se(ue(t), Ye(t))dt T5:.to iT
L
+
Sd(Ud(tk), Yd(t k ))]
kEZ[T,tO)
= Vr(to,xo) to and T_ > to such that x( T_) = 0 and x(T+) = O. Proof. Suppose 9 is lossless with respect to the hybrid supply rate (se, Sd). Since 9 is completely reachable to and from the origin it follows that, for every Xo E D, there exist T_, T+ > 0, ue(t) E Ue, t E [T_, T+J, Ud(tk) E Ud, k E such that x( T_) = 0, x(T+) = 0, and x(O) = Xo. Now, it follows that 0=
I: It;
Z[L,T+),
Se(ue(t), Ye(t))dt +

=
2':
Se(ue(t), Ye(t))dt +
L
Sd(Ud(tk), Yd(tk))
Z[T_,tO)
iT+ Se(ue(t), Ye(t))dt + L to
Z
inf (to (uc{·),Ud{·)), T90 iT
+
Sd(Ud(tk), Yd(tk))
Z[L,T+)

+
L
inf
(Uc(,),Ud{'))' T?to
Sd(Ud(tk), Yd(tk))
[to,T+)
Se(ue(t),Ye(t))dt +
Z
L
Sd(Ud(tk),Yd(tk))
[T,tO)
iT Se(ue(t),Ye(t))dt + L to
Z
Sd(Ud(tk),Yd(tk))
[to,T)
= Vr(to, xo)  Va(to, xo),
(3.65)
which implies that Vr(to, xo) ~ Va(to, xo), (to, xo) E ~ x D. However, since by definition 9 is dissipative with respect to the hybrid supply rate (se, Sd) it follows from Theorem 3.4 that Va(to, xo) ~ Vr(to, xo), (to, xo) E ~xD, and hence, every storage function Vs(to, xo), (to, xo) E ~ x D, satisfies Va(to, xo) = Vs(to, xo) = Vr(to, xo). Furthermore, it follows that the inequality in (3.65) is indeed an equality, which implies (3.64). Next, let to, t, T 2': 0 be such that to < t < T, x(T) = O. Hence, it follows from (3.64) that
0= Vs(to, x(to))
+
iT sc(ue(t), Ye(t))dt + L to
Z[to,T)
Sd(Ud(tk), Yd(tk))
102
= Vs(to, x(to)) +
+
iT
it
CHAPTER 3
Se(ue(t), Ye(t))dt +
to
= Vs(to, x(to))
+
it
Sd(Ud(tk), Yd(tk))
Z[to,t)
Se(ue(t), Ye(t))dt +
t
L
L
Sd(Ud(tk), Yd(tk))
Z[t,T)
Se( ue(t), Ye(t) )dt +
~
L
Sd( Ud (tk), Yd(tk))
z~~
 Vs(t, x(t)), which implies that (3.36) is satisfied as an equality. Conversely, if there exists a storage function Vs(t, x), (t, x) E lR x V, satisfying (3.36) as an equality it follows from Corollary 3.1 that 9 is dissipative with respect to the hybrid supply rate (se, Sd). furthermore, for every ue(t) E Ue, t ~ to, Ud(tk) E Ud, k E Z[to,t) and x(to) = x(t) = 0, it follows from (3.36) (satisfied as an equality) that
which implies that (Se,Sd).
9 is lossless with respect to the hybrid supply rate 0
Finally, as a direct consequence of Theorems 3.1 and 3.4, we show that the set of all possible storage functions of an impulsive dynamical system forms a convex set. An identical result holds for exponential storage functions.
Proposition 3.1 Consider the nonlinear impulsive dynamical system E lR x V, and required supply Vr(t,x), (t,x) E lR x V, and assume that 9 is
9 given by (3.7){3.10) with available storage Va(t, x), (t, x) completely reachable. Then
Vs(t, x) ~ aVa(t, x)
+ (1  a)Vr(t, x), a
E [0,1]'
(3.66)
is a storage function for g.
Proof. The result is a direct consequence of the complete reachability of 9 along with the dissipation inequality (3.36) by noting that if Va(t, x) and Vr(t, x) satisfy (3.36), then Vs(t, x) satisfies (3.36). 0
DISSIPATIVITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
103
3.3 Extended KalmanYakubovichPopov Conditions for Impulsive Dynamical Systems
In this section, we show that dissipativeness of an impulsive dynamical system can be characterized in terms of the system functions feO, GeO, heO, JeO, fdO, G d (·), hd('), and Jd(')' First, we concentrate on the theory for dissipative timedependent impulsive dynamical systems. Since in the case of dissipative statedependent impulsive dynamical systems it follows from Assumptions Al and A2 that, for S = [0,00) x Z, the resetting times are well defined and distinct for every trajectory of (3.27) and (3.28), the theory of dissipative statedependent impulsive dynamical systems closely parallels that of dissipative timedependent impulsive dynamical systems, and hence, many of the results are similar. In the cases where the results for dissipative statedependent impulsive dynamical systems deviate markedly from their timedependent counterparts, we present a thorough treatment of these results. For the results in this section we consider the special case of dissipative impulsive systems with quadratic hybrid supply rates and set Ue = jRmc and Ud = jRmd. Specifically, let Qe E §lc, Se E jRlcxmc, Re E §mc, Qd E §ld, Sd E jR1dxmd, and Rd E §md be given and assume se(ue, Ye) = y'[QeYe + 2y,[ Seue + U'[ Reue and Sd(Ud, Yd) = yJQdYd + 2yJ SdUd + RdUd· For simplicity of exposition, in the remainder of the chapter we assume that for timedependent impulsive dynamical systems the storage functions do not depend explicitly on time. This corresponds to the case in which 9 is time varying but the energy storage mechanism does not reflect this. However, this is not to say that system energy dissipation does not have a timevarying character. Furthermore, we assume that there exist functions /'i,e : jR1c + jRmc and /'i,d : jR1d + jRmd such that /'i,e(O) = 0, /'i,d(O) = 0, Se (/'i,e(Ye), Ye) < 0, Ye i 0, and Sd(/'i,d(Yd), Yd) < 0, Yd i 0, so that the storage function v;,(x), x E jRn, is positive definite, and we assume that Vs (.) is continuously differentiable.
ud
Theorem 3.6 Let Qe E §lc, Se E jR1cxmc, Re E §mc, Qd E §ld, Sd E jR1d xmd , and Rd E §md. If there exist functions Vs : jRn + jR, Le : jRn + jRPc, Ld : jRn + jRPd, We : jRn + jRPcxmc, Wd : jRn + jRPdxmd, P 1Ud : jRn + jR1xmd, and P2Ud : jRn + Nmd such that v;,(.) is continuously differentiable and positive definite, Vs(O) = 0, Vs(x
+ fd(x) + Gd(X)Ud) = v;, (x + fd(x)) + P 1Ud (X)Ud + ud P 2Ud (X)Ud, xE
jRn,
Ud E
jRmd,
(3.67)
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CHAPTER 3
and, for all x 0=
E
JR.n ,
v: (x )fe(x)  hJ (x )Qehe(x) + LJ (x )Le(x),
O=! V:(x)Ge(x) 
hJ(x)(QeJe(x) + Se) + LJ(x)We(x),
0= Rd + SJ Jd(X)
+ JJ(X)Sd + JJ(X)QdJd(X)  P2Ud (x)
(3.68) (3.69)
0= Re + S; Je(x) + J;(x)Se + J;(X)QeJe(x)  W;(x)We(x), (3.70) 0= Vs(x + fd(x))  Vs(x)  hJ (X)Qdhd(X) + LJ(x)Ld(X), (3.71) 0= !PIUd(X)  hJ(X)(QdJd(X) + Sd) + LJ(X)Wd(X), (3.72)
WJ(X)Wd(X),
(3.73)
then the nonlinear impulsive system g given by (3.14}(3.17) is dissipative with respect to the quadratic hybrid supply rate (se(u e, Ye), Sd(Ud, Yd)) = (yJ QeYe + 2yJ Seue + uJ Reue, yJ QdYd + 2yJ SdUd + uJ RdUd). If, alternatively, Ze(X) ~ Rc + S; Je(x)
+ i!(x)Se + J;(x)QeJe(x) > 0, x E JR. n , (3.74)
and there exist a continuously differentiable function Vs : JR.n  t JR. and matrix functions P 1Ud : JR.n  t JR.1xmd and P2Ud : JR.n  t Nmd such that Vs(·) is positive definite, Vs(O) = 0, (3.67) holds, and for all x E JR.n , Zd(X) ~ Rd + SJ Jd(X)
+ JJ(X)Sd + JJ(X)QdJd(X)  P2Ud (x) > 0, (3.75)
o~ V:(x)fe(x) 
hJ(x)Qehe(x) +[!V:(x)Ge(x)  hJ(x)(QeJe(X) + Se)] .Z;l(x)[!V:(x)Ge(x)  hJ(x)(QeJe(x) + Se)]T,
(3.76)
o~ Vs(x + fd(x)) 
Vs(x)  hJ(X)Qdhd(X) +[!PIUd(X)  hJ(X)(QdJd(X) + Sd)] ,Zci1(x)[!Plud(X)  hJ(X)(QdJd(X) + Sd)]T,
(3.77)
then g is dissipative with respect to the quadratic hybrid supply rate (se(ue,Ye), Sd(Ud,Yd)) = (yJQeYe + 2yJSeue + uJReue, yJQdYd + 2yJ SdUd + uJ RdUd). Proof. For any admissible input ue(t), t, i E JR., tk < t :S and k E Z+, it follows from (3.68)(3.70) that
Vs(x(i))  Vs(x(t)) =
1£
Vs(x(s))ds
i:s tk+1,
DISSIPATIVITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
105
:S li [Vs(X(S)) + [Lc(x(s)) + Wc(X(S))Uc(S)JT ·[Lc(x(s)) + Wc(X(S))UC(S)J] ds =l
i
[V:(X(S))(fC(X(S))
+ Gc(X(S))Uc(S))
+LJ(x(s))Lc(x(s)) + 2LJ(x(s))Wc(X(S))U c(S) +UJ (S )W; (X(S ))Wc(X(S ))Uc(S )Jds = li[hJ(X(S))QchC(X(S))
+ 2hJ(x(s))(Sc
+QcJc(X(S )))Uc(S) +UJ(S)(J;(X(S))QcJc(X(S)) + S; Jc(X(S)) +J;(X(S))Sc + Rc)uc(s)Jds =l
i
[yJ(S)QCYC(S)
+ 2yJ(s)ScUc(S)
+UJ (S )Rcuc(s )Jds = li sc(uc(s),Yc(s))ds,
(3.78)
where x(t), t E (tk' tk+1], satisfies (3.14) and Vs() denotes the total derivative of the storage function along the trajectories x(t), t E (tk' tk+1], of (3.14). Next, for any admissible input Ud(tk), tk E lR, and k E Z+, it follows that ~Vs(X(tk))
= Vs(X(tk) + fd(X(tk)) + Gd(X(tk))Ud(tk))  Vs(X(tk)), (3.79)
where ~ Vs (.) denotes the difference of the storage function at the resetting times tk, k E Z+, of (3.15). Hence, it follows from (3.71)(3.73), the structural storage function constraint (3.67), and (3.79), that for all x E lRn and Ud E lRmd , ~Vs(x)
= Vs(x + fd(X) + Gd(X)Ud)  Vs(x) = Vs(x + fd(X))  Vs(x) + P 1Ud (X)Ud + uJ P2Ud (X)Ud =hJ(X)Qdhd(X)  LJ(x)Ld(x) +2[hJ(x)(QdJd(X) + Sd)  LJ(X)Wd(X)JUd +UdlRd + SJ Jd(X) + Jl(x)Sd + Jl(x)Q and p ~ 1 such that allxll P ~ Vs(x) ~ (3llxII P , x E JR.n, then the undisturbed nonlinear impulsive dynamical system (3.14)(3.17) is exponentially stable. Next, we provide necessary and sufficient conditions for the case where 9 given by (3.14)(3.17) is loss less with respect to a quadratic hybrid supply rate (sc, Sd).
°
°
°
§lc, Sc E JR.1cxffic, Rc E §ffic, Qd E §ld, Sd E Then the nonlinear impulsive system 9 given by (3.14}(3.17) is lossless with respect to the quadratic hybrid supply rate (sc(uc,yc),Sd(Ud,Yd)) = (y'[Qcyc+2y'[Scuc+u'JRcuc, yJQdYd+ 2yJ SdUd + uJ RdUd ) if and only if there exist functions Vs : JR.n  t JR., P1Ud : JR.n  t JR.1Xffid, and P2Ud : JR.n  t Nffid such that Vs(·) is continuously differentiable and positive definite, Vs(O) = 0, and, for all x E JR. n , (3.67) holds and
Theorem 3.7 Let Qc E
JR.1dxffid,
and Rd
E §ffid.
0= V:(x)fc(x)  h'[(x)Qchc(x), 0= ~V:(x)Gc(x)  h'[(x)(QcJc(x) + Sc),
(3.87) (3.88)
109
DISSIPATIVITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
O=Rc + s'!Jc(X) + i!(x)Sc + i!(x)QcJc(x), 0= Ys(x + fd(X))  Ys(x)  hJ(X)Qdhd(X), 0= 1Plud (x)  hJ(X)(QdJd(X) + Sd), 0= Rd + SJ Jd(X)
(3.89) (3.90) (3.91)
+ JJ(X)Sd + JJ(X)QdJd(X) 
P 2Ud (x).
(3.92)
If, in addition, Ys(.) is twotimes continuously differentiable, then P1Ud (x)
= V;(x + fd(X))Gd(x),
P2Ud (x)
= 1GJ(x)V;'(x + fd(X))Gd(x).
Proof. Sufficiency follows as in the proof of Theorem 3.6. To show necessity, suppose that the nonlinear impulsive dynamical system 9 is lossless with respect to the quadratic hybrid supply rate (sc, Sd). Then, it follows from Theorem 3.2 that for all k E Z+,
Ys(x(f))  Ys(x(t)) =
it
tk < t::; f::; tk+1,
sc(uc(s),yc(s))ds,
(3.93)
and
Ys(X(tk)+ fd(X(tk))+Gd(X(tk))Ud(tk)) = Ys(X(tk))+Sd(Ud(tk), Yd(tk)). Now, dividing (3.93) by f  t+ and letting f to
Vs(x(t)) = V;(x(t)) [fc(x(t))
(3.94) t+, (3.93) is equivalent
t
+ Gc(x(t))uc(t)] = sc(uc(t), Yc (t)) , tk < t ::; tk+1'
Next, with t
= 0, it follows from (3.95) that
V;(xo)[fc(xo)
+ Gc(xo)uc(O)] = sc(uc(O), Yc(O)), Xo uc(O)
Since Xo E
jRn
V;(x)[Jc(x)
(3.95)
E jRn,
E jRmc.
(3.96)
is arbitrary, it follows that
+ Gc(x)uc]= y'fQcyc + 2y'f Scuc + u'f Rcuc = h'f(x)Qchc(x) + 2h'f(x) (QcJc(x) + Sc)uc +u'f(Rc + s'f Jc(x) + if(x)Sc +J;(x)QcJc(x))uc,
x
E jRn,
U
cE
Now, equating coefficients of equal powers yields (3.87)(3.89).
jRmc.
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Next, it follows from (3.94) with k
Vs(X(tl)
= 1 that
+ fd(X(tl)) + Gd(X(tl))Ud(tl)) = Vs(X(tl)) +Sd(Ud(tl), Yd(tl)). (3.97)
Now, since the continuoustime dynamics (3.14) are Lipschitz continuous on 1), it follows that for arbitrary x E jRn there exists Xo E jRn such that X(tl) = x. Hence, it follows from (3.97) that
Vs(x + fd(X) + Gd(X)Ud) = Vs(x) + yJQdYd + 2yJ SdUd + uJ RdUd = Vs(x) + hJ(X)Qdhd(X) +2hJ(x)(QdJd(X) + Sd)Ud +UJ(Rd + SJ Jd(X) + JJ(X)Sd +JJ(X)QdJd(X))Ud, x E jRn, Ud E jRmd. (3.98)
Since the righthand side of (3.98) is quadratic in Ud it follows that Vs(x + fd(X) + Gd(X)Ud) is quadratic in Ud, and hence, there exists P 1Ud : jRn t jR1xffid and P2Ud : jRn t Nffid such that
Vs(X + fd(X) + Gd(X)Ud) = Vs(x + fd(X)) + P 1Ud (X)Ud +uJ P2Ud (X)Ud'
(3.99)
Now, using (3.99) and equating coefficients of equal powers in (3.98) yields (3.90)(3.92). Finally, if Vs (.) is twotimes continuously differentiable, applying a Taylor series expansion on (3.99) about Ud = 0 yields P 1Ud (x)
=
8Vs(x + fd~) + Gd(X)Ud) Ud
I
=
V;(x + fd(X))Gd(X),
Ud=O
(3.100)
o
which proves the result.
The following result presents the statedependent analog of Theorem 3.6. Theorem 3.8 Let Qe E Sd E jRldXffid,
and
§le,
Rd E §ffid.
Se E jRleXffie,
Rc
E §ffie, Qd E §ld,
If there exist functions Vs :
jRn
t
jR,
DISSIPATIVITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
111
Le : ]Rn t ]RPc, Ld : ]Rn t ]RPd, We : ]Rn t ]RPcxmc, Wd : ]Rn t ]RPdxmd, PI Ud :]Rn t ]R1xmd, and P2Ud :]Rn t Nmd such that Vs(·) is continuously differentiable and positive definite, Vs(O) = 0, Vs(x + fd(X) + Gd(X)Ud) = Vs(x + fd(X)) + P1Ud (X)Ud + uI P2Ud (X)Ud, x E Zx, Ud E ]Rmd, (3.102) and 0= V:(x)fe(x)  h~(x)Qehe(x) + L~(x)Le(x), x ¢ Zx, (3.103) 0= !V:(x)Ge(x)  h~(x)(QeJe(X) + Se) + L~(x)We(X), x ¢ Zx, (3.104)
O=Rc + SJJe(x) + i!(x)Se + i!(x)QeJe(x)  WJ(x)We(x), x ¢ Zx, (3.105) 0= Vs(x + fd(X))  Vs(x)  hI(x)Qdhd(X) + LI(x)Ld(X), x E Zx, (3.106)
0= !P1Ud(X)  hI(x)(QdJd(X) + Sd)
+ LI(x)Wd(X),
x
E
Zx, (3.107)
o=Rd + SJ Jd(X) + JJ(X)Sd + JJ(X)QdJd(X) WJ(X)Wd(X),
x
E
P2Ud (x)
Zx,
(3.108)
then the nonlinear impulsive system 9 given by (3.27}(3.30) is dissipative with respect to the quadratic hybrid supply rate (se(Ue, Ye), Sd( Ud, Yd)) = (y;QeYe + 2y;Seue + u; Reue, yJQdYd + 2yISdud + uI RdUd). If, alternatively, Ze(x) ~Re + s'!Je(x) + JJ(x)Se + J;(X)QeJe(x) > 0, x ¢ Zx, (3.109)
and there exist a continuously differentiable function Vs : ]Rn t ]R and matrix functions P1Ud : ]Rn t ]R1xmd and P2Ud : ]Rn t Nmd such that Vs(·) is positive definite, Vs(O) = 0, (3.102) holds, and Zd(X) ~ Rd + SJ Jd(X) + JJ(X)Sd + JJ(X)QdJd(X)  P2Ud (x) > 0, x E Zx, (3.110) o~ V:(x)fe(x)  h~(x)Qehe(x) +[!V:(x)Ge(x)  h~(x)(QeJe(x) + Se)] .Z;l(x)[! V:(x)Ge(x)  h~(x)(QeJe(x)
+ Se)]T,
x ¢ Zx, (3.111)
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CHAPTER 3
+[~PIUd (X)  hJ(X)(QdJd(X)
+ Sd)]
.Zdl(X)[~Plud(X)  hI(x)(QdJd(x)
+ Sd)]T,
X E Zx, (3.112)
then 9 is dissipative with respect to the quadratic hybrid supply rate (se(u e, Ye), Sd(Ud, Yd)) = (yJQeYe + 2yJ Seue + uJ Rcue, YdQdYd + 2Yd SdUd + uI RdUd)· Proof. The proof is similar to the proof of Theorem 3.6.
0
Next, we provide two definitions of nonlinear impulsive dynamical systems which are dissipative (respectively, exponentially dissipative) with respect to hybrid supply rates of a specific form.
Definition 3.5 An impulsive dynamical system 9 of the form (3.7)(3.10) with me = Ie and md = ld is passive (respectively, exponentially passive) if 9 is dissipative (respectively, exponentially dissipative) with respect to the hybrid supply rate (se(ue,Ye),Sd(Ud,Yd)) = (2uJ Ye, 2uI Yd). Definition 3.6 An impulsive dynamical system 9 of the form (3.7)(3.10) is nonexpansive (respectively, exponentially nonexpansive) if 9 is dissipative (respectively, exponentially dissipative) with respect to the hybrid supply rate (se(u e, Ye), Sd(Ud, Yd)) = (,,!~uJ ueyJYe, ,,!JUIUd  YdYd), where "!e,"!d > are given.
°
A mixed passivenonexpansive formulation of 9 can also be considered. Specifically, one can consider impulsive dynamical systems 9 which are dissipative with respect to hybrid supply rates of the form (se(ue,Ye),Sd(Ud,Yd)) = (2uJYe, ,,!JUIUd  YdYd) , where "!d > 0, and vice versa. Furthermore, supply rates for input strict passivity, output strict passivity, and inputoutput strict passivity can also be considered [74]. However, for simplicity of exposition we do not do so here. The following results present the nonlinear versions of the KalmanYakubovichPopov positive real lemma and the bounded real lemma for nonlinear impulsive systems 9 of the form (3.14)(3.17).
Corollary 3.2 Consider the nonlinear impulsive system 9 given by (3.14)(3.17). If there exist functions Vs : jRn t jR, Le : jRn t jRPc, Ld : jRn t jRPd, We : jRn t jRPcxmc, Wd : jRn t jRPdxmd, P1Ud :
DISSIPATIVITY THEORY FOR NONLINEAR IMPULSIVE DYNAMICAL SYSTEMS
113
JR.n t JR.l xmd , and P2Ud : JR.n t Nmd such that Vs (.) is continuously differentiable and positive definite, Vs(O) = 0, Vs(x
+ fd(X) + Gd(X)Ud) = Vs(x + fd(X)) + P1Ud (X)Ud + uJ P2Ud (X)Ud, x E JR.n ,
and, for all x
E
Ud E JR.md,
(3.113)
JR.n ,
0= V;(x)fe(x) + LJ(x)Le(x), O=! V;(x)Ge(x)  hJ(x) + LJ(x)We(x),
(3.114) (3.115)
O=Je(x) + JJ'(x)  WJ'(x)We(x), 0= Vs(x + fd(X))  Vs(x) + LJ(x)Ld(x), 0= !P1Ud(X)  hJ(x) + LJ(X)Wd(X),
(3.116) (3.117) (3.118)
0= Jd(X)
+ JJ(x)
 P2Ud (x)  WJ(X)Wd(X),
(3.119)
then g is passive. If, alternatively, Je(x) + JJ' (x) > 0, x E JR.n, and there exist a continuously differentiable function Vs : JR.n t JR. and matrix functions P1Ud : JR.n t JR.lxmd and P2Ud : JR.n t Nmd such that Vs() is positive definite, Vs(O) = 0, {3.113} holds, and for all x E JR. n ,
+ JJ (x)  P2Ud (x), (3.120) o~ V;(x)fe(x) + [!V;(x)Ge(x)  hJ(x)] (3.121) ·[Je(x) + JJ'(x)]l[!V;(x)Ge(x)  hJ(x)]T, o ~ Vs(x + fd(X))  Vs(x) + [~Plud (x)  hJ (x)] ·[Jd(X) + JJ(x)  P2Ud(X)rl[!Plud(X)  hJ(x)]T,
0< Jd(X)
(3.122)
then g is passive. Proof. The result is a direct consequence of Theorem 3.6 with le = me, ld = md, Qe = 0, Qd = 0, Se = I mc ' Sd = I md , Rc = 0, and Rd = O. Specifically, with Ke(Ye) = Ye and Kd(Yd) = Yd, it follows that Se(Ke(Ye), Ye) = 2yJYe < 0, Ye =1= 0, and Sd(Kd(Yd), Yd) = 2yJ Yd < 0, Yd =1= 0, so that all of the assumptions of Theorem 3.6 0 are satisfied. Corollary 3.3 Consider the nonlinear impulsive system g given by {3.14}{3.11}. If there exist functions Vs: JR.n t JR., Le: JR.n t JR.Pc, Ld : JR.n t JR.Pd, We : JR.n t JR.Pcxmc, Wd : JR.n t JR.Pdxmd, P1Ud :
CHAPTER 3
114
IR n t 1R 1xmd , and P2Ud : IR n t fi:i[md such that Vs(·) is continuously differentiable and positive definite, Vs(O) = 0, Vs(x
+ fd(X) + Gd(X)Ud) = Vs(x + fd(X)) + P1Ud (X)Ud +UJp2Ud (X)Ud,
and, for all x
E
x E IR n ,
Ud E
IRmd , (3.123)
IR n ,
0= V:(x)fc(x) + hZ(x)hc(x) + LZ(x)Lc(x), 0= ~V:(x)Gc(x) + hZ(x)Jc(x) + LZ(x)Wc(x),
(3.124) (3.125)
O="(~Ime  i[(x)Jc(x)  W::r(x)Wc(x), 0= Vs(x + fd(X))  Vs(x) + hJ(x)hd(x) + LJ(x)Ld(x),
(3.126) (3.127)
o= ~P1Ud (x) + hJ(X)Jd(X) + LJ(X)Wd(X),
(3.128)
O="(~Imd  JJ(X)Jd(X)  P2Ud (X)  WJ(X)Wd(X),
(3.129)
then 9 is nonexpansive. If, alternatively, "(;Ime J;[(x)Jc(x) > 0, x E IR n , and there exist a continuously differentiable function Vs : IR n t IR and matrix functions P1Ud : IR n t 1R1xmd and P2Ud : IR n t fi:i[md such that Vs() is positive definite, Vs(O) = 0, {3.123} holds, and for all x E IR n , Ooo eWt = O.
Definition 5.2 The largescale impulsive dynamical system g given by (5.1}(5.4) is vector dissipative (respectively, exponentially vector dissipative) with respect to the vector hybrid supply rate (Se, Sd) if
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there exist a continuous, nonnegative definite vector function Vs = T q . [vs!, ... ,Vsq] : V t lR+, called a vector storage functIOn, and an essentially nonnegative dissipation matrix WE lR qxq such that Vs(O) = 0, W is semistable (respectively, asymptotically stable), and the vector hybrid dissipation inequality Vs(x(T))::;::; eW(Tto)Vs(x(to))
+
L
+
T eW(Tt) Sc(uc(t), yc(t))dt r ito
eW(Ttk)Sd(Ud(tk),Yd(tk)),
T ~ to, (5.9)
kEZ[to,T)
is satisfied, where x(t), t ~ to, is the solution to (5.1)(5.4) with (uc(t), Ud(tk)) E Uc x Ud and x(to) = Xo. The largescale impulsive dynamical system Q given by (5.1)(5.4) is vector lossless with respect to the vector hybrid supply rate (Sc, Sd) if the vector hybrid dissipation inequality is satisfied as an equality with W semistable. Note that if the subsystems Qi ofQ are disconnected, that is, Ici(X) == o and Idi(X) == 0 for all i = 1, ... ,q, and W E lR qxq is diagonal and nonnegative definite, then it follows from Definition 5.2 that each of the disconnected subsystems Qi is dissipative or exponentially dissipative in the sense of Definition 3.1. A similar remark holds in the case where q = l. Next, define the vector available storage of the largescale impulsive dynamical system Q by Va(xo)
~
inf
(u c ('), UdO), T~to
[
r
T eW(tto)Sc(uc(t),Yc(t))dt ito
+
L
eW(tktO)Sd(Ud(tk),Yd(tk))],
kEZ[to,T)
(5.10) where x(t), t ~ to, is the solution to (5.1)(5.4) with x(to) = Xo and admissible inputs (u c (), Ud(')) E Uc x Ud. The infimum in (5.10) is taken componentwise which implies that for different elements of Va (·) the infimum is calculated separately. Note that Va(xo) ~~ 0, Xo E V, since Va (xo) is the infimum over a set of vectors containing the zero vector (T = to). To state the main results of this section recall the definitions of complete reachability and zerostate observability given in Section 3.2.
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LARGESCALE IMPULSIVE DYNAMICAL SYSTEMS
Theorem 5.1 Consider the largescale impulsive dynamical system 9 given by (5.1}(5.4) and assume that 9 is completely reachable. Then 9 is vector dissipative (respectively, exponentially vector dissipative) with respect to the vector hybrid supply rate (Se, Sd) if and only if there exist a continuous, nonnegativedefinite vector function Vs : V  t R~ and an essentially nonnegative dissipation matrix WE Rqxq such that Vs(O) = 0, W is semistable (respectively, asymptotically stable), and for all k E Z+, Vs(x(i))
Vs(X(tk)
~~ eW(tt)Vs(x(t)) + +
it
eW(tS)Se(Ue(S), Ye(s))ds, tk
0, x E V, x =1= 0. Proof. It follows from Theorem 5.3 that va(x), x E V, is a storage function for g that satisfies (5.28). Next, suppose, ad absurdum, that there exists x E V such that va(x) = 0, x =1= 0. Then it follows from the definition of va(x), x E V, that for x(to) = x,
iT to
ea(ttO)sc(uc(t),Yc(t))dt+
L
ea(tktO)Sd(Ud(tk),Yd(tk)) 2:: 0,
kEZ[to,T)
T 2:: to,
(uc(t), Ud (tk)) E Uc X Ud·
°
(5.32)
However, for Uci = "'ci(Yci) and Udi = "'di(Ydi) we have Sci ("'ci (Ycd , Yci) < 0, Sdi("'di(Ydi), Ydi) < 0, Yci =1= 0, Ydi =1= for all i = 1, ... , q, and since p > > 0, it follows that Yci(t) = 0, tk < t S tk+l, Ydi(tk) = 0, k E Z+, i = 1, ... ,q, which further implies that Uci(t) = 0, tk < t S tk+1, and Udi(tk) = 0, k E Z+, i = 1, ... ,q. Since g is zerostate observable it follows that x = 0, and hence, va(x) = if and only if x = 0. The
°
162
CHAPTER 5
result now follows from (5.27). Finally, for the exponentially vector dissipative case it follows from Lemma 5.2 that p > > 0, with the rest of the proof being identical to that above. 0 Next, we introduce the concept of vector required supply of a largescale impulsive dynamical system. Specifically, define the vector required supply of the largescale impulsive dynamical system 9 by
Vr(xo)
~
inf
(uc(·),Ud()), TSio
[
(to eW(tto) Sc(uc(t), yc(t))dt
iT
+
L
eW(tktO) Sd(Ud(tk), Yd(t k ))] ,
kEZ[T,tO)
(5.33) where x(t), t 2: T, is the solution to (5.1)(5.4) with x(T) = 0 and x(to) = Xo. Note that since, with x(to) = 0, the infimum in (5.33) is the zero vector it follows that Vr(O) = O. Moreover, since 9 is completely reachable it follows that Vr(x) « 00, x E 'D. Using the notion of the vector required supply we present necessary and sufficient conditions for vector dissipativity of a largescale impulsive dynamical system with respect to a vector hybrid supply rate. Theorem 5.5 Consider the largescale impulsive dynamical system 9 given by (5.1}(5.4) and assume that 9 is completely reachable. Then 9 is vector dissipative (respectively, exponentially vector dissipative) with respect to the vector hybrid supply rate (Sc(u c, Yc), Sd(Ud, Yd)) if
and only if
0::;::; Vr(x) «00,
x E 'D.
(5.34)
Moreover, if (5.34) holds, then Vr(x), x E'D, is a vector storage function for g. Finally, if the vector available storage Va(x), x E 'D, is a vector storage function for g, then 0::;::; Va(x) ::;::; Vr(x) «00,
x E 'D.
(5.35)
Proof. Suppose (5.34) holds and let x(t), t E JR., satisfy (5.1)(5.4) with admissible itlputs (uc(t),Ud(tk)) E Uc x Ud, t E JR., k E Z+, and x(to) = Xo. Then, it follows from the definition of Vr() that for T::; tf ::; to, u c() E Uc, and Ud(') E Ud,
(to
Vr(xo)::;::; iT eW(tto) Sc(uc(t), yc(t))dt
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LARGESCALE IMPULSIVE DYNAMICAL SYSTEMS
+
L
eW(tktO)Sd(Ud(tk),Yd(tk))
kEZ[T,tO)
= !rtr eW(ttO)SC(UC(t),Yc(t))dt
+
L
eW(tktO)Sd(Ud(tk),Yd(tk))
kEZ[T,tr)
+ ito eW(tto) Sc( uc(t), Yc(t))dt tr
+
L
eW(tktO) Sd(Ud(tk), Yd(tk)),
(5.36)
kEZ[tr,to)
and hence,
Vr(xo) :::;:::;eW(totr)
+
inf
(uc(·),Ud(·», TStr
L
[ rtr eW(ttdSc(uc(t),yc(t))dt
iT
eW(tktdSd(Ud(tk),Yd(tk))]
kEZ[T,tr)
.
+ ito eW(tto) Sc( uc(t), Yc(t))dt tr
+
L
eW(tktO)Sd(Ud(tk),Yd(tk))
kEZ[tr,to)
eW(totdVr(X(tf)) + ito eW(tto) Sc( uc(t), Yc(t))dt
+
L
tr
eW(tkto)Sd(Ud(tk),Yd(tk)),
(5.37)
kEZ[tr,to)
which shows that Vr(x), x E V, is a vector storage function for g, and hence, g is vector dissipative with respect to the vector hybrid supply rate (Sc(u c, Yc), Sd(Ud, Yd)). Conversely, suppose that g is vector dissipative with respect to the vector hybrid supply rate (Sc(uc,Yc),Sd(Ud,Yd)). Then there exists a nonnegative vector storage function Vs(x), x E V, such that Vs(O) = O. Since g is completely reachable it follows that for x(to) = Xo there exist T < to and uc(t), t E [T, to], and Ud(tk), k E Z[T,to]' such that x(T) = o. Hence, it follows from the vector hybrid dissipation
164
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inequality (5.9) that
o ::;::; Vs(x(to)) ::;::; eW(toT)Vs(x(T)) + lt~ eW(tot) Sc( uc(t), Yc(t))dt
L
+
eW(totk)Sd(Ud(tk),Yd(tk)),
(5.38)
kEZ[T,to) which implies that for all T ::; to, uc(t) E Uc, and Ud(tk) E Ud,
0::;::;
[0 +
eW(tot) Sc( uc(t), Yc(t))dt
L
eW(totk)Sd(Ud(tk),Yd(tk))
(5.39)
kEZ[T,to) or, equivalently, 0::;::;
inf
(uc(·),Ud(·)), T90
L
+
[
(to eW(tot)Sc(uc(t),Yc(t))dt
iT
eW(totk)Sd(Ud(tk),Yd(tk))]
kEZ[T,to) = Vr(xo).
(5.40)
Since, by complete reach ability, Vr(x) « 00, x E V, it follows that (5.34) holds. Finally, suppose that Va(x), x E V, is a vector storage function. Then for x(T) = 0, x(to) = Xo, uc(t) E Uc, and Ud(tk) E Ud, it follows that
Va(x(to)) ::;::; eW(toT)Va(x(T)) +
L
+
£0
eW(tot) Sc( uc(t), Yc(t))dt
eW(totk) Sd(Ud(tk), Yd(tk)),
(5.41)
kEZ[T,tO) which implies that
O::;::;Va(x(to))::;::;
+
L
kEZ[T,tO) = Vr(x(to)),
inf
(uc(),Ud(·)), T90
[
(to eW(tot)Sc(uc(t),Yc(t))dt
iT
eW(totk) Sd(Ud(tk), Yd(t k ))] x
E
V.
(5.42)
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LARGESCALE IMPULSIVE DYNAMICAL SYSTEMS
Since x(to) = Xo E V is arbitrary and, by complete reachability, \'rex) « 00, x E V, (5.42) implies (5.35). 0 The next result is a direct consequence of Theorems 5.2 and 5.5. Proposition 5.1 Consider the largescale impulsive dynamical system g given by (5.1}(5.4) and assume g is completely reachable. Let M = diag [/tI, ... ,/tq] be such that 0 :$ /ti :$ 1, i = 1, ... , q. If Va(x), x E V, and \'rex), x E V, are vector storage functions for g, then
Vs(x) = MVa(x)
+ (Iq 
M)\'r(x),
x E V,
(5.43)
is a vector storage function for g.
Proof. First note that M 2':2': 0 and Iq  M 2':2': 0 if and only if M = diag [/tI, ... ,/tq] and /ti E [0,1], i = 1, ... , q. Now, the result is a direct consequence of the complete reachability of g along with vector hybrid dissipation inequality (5.9) by noting that if Va(x) and \'rex) satisfy (5.9), then Vs(x) satisfies (5.9). 0
Next, recall that if g is vector dissipative (respectively, exponentially vector dissipative), then there exist p E p =f 0, and a 2': 0 (respectively, p E R~ and a > 0) such that (5.24) and (5.25) hold. Now, define the (scalar) required supply for the largescale impulsive dynamical system g by
iRt,
Vr(XO)~ (uc(·),Ud(·)), inf [ (to pTeW(tto)Sc(uc(t),yc(t))dt T$to iT +
L
eW(tktO)Sd(Ud(tk),Yd(tk))]
kEZ[T,tO) =
inf
(Uc(·),Ud(·)), T$to
+
L
[ (to ea(ttO)sc(uc(t),Yc(t))dt iT
ea(tktO)Sd(Ud(tk),Yd(tk))] ,
Xo E V, (5.44)
kEZ[T,tO) where sc(uc,Yc) = pTSc(uc,Yc), Sd(Ud,Yd) = pTSd(Ud,Yd), and x(t), t 2': T, is the solution to (5.1)(5.4) with x(T) = 0 and x(to) = Xo. It follows from (5.44) that the required supply of a largescale impulsive
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dynamical system is the minimum amount of generalized energy which can be delivered to the largescale system in order to transfer it from an initial state x(T) = to a given state x(to) = Xo. Using the same arguments as in case of the vector required supply, it follows that vr(O) = and vr(x) < 00, x E 'D. Next, using the notion of the required supply, we show that all storage functions of the form vs(x) = pTVs(x), where p E R~, p =1= 0, are bounded from above by the required supply and bounded from below by the available storage. Hence, a dissipative largescale impulsive dynamical system can deliver to its surroundings only a fraction of all of its stored subsystem energies and can store only a fraction of the work done to all of its subsystems.
°
°
Corollary 5.1 Consider the largescale impulsive dynamical system
9 given by (5.1}(5.4). Assume that 9 is vector dissipative with re
spect to the vector hybrid supply rate (Se (u e, Ye), Sd (Ud, Yd)) and with vector storage function Vs : 'D t R~. Then vr(x), x E 'D, is a storage function for g. Moreover, if vs(x) ~ pTVs(x), x E 'D, where q p E R+, p =1= 0, then (5.45) Proof. It follows from Theorem 5.3 that if 9 is vector dissipative with respect to the vector hybrid supply rate (Se(u e, Ye), Sd(Ud, Yd)) and with a vector storage function Vs : 'D t R~, then there exists p E R~, p =1= 0, such that 9 is dissipative with respect to the hybrid supply rate (se(ue,Ye),Sd(Ud,Yd)) = (pTSe(ue,Ye),pTSd(Ud,Yd)) and with storage function vs(x) = pTVs(x), x E 'D. Hence, it follows from (5.28), with x(T) = and x(to) = Xo, that
°
(5.46)
which implies that vr(xo) ~ 0, Xo E'D. Furthermore, it is easy to see from the definition of the required supply that vr(x), x E 'D, satisfies the dissipation inequality (5.28). Hence, vr(x), x E 'D, is a storage function for g. Moreover, it follows from the dissipation inequality (5.28), with x(T) = 0, x(to) = Xo, ue(t) E Ue, and Ud(tk) E Ud, that
167
LARGESCALE IMPULSIVE DYNAMICAL SYSTEMS
+
L
eatksd(Ud(tk), Yd(tk))
kEZ[T,tO)
= ito eatsc(uc(t), Yc(t))dt
+
L
eatkSd(Ud(tk), Yd(tk)),
(5.47)
kEZ[T,to)
which implies that
vs(x(to))::;
inf
(uC('),Ud('», T90
+
L
[ (to ea(tto)sc(uc(t),Yc(t))dt iT
ea(tktO) Sd(Ud(tk), Yd(tk))]
kEZ[T,to) =vr(x(to)).
(5.48)
Finally, it follows from Theorem 5.3 that va(x), x E V, is a storage function for g, and hence, using (5.27) and (5.48), (5.45) holds. D It follows from Theorem 5.5 that if g is vector dissipative with respect to the vector hybrid supply rate (Sc(uc,Yc),Sd(Ud,Yd)), then Vr(x), x E V, is a vector storage function for g and, by Theorem 5.3, there exists p E ~, p =1= 0, such that vs(x) ~ pTVr(x), x E V, is a storage function for g satisfying (5.28). Hence, it follows from Corollary 5.1 that pTVr(x) ::; vr(x), x E V. The next result relates the vector (respectively, scalar) available storage and the vector (respectively, scalar) required supply for vector loss less largescale impulsive dynamical systems. Theorem 5.6 Consider the largescale impulsive dynamical system g given by (5.1)(5.4). Assume that g is completely reachable to and from the origin. If g is vector loss less with respect to the vector hybrid supply rate (Sc(u c, Yc), Sd(Ud, Yd)) and Va(x), x E V, is a vector storage function, then Va(x) = Vr(x), x E V. Moreover, if Vs(x), x E V, is a vector storage junction, then all (scalar) storage functions of the form vs(x) = pTVs(x) , x E V, where p E~, P =1= 0, are given by
vs(xo)
= va(xo) = vr(xo) = IT+ ea(tto)sc(uc(t), Yc(t))dt to
L
kEZ[to,T+)
ea(tktO) Sd(Ud(tk), Yd(tk))
168
=
£: +
CHAPTER 5
eo«tto) Sc( Uc(t),
L
Yc(t))dt
eo«tktO)Sd(Ud(tk),Yd(tk)),
kEZ[T_,tO) (5.49)
where x(t), t ~ to, is the solution to (5.1}(5.4) with u c() E Uc, Ud() E Ud, x(to) = Xo E V, sc(uc,Yc) = pTSc(uc,Yc), andsd(ud,Yd) = pTSd(Ud, Yd), for any T+ > to and T_ < to such that x(T+) = 0 and x(T_) = O. Proof. Suppose 9 is vector lossless with respect to the vector hybrid supply rate (Sc(u c, Yc), Sd(Ud, Yd)). Since 9 is completely reachable to and from the origin it follows that for every Xo = x(to) E V there exist T+ > to, T_ < to, uc(t) E Uc, and Ud(tk) E Ud, t E [T_, T+l, k E Z[T_,T+], such that x(T_) = 0, x(T+) = 0, and x(to) = Xo. Now, it follows from the dissipation inequality (5.9) which is satisfied as an equality that
0= (T+ eW(T+t)Sc(uc(t),Yc(t))dt
IT_
+
L
eW(T+tk)Sd(Ud(tk),Yd(tk)),
kEZ[T_,T+) or, equivalently,
o=
£:+ +
£:
eW(tto)Sc(uc(t),Yc(t))dt
L
eW(tkto)Sd(Ud(tk),Yd(tk))
kEZ[T_,T+)
=
eW(tto) Sc(uc(t), Yc(t))dt
+
L
eW(tkto)Sd(Ud(tk),Yd(tk))
kEZ[T_,tO)
+ IT+ eW(tto) Sc( uc(t), Yc(t) )dt +
to
L
kEZ[to,T+)
eW(tktO) Sd(Ud(tk), Yd(tk))
(5.50)
LARGESCALE IMPULSIVE DYNAMICAL SYSTEMS
22
inf
(Uc(·),Ud(·», T9o
L
+
169
[ (to eW(tto)Sc(uc(t),Yc(t))dt
ir
eW(tktO)Sd(Ud(tk),Yd(tk))]
kEZ[T_,to)
+
inf
(uc(,),Ud('»' T+2 to
L
+
[ (T+ eW(tto) Sc(uc(t), Yc(t))dt
ito
eW(tktO)Sd(Ud(tk),Yd(tk))]
kEZ[to,T+)
 Vr(xo)  Va(Xo),
(5.51)
which implies that Vr(xo) ::;::; Va(xo), Xo E V. However, it follows from Theorem 5.5 that if 9 is vector dissipative and Va(x), x E V, is a vector storage function, then Va(x) ::;::; Vr(x), x E V, which along with (5.51) implies that Va(x) = Vr(x), x E V. Next, since 9 is vector lossless there exist a nonzero vector p E ~~ and a scalar a 2 0 satisfying (5.24). Now, it follows from (5.50) that
0= l~+ pTeW(tto)Sc(uc(t),Yc(t))dt
L
+
pTeW(tktO)Sd(Ud(tk),Yd(tk))
kEZ[L,T+)
=
£:+ ea(tto) sc(uc(t), Yc(t))dt L
+
ea(tktO)Sd(Ud(tk),Yd(tk))
kEZ[T_,T+)
=
r ir
o ea(tto) sc( uc(t), Yc(t) )dt
+
L
ea(tktO)Sd(Ud(tk),Yd(tk))
kEZ[L,to)
+
+
(T+ ea(tto)sc(uc(t), Yc(t))dt
ito
L
kEZ[to,T+)
ea(tkto)Sd(Ud(tk),Yd(tk))
CHAPTER 5
170
~ (Uc(·),UdO), inf [ (to eCt(tto)sc(uc(t),Yc(t))dt T9o JT_ +
L
eCt(tktO) Sd(Ud(tk), Yd(t k))]
kEZ[L,to)
+ +
inf
(u c ('),udO),T+2: to
L
[ {T+ eCt(tto)sc(uc(t),Yc(t))dt
Jto
eCt(tktO)Sd(Ud(tk),Yd(tk))]
kEZ[to,T+)
=
Vr(XO)  Va(XO),
Xo ED,
(5.52)
which along with (5.45) implies that for any (scalar) storage function of the form vs(x) = pTVs(x), xED, the equality va(x) = vs(x) = Vr (x), xED, holds. Moreover, since 9 is vector lossless the inequalities (5.28) and (5.47) are satisfied as equalities and
Vs(XO) = 
(T+
Jto
eCt(tto)sc(uc(t),Yc(t))dt
L
eCt(tkto)Sd(Ud(tk),Yd(tk))
kEZ[to,T+)
=
rJT_o eCt(tto)sc(uc(t),Yc(t))dt +
L
eCt(tktO) Sd(Ud(tk), Yd(tk)),
(5.53)
kEZ[L,to)
where x(t), t ~ to, is the solution to (5.1)(5.4) with uc(t) E Uc, Ud(tk) E Ud, x(T_) = 0, x(T+) = 0, and x(to) = Xo ED. 0 The next proposition presents a characterization for vector dissipativity of largescale impulsive dynamical systems in the case where Vs (.) is continuously differentiable.
Proposition 5.2 Consider the largescale impulsive dynamical system 9 given by (5.1)(5.4), assume Vs = [Vsl,"" VSq]T : D  t lR~ is a continuously differentiable vector storage function for g, and assume 9 is completely reachable. Then 9 is vector dissipative with respect to
171
LARGESCALE IMPULSIVE DYNAMICAL SYSTEMS
the vector hybrid supply rate (Sc(u c, Yc), Sd(Ud, Yd)) if and only if Vs(x(t)) ::;::; WVs(x(t)) + Sc(uc(t), yc(t)), tk < t ::; tk+1, (5.54) Vs(X(tk) + Fd(x(tk), Ud(tk))) ::;::; Vs(X(tk)) + Sd(Ud(tk), Yd(tk)), k E Z+, (5.55) where Vs (x( t)) denotes the total time derivative of each component of Vs(·) along the state trajectories x(t), tk < t::; tk+1, ofg. Proof. Suppose g is vector dissipative with respect to the vector hybrid supply rate (Sc(u c, Yc), Sd(Ud, Yd)) and with a continuously differentiable vector storage function Vs : V + lR~. Then, with T = i and to = t, it follows from (5.11) that there exists a nonnegative vector function l(t,i,xo,u c()) 2:2: O,tk+l 2: i 2: t > tk,xo E V, uc() E Uc,
such that
Vs(x(i)) = eW(tt)Vs(x(t)) +
1t
eW(tu) Sc( uc(O"), Yc(O") )dO"
l(t, i, xo, u c ()),
(5.56)
or, equivalently,
eWtVs(x(i))  eWtVs(x(t)) =
1t
e wu Sc(uc(O"), Yc(O"))dO"
eWiZ(t, i, xo, uJ)). Now, dividing (5.57) by
i
t and letting
(5.57)
i + t+, (5.57) is equivalent
to
:0"
[eWUVs(x(O")) ] lu=t = e Wt Sc( uc(t), yc(t))
:0,
_eWt lim l(t, i, u c(·)) , (5.58) t>t+ t  t where the limit in (5.58) exists since Vs(·) is assumed to be continuously differentiable. Next, premultiplying (5.58) by eWt , t 2: 0, yields
. ( Vs(x(t))  WVs(x(t)) = Sc(uc(t), Yc t)) 
Z(t,i,xo,uc()) ( ) 5.59 t>t+ t t . JIm
A
,
which, since limt>t+ l(t,t'i~~c(.)) 2:2: 0 and t is arbitrary, gives (5.54). Inequality (5.55) is a restatement of (5.12). The converse is immediate from Theorem 5.l. 0
172
CHAPTER 5
Recall that if a disconnected subsystem (h (Le., Ici(X) == 0 and Idi(X) == 0, i E {I, ... ,q}) of a largescale impulsive dynamical system 9 is exponentially dissipative (respectively, dissipative) with respect to a hybrid supply rate (Sci (Uci, Yci) , Sdi ( Udi, Ydd), then there exist a storage function Vsi : R.ni + JR+ and a constant Ci > 0 (respectively, Ci = 0), i = 1, ... ,q, such that the dissipation inequality
eeiTvsi(X(T)) :::;eeitOvSi(X(tO)) +
+
2:
T r eeitsci(uci(t),Yci(t))dt lto
eeitkSdi(Udi(tk)' Ydi(tk)) ,
T 2: to, (5.60)
kEZ[to,T)
holds. In the case where Vsi : JRni + JR+ is continuously differentiable and 9 is completely reachable, (5.60) yields V~i(Xi)(fci(Xi)
+ Gci(Xi)Uci) :::; ciVsi(Xi) + Sci (Uci, Yci) ,
x (j. Zi, Uci E Uci , Vsi(Xi + fdi(xi) + Gdi(Xi)Udi):::; Vsi(Xi) + Sdi (Udi' Ydi) , x E Zi, Udi E Udi,
(5.61) (5.62)
where Zi ~ JRnl x ... x JRnil x ZXi X JRnHl x ... x JRq c JRn and ZXi C JRni, i = 1, ... , q. The next result relates exponential dissipativity with respect to a scalar hybrid supply rate of each disconnected subsystem gi of 9 with vector dissipativity (or, possibly, exponential vector dissipativity) of 9 with respect to a vector hybrid supply rate. Proposition 5.3 Consider the largescale impulsive dynamical system 9 given by (5.1}(5.4) with Zx = U{=lZi. Assume that 9 is completely reachable and each disconnected subsystem gi of 9 is exponentially dissipative with respect to the hybrid supply rate (Sci(Uci,Yci),Sdi (Udi' Ydi)) and with a continuously differentiable storage function Vsi : JRni + JR+, i = 1, ... , q. Furthermore, assume that the interconnection functions Ici : D + JRni and Idi : V + JRni, i = 1, ... ,q, of 9 are such that v~i(xdIci(X) :::;
Vsi(Xi
2:J=1 €ij (x)vsj (Xj),
x (j. Zx,
(5.63)
+ fdi(xi) + Idi(X) + Gdi(Xi)Udi) x E Zx,
+Gdi(Xi)Udi),
:::; Vsi(Xi + fdi(xi) Udi E Udi, i = 1, ... ,q, (5.64)
where €ij : D + JR, i,j = 1, ... ,q, are given bounded functions. If WE JRqxq is semistable (respectively, asymptotically stable), with W(i,j)
=
{
Ci
+ aii,
aij,
i = j, L . i I]'
(5.65)
LARGESCALE IMPULSIVE DYNAMICAL SYSTEMS
173
where Ci > 0 and 0ij ~ max{O, SUPxE'D ~ij(X)}, for all i,j = 1, ... , q, then Y is vector dissipative (respectively, exponentially vector dissipative) with respect to the vector hybrid supply rote (Se(u e, Ye), Sd(Ud, Yd) ) ~ ([Sel (Ueb Yel), ... ,Seq(Ueq , Yeq)F, [Sdl (Udl' Ydl), ... ,Sdq(Udq, Ydq)JT ) and with vector storoge function Vs (x) ~ [VsI (Xl), ... , Vsq (Xq) JT, X E
'D.
Proof. Since each disconnected subsystem Yi of Y is exponentially dissipative with respect to the hybrid supply rate Sei(Uei, Yei), i = 1, ... ,q, it follows from (5.61)(5.64) that, for all Uci E Uei and i = 1, ... ,q,
Vsi (Xi(t)) = V~i(Xi(t) )[fei (Xi(t))
+ Iei(X(t)) + Gei (Xi (t) )Uci (t)] q
5 ciVsi(Xi(t)) + Sci(Uei(t), Yci(t)) + L ~ij(X(t))vsj(Xj(t)) j=l q
5 ciVsi(Xi(t)) + Sei(Uei(t),Yci(t» + LOijVSj(Xj(t)), j=l
tk < t 5 tk+b (5.66) and
Vsi(Xi(tk) + fdi(Xi(tk)) + Idi(X(tk)) + Gdi(Xi(tk))Udi(tk)) 5 Vsi(Xi(tk) + fdi(Xi(tk)) + Gdi(Xi(tk))Udi(tk)) 5 Vsi(Xi(tk)) + Sdi(Udi(tk), Ydi(tk)), k E Z+. (5.67) Now, the result follows from Proposition 5.2 by noting that for all subsystems Yi of y,
+ Se(Ue(t) , Ye(t)),
tk < t 5 tk+b ue(t) E Ue, (5.68) Vs(X(tk) + Fd(X(tk), Ud(tk») 55 Vs(X(tk)) + Sd(Ud(tk), Yd(tk)), k E Z+, Ud(tk) E Ud, (5.69) Vs(x(t)) 55 WVs(x(t))
where W is essentially nonnegative and, by assumption, semistable (respectively, asymptotically stable), and the vector function Vs(x) ~ [VsI (Xl), ... ,Vsq(xq)F, for all X E 'D, is a vector storage function for y. 0 As a special case of vector dissipativity theory we can analyze the stability of largescale impulsive dynamical systems. Specifically, assume that the largescale impulsive dynamical system y is vector dissipative (respectively, exponentially vector dissipative) with respect to
174
CHAPTER 5
the vector hybrid supply rate (Se(ue,Ye),Sd(Ud,Yd)) and with a continuously differentiable vector storage function Vs : V t R~. Moreover, assume that the conditions of Theorem 5.4 are satisfied. Then it follows from Proposition 5.2, with ue(t) == 0, Ud(tk) == 0, Ye(t) == 0, and Yd(tk) == 0, that
Vs(x(t)) ::;::; WVs(x(t)), tk < t ::; tk+1 Vs(X(tk) + fd(X(tk)) + Id(X(tk))) ::;::; Vs(X(tk)), k
E
Z+,
(5.70) (5.71)
where x(t), t ~ to, is a solution to (5.1)(5.4) with x(to) = XO, ue(t) == 0, and Ud(tk) == O. Now, it follows from Theorem 2.11, with we(z) = Wz and Wd(Z) = 0, that the zero solution x(t) == 0 to (5.1)(5.4), with ue(t) == 0 and Ud(tk) == 0, is Lyapunov (respectively, asymptotically) stable. More generally, the problem of control system design for largescale impulsive dynamical systems can be addressed within the framework of vector dissipativity theory. In particular, suppose that there exists a continuously differentiable vector function Vs : V t R~ such that Vs(O) = 0 and
Vs(x(t)) ::;::; Fe(Vs(x(t)), ue(t)), Vs(X(tk)
tk < t ::; tk+b
+ Fd(X(tk), Ud(tk))) ::;:::; Vs(X(tk)),
k E
ue(t) E Ue, (5.72) Z+, Ud(tk) E Ud, (5.73)
where Fe : R~ X Rmc t Rq and Fe(O,O) = o. Then the control system design problem for a largescale impulsive dynamical system reduces to constructing a hybrid energy feedback control law (. Zdi E jRSdiXffid, i = 1, ... , q, such that W is essentially nonnegative and semistable (respectively, asymptotically stable), and, for all i
= 1, ... ,q,
0= xT(AJ Pi
+ ~Ae 
q
C'!QeiCe 
L W(i,j)Pj + L;Lei)x,
x ¢ Zx,
j=l
(5.122)
187
LARGESCALE IMPULSIVE DYNAMICAL SYSTEMS
(5.123) (5.124) (5.125) (5.126) (5.127) Proof. The proof follows from Theorem 5.7 with Fc(x) = Acx, Gc(x) = Bc, hc(x) = Ccx, Jc(x) = Dc, wc(r) = Wr, Rci(X) = Lcix, Zci(X) = ZCi, Fd(X) = Adx, Gd(X) = Bd, hd(x) = CdX, Jd(X) = Dd, Rdi(X) = LdiX, Zdi(X) = Zdi, P1i(X) = 2xT AJ [{Bd, P2i(X) = BJ PiBd, and Vsi(X) = xT [{x, i = 1, ... , q. D
Note that (5.122)(5.127) are implied by
Zci ] ::; 0,
(5.128) i
where, for all i
= 1, ... ,q, (5.129)
= 1, ... ,q, (5.130) (5.131) (5.132) (5.133) (5.134) (5.135)
Hence, vector dissipativity of largescale linear impulsive dynamical systems with respect to quadratic hybrid supply rates can be characterized via (cascade) linear matrix inequalities (LMls) [27J. A similar remark holds for Theorem 5.11 below. The next result presents sufficient conditions guaranteeing vector dissipativity of 9 with respect to a quadratic hybrid supply rate in case where the vector storage function is component decoupled. Theorem 5.11 Consider the largescale linear impulsive dynamical system 9 given by (5. 118}(5. 121}. Let Rei E §mci, Sci E lRlcixmci, Rz. E §mdi ,dz S· E ll'\o.ldi xmdi , and Q dz. E §ldi , ,;•  1, ... , q, be Qcz. E §lci , d l1])
188
CHAPTER 5
given.
Assume there exist matrices W E jRqx q , Pi E Nn i , Lcii E jRsCiixni, Zcii E jRsciixmCi, Ldii E jRsdiixni, Zdii E jRsdiixmci, i = 1 , ... , q , L·· E jRScij xni , Z·· E jRScij xnj Ld" E jRSdij xni , and Zd" E jRSdij xnj, i, j = 1, ... , q, i =I j, such that W is essentially nonnegative and semistable (respectively, asymptotically stable), and, for all i = 1, ... ,q, c~3
c~3
'~3
0::> X[ ( A;;P; + P;A,,;;  C;Qo;Co; 
.t .
~3
W(;,;)p;
+ L;,Lo;; (5.136)
X ¢ Zx,
L;jLCi j ) Xi,
3=1,3i=~
O=x[(PiBci  C~Sci  C~QciDci
+ L~iZcii)'
o~ Rei + D~Sci + S~DCi + D~QciDci 
t .
Z~iZcii'
(5.139)
X E Zx,
LJijLdij) Xi,
(5.137) (5.138)
P; + LJ;;Ld;;
0::> x[ ( AJ;;P;A.i;;  c;r.QdiCd; 
+.
X ¢ Zx
3=1,3i=~
0= X[ (AJiiPiBdi  C;£Sdi  C;£QdiDdi
+ LJiiZdii) ,
o ~ Rdi + D;£Sdi + S;£Ddi + DlQdiDdi  Bl~Bdi and for j = 1, ... ,q, 1 = 1, ... ,q, j
o~ XJ(AJijPiAdil)Xl , o~x[(Ali~Adij
X¢
Z~jZCij)Xj,
O=XJ(AJij~Bdd,
zx, X E zx,
zx, X¢
(5.142)
zx,
X E Zx.
(5.143) (5.144)
XE
+ LJijZdij)Xj,
O=xJ(Z;£jZdij)Xj,
Z;£iZdii, (5.141)
=I i, 1 =I i,
O=x[(PiAcij +L;jZcij),
o~ XJ(W(i,j)Pj 
X E Zx, (5.140)
(5.145) XE
zx,
(5.146) (5.147)
Then g is vector dissipative (respectively, exponentially vector dissipative) with respect to the vector hybrid supply rate (Sc(u c , yc), Sd(Ud, Yd)), where Sci(Uci,Yci) = U~RciUci+2y~SciUci+y;QciYCi and Sdi(Udi, Ydi) = UJiRdiUdi + 2ylSdiudi + ylQdiYdi, i = 1, ... , q.
189
LARGESCALE IMPULSIVE DYNAMICAL SYSTEMS
Proof. Since Pi is nonnegative definite, the function Vsi(Xi) ~ X[ PiXi, Xi E jRni, is nonnegative definite and Vsi(O) = O. Moreover, since Vsi() is continuously differentiable it follows from (5.136)(5.147) that for all Uci(t) E jRmci, Udi(tk) E ~mdi, i = 1, ... ,q, and tk < t::;
Z+,
tk+1, k E
Vsi(Xi(t))
= 2x[ (t)Pi
[t
J=l
AcijXj(t) + BCiUCi(t)]
5: X[(t) [WCi,i)Pi
+
C~QciCci  L;iLCii 
.
t.
L;jLCij] Xi(t)
J=l,J#~
q
L:
j=l,#i
2x[(t)L;j ZcijXj(t)
+2X[(t)C~SciUci(t)
+ 2X[(t)C~QciDciUci(t)
2x[(t)L;i ZciiUci(t) +
q
L:
xJ(t)[W(i,j)Pj  Z~jZCij]Xj(t)
j=l,#i +U;(t)RciUci(t) + 2u;(t)D~SciUci(t)
+ u;(t)D;QciDciUci(t)
u;(t)Z~iZciiUci(t) q
=
L WCi,j)Vsj(Xj(t)) + U~(t)RciUCi(t) + 2y~(t)SCiUci(t) j=l +y;(t)QciYci(t) [LciiXi(t) + ZciiUci(t)]T[LciiXi(t) q
 L:
j=l,#i
(LcijXi(t)
5: Sci (Uci (t), Yci(t)) + Furthermore,
+ ZciiUci(t)]
+ ZcijXj(t))T(LcijXi(t) + ZcijXj(t)) q
L: WCi,j)Vsj(Xj(t)). j=l
(5.148)
190
·Pi
[t
CHAPTER 5
AdijXj(tk) + BdiUdi(tk)]
J=1
~ x; (tk) [~+ C~QdiCdi  LJiiLdii q
L
j=l,jii
2X;(tk)LJij ZdijXj(tk)
+2x; (tk)C~QdiDdiUdi(tk) 2X;(tk)LJii ZdiiUdi(tk) 
.
t.
LJijLdij] Xi(tk)
J=I,Jf~
+ 2X;(tk)C~SdiUdi(tk)
q
L
j=l,jii
XJ(tk)Z~jZdijXj(tk)
+UJi(tk)RdiUdi(tk) + 2UJi (tk)DISdiudi (tk) +uJi(tk)DIQdiDdiudi(tk)  UJi(tk)Z~iZdiiUdi(tk) = Vsi(Xi(tk)) + UJi(tk)RdiUdi(tk) + 2yI(tk)Sdiudi(tk) +yI(tk)QdiYdi(tk)  [LdiiXi(tk) + ZdiiUdi(tk)]T[LdiiXi(tk) q
+ZdiiUdi(tk)] 
L
[LdijXi(tk) j=l,jii +ZdijXj(tk)]T[LdijXi(tk) + ZdijXj(tk)] ~ Sdi(Udi(tk), Ydi(tk)) + Vsi(Xi(tk))·
(5.149)
Writing (5.148) and (5.149) in vector form yields
Vs(x) ~~ WVs(x)
+ Se(u e, Ye),
Ue E
]R.mc,
X rt Zx, (5.150)
Vs(AdX + BdUd)
~~ Vs(x)
+ Sd(Ud, Yd),
Ud E ]R.md,
x E Zx, (5.151)
where Vs(x) ~ [Vsl(Xl), ... ,vSq (x q )]T, x E ]R.n. Now, it follows from Definition 5.3 that g is vector dissipative (respectively, exponentially vector dissipative) with respect to the vector hybrid supply rate (Se (u e, Ye), Sd (Ud, Yd)) and with vector storage function Vs (x), x E ]R.n. 0
Chapter Six Stability and Feedback Interconnections of Dissipative Impulsive Dynamical Systems
6.1 Introduction
In Chapters 2 and 3 stability and dissipativity theory for nonlinear impulsive dynamical systems was developed. Using the concepts of .dissipativity and exponential dissipativity for impulsive systems, in this chapter we develop feedback interconnection stability results for nonlinear impulsive dynamical systems. The feedback system can be impulsive, nonlinear, and either dynamic or static. General stability criteria are given for Lyapunov, asymptotic, and exponential stability of feedback impulsive systems. In the case of quadratic supply rates involving net system power and inputoutput energy, these results generalize the positivity and smallgain theorems [53,74,142,165,171 J to the case of nonlinear impulsive dynamical systems. In particular, we show that if the nonlinear impulsive dynamical systems Q and Qc are dissipative (respectively, exponentially dissipative) with respect to quadratic supply rates corresponding to net system power, or weighted input and output energy, then the negative feedback interconnection of Q and Qc is Lyapunov (respectively, asymptotically) stable.
6.2 Stability of Feedback Interconnections of Dissipative Impulsive Dynamical Systems
In this section, we consider feedback interconnections of dissipative impulsive dynamical systems. Specifically, using the notion of dissipative and exponentially dissipative impulsive dynamical systems introduced in Chapter 3, with appropriate storage functions and supply rates, we construct Lyapunov functions for interconnected impulsive dynamical systems by appropriately combining storage functions for each subsystem. Here, we restrict our attention to input/statedependent impulsive dynamical systems. Analogous results, with the exception of results requiring the impulsive invariance principle, hold
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for timedependent impulsive dynamical systems. We begin by considering the nonlinear dynamical system Q given by
x(t) = fe(x(t))
+ Ge(x(t))ue(t),
x(O) = Xo,
(x(t),Ue(t))
rt z, (6.1)
fd(X(t)) + Gd(X(t))Ud(t), (x(t), ue(t)) E Z, Ye(t) = he(x(t)) + Je(x(t))ue(t), (x(t), ue(t)) rt z, Yd(t) = hd(x(t)) + Jd(X(t))Ud(t), (x(t),ue(t)) E Z,
(6.2) (6.3) (6.4)
~x(t) =
where t ~ 0, x(t) E V ~ ~n, V is an open set with 0 E V, ~x(t) ~ x(t+)  x(t), ue(t) E Ue ~ ~mc, Ud(tk) E Ud ~ ~md, tk denotes the kth instant of time at which (x (t), U e (t)) intersects Z for a particular trajectory x(t) and input ue(t), Ye(t) E ~lc, Yd(tk) E ~ld, fe : V  t ~n is Lipschitz continuous on V and satisfies fe(O) = 0, G e : V  t ~nxmc, fd : V  t ~n is continuous on V, Gd : V  t ~nxmd, he : V  t ~lc and satisfies he(O) = 0, Je : V  t ~lcxmc, hd : V  t ~ld, Jd : V  t ~ldxmd, and Z ~ Zx x ZU c, where Zx C V and ZUc cUe, is the resetting set. Furthermore, consider the impulsive nonlinear feedback system Qe given by
+ Gee (uee (t), xe(t))uee(t), xe(O) = XeO, (Xe(t), Uee(t)) rt Ze, (6.5) ~xe(t) = fde(Xe(t)) + Gde( Ude(t), xe(t))Ude(t), (xe(t), uee(t)) E Ze, Xe(t) = fcc (xc (t))
Yee(t) = hee(xe(t)) + Jee(uee(t), xe(t))uee(t), Yde(t) = hde(Xe(t)) + Jde(Ude(t), xe(t))Ude(t),
(6.6)
(xe(t), uee(t)) (Xe(t), uee(t))
rt Ze, E
(6.7) Ze, (6.8)
where t ~ 0, ~xe(t) = xe(t+)  xe(t), xe(t) E ~nc, uee(t) E Uee ~ ~mcc,
Ude(tk)
E
Ude ~ ~mdc, Yee(t)
E ~lcc,
Yde(tk)
E ~ldc,
fcc : ~nc
t
~nc and satisfies fee(O) = 0, Gee : fde : ~nc  t ~nc is continuous on ~nc, Gde : ~mdc X ~nc  t ~ncxmdc, J ee : ~mcc X ~nc  t ~lccxmcc, hee : ~nc  t ~lcc and satisfies hcc(O) = 0, Jde : ~mdc X ~nc  t ~ldcxmdc, hde : ~nc  t ~ldc, mee = le, mde = ld, lee = me, lde = md, and Ze ~ Zexc X Zeucc' where Zcxc C ~nc and Zeucc C Uee, is such that Assumptions Al and ~nc
is Lipschitz continuous on
~mcc X ~nc  t ~ncxmcc,
A2 of Chapter 2 hold. Note that with the feedback interconnection given by Figure 6.1, (uee , Ude) = (Ye, Yd) and (Yee, Yde) = (u e, Ud).
STABILITY OF FEEDBACK IMPULSIVE DYNAMICAL SYSTEMS
Q
193
I
+ Figure 6.1 Feedback interconnection of (I and (Ie.
Furthermore, note that, for generality, we allow the feedback system Qc to be of dimension nc which may be less than the plant order n. Even though the inputoutput pairs of the feedback interconnection shown on Figure 6.1 consist of twovector inputs/twovector outputs, at any given instant of time a singlevector input/singlevector output is active. Here, we assume that the negative feedback interconnection of Q and Qc is well posed, that is, det[Ime + Jcc(Yc, xc)Jc(x)] i= 0 and det[Imd + Jdc(Yd,Xc)Jd(X)] i= 0 for all Yc,Yd,X, and xc' The following results give sufficient conditions for Lyapunov, asymptotic, and exponential stability of the negative feedback interconnection given by Figure 6.1. In contrast to Chapter 2, in this chapter we represent the resetting time Tk (xo) for a statedependent impulsive dynamical system by tk' This minor abuse of notation considerably simplifies the presentation. For the results of this section we define the closedloop resetting set I:>. Zx = Zx x ZCXe U {(X, xc) : (Fcc(x),Fc(xc)) E ZCUee x Zue}, where FccO and FcO are functions of X and Xc arising from the algebraic loops due to U cc and U c, respectively. Note that since the feedback interconnection of Q and Qc is well posed, it follows that Zx is well defined and depends on the closedloop states x ~ [xT x;]T. In the special case where Jc(x) == 0 and Jcc (U cc , xc) == 0 it follows that Zx = Zx x ZCXe U {(x, xc) : (hc(x), hcc(xc)) E ZCUee x ZuJ. Furthermore, note that in the case where Z = 0, that is, the plant is a continuoustime dynamical system without any resetting, it follows that Zx = ZCXe U {(x,x c ) : hc(x) E ZcueJ, and hence, knowledge of Xc and Yc is sufficient to determine whether or not the closedloop state vector is in the set Zx. Here we assume that the solution s(t, xo) of the dynamical system resulting from the feedback interconnection of Q and Qc is such that Assumption 2.1 is satisfied. For the statement of
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the results of this section let ~co,uc denote the set of resetting times of g, let Txo,uc denote the complement of Tio,u c ' that is, [0,00) \ ~~,uc' let T.xc u denote the set of resetting times of gc, and let Txco ,Ucc denote the complement of ~cco,ucc' that is, [0, oo)\~~o,ucc' ~,
~
Theorem 6.1 Consider the closedloop system consisting of the nonlinear impulsive dynamical systems 9 given by (6.1}(6.4) and gc given by (6.5}(6.8) with inputoutput pairs (u c , Ud; Yc, Yd) and (u cc , udc;Ycc,Ydc), respectively, and with (Ucc,Udc) = (Yc,Yd) and (Ycc,Ydc)
=
(u c , Ud). Assume 9 and gc are zerostate observable, dissipative with respect to the hybrid supply rates (sc(uc,yc),Sd(Ud,Yd)) and (scc( Ucc , ycc), Sdc( Udc, Ydc)), respectively, and with continuously differentiable positive definite, radially unbounded storage functions Vs (.) and VscO, respectively, such that Vs(O) = 0 and Vsc(O) = O. Furthermore, assume there exists a scalar (J' > 0 such that Sc (Uc , Yc) +(J' Scc (Ucc , Ycc) ~ and Sd (Ud, Yd) + (J'Sdc (Udc, Ydc) ~ O. Then the following statements hold:
o
i) The negative feedback interconnection of 9 and gc is Lyapunov stable.
ii) If 9 is strongly zerostate observable, gc is exponentially dissipative with respect to the hybrid supply rate (scc( Ucc , Ycc), Sdc( Udc, Ydc)), and rank [Gcc (ucc , O)J = m cc , Ucc E Ucc , then the negative feedback interconnection of stable.
9 and gc is globally asymptotically
iii) If 9 and gc are exponentially dissipative with respect to the supply rates (sc(u c , Yc), Sd(Ud, Yd)) and (scc(ucc , Ycc) , Sdc(Udc, Ydc)), respectively, and Vs(·) and VscO are such that there exist constants a, a c, f3, f3c > 0 such that
allxl1 2 :::; Vs(x) ~ f3llxI1 2, x E ]Rn, a cllxcl1 2 :::;Vsc(x c ):::; f3cllxcI12, Xc E ]Rnc,
(6.9) (6.10)
then the negative feedback interconnection of 9 and exponentially stable.
yc is globally
Proof. Let TC ~ ~co,uc U ~cco,ucc and tk ETc, k E Z+. First, note that it follows from Assumptions Al and A2 of Chapter 2 that the resetting times tk(= 'Tk(XO)) for the feedback system are well defined and distinct for every dosedloop system trajectory.
STABILITY OF FEEDBACK IMPULSIVE DYNAMICAL SYSTEMS
195
i) Consider the Lyapunov function candidate V(x, xc) = Vs(x) + O"Vsc(xc). Now, the corresponding Lyapunov derivative of V(x, xc) along the state trajectories (x(t),xc(t)), t E (tk' tk+1], is given by
V(x(t), xc(t)) = Vs(x(t)) + O"v;'c(xc(t)) :::; sc(uc(t), Yc(t)) + O"scc(ucc(t), Ycc(t)) :::; 0, (x(t), xc(t)) ¢ ix,
(6.11)
and the Lyapunov difference of V(x, xc) at the resetting times tk, k E Z+, is given by ~ V(X(tk),
Xc(tk)) = ~ Vs(X(tk)) + O"~ Vsc(Xc(tk) :::; Sd(Ud(tk), Yd(tk)) + O"Sdc(Udc(tk), Ydc(tk)) :::; 0, (X(tk), Xc(tk)) E i x. (6.12)
Now, Lyapunov stability of the negative feedback interconnection of g and gc follows as a direct consequence of Theorem 2.l. ii) Next, if gc is exponentially dissipative it follows that for some scalar Ccc > 0,
V(x(t), xc(t)) = Vs(x(t)) + O"v;'c(xc(t)) :::; O"Ccc Vsc(xc(t)) + sc(uc(t), Yc(t)) + O"scc(ucc(t), Ycc(t)) :::; O"Ccc Vsc(xc(t)) , (x(t), xc(t)) ¢ ix, tk < t :::; tk+1, (6.13)
and ~ V(X(tk),
Xc(tk)) = ~ Vs(X(tk)) + O"~ Vsc(Xc(tk)) :::; Sd(Ud(tk), Yd(tk)) + O"Sdc(Udc(tk), Ydc(tk)) :::; 0, (X(tk), Xc(tk)) E ix, k E Z+. (6.14)
Let R ~ {(x, xc) E ~n x ~nc : (x, xc) ¢ ix, V(x, xc) = O} U {(x, xc) E ~n x ~nc : (x, xc) E ix, ~ V(x, xc) = O}, where V(x, xc) and ~ V(x, xc) denote the total derivative and difference of V(x, xc) of the closedloop system for all (x, xc) ¢ ix and (x, xc) E ix, respectively. Since Vsc(x c) is positive definite, note that V(x, xc) = 0 for all (x, xc) E ~n X ~nc\ix only if Xc = O. Now, since rank [G cc (u cc , 0)] = m cc , Ucc E Ucc , it follows that on every invariant set M contained in R, ucc(t) = Yc(t) == 0, and hence, Ycc(t) == uc(t) == 0 so that x(t) = fc(x(t)). Now, since g is strongly zerostate observable it follows that R = {(O, O)} U {(x, xc) E ~n x ~nc : (x, xc) E ix, ~ V(x, xc) = O} contains no solution other than the trivial solution (x(t), xc(t)) == (0,0).
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Hence, it follows from Theorem 2.3 that (x(t), xc(t))  M = {(O, On as t  00. Now, global asymptotic stability of the negative feedback interconnection of g and gc follows from the fact that Vs(·) and VscO are, by assumption, radially unbounded. iii) Finally, if g and gc are exponentially dissipative and (6.9) and (6.10) hold, it follows that that for all t E (tk, tk+1), V(x(t), xc(t)) = Vs(x(t))
+ O"Vsc(xc(t))
S ccVs(x(t))  O"cccVsc(xc(t)) + sc(uc(t),yc(t)) +O"scc( u cc (t), Ycc (t))
S  min{cc, ccc}V(x(t), xc(t)),
(x(t), xc(t)) ¢
Zx, (6.15)
and ~V(X(tk),Xc(tk))' (X(tk),Xc(tk)) E Zx, k E Z+, satisfies (6.14). Now, Theorem 2.1 implies that the negative feedback interconnection of g and gc is globally exponentially stable. 0 The next result presents Lyapunov, asymptotic, and exponential stability of dissipative feedback systems with quadratic supply rates. Theorem 6.2 Let Qc E
§lc, Sc E ~lcxmc, Rc E §mc, Qd E §ld, Sd E ~ldxmd, ~ E §md, Qcc E §lcc, Sec E ~lccxmcc, Rcc E §mcc, Qdc E §ldc, Sdc E ~ldcxmdc, and Rdc E §mdc. Consider the closed
loop system consisting of the nonlinear impulsive dynamical systems g given by (6.1}(6.4) and gc given by (6.5}(6.8), and assume g and gc are zerostate observable. Furthermore, assume g is dissipative with respect to the quadratic hybrid supply rate (sc(uc,yc),Sd(Ud,Yd)) = (y'[QcYc+2Y'[Scuc+u'[Rcuc, yJQdYd+2yJSdUd+UJRdUd) and has a radially unbounded storage function Vs (. ), and gc is dissipative with respect to the quadratic hybrid supply rate (scc(ucc,Ycc),Sdc(Udc,Ydc)) = (yJcQccYcc+2yJcSccucc+uJcRccucc, yJcQdcYdc+2yJcSdcUdc+UJcRdcUdc) and has a radially unbounded storage function Vsc(·). Finally, assume there exists a scalar 0" > 0 such that
(6.16) (6.17) Then the following statements hold:
i) The negative feedback interconnection of g and gc is Lyapunov stable.
197
STABILITY OF FEEDBACK IMPULSIVE DYNAMICAL SYSTEMS
ii) If 9 is strongly zerostate observable,
9c is exponentially dissipa
tive with respect to the hybrid supply rate (scc(ucc,Ycc), Sdc(Udc, Ydc)), and rank[Gcc(ucc , 0)] = m cc , Ucc E Ucc , then the negative feedback interconnection of 9 and 9c is globally asymptotically stable.
iii) If 9 and 9c are exponentially dissipative with respect to the sup
ply rates (sc( u c, Yc), Sd(Ud, Yd)) and (scc( Ucc , Ycc) , Sdc( Udc, Ydc)), respectively, and there exist constants a, a c, /3, /3c > 0 such that (6.9) and (6.10) hold, then the negative feedback interconnection of 9 and 9c is globally exponentially stable.
iv) IfQc < 0 and Qd < 0, then the negative feedback interconnection of 9 and 9c is globally asymptotically stable. Proof. Statements i)iii) are a direct consequence of Theorem 6.1 by noting
sc(uc,Yc) +oscc(ucc,Ycc) = [ Sd(Ud,Yd)
+ oSdc(Udc,Ydc) =
[
~: ]T Qc [ :: ],
:! ]T Qd [ ::c ],
(6.18) (6.19)
and hence, sc(uc, Yc)+oscc(u cc , Ycc) ::; 0 and Sd(Ud, Yd)+oSdc(Udc, Ydc)
::; o.
To show iv) consider the Lyapunov function candidate V(x, xc) = + oVsc(x). Noting that Ucc = Yc and Ycc = Uc, it follows that the corresponding Lyapunov derivative satisfies
Vs(x)
V(x(t), xc(t)) = Vs(x(t)) + oVac(xc(t)) ::; sc(uc(t), Yc(t)) + oScc (ucc (t) , Ycc(t)) = yJ(t)QcYc(t) + 2yJ(t)Scuc(t) + uJ(t)Rcuc(t) +o[yJc(t)QccYcc(t) + 2yJc(t)Sccucc(t) +uJc (t )Rccucc (t) 1 = [ Yc(t)
Ycc(t)
]T Qc [ ycc(t) yc(t) ]
(x(t), xc(t)) ¢ ix,
::; 0,
tk < t ::; tk+l, (6.20)
and, similarly, the Lyapunov difference satisfies
Yd(tk) ] T ~ [Yd(tk)] ~V(X(tk),Xc(tk)) = [ Ydc(tk) Qd Ydc(tk) ::; 0,
(X(tk), Xc(tk)) E
ix,
k E
Z+,
(6.21)
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which implies that the negative feedback interconnection of 0 and Oc is Lyapunov stable. Next, let 'R ~ {(x, xc) E Rn x Rnc : (x, xc) ¢ ix, V(x, xc) = o} U ((x,x c ) E RnxRnc : (x,x c ) E ix, LlV(x,xc ) = O}, where V(x, xc) and Ll V(x, xc) denote the total derivative and difference of V(x, xc) of the closedloop system for all (x, xc) ¢ ix and (x, xc) E ix, respectively. for all (x,x c ) E Rn X Rnc\ix if and Now, note that V(x,x c ) = only if (Yc, Ycc) = (0,0) and Ll V(x, xc) = for all (x, xc) E ix if and only if (Yd, Ydc) = (0,0). Since 0 and Oc are zerostate observable it follows that 'R ~ {(x, xc) ERn xRnc : (x, xc) ¢ ix, (hc(x),hcc(x c = (O,O)} U {(x, xc) E ix, (hd(x),hdc(x c» = (O,O)} which contains no solution other than the trivial solution (x(t), xc(t» == (0,0). Hence, it follows from Theorem 2.3 that (x(t), xc(t»  t M = {(O,O)} as t  t 00. Finally, global asymptotic stability follows from the fact that Vs (.) and Vsc (.) are, by assumption, radially unbounded. 0
°
°
»
The following result generalizes the classical positivity and smallgain theorems to the case of impulsive systems. For this result note that if a nonlinear dynamical system 0 is dissipative (respectively, exponentially dissipative) with respect to the hybrid supply rate (sc (Uc , Yc), Sd(Ud, Yd» = (2u;yc, 2ud Yd), then, with (Kc(Yc), Kd(Yd» = (kcYc, kdYd), where kc, kd > 0, it follows that (sc(uc, Yc), Sd(Ud, Yd» = (2kcY'Iyc,  2kdyJYd) < (0,0), (Yc, Yd) =I (0,0). Alternatively, if a nonlinear dynamical system 0 is dissipative (respectively, exponentially dissipative) with respect to the hybrid supply rate (sc( u c , Yc), Sd ( Ud,Yd» = (y;u;uc  Y;Yc,'Y~UJUd  yJYd), where 'Yc,'Yd > 0, then, with (Kc(Yc), Kd(Yd» = (0,0), it follows that (sc(u c , Yc), Sd(Ud, Yd» = (Y'[Yc, yJYd) < (0,0), (Yc,Yd) =I (0,0). Hence, if 0 is zerostate observable it follows from Theorem 3.3 that all storage functions of 0 are positive definite.
Corollary 6.1 Consider the closedloop system consisting of the nonlinear impulsive dynamical systems 0 given by {6.1}{6.4} and Oc given by {6.5}{6.8}. Assume 0 and Oc are zerostate observable. Then the following statements hold:
i) If 0 is passive and strongly zerostate observable, Oc is exponentially passive, and rank[Gcc(ucc,O)] = m cc , Ucc E Ucc , then the
negative feedback interconnection of 0 and Oc is asymptotically stable.
ii) If 0 and Oc are exponentially passive with storage functions Vs(·) and Vsc(), respectively, such that {6.9} and {6.10} hold, then the
STABILITY OF FEEDBACK IMPULSIVE DYNAMICAL SYSTEMS
199
negative feedback interconnection of Q and Qe is exponentially stable.
°
iii) If Q is nonexpansive with gains "Ie, 'Yd > and strongly zerostate observable, Qe is exponentially nonexpansive with gains "Icc> 0, "Ide> 0, rank [Gee (u ee , 0)] = m ee, U ee E Uee , "Ie "Icc ~ 1, and 'Yd'Yde ~ 1, then the negative feedback interconnection of Q and Qe is asymptotically stable.
iv) If Q and Qe are exponentially nonexpansive with storage functions Vs() and Vse(), respectively, such that {6.9} and {6.10} hold, and with gains "Ie, 'Yd > and "Icc, "Ide > 0, respectively, such that 'Ye'Yee ~ 1 and 'Yd'Yde ~ 1, then the negative feedback interconnection of Q and Qe is exponentially stable.
°
Proof. The proof is a direct consequence of Theorem 6.2. Specifically, statements i) and ii) follow from Theorem 6.2 with Qe = 0, Qd = 0, Qee = 0, Qde = 0, Se = Imc ' Sd = I md , Sec = I mec ' Sde = I mde , Rc = 0, Rd = 0, Rce = 0, and ~e = 0. Statements iii) and iv) follow from Theorem 6.2 with Qe = llc' Qd = lid' Qee = Ilec' Qde = Ildc ' Se = 0, Sd = 0, Sec = 0, Sde = 0, and Rc = 'Y;Imc ' Rd = 'YJlmd' Rce = "I! Imec , and ~e = 'YJJmdC· D Global asymptotic stability of the negative feedback interconnection of Q and Qe is also guaranteed if the nonlinear impulsive system Q is input strict passive (respectively, output strict passive) and the nonlinear impulsive compensator Qe is input strict passive (respectively, output strict passive) [74].
6.3 Hybrid Controllers for Combustion Systems
In this section, we apply the concepts developed in Section 6.2 to the control of thermoacoustic instabilities in combustion processes. Engineering applications involving steam and gas turbines and jet and ramjet engines for power generation and propulsion technology involve combustion processes. Due to the inherent coupling between several intricate physical phenomena in these processes involving acoustics, thermodynamics, fluid mechanics, and chemical kinetics, the dynamic behavior of combustion systems is characterized by highly complex nonlinear models [9, 10, 43, 86]. The unstable dynamic coupling between heat release in combustion processes generated by reacting
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mixtures releasing chemical energy and unsteady motions in the combustor develop acoustic pressure and velocity oscillations which can severely impact operating conditions and system performance. These pressure oscillations, known as thermoacoustic instabilities, often lead to high vibration levels causing mechanical failures, high levels of acoustic noise, high burn rates, and even component melting. Hence, the need for active control to mitigate combustioninduced pressure instabilities is critical. To design hybrid controllers for combustion systems we concentrate on a twomode, nonlinear timeaveraged combustion model with nonlinearities present due to the secondorder gas dynamics. This model is developed in [43] and is given by
,B(Xl(t)X3(t) + X2(t)X4(t)) + Usl(t), Xl(O) = XlO, (6.22) X2(t) = (hXl(t) + alx2(t) + ,B(X2(t)X3(t)  Xl(t)X4(t)) + Us2(t), X2(0) = X20, (6.23) X3(t) = a2x3(t) + (hX4(t) + ,B(xr(t)  x~(t)) + Us3(t), X3(0) = X30,
Xl(t)
=alxl(t)
+ (hX2(t) 
(6.24)
X4(t) = (hX3(t) + a2x4(t)
+ 2,Bxl (t)X2(t) + Us4(t),
X4(0)
= X40, (6.25)
where aI, a2 E R represent growth/decay constants, 8l ,(h E R represent frequency shift constants, ,B = (b + 1)/8,), )w!, where,), denotes the ratio of specific heats, WI is the frequency of the fundamental mode, and Usi, i = 1, ... ,4, are control input signals. For the data parameters al = 5, a2 = 55, 81 = 4, 82 = 32, ')' = 1.4, WI = 1, and Xo = [111 the openloop (Le., Usi(t) == 0, i = 1, ... ,4) dynamics (6.22)(6.25) result in a limit cycle instability. Figures 6.2, 6.3, and 6.4 show, respectively, the phase portrait, state response, and plant energy
IF,
(6.26)
versus time. To design a stabilizing timedependent hybrid controller for (6.22)(6.25) we first design a continuoustime control law (6.27)
where Ks ~ diag[ks !' ks2, ks3, ks4], x ~ [Xl, X2, X3, x4F, Us ~ [Us!' Us2, Us3, us4F, and Uc ~ [u e!' Ue2, Uc3, Uc4]T. In this case, (6.22)
201
STABILITY OF FEEDBACK IMPULSIVE DYNAMICAL SYSTEMS 1001''==~=,===~=..rl 50
,r
0
50
_100!:::~::L~:;===;:==L=:=;:::===~''''~ 100 SO 60 40 20 0 20 40 60 80 100 Xl
30r..=====~===.r~
20 10 x"
0
10 20
_30_~____:~======L::=====L.~ 30 20 10 0 10 20 30 x3
Figure 6.2 Phase portrait.
(6.25) are given by (6.1) and (6.2) with Z = 0 and
[
aIxI + (h X2  (3(XIX3 + X2 X4)  kslXI (hXI + aIX2 + (3(X2X3  XIX4)  ks2X2 a2x3 + (hX4 + (3(x~  x~)  ks3X3 (J2X3 + a2X4 + 2{3xIx2  ks4X4
1 ' (6.28) (6.29)
Now, with Yc = X, ksl = ks2 = aI, and ks3 = ks4 = 0, it follows that (6.1) and (6.3), with fc(x) and Gc(x) given by (6.28) and hc(x) = x and Jc(x) = 0, is passive with input u c, output Yc, and plant energy function, or storage function, Vs(x). Hence, V:(x)fc(x) ~ 0, x E R4. Furthermore, (6.1) and (6.3), with fc(x) and Gc(x) given by (6.28) and hc(x) = x and Jc(x) = 0, is zerostate observable. Figures 6.5, 6.6, and 6.7 show, respectively, the phase portrait, state response, and plant energy of the controlled system (6.1) and (6.3) with Us = Ksx + U c and U c == 0. To improve the performance of the above controller, we use the flexibility in U c to design a hybrid controller. Specifically, consider the hybrid controller emulating the plant structure given by (6.5)
202
CHAPTER 6
time
time
time
time
Figure 6.3 State response.
(6.31)
(6.32)
(6.33) (6.34) where kcl > aI, kc2 > aI, kc3 > a2, and kc4 > a2. It can be easily shown using the results of Chapter 3 that the hybrid controller (6.5)(6.8) with dynamics given by (6.30)(6.34), resetting set Sc =
203
STABILITY OF FEEDBACK IMPULSIVE DYNAMICAL SYSTEMS
8000
7000
6000
>. 5000
j
4000
3000
2000
)
1000
00
4
2
5
6
time
10
8
Figure 6.4 Plant energy versus time.
T x Rnc X Rmcc, input Yc, and output Uc is exponentially passive with controller energy, or storage function, Vsc(xc) ~ x~l + x~2 + x~3 + X~4· Hence, V;c(xc)fcc(xc) :s; cVsc(xc), Xc E R 4 , where c = min{ 0:1  kcb 0:1  kc2, 0:2  kc3, 0:2  k c4}. Furthermore, note that rank [Gcc(O)] = 4. Hence, stability of the closedloop system (6.1), (6.3), and (6.5)(6.8) is guaranteed by Theorem 6.1. Finally, we note that the total energy of the closedloop system (6.1), (6.3), and (6.5)(6.8) is given by [:,. TT V( X) = Vs (X)
+ vsc TT
Xc
()
2 2 = Xl2 + X22 + X32 + X42 + XcI + Xc22 + Xc32 + xc4'
(6.35)
xrF.
where x ~ [xT The effect of the resetting law (6.6) with fdc(X c) and Gdc(xc) given by (6.32), is to cause all the controller states to be instantaneously reset to zero, that is, the resetting law (6.6) implies Vsc (xc + Llxc) = o. The closedloop resetting law is thus given by Llx = [0 0 0 0 XcI
Xc2
Xc3
Xc4
f.
(6.36)
,
(6.37)
Note that since x
+ Llx =
[Xl
X2
X3
X4
0 0 0 0] T
204
CHAPTER 6 1.51,~::::::::::::======:::::::!=,1
0.5
.,0'
0
0.5 1
_1.5L'=;::::::::::::===::::::::::::;='~ 1.5
1
0.5
0
0.5
1.5
Xl
1.2..,.,.__,,.__,~__,_____"l
0.8 0.6 0.4 0.2
o 0.2 '~'''''''' 0.2 0 0.2 0.4 0.6 0.8 1.2
Figure 6.5 Phase portrait.
it follows that V{x
+ ~x) =
Vs{X)
(6.38)
and (6.39)
Now, from (6.39) it follows that the resetting law (6.6) causes the total energy to instantaneously decrease by an amount equal to the accumulated controller energy. To illustrate the dynamic behavior of the closedloop system, let al = 5, a2 = 55, ksl = aI, ks2 = a!, ks3 = 0, ks4 = 0, kc1 = al +0.1, kc2 = al + 0.1, kc3 = 0, kc4 = 0, and T = {2,4, 6, ... }, so that the controller resets periodically with a period of 2 seconds. The response of the controlled system (6.1) and (6.3) with the resetting controller (6.5){6.8) and initial condition Xo = [1 1 1 1 0 0 0 ojT is shown in Figure 6.8. Note that the control force versus time is discontinuous at the resetting times. A comparison of the plant energy, control energy, and total energy is given in Figure 6.9. In this example the resetting times were chosen arbitrarily. However, with the same choice of controller parameters we can choose a resetting time to achieve finitetime stabilization. Specifically, this
205
STABILITY OF FEEDBACK IMPULSIVE DYNAMICAL SYSTEMS 1.2r~'
0.8
0.5
,r
0
x'"
0.5
0.4
0.2
1
1.5
0.6
o
v V V 2
4 lime
V 6
V
O~~
V 8
10
1.5rA~A~A~i\A~A'
_0.2L~~~~1
o
2
4
lime
6
8
10
1.2~~,
0.8
0.5
0.6 0.4
0.5
0.2
1
1.5 L V,",,V=V~..:..........V=V~,V1
o
2
4
lime
6
8
10
O~~ _0.2L~~~~.J
o
2
4
time
6
8
10
Figure 6.6 State response.
resetting time will correspond to the time at which all of the energy of the plant is drawn to the controller. This resetting time can be obtained from the energy history of the closedloop system without resetting. In particular, the time instant when the plant and controller energy interchange is such that the plant energy is at zero corresponds to the resetting time that achieves finitetime stabilization. For this example, finitetime stability is achieved by choosing the resetting instant at t = 1.6223 sec. Next, we describe the mathematical setting and design of an input/statedependent resetting controller. We consider the plant and resetting controller as given above with Sc = [0, 00) x Zcxc x Zcucc' where
ZCXc x Zcucc = {(xc, u cc ) : !dc(Xc) i= 0 and V:c(xc)[!cc(xc) +Gcc(xc)uccl :S o}. (6.40) The resetting set (6.40) is thus defined to be the set of all controller states and input points that represent nonincreasing control energy, except for those points that satisfy !dc(Xc) = O. The states Xc that satisfy !dc(Xc) = 0 are states that do not change under the action of the resetting law, and hence, we need to exclude these states from
CHAPTER 6
206
4.,,,,,,,,,,
3.5
3
2.5
1.50L'.l.2.l..4':'5::.=:a::'9::10 time
Figure 6.7 Plant energy versus time.
Yl Y2 Y3
Y4
1.5
0
3
4
5
time
6
10
7
1.5 Uc1 Uc2
t:
0.5
eE
0
~
Uc3
.0, 0, fi,di > 0, i = 1, ... ,q, it follows that Sci (fi,ci (Yci) , Yci) = 2fi,ciY;Yci < and Sdi(fi,di(Ydi)' Ydi) = 2fi,diyIYdi < 0, Yci =1= 0, Ydi =1= 0, i = 1, ... , q. Alternatively, if 9 is vector dissipative with respect to the vector hybrid supply rate (Sc(uc, Yc), Sd(Ud, Yd)), where Sci (Uci' Yci) = l'~iU~Uci  Y;Yci and Sdi( Udi, Ydi) = l'~iUriUdi yIYdi, where I'ci > 0, I'di > 0, i = 1, ... ,q, then, with fi,ci(Ycd = and fi,di(Ydi) = 0, it follows that Sci (fi,ci (Yci) , Yci) = Y;Yci < and Sdi(fi,di(Ydi), Ydi) = yIYdi < 0, Yci =1= 0, Ydi =1= 0, i = 1, ... , q. Hence, if 9 is zerostate observable and the dissipation matrix W is such that there exist a 2: and p E ]R~ such that (5.24) holds, then it follows from Theorem 5.4 that (scalar) storage functions of the form vs(x) = pTv;,(x), X E ]Rn, where v;,(.) is a vector storage function for g, are positive definite. If 9 is exponentially vector dissipative, then p is positive.
°
° °
°
Corollary 6.3 Consider the largescale impulsive dynamical systems
9 and gc given by (6.57}(6.60) and (6.65}(6.68), respectively. Assume that 9 and gc are zerostate observable and the dissipation matrices WE ]Rqxq and Wc E ]Rqxq are such that there exist, respectively, a 2: 0, p E ]R~, a c 2: 0, and Pc E ]R~ such that (5.24) is satisfied. Then the following statements hold:
220
CHAPTER 6
i) If g and gc are vector passive and W E lR qxq is asymptotically stable, where W(i,j) ~ max{W(i,j)' WC(i,j)}' i, j = 1, ... ,q, then the negative feedback interconnection of g and gc is asymptotically stable.
ii) If g and gc are vector nonexpansive and W E lRqxq is asymptotically stable, where W(i,j) ~ max{W(i,j) , Wc(i,j)}, i,j = 1, ... , q, then the negative feedback interconnection of g and gc is asymptotically stable.
Proof. The proof is a direct consequence of Theorem 6.4. Specifically, statement i) follows from Theorem 6.4 with Rei = 0, Sci = Imci' Qci = 0, Rdi = 0, Sdi = Imdi' Qdi = 0, Reci = 0, Scci = Imci' Qcci = 0, Rdci = 0, Sdci = Imdi' Qdci = 0, i = 1, ... ,q, and 'E = I q . Statement ii) follows from Theorem 6.4 with Rei = 'Y;Jmci' Sci = 0, Qci = Ilci' Rdi = 'Y~Jmdil Sdi = 0, Qdi = Ildi' Reci = 'Y;ciIlci' Scci = 0, Qcci = Imci' Rdci = 'Y~cJldi' Sdci = 0, Qdci = Imdi' i = 1, ... ,q, and 'E = I q . 0
Chapter Seven EnergyBased Control for Impulsive PortControlled Hamiltonian Systems
7.1 Introduction
In a recent series of papers [136138] a passivitybased control framework for portcontrolled Hamiltonian systems is established. Specifically, the authors in [136138] develop a controller design methodology that achieves stabilization via system passivation. In particular, the interconnection and damping matrix functions of the portcontrolled Hamiltonian system are shaped so that the physical (Hamiltonian) system structure is preserved at the closedloop level, and the closedloop energy function is equal to the difference between the physical energy of the system and the energy supplied by the controller. Since the Hamiltonian structure is preserved at the closedloop level, the passivitybased controller is robust with respect to unmodeled passive dynamics. Furthermore, passivitybased control architectures are extremely appealing since the control action has a clear physical energy interpretation which can considerably simplify controller implementation. Modern complex engineering systems involve multiple modes of operation, placing stringent demands on controller design and implementation of increasing complexity. As discussed in Chapter 1, such systems typically possess a multiechelon hierarchical hybrid control architecture characterized by continuoustime dynamics at the lower levels of the hierarchy and discretetime dynamics at the higher levels of hierarchy. The mathematical description of many of these systems can be characterized by impulsive differential equations. Furthermore, since certain dynamical systems such as telecommunications, transportation, biological, physiological, power, and network systems involve highlevel, abstract hierarchies with inputoutput properties related to conservation, dissipation, and transport of mass and/or energy, these systems can be modeled as impulsive portcontrolled Hamiltonian systems. In this chapter, we use the stability and dissipativity framework
222
CHAPTER 7
for impulsive dynamical systems developed in Chapters 2 and 3 to extend the results in [136138] to nonlinear impulsive portcontrolled Hamiltonian systems. Specifically, we develop an energybased hybrid feedback control framework for nonlinear impulsive portcontrolled Hamiltonian systems that preserves the physical hybrid Hamiltonian structure at the closedloop level. In particular, we present sufficient conditions for hybrid feedback stabilization that preserve the physical hybrid Hamiltonian structure at the closedloop level while providing a shaped Hamiltonian energy function as a Lyapunov function for the closedloop impulsive system. These sufficient conditions consist of a hybrid system of two partial differential equations involving the continuoustime dynamics and the resetting (discretetime) dynamics. We emphasize that our approach is constructive in nature providing a hybrid system of partial differential equations whose solutions, when they exist, characterize the set of all desired shaped Hamiltonian energy functions that can be assigned while preserving the hybrid Hamiltonian structure at the closedloop system level. Unlike the passivitybased control framework developed in [136138] for portcontrolled Hamiltonian systems with continuous flows, our approach does not achieve stabilization via hybrid system passivation in the sense of [56,61]. However, under certain conditions on the open and closedloop dissipation matrix functions, the closedloop energy function over the continuoustime trajectories is equal to the difference between the physical energy of the hybrid system and the energy supplied by hybrid controller. Furthermore, the closedloop energy function at the resetting instants is nonincreasing.
7.2 Impulsive PortControlled Hamiltonian Systems
In this section, we introduce nonlinear impulsive portcontrolled Hamiltonian systems. We begin by considering an input/statedependent impulsive portcontrolled Hamiltonian system 9 given by
x(t) = [J"c(x(t))  'Rc(x(t))]
(~: (x(t))) T + Gc(x(t))uc(t),
x(O) = Xo,
~x(t) = [J"d(X(t)) yc(t) = hc(x(t))
'Rd(X(t))]
(x(t), uc(t))
tt z,
(7.1)
(~: (X(t))) T + Gd(X(t))Ud(t),
+ Jc(x(t))uc(t),
(x(t),uc(t)) E Z, (x(t), uc(t)) tt z,
(7.2) (7.3)
223
ENERGYBASED CONTROL FOR IMPULSIVE HAMILTONIAN SYSTEMS
where t 2:: 0, x(t) E V ~ ~n, V is an open set, ~x(t) ~ x(t+)  x(t), uc(t) E Uc ~ ~mc, Ud(tk) E Ud ~ ~md, tk denotes the kth instant of time at which (x( t), U c (t)) intersects Z for a particular trajectory x(t) and input uc(t), yc(t) E Yc ~ ~lc, Yd(tk) E Yd ~ ~ld, 1{ : V t ~ is a continuously differentiable Hamiltonian function for the impulsive system (7.1)(7.4), Jc : V t ~nxn is such that Jc(x) = J?(x), Rc : V t §n is such that Rc(x) 2:: 0, x E V, [Jc(x)  Rc(x)] (x)) T, x E V, is Lipshitz continuous, C c : V t ~nxmc, Jd : V t ~nxn is such that Jd(X) = Jl(x), Rd : V t §n
(a;;;
(a;;;
is such that Rd(X) 2:: 0, x E V, [Jd(X) Rd(X)] (x))T ,x E V, is continuous, Cd : V t ~nxmc, hc : V t ~lc, J c : V t ~lc xmc , hd : V t ~ld, Jd : V t ~ldxmd, and Z ~ (Zx x Uc ) U (~n X ZuJ c V x Uc is the resetting set. The skewsymmetric matrix functions Jc(x) and Jd(X), x E V, capture the internal hybrid system interconnection structure, the input matrix functions Cc(x) and Cd(x), x E V, capture hybrid interconnections with the environment, and the symmetric nonnegativedefinite matrix functions Rc(x) and Rd(X), x E V, capture hybrid system dissipation. Here, we assume that U c ( .) and Ud (.) are restricted to the class of admissible inputs consisting of measurable functions such that (uc(t), Ud(tk)) E Uc x Ud for all t 2:: and k E Z[O,t) ~ {k : ~ tk < t}. We denote the solution to (7.1) and (7.2) with initial condition Xo E V by s(t, xo), t ~ 0, and the set of the resetting times tk == Tk(XO) for a particular trajectory s(,xo) by [O,oo)\Txo,uc ~ {tl,t2, ... }, where Txo,uc is a dense subset of the semiinfinite interval [0,(0) such that 'Ixco,uc ~ [0, oo)\Txo,uc is (finitely or infinitely) countable. For notational convenience we write T and TC for Txo,uc and 'Ixco,uc' respectively. Note that the solution x(t), t 2:: 0, of (7.1) and (7.2) is leftcontinuous. Furthermore, as shown in Chapter 2, if the resetting set is such that it removes X(tk) from the resetting set and if no trajectory can intersect the interior of Z, then the resetting times tk, k E Z+, are well defined and distinct. Since the resetting times are well defined and distinct, and since the solution to (7.1) exists and is unique, it follows that the solution of the impulsive portcontrolled Hamiltonian system (7.1) and (7.2) also exists and is unique over a forward time interval. However, as discussed in Chapter 2, the analysis of impulsive dynamical systems can be quite involved. In particular, such systems can exhibit Zenoness and beating, as well as confluence. Furthermore,
°
°
224
CHAPTER 7
due to Zeno solutions, not every bounded solution of an impulsive dynamical system over a forward time interval can be extended to infinity. Here, we assume that Assumptions Al and A2 established in Chapter 2 hold, and hence we allow for the possibility of confluence and Zeno solutions, however, we preclude the possibility of beating. It is important to note that in our impulsive system formulation (7.1) and (7.2) we assume that the impact model dynamics (7.2) is Hamiltonian. For mechanical systems with collisions this is without loss of generality. To see this, let x = [qT, qTF, where q E ]Rn represents generalized positions and q E ]Rn represents generalized velocities, and n = ~, and note that the impact dynamics are given by (7.5) (7.6) where T(q, q) = ~qT M(q)q is the system kinetic energy, M(q) > 0, q E ]Rn, is the system inertia matrix function, 'I : ]Rn x ]Rn  t ]Rnxn is an impact matrix function, and tk, tt are the instants before and after collisions, respectively. The impact function I(·,·) can be quite difficult to characterize since solid impacts can involve stress waves, expansions in colliding solids, and reflections from solid boundaries. To capture the dynamics of these waves it is often necessary to use partial differential equations. For an additional discussion on impact dynamics see [33,34]. However, assuming that across a collision event the generalized system velocities change according to the law of conservation of momentum, and the generalized velocities account for the loss of kinetic energy in a collision, (7.5) and (7.6) can be rewritten as
where H(q, q) = T(q, q) + V(q) denotes the total system energy and V (q) is the system potential energy. Next, note that the matrix function M 1 (q)I(q, q)_M 1 (q), (q, q) E ]Rn x ]Rn, can be represented as a sum of a skewsymmetric matrix
225
ENERGYBASED CONTROL FOR IMPULSIVE HAMILTONIAN SYSTEMS
function and a negativesemidefinite matrix function if and only if
(IT (q, q)  I ii )M 1(q)
+ M1(q)(I(q, q) 
Iii) ::; 0,
(q,q) E ~ii x ~ii. (7.8) Now, assuming that the kinetic energy after the impact is less than or equal to the kinetic energy before the impact, that is, (7.9) it follows from (7.7) and (7.9), since (7.9) holds for arbitrary q, that
M(q)IT (q, q)M 1(q)I(q, q)M(q) ::; M(q),
q E ~ii,
(q, q) E ~ii x ~ii, (7.10)
which is equivalent to
where O"max() denotes the maximum singular value. Now, it follows from (7.11) that
M~(q)I(q,q)M~(q)
+ M~(q)IT(q,q)M~(q) 1
1
::; O"max[M'2 (q)I(q, q)M'2 (q)
+M~ (q)I T (q, q)M~ (q)]Jn 1
1
::; 20"max[M'2 (q)I(q, q)M'2 (q)]Jn ::; 21ii, (q, q) E ~ii X ~ii, (7.12) where (.)1/2 denotes the (unique) positivedefinite square root. Hence,
M~ (q)I(q, q)M~ (q)
+ M~ (q)IT (q, q)M~ (q) (q, q)
21ii ::; 0,
E ~ii X ~ii,
(7.13)
which is equivalent to (7.8), and hence, the impact dynamics (7.7) can be written in a Hamiltonian form
(7.14) Finally, we note that Gd(X)Ud in (7.2) provides the additional flexibility of including an impulsive control to the impact dynamics. See [168] for additional details. Assuming that the Hamiltonian energy function 1t (.) is lower bounded, it can be shown (with an additional structural constraint on 1t(.))
226
CHAPTER 7
that impulsive portcontrolled Hamiltonian systems provide a hybrid energy balance in terms of the stored or accumulated energy, hybrid supplied system energy, dissipated energy over the continuoustime dynamics, and dissipated energy at the resetting instants. To see this, let the hybrid inputs and hybrid outputs be dual (conjugated) variables so that Yc(t)
= GJ(x(t))
(a;; (x(t)))T ,(x(t),uc(t))
Yd(t) = GJ(x(t)) (~~(x(t)))T, (x(t),uc(t))
E
rf
Z,
Z, and assume ft(·)
is such that 1
11 (x + [Jd(X)  nd(X)] ( ' : (X)f + Gd(X)Ud)
+ 8ft 8x (x) [Jd(X)
 Rd(X)] (8ft 8x (x) X E
V,
~ 1I(x)
)T + (8ft8x (x) ) Ud E Ud •
Gd(X)Ud,
(7.15)
Now, computing the rate of change of the Hamiltonian along the system state trajectories x(t), t E (tk' tk+l], and the Hamiltonian difference at the resetting times tk, k E Z+, yields the set of energy conservation equations given by2
(7.16)
(7.17) Equation (7.16) shows that the rate of change in energy, or power, over the time interval t E (tk' tk+l] is equal to the system power input minus the internal system power dissipated, while (7.17) shows that the change of energy at the resetting times tk, k E Z+, is equal to the supplied system energy at the resetting times minus the dissipated energy at the resetting times. Using Theorem 3.2, (7.16) and (7.17) IThe structural constraint on the Hamiltonian given by (7.15) is natural for nonnegative and compartmental dynamical systems where the state vector is restricted to the nonnegative orthant of the state space [57,63]. For these systems the Hamiltonian represents the total mass/energy in the system and is a linear function of the state. For details, see Section 4.3. 2Note that (7.16) holds even if H(·) does not satisfy the structural constraint (7.15).
ENERGYBASED CONTROL FOR IMPULSIVE HAMILTONIAN SYSTEMS
227
can be equivalently written as
t 2:: O. (7.18)
Equation (7.18) shows that the stored or accumulated system energy is equal to the energy supplied to the system via the hybrid external inputs U c and Ud minus the energy dissipated over the continuoustime dynamics and the resetting instants. Since Rc (x) and Rd (x) are nonnegative definite for all x E V, it follows from (7.18) that (7.19) which shows that the energy that can be extracted from the impulsive portcontrolled Hamiltonian system through the hybrid inputoutput ports is less than or equal to the initial energy stored in the system. Hence, impulsive portcontrolled Hamiltonian systems with the structural constraint (7.15) are passive systems in the sense of Definition 3.5.
7.3 EnergyBased Hybrid Feedback Control
In this section, we present an energybased hybrid feedback control framework for nonlinear impulsive portcontrolled Hamiltonian systems that preserves the Hamiltonian structure at the closedloop level. In particular, we obtain constructive sufficient conditions for feedback stabilization of an arbitrary equilibrium point in V that provide a shaped energy function for the closedloop system while preserving a hybrid Hamiltonian structure at the closedloop level. To address the energybased hybrid feedback control problem let cPc : V  Uc and cPd : V  Ud. If (uc(t),Ud(tk)) = (cPc(x(t)),cPd(X(tk))), then (ucO, Ud(·)) is a hybrid feedback control. Note that with the hybrid
228
CHAPTER 7
feedback control law (uc(t), Ud(tk)) = ( 0 such that sup IX('ljJ(t, x))  X('ljJ(t, xo))1
< c,
x
B8(xo),
E
(8.12)
09::;£2
which implies that X ('ljJ(il , x)) > 0 and X('ljJ(i2' x)) < 0, x E B8(xo), Hence, it follows that il < Tl (x) < i 2, x E B8 (xo). The continuity of Tl (.) at Xo now follows immediately by noting that e can be chosen arbitrarily small. Finally, let Xo E Z\Z be such that limi+CXl Xi = Xo for some sequence {Xi}~1 E Z\Z. Then using similar arguments as above it can be shown that limi+CXl Tl (Xi) = Tl (xo). Alternatively, if Xo E Z\Z is such that limi+CXl Xi = Xo and limi+CXl Tl (Xi) > 0 for some sequence {Xi}~1 cj Z, then it follows that there exists sufficiently small i> 0 and 1 E Z+ such that s(i, Xi) = 'ljJ(i, Xi), i = 1,1 + 1, ... , which implies that limi+CXl s(t, Xi) = s(t, xo). Next, define Zi = 'ljJ(t, xd, i = 0,1, ... , so that limi+CXl Zi = Zo and note that it follows from the transversality assumption that Zo cj Z, which implies that TI0 is continuous at ZOo Hence, limitCXl Tl (Zi) = Tl (zo). The result now follows by noting that Tl(Xi) = i + Tl(Zi), i = 1,2,.... 0 A
A
~
A
Note that if Xo cj Z is such that limi+CXl Tl (Xi) =I Tl (xo) for some sequence {xd~1 cj Z, then it follows from Proposition 8.2 that limi+CXl Tl(Xi) = O. The next result characterizes impulsive dynamical system limit sets for impulsive dynamical systems satisfying the weak quasicontinuous dependence Assumption 8.1 in terms of continuously differentiable functions. In particular, we show that the system trajectories of a statedependent impulsive dynamical system converge to an invariant set contained in a union of level surfaces characterized by the continuoustime system dynamics and the resetting system dy
257
ENERGY AND ENTROPYBASED HYBRID STABILIZATION
namics. For the next set of results we assume that feO, fdO, and Z are such that the dynamical system 9 given by (8.6) and (8.7) satisfies Assumptions AI, A2, and 8.1, and Z n {x: fd(x) = x} is empty. Theorem 8.1 Consider the impulsive dynamical system (8.6) and (8.7), assume 'Dei C 'D is a compact positively invariant set with respect to (8.6) and (8.7), assume that if Xo E Z then Xo + fd(xo) E Z\Z, and assume that there exists a continuously differentiable function V : 'Dei t lR such that V'(x)fe(x) :::; 0, V(x
x E'Dei,
+ fd(x)):::; V(x),
X
x E'Dei,
tf. z, X E Z.
(8.13) (8.14)
Let R ~ {x E 'Dei : x tf. Z, V'(x)fe(x) = O} U {x E 'Dei : x E Z, V(x + fd(x)) = V(x)} and let M denote the largest invariant set contained in R. If Xo E 'Dei, then x(t) t M as t t 00. Furthermore, if
oE
o
'Dei, V(O) = 0, V(X) > 0, X i= 0, and the set R contains no invariant set other than the set {O}, then the zero solution x(t) == 0 to (8.6) and (8.7) is asymptotically stable and 'Dei is a subset of the domain of attraction of (8.6) and (8.7).
Proof. The proof is similar to the proof of Theorem 2.3 and, hence, is omitted. 0 Setting 'D = lRn and requiring V(x) t 00 as Ilxll t 00 in Theorem 8.1, it follows that the zero solution x(t) == 0 to (8.6) and (8.7) is globally asymptotically stable. A similar remark holds for Theorem 8.2 below. Theorem 8.2 Consider the impulsive dynamical system (8.6) and (8.7), assume 'Dei C 'D is a compact positively invariant set with reo
spect to (8.6) and (8.7) such that 0 E 'Dei, assume that if Xo E Z then Xo + fd(xo) E Z\Z, and assume that for all Xo E 'Dei, Xo i= 0, there exists T ~ 0 such that X(T) E Z, where x(t), t ~ 0, denotes the solution to (8.6) and (8.7) with the initial condition Xo. Furthermore, assume there exists a continuously differentiable function V : 'Dei t lR such that V(O) = 0, V(x) > 0, xi= 0, V'(x)fe(x):::; 0,
x E'Dei , X V(x + fd(x)) < V(x), x E'Dei,
tf. z, X E Z.
(8.15) (8.16)
Then the zero solution x(t) == 0 to (8.6) and (8.7) is asymptotically stable and 'Dei is a subset of the domain of attraction of (8.6) and
(8.7).
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Proof. The proof is identical to the proof of Corollary 2.3 with Theorem 8.1 invoked in place of Corollary 2.1. D
8.3 Hybrid Control Design for Dissipative Dynamical Systems
In this section, we present a hybrid controller design framework for dissipative dynamical systems [165]. Specifically, we consider nonlinear dynamical systems Qp of the form
Xp(t) = fp(xp(t),u(t)), y(t) = hp(xp(t)),
xp(O) = xpo,
t ~ 0,
(8.17) (8.18)
0, xp(t) E 'Dp ~ ]Rnp , 'Dp is an open set with 0 E 'Dp, u(t) E ]Rm, y(t) E ]Rl, fp : 'Dp x ]Rm + ]Rnp is smooth on 'Dp x ]Rm and satisfies fp(O, 0) = 0, and hp : 'Dp + ]Rl is continuous and satisfies hp(O) = O. Furthermore, for the nonlinear dynamical system Qp we assume that the required properties for the existence and uniqueness of solutions are satisfied, that is, u(·) satisfies sufficient regularity conditions such that (8.17) has a unique solution forward in time. Next, we consider hybrid resetting dynamic controllers Qc of the form where t
~
Xc(t) = fcc(xc(t), y(t)), ~xc(t) = 'fJ(y(t))  xc(t), yc(t) = hcc(xc(t), y(t)),
xc(O) = Xco, (xc(t), y(t)) ¢ Zc, (8.19) (8.20) (xc(t), y(t)) E Zc, (8.21)
where xc(t) E 'Dc ~ ]Rnc, 'Dc is an open set with 0 E 'Dc, y(t) E ]Rl, yc(t) E ]Rm, fcc: 'Dc x ]Rl + ]Rnc is smooth on 'Dc x ]Rl and satisfies fcc (0, 0) = 0, 'fJ : ]Rl + 'Dc is continuous and satisfies 'fJ(0) = 0, and hcc : 'Dc X ]Rl + ]Rm is continuous and satisfies hcc(O, 0) = O. Recall that for the dynamical system Qp given by (8.17) and (8.18), a function s(u,y), where s:]Rm x]Rl +]R is such that s(O,O) = 0, is called a supply rate [165] if it is locally integrable for all inputoutput pairs satisfying (8.17) and (8.18), that is, for all inputoutput pairs u(·) E U and y(.) E Y satisfying (8.17) and (8.18), s(·,·) satisfies ri Jt Is(u(a), y(a))lda < 00, t, t ~ O. Here, U and Yare input and output spaces, respectively, that are assumed to be closed under the shift operator. Furthermore, we assume that Qp is dissipative with respect to the supply rate s( u, y), and hence, there exists a continuous, nonnegativedefinite storage function Vs : 'Dp + ]R+ such that Vs(O) = A
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ENERGY AND ENTROPYBASED HYBRID STABILIZATION
y
Yc
9c
Figure 8.1 Feedback interconnection of 9p and 9c.
°
and
Vs(xp(t))
=
Vs(xp(to))
+
t r [S(U(C5), y(C5))  d(xp(C5))]dC5, ltD
t
~ to,
for all to, t ~ 0, where xp(t), t ~ to, is the solution to (8.17) with u(·) E U and d : Vp ~ lR+ is a continuous, nonnegativedefinite dissipation rate function. In addition, we assume that the nonlinear dynamical system 9p is completely reachable [165] and zerostate observable [165], and there exists a function I'\, : lRl ~ lRm such that 1'\,(0) = and s(I'\,(Y) , y) < 0, y i= 0, so that all storage functions Vs(xp), xp E V p , of 9p are positive definite [75]. Finally, we assume that Vs() is continuously differentiable. Consider the negative feedback interconnection of 9p and Qc given in Figure 8.1 such that y = Uc and u = Yc. In this case, the closedloop system 9 is given by
°
x(t) = fc(x(t)), ~x(t) = fd(X(t)),
x(o) = Xo, x(t) E Z,
where t ~ 0, x(t) ~ [x~(t), xJ(t)F, Z
fc(x) = [ fp(x p, hcc(xc, hp(x p))) ], fcc (xc, hp(xp))
x(t) (j. Z,
(8.22)
t ~ 0,
(8.23)
= {x
E
V : (xc, hp(xp))
fd(X) = [
°
E
Zc},
'f/(hp(xp))  Xc
].
(8.24)
°
Assume that there exists an infinitely differentiable function Vc : Vc x l I lR ~ lR+ such that Vc(x c, y) ~ 0, Xc E Vc, Y E lR , and Vc(xc, y) = if and only if Xc = 'f/(y) and
Vc(xc(t), y(t)) = sc(uc(t), yc(t)),
(xc(t), y(t)) (j. Z,
t ~ 0, (8.25)
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where Sc : ]Rl X ]Rm + ]R is such that sc(O,O) = 0 and is locally integrable for all inputoutput pairs satisfying (8.19)(8.21). We associate with the plant a positivedefinite, continuously differentiable function Vp(xp) ~ Vs(xp), which we will refer to as the plant energy. Furthermore, we associate with the controller a nonnegativedefinite, infinitely differentiable function Vc(x c , y) called the controller emulated energy. Finally, we associate with the closedloop system the function
(8.26) called the total energy. Next, we construct the resetting set for the closedloop system 9 in the following form Z
= {(xp, xc)
E Vp x Vc : Llc Vc(x c , hp(xp))
and Vc(x c , hp(xp)) > 0 } .
=0 (8.27)
The resetting set Z is thus defined to be the set of all points in the closedloop state space that correspond to decreasing controller emulated energy. By resetting the controller states, the plant energy can never increase after the first resetting event. Furthermore, if the continuoustime dynamics of the closedloop system are lossless and the closedloop system total energy is conserved between resetting events, then a decrease in plant energy is accompanied by a corresponding increase in emulated energy. Hence, this approach allows the plant energy to flow to the controller, where it increases the emulated energy but does not allow the emulated energy to flow back to the plant after the first resetting event. This energydissipating hybrid controller effectively enforces a oneway energy transfer between the plant and the controller after the first resetting event. For practical implementation, knowledge of Xc and y is sufficient to determine whether or not the closedloop state vector is in the set Z. The next theorem gives sufficient conditions for asymptotic stability of the closedloop system 9 using statedependent hybrid controllers.
Theorem 8.3 Consider the closedloop hybrid dynamical system 9 given by {8.22} and {8.23} with the resetting set Z given by {8.27}. Assume that Vei C V is a compact positively invariant set with reo
spect to 9 such that 0 E Vei, assume that gp is lossless with respect to the supply rate s(u, y) (i.e., d(xp) == O} and with a positive definite, continuously differentiable storage function Vp(xp), Xp E V p,
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ENERGY AND ENTROPYBASED HYBRID STABILIZATION
and assume there exists a smooth {i.e., infinitely differentiable} func. twn Vc : 'Dc x jRI t jR+ such that Vc(x c, y) ~ 0, Xc E 'Dc, y E jR,I and Vc(x c, y) = 0 if and only if Xc = 'f/(y) and {8.25} holds. Furthermore, assume that every Xo E 2 is transversal to {8.22} and (8.28)
where y = U c = hp(xp), u = Yc = hcc(xc, hp(xp)), and 2 is given by {8.27}. Then the zero solution x(t) == 0 to the closedloop system Q is asymptotically stable. In addition, the total energy function V(x) of Q given by {8.26} is strictly decreasing across resetting events. Alternatively, assume Qp is dissipative with respect to the supply rate s( u, y) and the largest invariant set contained in R ~ {(xp, xc) E 'Dei : d(xp) = O} is M = {(O, On. Then the zero solution x(t) == 0 to Q is asymptotically stable. Finally, if'Dp = jRnp , 'Dc = jRnc, and V(·) is radially unbounded, then the above asymptotic stability results are global.
Proof. First we consider the case where Qp is lossless with respect to the supply rate s(u, y). Note that since Vc(x c, y) ~ 0, Xc E 'Dc, y E jRl, it follows that 2={(xp,xc) E'Dp x'Dc: LlcVc(xc,hp(xp)) =0 and Vc(x c, hp(xp)) ~ O} = {(xp, xc) E'Dp x 'Dc : X(x) = O},
(8.29)
where X(x) = Llc Vc(x c, hp(xp)). Next, we show that if the transversality condition (8.11) holds, then Assumptions A1, A2, and 8.1 hold, and, for every Xo E 'Dei, there exists r ~ 0 such that x(r) E 2. Note that if Xo E 2\2, that is, Vc(xc(O), hp(xp(O))) = 0 and Llc Vc(xc(O), hp (xp(O))) = 0, it follows from the transversality condition that there exists 8 > 0 such that for all t E (0,8], Llc Vc(xc(t), hp(xp(t))) =1= O. Hence, since Vc(xc(t) , hp(xp(t))) = Vc(xc(O), hp(xp(O)))+tLlc Vc(xc(r) ,hp(xp(r))) for some r E (O,t] and Vc(xc,y) ~ 0, Xc E 'Dc, y E jRl, it follows that Vc(xc(t), hp(xp(t))) > 0, t E (0,8], which implies that Assumption A1 is satisfied. Furthermore, if X E 2 then, since Vc(x c, y) = 0 if and only if Xc = 'f/(y) , it follows from (8.23) that X + fd(x) E 2\2. Hence, Assumption A2 holds. Next, consider the set
M')'
~ {x E 'Dei : Vc(x c, hp(xp))
= 'Y} ,
(8.30)
where 'Y ~ O. It follows from the transversality condition that for every 'Y ~ 0, M')' does not contain any nontrivial trajectory of Q. To
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see this, suppose, ad absurdum, there exists a nontrivial trajectory x(t) E My, t 2: 0, for some 'Y 2: O. In this case, it follows that Vc(xe(t), hp(xp(t))) = Ljc Vc(xe(t), hp(xp(t))) = 0, k = 1,2, ... , which contradicts the transversality condition. Next, we show that for every Xo ¢ Z, Xo i= 0, there exists 7 > 0 such that X(7) E Z. To see this, suppose, ad absurdum, x(t) ¢ Z, t 2: 0, which implies that
Jt:
(8.31) or (8.32) If (8.31) holds, then it follows that Vc(xe(t), hp(xp(t))) is a (decreasing or increasing) monotonic function of time. Hence, Vc (xc (t), hp (xp (t))) t 'Y as t t 00, where 'Y 2: 0 is a constant, which implies that the positive limit set of the closedloop system is contained in My for some 'Y 2: 0, and hence, is a contradiction. Similarly, if (8.32) holds then Mo contains a nontrivial trajectory of 9 also leading to a contradiction. Hence, for every Xo ¢ Z, there exists 7> 0 such that X(7) E Z. Thus, it follows that for every Xo ¢ Z, 0 < 71(XO) < 00. Now, it follows from Proposition 8.2 that 71() is continuous at Xo ¢ Z. Furthermore, for all Xo E Z\Z and for every sequence {Xi}~1 E Z\Z converging to Xo E Z\Z, it follows from the transversality condition and Proposition 8.2 that limhoo 71 (Xi) = 71(XO). Next, let Xo E Z\Z and let {Xi}~1 E 'Dei be such that limi>oo Xi = Xo and limi>oo 71 (Xi) exists. In this case, it follows from Proposition 8.2 that either limi>oo 71 (Xi) = 0 or limi>oo 71 (Xi) = 71 (xo). Furthermore, since Xo E Z\Z corresponds to the case where Vc(xeo, hp(xpo)) = 0, it follows that XeO = 7}(hp(xpo)), and hence, fd(XO) = O. Now, it follows from Proposition 8.1 that Assumption 8.1 holds. 0 to 9 is asymptotically To show that the zero solution x(t) stable, consider the Lyapunov function candidate corresponding to the total energy function V(x) given by (8.26). Since gp is lossless with respect to the supply rate s(u, y), and (8.25) and (8.28) hold, it follows that
=
V(X(t)) = s(u(t),y(t))
+ se(ue(t),Ye(t)) = 0,
x(t) ¢ Z. (8.33)
Furthermore, it follows from (8.24) and (8.27) that ~V(X(tk))
= Vc(xe(tt), hp(xp(tt)))  Vc(Xe(tk), hp(Xp(tk)))
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ENERGY AND ENTROPYBASED HYBRID STABILIZATION
Vc(',,(hp(Xp(tk))) , hp(Xp(tk)))  Vc(Xc(tk), hp(Xp(tk))) =  Vc(Xc(tk), hp(Xp(tk))) < 0, X(tk) E Z, k E Z+. (8.34) =
Thus, it follows from Theorem 8.2 that the zero solution x(t) = 0 to y is asymptotically stable. If 'Dp = ]Rnp, 'Dc = ]Rnc, and V (.) is radially unbounded, then global asymptotic stability is immediate. If Yp is dissipative with respect to the supply rate s( u, y) and for every Xo z, Xo = 0, there exists T > 0 such that X(T) E Z, then the proof is identical to the proof for the lossless case. Alternatively, if there exists kmax 2: 0 such that k ~ kmax , that is, the closedloop system trajectory intersects the resetting set Z a finite number of times, then the closedloop impulsive dynamical system possesses a continuous flow for all t > tkmax • In this case, since the largest invariant set contained in R is {(O, On, closedloop asymptotic stability of y follows from standard Lyapunov and invariant set arguments. Finally, if'Dp = ]Rnp , 'Dc = ]Rnc, and V(·) is radially unbounded, then global 0 asymptotic stability is immediate.
rt
If Vc = Vc(x c, y) is only a function of Xc and Vc(xc) is a positivedefinite function, then we can choose 'f/(y) = O. In this case, Vc(xc) = 0 if and only if Xc = 0, and hence, Theorem 8.3 specializes to the case of a negative feedback interconnection of a dissipative dynamical system yp and a hybrid lossless controller Yc. In the proof of Theorem 8.3, we assume that Xo Z for Xo =/: O. This proviso is necessary since it may be possible to reset the states of the closedloop system to the origin, in which case x(s) = 0 for a finite value of s. In this case, for t > s, we have V(x(t)) = V(x(s)) = V(O) = O. This situation does not present a problem, however, since reaching the origin in finite time is a stronger condition than reaching the origin as t t 00. Finally, we specialize the hybrid controller design framework just presented to portcontrolled Hamiltonian systems. Specifically, consider the portcontrolled Hamiltonian system given by
rt
y(t) =
G~(xp(t)) (~~: (Xp(t))) T ,
where xp(t) E 'Dp
~ ]Rnp,
(8.36)
'Dp is an open set with 0 E 'Dp, u(t) E
]Rm,
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CHAPTER 8
y(t) E jRm, Hp : 'Dp t jR is an infinitely differentiable Hamiltonian function for the system (8.35) and (8.36), .Jp : 'Dp t jRnpxnp is such that .Jp(xp) = (Xp), Rp : 'Dp t jRnpxnp is such that
.11
Rp(xp) = RJ(xp) 2:: 0, xp E 'Dp , [.Jp(xp)  Rp(xp)](~: (xp))T, xp E 'Dp, is smooth on 'Dp, and G p : 'Dp t jRnpxm. The skewsymmetric matrix function .Jp(xp), xp E 'Dp , captures the internal system interconnection structure and the symmetric nonnegative definite matrix function Rp(xp), xp E 'Dp, captures system dissipation. Furthermore, we assume that Hp(O) = 0 and Hp(xp) > 0 for all xp f: 0 and xp E'Dp. Next, consider the fixedorder, energybased hybrid controller
xc(t) = .Jcc(xc(t))
(~~: (xc(t))) T + Gcc(xc(t))y(t),
xc(O) = XcO,
(xp(t), xc(t)) t/ Z, D..xc(t) = xc(t),
(xp(t), xc(t))
u(t) = GJc(xc(t))
E
Z,
(~~: (xc(t)))
(8.37) (8.38) (8.39)
T ,
where t 2:: 0, xc(t) E 'Dc
~ jRnc, 'Dc is an open set with 0 E 'Dc, D..xc(t) ~ xc(t+)  xc(t), Hc : 'Dc t jR is an infinitely differentiable Hamiltonian function for (8.37), .Jcc : 'Dc t jRncxnc is such that .Jcc(xc) = .Jc~(xc), Xc E 'Dc, .Jcc(xc)(~c(xc))T, Xc E 'Dc, is uxc smooth on 'Dc, G cc : 'Dc t jRnc xm , and the resetting set Z c 'Dp x 'Dc
is given by
Z
~ { (xp, xc) E 'Dp x 'Dc :
:t
Hc(xc) = 0 and Hc(x c) > O}, (8.40)
it
where Hc (xc (t)) ~ limT>t t~T [Hc (xc (t)) Hc (xc ('T)) 1whenever the limit on the righthand side exists. Here, we assume that Hc(O) = 0 and Hc(xc) > 0 for all Xc f: 0 and Xc E 'Dc. Note that Hp(xp), xp E 'Dp, is the plant energy and Hc(x c), Xc E 'Dc, is the controller emulated energy. Furthermore, the closedloop system energy is given by H(xp, xc) ~ Hp(xp) + Hc(x c). Next, note that the total energy function H(xp, xc) along the trajectories of the closedloop dynamics (8.35) (8.39) satisfies
:t
H(xp(t), xc(t)) = 
~~: (xp(t))Rp(xp(t)) (~~: (Xp(t))) T ~ 0, (xp(t), xc(t)) t/ Z,
tk < t
~
tk+1,
(8.41)
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ENERGY AND ENTROPYBASED HYBRID STABILIZATION
6.'H(Xp(tk) , Xc(tk)) = 'Hc(Xc(tk)),
(Xp(tk), Xc (tk))
E
Z,
k
E
Z+.
(8.42) Here, we assume that every (xpQ, xcQ) E Z is transversal to the closedloop dynamical system given by (8.35)(8.39). Furthermore, we assume 'Dei C 'Dp x 'Dc is a compact positively invariant set with respect
°
o
to the closedloop dynamical system (8.35)(8.39) such that E 'Dei. In this case, it follows from Theorem 8.3, with Vs(xp) = 'Hp(Xp), Vc(xc,y) = 'Hc(xc), s(u,y) = uTy, and sc(uc,Yc) = u;Yc, that if Rp(xp) == 0, then the zero solution (xp(t), xc(t)) == (0,0) to the closedloop system (8.35)(8.39), with Z given by (8.40), is asymptotically stable. Alternatively, if Rp(xp) I 0, xp E 'Dp , and the largest invariant set contained in
'R
~ { (xp, xc) E 1)", '::: (xp)'Rp(xp)
Ci::
(x p )
r o} ~
(8.43)
is M = {(O,O)}, then the zero solution (xp(t),xc(t)) == (0,0) of the closedloop system (8.35)(8.39), with Z given by (8.40), is asymptotically stable. 8.4 Lagrangian and Hamiltonian Dynamical Systems
Consider the governing equations of motion of an npdegreeoffreedom dynamical system given by the EulerLagrange equation d [aC dt aq (q(t),q(t))
]T 
[aC aq (q(t),q(t))
]T + [aR aq (q(t)) ]T =u(t),
q(O) = qQ,
q(O) = qQ,
(8.44)
where t ~ 0, q E lRnp represents the generalized system positions, q E lRnp represents the generalized system velocities, C : lRnp x lRnp + lR denotes the system Lagrangian given by C(q, q) = T(q, q)  U(q), where T : lRnp x lRnp + lR is the system kinetic energy and U : lRnp + lR is the system potential energy, R : lRnp + lR represents the Rayleigh dissipation function satisfying (q)q ~ 0, q E lRnp , and u E lRnp is the vector of generalized control forces acting on the system. Furthermore, let 'H : lRnp x lRnp + lR denote the Legendre transformation of the Lagrangian function C(q, q) with respect to the generalized velocity q defined by
a;:
'H(q,p) ~ qTp  C(q, q),
(8.45)
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where p denotes the vector of generalized momenta given by (8.46)
where the map from the generalized velocities q to the generalized momenta p is assumed to be bijective (Le., onetoone and onto). Now, if 1t(q,p) is lower bounded, then we can always shift 1t(q,p) so that, with a minor abuse of notation, 1t(q,p) ~ 0, (q,p) E ]Riip x ]Riip. In this case, using (8.44) and the fact that
d[p( .)] dt '' q, q
.). 8C( .) .. = 8C( 8q q, q q + 8q q, q q,
(8.47)
it follows that
d 1t( q, p) = u T q.  8n .) . 8q (qq.
dt
(8.48)
Next, we transform the EulerLagrange equations to the Hamiltonian equations of motion. To reduce the EulerLagrange equations (8.44) to a Hamiltonian system of equations consider the Legendre transformation 1t(q,p) given by (8.45) and note that it follows from (8.44)(8.46) that
]T 81t q(t) = [ 8p (q(t),p(t)) , 81t p(t) =  [aq(q(t),p(t))
]T 
q(O) = qo, [8n 8q (q(t))
t
~ 0,
]T + u(t),
(8.49)
p(O) = Po, (8.50)
where p E ]Riip , q E ]Riip , and 1t(.,') is a lower bounded Hamiltonian function. These equations provide a fundamental structure of the mathematical description of numerous physical dynamical systems by capturing energy conservation and energy dissipation, as well as internal interconnection structural properties of physical dynamical systems. It is well known that the Hamiltonian system dynamics (8.49) and (8.50) is equivalent to the Lagrangian system dynamics (8.44). Thus, a stabilizing controller for (8.49) and (8.50) with output y = [hf(q) , hi(q)F serves as a stabilizing controller for (8.44) with the same output. Hence, in the next section we only consider controller designs for Lagrangian systems of the form (8.44) since these controllers can be equivalently applied, via a suitable transformation, to Hamiltonian systems of the form (8.49) and (8.50).
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ENERGY AND ENTROPYBASED HYBRID STABILIZATION
8.5 Hybrid Control Design for EulerLagrange Systems
In this section, we present a hybrid feedback control framework for EulerLagrange dynamical systems. grangian system (8.44) with outputs
Specifically, consider the La
where hI : ]Rnp + ]Rh and h2 : ]Rnp + ]Rlh are continuously differentiable, hl(O) = 0, h2 (0) = 0, and hl(q) =t 0. We assume that the system kinetic energy is such that T(q,q) = ~qT[~(q,q)jT, T(q,O) = 0, and T(q, q) > 0, q =I 0, q E ]Rnp. We also assume that the system potential energy U(·) is such that U(O) = and U(q) > 0, q =I 0, q E Vq ~ ]Rnp , which implies that 1t(q,p) = T(q,q) + U(q) > 0, (q,q) =I 0, (q, q) E Vq x ]Rnp. Next, consider the energybased hybrid controller
°
(8.54) where t 2': 0, qe E ]Rnc represents virtual controller positions, qe E ]Rnc represents virtual controller velocities, Yq ~ hl(q), Le : ]Rnc x ]Rnc x ]Rh + ]R denotes the controller Lagrangian given by
where Te : ]Rnc x ]Rnc + ]R is the controller kinetic energy and Ue : ]Rnc x]Rh + ]R is the controller potential energy, 'T}(.) is a continuously differentiable function such that 'T}(O) = 0, Ze C ]Rnc x ]Rnc x ]Rl is the resetting set, D..qe(t) ~ qe(t+)  qe(t), and D..qe(t) ~ qe(t+)  qe(t). We assume that the controller kinetic energy Te(qe, qe) is such that Te(qe, qe) = ~qJ[~(qe, qe)jT, with Te(qe,O) = and Te(qe, qe) > 0, qe =I 0, qe E ]Rnc. Furthermore, we assume that Ue('T}(Yq) , Yq) = and Ue(qe,Yq) > for qe =I 'T}(Yq), qe E Vqc ~ ]Rnc.
°
°
°
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CHAPTER 8
As in Section 8.3, note that Vp(q, (1) ~ T(q, (1) + U(q) is the plant energy and Vc(qe, £1e, Yq) ~ Te(qe, £1e) + Ue(qe, Yq) is the controller emulated energy. Finally,
is the total energy of the closedloop system. It is important to note that the Lagrangian dynamical system (8.44) is not dissipative with outputs Yq or y. Next, we study the behavior of the total energy function V(q, £1, qe, £1e) along the trajectories of the closedloop system dynamics. For the closedloop system, we define our resetting set as (8.57) Note that
dv,( .) dt p q, q
d'H(
= dt
q, P) = uT·q  8R(.). 8£1 q q,
it
To obtain an expression for Vc(qe, £1e, Yq) when (q, £1, qe, £1e) define the controller Hamiltonian by
'He (qe, £1e,Pe, Yq) ~ £1JPe  £e(qe, £1e, Yq),
rt z, (8.59)
r·
where the virtual controller momentum Pc is given by Pc (qe, £1e, Yq)
[~~: (qe, £1e, Yq)
=
Next, note that the controller (8.52) and (8.54) can be written in Hamiltonian form. Specifically, it follows from (8.52) and (8.59) that
'Pc(t) = 
[~:e (qe(t), £1e(t),Pe(t), yq(t))] T , (q(t), £1(t), qe(t), £1e(t))
£1e(t) =
rt z,
(8.60)
rt z,
(8.61)
[~;ee (qe(t), £1e(t),Pe(t), yq(t))] T , (q(t), £1(t), qe(t), £1e(t))
u(t) = 
[8~e (qe(t), £1e(t),Pe(t), yq(t))] T ,
(8.62)
where 'He (qe, £1e,Pe, Yq) = Te(qe, £1e) + Ue(qe, Yq)· Now, it follows from (8.52) and the structure of Te(qe, £1e) that, for t E (tk' tk+1],
269
ENERGY AND ENTROPYBASED HYBRID STABILIZATION
d
= dt [p~(qc(t),qc(t),Yq(t))qc(t)J
 p~(qc(t),qc(t),Yq(t))qc(t)
+ ~~c (qc(t), qc(t), Yq(t))qc(t) + 8fc (qc(t), qc(t), Yq(t))q(t) uq
u~
d .  dtLc(qc(t), qc(t), Yq(t)) d
= dt [p~ (qc( t),
qc( t), Yq (t) )qc (t)  Lc (qc( t), qc( t), Yq (t)) 1
+ 8~c (qc(t), qc(t), Yq(t) )q(t) = :t 'Hc(qc(t) , qc(t),Pc(t), Yq(t)) + 8~c (qc(t), qc(t), Yq(t))q(t) = :t Vc(qc(t), qc(t), Yq(t)) + 8~c (qc(t), qc(t), Yq(t))q(t), (q(t), q(t), qc(t), qc(t))
~
Z.
(8.63)
Hence,
:t V(q(t), q(t), qc(t), qc(t)) = u(t)T q(t) 
8~c (qc(t), qc(t), Yq(t))q(t)

~~ (q(t))q(t)
= 
~~ (q(t))q(t)
::; 0,
(q(t), q(t), qc(t), qc(t)) ~ Z, tk < t ::; tk+l,
(8.64)
which implies that the total energy of the closedloop system between resetting events is nonincreasing. Alternatively, if R(q) == 0, then V(q, q, qc, qc) = 0, (q, q, qc, qc) ~ Z, which implies that the total energy of the closedloop system is conserved between resetting events. The total energy difference across resetting events is given by
Jt
.6. V(q(tk), q(tk), qc(tk), qc(tk)) = Tc(qc(tt), qc(tt))
+ Uc(qc(tt), Yq(tk))
 Vc(qc(tk), qc(tk), Yq(tk)) = Tc(17(Yq(tk)), 0) +Uc(17(Yq(tk)), Yq(tk))  Vc (qc( tk), qc (tk), Yq( tk)) = Vc(qc(tk), qc(tk), Yq(tk)), < 0, (q(tk), q(tk), qc(tk), qc(tk)) E Z, kE
Z+,
(8.65)
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CHAPTER 8
which implies that the resetting law (8.53) ensures the total energy decrease across resetting events by an amount equal to the accumulated emulated energy. Here, we concentrate on an energydissipating statedependent resetting controller that affects a oneway energy transfer between the plant and the controller. Specifically, consider the closedloop system (8.44), (8.51)(8.54), where Z is defined by Z
~ { (q, q, qe, qe) : 1t Ve(qe, qe, Yq) =
0 and Vc(qe, qe, Yq) >
o} . (8.66)
Since Yq
1t
= hl(q)
and
Vc(qe,qe,Yq)
=
[8~e(qe,qe,yq)] q = [8~e(qe,yq)] q, (8.67)
it follows that (8.66) can be equivalently rewritten as Z
=
{(q,q,qe,qe):
[8~e(qe,hl(q))] q =
0 and Vc(qe,qe,h 1 (q))
>
o}.
(8.68) Once again, for practical implementation, knowledge of qe, qe, and Yq is sufficient to determine whether or not the closedloop state vector is in the set Z. The next theorem gives sufficient conditions for stabilization of EulerLagrange dynamical systems using statedependent hybrid controllers. For this result define the closedloop system states x !E:. [qT, qT, qJ,qJlT.
Theorem 8.4 Consider the closedloop dynamical system g given by (8.44), (8. 51}(8. 54}, with a~~q)q == 0 and the resetting set Z given by (8. 66}. Assume that Dei C Dq x jRnp x Dqc x jRnc is a compact o
positively invariant set with respect to g such that 0 E Dei. Furthermore, assume that the transversality condition (8.11) holds with X(x) = Vc(qe, qe, Yq)· Then the zero solution x(t) == 0 to g is asymptotically stable. In addition, the total energy function V(x) of g given by (8.56) is strictly decreasing across resetting events. Alterna~ively, if a~~q) q i 0, q E jRnp , and the largest invariant set contained
1t
zn
(8.69)
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ENERGY AND ENTROPYBASED HYBRID STABILIZATION
is M = {O}, then the zero solution x(t) == 0 to 9 is asymptotically stable. Finally, ifVq = jRnp , Vqc = jRnc, and the total energy function V (x) is radially unbounded, then the above asymptotic stability results are global.
Proof. The proof is a direct consequence of Theorem 8.3 with Vp(x p) = Vp(q, q), Vc(x c , y) = Vc(qc, qc, yq), y = U c = x p, u Yc = a~c, s(u,y) = uTp(y), sc(uc,Yc) = y;p(u c), where p(y) p ([
~ ])
= q, and TJ(Y) replaced by [TJ(6 q )
D
].
8.6 Thermodynamic Stabilization
In this section, we use the recently developed notion of system thermodynamics [65] to develop thermodynamically consistent hybrid controllers for lossless dynamical systems. Specifically, since our energybased hybrid controller architecture involves the exchange of energy with conservation laws describing transfer, accumulation, and dissipation of energy between the controller and the plant, we construct a modified hybrid controller that guarantees that the closedloop system is consistent with basic thermodynamic principles after the first resetting event. To develop thermodynamically consistent hybrid controllers consider the closedloop system 9 given by (8.22) and (8.23), with Z given by Z
~ {x
E V: ¢(x)(Vp(x)  Vc(x))
= 0 and
Vc(x) > O}, (8.70)
where ¢(x) ~ Vc(x), x ~ Z. It follows from (8.33) that ¢(.) is the net energy flow from the plant to the controller, and hence, we refer to ¢(.) as the net energy flow function. We assume that the energy flow function ¢(x) is infinitely differentiable and the transversality condition (8.11) holds with X(x) = ¢(x)(Vp(x)  Vc(x)). To ensure a thermodynamically consistent energy flow between the plant and controller after the first resetting event, the controller resetting logic must be designed in such a way so as to satisfy three key thermodynamic axioms on the closedloop system level. Namely, between resettings the energy flow function ¢(.) must satisfy the following two axioms [64, 65]:
Axiom i) For the connectivity matrix C E
jR2x2
[65, p. 56] associ
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CHAPTER 8
ated with the closedloop system C
(i,j)
6.
=
g defined by
{O, if4>(x(t)) == 0, 1, otherwise,
i =/=j,
i,j
= 1,2, t 2:: tt, (8.71)
= C(k,i)'
i =/= k,
i, k
= 1,2,
and C(i,i) rankC
=
1, and for C(i,j)
=
1, i =/= j, 4>(x(t))
Vp(x(t)) = Vc(x(t)), x(t) ¢ Z, t 2:: tt.
=
°
(8.72) if and only if
Axiom ii) 4>(x(t))(Vp(x(t))  Vc(x(t))) :::; 0, x(t) ¢ Z, t 2:: tt· Furthermore, across resettings the energy difference between the plant and the controller must satisfy the following axiom [68,69]:
Axiom iii) [Vp(x x EZ.
+ fd(x))
 Vc(x
+ fd(x))][Vp(x)
 Vc(x)] 2:: 0,
°
The fact that 4>(x(t)) = if and only if Vp(x(t)) = Vc(x(t)), x(t) ¢ Z, t 2:: tt, implies that the plant and the controller are connected; alternatively, 4>( x( t)) == 0, t 2:: tt, implies that the plant and the controller are disconnected. Axiom i) implies that if the energies in the plant and the controller are equal, then energy exchange between the plant and controller is not possible unless a resetting event occurs. This statement is consistent with the zeroth law of thermodynamics, which postulates that temperature equality is a necessary and sufficient condition for thermal equilibrium of an isolated system. Axiom ii) implies that energy flows from a more energetic system to a less energetic system and is consistent with the second law of thermodynamics, which states that heat (energy) must flow in the direction of lower temperatures. Finally, Axiom iii) implies that the energy difference between the plant and the controller across resetting instants is monotonic, that is, [Vp(x(tt))  Vc(x(tt))][Vp(X(tk))  Vc(X(tk))] 2:: for all Vp(x) =/= Vc(x), x E Z, k E Z+. With the resetting law given by (8.70), it follows that the closedloop dynamical system g satisfies Axioms i)iii) for all t 2:: it. To see this, note that since 4>(x) ¢. 0, the connectivity matrix C is given by
°
C=
[1 1] 1
1
°
'
(8.73)
and hence, rank C = 1. The second condition in Axiom i) need not be satisfied since the case where 4>(x) = or Vp(x) = Vc(x) corresponds
273
ENERGY AND ENTROPYBASED HYBRID STABILIZATION
to a resetting instant. Furthermore, it follows from the definition of the resetting set (8.70) that Axiom ii) is satisfied for the closedloop system for all t 2: tt. Finally, since Vc(x + fd(x)) = 0 and Vp(x + fd(x)) = Vp(x), x E Z, it follows from the definition of the resetting set that
[Vp(x
+ fd(x)) 
Vc(x + fd(x))][Vp(x)  Vc(x)J = Vp(x)[Vp(x)  Vc(x)J 2: 0, x
E Z,
(8.74)
and hence, Axiom iii) is satisfied across resettings. Hence, the closedloop system 9 is thermodynamically consistent after the first resetting event in the sense of [64,65,68, 69J. Next, we give a hybrid definition of entropy for the closedloop system 9 that generalizes the continuoustime and discretetime entropy definitions established in [64,65,68, 69J.
Definition 8.2 For the impulsive closedloop system 9 given by (8.22) and (8.23), a function S : lR! ~ lR satisfying
S(E(x(T))) 2: S(E(X(tl))) 
L
1 c
)) dt itlr c +d(i?)/ x t T
P
Vc(X(tk)),
T 2: tl,
(8.75)
kEZ[tl,T)
where k E Z[tl,T) ~ {k : tl ~ tk < T}, E ~ [Vp, VcJ T , d: Vp ~ lR is a continuous, nonnegativedefinite dissipation rate junction, c > 0, is called an entropy function of g. The next result gives necessary and sufficient conditions for establishing the existence of an entropy function of 9 over an interval t E (tk' tk+1J involving the consecutive resetting times tk and tk+l, k E Z+.
Theorem 8.5 For the impulsive closedloop system 9 given by (8.22) and (8.23), a function S : lR! ~ lR is an entropy function of 9 if and only if
S(E(x(t))) 2: S(E(x(t)))
it :(~f=~~)) c
ds,
tk < t ~ t ~ tk+1, S(E(X(tk)
+ fd(x(tk)))) 2: S(E(X(tk))) 
Vc(x(t k)) , c
(8.76)
k E Z+. (8.77)
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CHAPTER 8
Proof. Let k E Z+ and suppose S(E) is an entropy function of Then, (8.75) holds. Now, since for tk < t ::; £ ::; tk+1, Z[t,i) = 0, (8.76) is immediate. Next, note that
g.
S(E(x(tt)))
~ S(E(X(tk))) _itt tk
d(xp(s))
C
+ Vp(X(S))
ds _ Vc(X(tk)) , C
(8.78) which, since Z[tk,tt)
= k,
implies (8.77).
Conversely, suppose (8.76) and (8.77) hold, and let £ ~ t ~ it and + 1, ... ,j}. (Note that if Z[t,i) = 0 the converse result is a direct consequence of (8.76).) If Z[t,i) i= 0, it follows from (8.76) and (8.77) that
Z[t,i) = {i, i
S(E(x(£)))  S(E(x(t))) = S(E(x(i)))  S(E(x(tj))) jil
+
L
S(E(x(tj_m)))  S(E(x(tj_m_l)))
m=O +S(E(x(tt)))  S(E(x(t))) = S(E(x(i)))  S(E(x(tj))) ji + S(E(x(tj_m) + fd(X(tjm))))  S(E(x(tj_m))) m=O
L
jil
+
L
S(E(x(tj_m)))  S(E(x(tj_m_l))) m=O +S(E(X(ti)))  S(E(x(t)))
~
1+
d(xp(s))
t
tJ
C
P
x S
1ti  m  ji1 L m=O tJml = 
1
1 ji
L Vc(x(tjm))
v, ( ( )) ds  
m=O
C
d(x (s)) p
C
+ Vp(x(s))
ds 
iti t
d(x (s)) p
C
+ Vp(x(s))
d(xp(s)) 1 "'" v, ( ( )) ds   ~ Vc(X(tk)), tC+pxs C t
ds (8.79)
kEZ[t,i)
which implies that S(E) is an entropy function of g.
o
The next theorem establishes the existence of an entropy function
275
ENERGY AND ENTROPYBASED HYBRID STABILIZATION
for the impulsive closedloop system g.
Theorem 8.6 Consider the impulsive closedloop system g given by (8.22) and (8.23), with Z given by (8.70). Then the function S : 2 lR.+ t lR. given by
S(E) = loge(c + Vp) + loge(c + Vc)  2 loge c,
E
2 E lR.+, (8.80)
where c > 0, is a continuously differentiable entropy function of g. In addition, . d(xp(t)) S(E(x(t))) >  c + Vp(x(t)) '
x(t) ¢ Z,
tk < t ::; tk+l, (8.81)
 Vc(X(tk)) < baS(E(X(tk))) < _ Vc(X(tk)) , c c + Vc(X(tk))
X(tk)
E
Z,
k E Z+. (8.82)
Proof. Since Vp(x(t))
= cjJ(x(t))d(xp(t)) and "Vc(x(t)) = cjJ(x(t)),
x(t) ¢ Z, t E (tk' tk+1J, k E Z+, it follows that
S(E(x(t))) = cjJ(x(t))(Vc(x(t))  Vp(x(t))) _ d(xp(t)) (c + Vp(x(t)))(c + Vc(x(t))) c + Vp(x(t)) > _ d(xp(t)) c + Vp(x(t)) ' x(t) ¢ Z, tk < t ::; tk+l. (8.83) Furthermore, since Vc(X(tk) + fd(x(tk))) = 0 and Vp(X(tk) + fd(x(tk))) E Z, k E Z+, it follows that
= Vp(X(tk)), X(tk)
b.S(E(x(t ))) = log [1  Vc(X(tk)) ] >  Vc(X(tk)) x(t) k e c+ Vc(X(tk)) c' k k E Z+,
E
Z
,
(8.84)
and
Vc(x(t k ))] Vc(X(tk)) b.S(E(X(tk))) = loge [1  c + Vc(X(tk)) <  c + Vc(X(tk)) ' X(tk)
E
Z,
k E Z+,
(8.85)
where in (8.84) and (8.85) we used the fact that l~x < loge(1+x) < x, x > 1, x =1= O. The result is now an immediate consequence of 0 Theorem 8.5. Note that in the absence of energy dissipation into the environment (Le., d(xp(x)) == 0) it follows from (8.81) that the entropy of the
276
CHAPTER 8
closedloop system strictly increases between resetting events, which is consistent with thermodynamic principles. This is not surprising since in this case the closedloop system is adiabatically isolated (i.e., the system does not exchange energy (heat) with the environment) and the total energy of the closedloop system is conserved between resetting events. Alternatively, it follows from (8.82) that the entropy of the closedloop system strictly decreases across resetting events since the total energy strictly decreases at each resetting instant, and hence, energy is not conserved across resetting events. Using Theorem 8.6, the resetting set Z given by (8.70) can be rewritten as 6.
{d
Z = xED: dt S (E( x))
d(xp)
+ c + Vp ( x) = 0 and Vc(x) >
o} ,
(8.86)
where X(x) ~ JtS(E(x)) + e~~:(~) is a continuously differentiable function that defines the resetting set as its zero level set. The resetting set (8.70) or, equivalently, (8.86) is motivated by thermodynamic principles and guarantees that the energy of the closedloop system is always flowing from regions of higher to lower energies after the first resetting event, which is consistent with the second law of thermodynamics. As shown in Theorem 8.6, this guarantees the existence of an entropy function S(E) for the closedloop system that satisfies the Clausiustype inequality (8.81) between resetting events. If ¢(x) = 0 or Vp(x) = Vc(x), then inequality (8.81) would be subverted, and hence, we reset the compensator states in order to ensure that the second law of thermodynamics is not violated. Finally, if Dei c D is a compact positively invariant set with respect to the closedloop dynamical system 9 given by (8.22) and (8.23) such o
== 0, and the transversality condition (8.11) holds with X(x) = JtS(E(x)) + e~~;(~)' then it follows from Theorem 8.3 that the zero solution x(t) == 0 of the closedloop system g, with that 0 E Dei, d(x p )
resetting set Z given by (8.70), is asymptotically stable. Alternatively, if d(xp) f. 0, xp E D p , and the largest invariant set contained in n ~ {x E Dei: d(xp) = O} is {O}, then the zero solution x(t) == 0 of the closedloop system 9 is asymptotically stable. Furthermore, in this case, the hybrid controller (8.52) and (8.53), with resetting set (8.70), is a thermodynamically stabilizing compensator. Analogous thermodynamically stabilizing compensators can be constructed for
277
ENERGY AND ENTROPYBASED HYBRID STABILIZATION
portcontrolled Hamiltonian and EulerLagrange dynamical systems. 8.7 EnergyDissipating Hybrid Control Design
In this section, we apply the energy dissipating hybrid controller synthesis framework developed in Sections 8.5 and 8.6 to two examples. For the first example, consider the vector secondorder nonlinear Lienard system given by
ij(t) + f(q(t)) = u(t),
q(O) = qQ,
q(O) = qQ,
t
~
C1q(t) ] y(t) = [ C2 q(t) ,
(8.87) (8.88)
where q E ]Rnp , f : ]Rnp t ]Rnp is infinitely differentiable, and only if q = 0, C 1 E ]Rl! xn p , C2 E ]R(llr)xnp , and i,j = 1, ...
0,
,np.
f (q) = 0 if (8.89)
The plant energy of the system is given by
(8.90) where T(q, q) = ~qTq and U(q) = JQ~path fT(CT)dCT. Note that the path integral in (8.90) is taken over any path joining the origin to q E ]Rnp. Furthermore, the path integral in (8.90) is well defined since fO is such that is symmetric, and hence, f(·) is a gradient of a realvalued function [8, Theorem 1O37J. Here, we assume that U(O) = 0 and U(q) > 0 for q i 0, q E ]Rnp. Note that defining p ~ q and
*
(8.91)
278
CHAPTER 8
it follows that (8.87) can be written in Hamiltonian form
q(t)
= [
81t
8p (q(t),p(t))
]
T
,
]T 81t p(t) =  [ 8q (q(t),p(t))
q(O)
+ u,
=
qo,
(8.92)
t ~ 0,
(8.93)
p(O) = PO·
To design a statedependent hybrid controller for the Lienard system (8.87), let C 1 = C2 = Inp ' let
(8.94) (8.95) where qc E ]Rnp , 9 : ]Rnp  t ]Rnp is infinitely differentiable, g( x) and only if x = 0, and g'(O) is positive definite, and let
= 0 if (8.96)
so that
(8.97)
J;'
Here, we assume that path gT (0' )dO' > 0 for all x # 0, x E ]Rnp. In this case, the statedependent hybrid controller has the form
Qc(t) + g(qc(t)  q(t)) = 0,
(q(t), q(t), qc(t), qc(t)) fi Z,
t ~ 0,
(8.98)
] = [ q(t)  qc(t) ] [ D.qc(t) D.qc(t) qc(t) '
(( q t), q(t), qc(t), qc(t))
E
Z,
t ~ 0,
(8.99)
u(t) = g(qc(t)  q(t)),
(8.100)
with the resetting set (8.66) taking the form
Z = {(q, q, qc, qc) : [g(qc  q)]T q = 0 and [ q~q:c ]
# O}.
(8.101)
Here, we consider the case where np = qr = 1. To show that Al holds in this case, we show that upon reaching a nonequilibrium point x(t) ~ [q(t), q(t), qc(t), £ic(t)]T fi Z that is in the closure of Z,
279
ENERGY AND ENTROPYBASED HYBRID STABILIZATION
the continuoustime dynamics x = fc(x) remove x(t) from Z, and hence, necessarily move the trajectory a finite distance away from Z. If x(t) ¢ Z is an equilibrium point, then x(s) ¢ Z, s ~ t, which is also consistent with AI. The closure of Z is given by (8.102)
Furthermore, the points x* satisfying [q*  q~, £1~jT = 0 have the form
0] T ,
x* ~ [q £1 q
(8.103)
that is, qc = q and £1c = O. It follows that x* ¢ Z, although x* E Z. To show that the continuoustime dynamics x = fc(x) remove x* from Z, note that (8.104)
and
::2 Vp(q, £1) = q[g(qc  q)]
+ £1[g' (qc 
:t33 Vp(q,£1) = q(3)[g(qc  q)]
q)](£1c  £1),
+ [g'(qc 
q)](£1qc + 2£1cq  3£1q)
+[g"(qc  q)](£1c  £1)2£1, :t: Vp(q,£1) = q(4)[g(qc  q)]
(8.105)
(8.106)
+ [g'(qc  q)](3qcq(3)  4qq(3) + 3ijijc
_ 3(2) +[g"(qc  q)](3£1£1cqc + 3£1;q  9£1Qcq  3£12qc + 6£12q) +g(3)(qc  q)(£1c  q)3£1, (8.107) +£1q~3)
where g(n)(t) ~ d~~~t). Since (8.108)
it follows that if £1 =1= 0, then the continuoustime dynamics x = fc(x) remove x* from Z. If £1 = 0, then it follows from (8.105)(8.107) that 2
Ix=x',q=O = 0,
(8.109)
3
I
(8.110)
d Vp(q, £1) dt2
_ * ,q'0 = 0, ddt 3Vp(q, £1) xx
280
CHAPTER 8
(8.111) where in the evaluation of (8.110) and (8.111) we use the fact that if qc = q and qc = 0, then ijc = 0, which follows immediately from the continuoustime dynamics. Since if q = and ij i 0, then the lowestorder nonzero time derivative of Vp(xp) is negative, it follows that the continuoustime dynamics remove x* from Z. However, if q = and ij = 0, then it follows from the continuoustime dynamics that x* is necessarily an equilibrium point, in which case the trajectory never again enters Z. Therefore, we can conclude that Al is indeed valid for this system. Also, since fd(X + fd(X)) = 0, it follows from (8.101) that if x E Z, then x + fd(X) ~ Z, and thus A2 holds. For thermodynamic stabilization, the resetting set (8.70) is given by
°
°
Z = { (q, q, qc, qc) : qT[g(qc  q)] [Vp(q, q)  Vc(qc, qc, q)] = and [ q
~q:c
]
i
°}.
°
(8.112)
Furthermore, the entropy function S(E) is given by (8.113)
To illustrate the behavior of the closedloop impulsive dynamical system, let np = = 1, f(x) = x + x 3 , and g(x) = 3x with initial conditions q(O) = 0, q(O) = 1, qc(O) = 0, and qc(O) = 0. For this system, the transversality condition is sufficiently complex that we have been unable to show it analytically. This condition was verified numerically, and hence, Assumption 8.1 holds. Figure 8.2 shows the controlled plant position and velocity states versus time, while 8.3 shows the virtual position and velocity compensator states versus time. Figure 8.4 shows the control force versus time. Note that the compensator states are the only states that reset. Furthermore, the control force versus time is discontinuous at the resetting times. A comparison of the plant energy, controller energy, and total energy is shown in Figure 8.5. Figures 8.68.9 show analogous representations for the thermodynamically stabilizing compensator. Finally, Figure 8.10 shows the closedloop system entropy versus time. Note that the entropy of the closedloop system strictly increases between resetting events. As our next example, we consider the rotational/translational proofmass actuator (RTAC) nonlinear system studied in [36]. The system
i
281
ENERGY AND ENTROPYBASED HYBRID STABILIZATION 0.6,,...,r,,,...,r,,
o 0.1 '_'_ _'_ _..L._ _'   _     ' _ _ '  _ _ '  _    '_ _ L_ _ o 2 3 4 5 6 7 8 9 10 Time ~
~ 0.2 0.4
0
2
3
4
5 TIme
6
7
8
9
10
Figure 8.2 Plant position and velocity versus time. 0.6 0.5 0.4
2
,,'"
0.3 0.2 0.1 0 0.1
0
2
3
4
5 Time
6
7
8
9
10
4
5 Time
6
7
8
9
10
0.8 0.6 0.4
~v
0.2 0 0.2 0.4 0.6
j~~ 0
2
3
Figure 8.3 Controller position and velocity versus time.
282
CHAPTER 8 0.6 0.4 0.2
0 ·0.2
~ 0.4
0.6
0.8 1
1.2
0
2
3
4
5
6
7
8
9
10
Time
Figure 8.4 Control signal versus time. 0.7i,,.....,,,,r;===:::::c===='===::;  Plant Energy   Emulated Energy    Total Energy
0.6
0.5 \ , \
I
0.4
0.3
0.2
,
.......        ,
0.1
\
\.
2
3
4
5 Time
6
7
8
9
Figure 8.5 Plant, emulated, and total energy versus time.
10
283
ENERGY AND ENTROPYBASED HYBRID STABILIZATION
o 0.2 OL''2L'3'4'S'6'7'89'..1 10 TIme
S
x'"
0.4 0.2 0 0.2 0.4
0
3
5
4
6
Time
7
8
9
10
Figure 8.6 Plant position and velocity versus time for thermodynamic controller. 0.6.__ ~=~:;r____._,_.,.._y_,_._,..__, 0.4
S
x"
0.2 G
ol/
0..
~
G'
G
G~
0.2
0
2
3
4
5 Time
7
6
8
9
10
0.4 0.2
~
V/\\~""~'
0 0.2
0.4 L_'_ _ L_ _ o 2 3
'_'~
4
_
~
~
_'__ __'___ _:':_
5 Time
6
7
___'_=_..J 8 9 10
Figure 8.7 Controller position and velocity versus time for thermodynamic controller.
284
CHAPTER 8
0.2
0.4
0.6
0.8
1
1.2 '     '      '      '      '    '     "    '      '      ' '        ' 4 7 8 9 10 o 2 3 5 6 Time
Figure 8.8 Control signal versus time for thermodynamic controller.
(see Figure 8.11) involves an eccentric rotational inertia, which acts as a proofmass actuator mounted on a translational oscillator. The oscillator cart of mass M is connected to a fixed support via a linear spring of stiffness k. The cart is constrained to onedimensional motion and the rotational proofmass actuator consists of a mass m and mass moment of inertia 1 located a distance e from the center of mass of the cart. In Figure 8.11, N denotes the control torque applied to the proof mass. Since the motion is constrained to the horizontal plane the gravitational forces are not considered in the dynamic analysis. Letting q, q, 0, and 0 denote the translational position and velocity of the cart and the angular position and velocity of the rotational proof mass, respectively, and using the energy function
Vs(q, q, 0, 0) =
~[kq2 + (M + m)q2 + (1 + me2)02 + 2meqO cos 0], (8.114)
the nonlinear dynamic equations of motion are given by
(M + m)ij + kq= me(OcosO  02 sinO), (1 + me2)O= meijcosO + N,
(8.115) (8.116)
285
ENERGY AND ENTROPYBASED HYBRID STABILIZATION
0.7i.,,...,..r;:==::::::!::==:=::!:==:::;]   Plant Energy   Emulated Energy    Total Energy
0.6
0.5
'.
I
0.4
, ,, ,
,
I
,
I,
'I'
I' 0.3
"
~~~
"":,1 , , '\
0.1
, ','
2
3
4
5
Time
6
7
8
9
10
Figure 8.9 Plant, emulated, and total energy versus time for thermodynamic controller.
with problem data given in Table 8.1 and output y = [O,ojT. The physical configuration of the system necessitates the constraint Iql ~ 0.025 m. In addition, the control torque is limited by INI ~ 0.100 N m [36]. With the normalization
A~
r=v~t,
u
A
=
M+m
k(1
+ me2 ) N,
(8.117)
the equations of motion become ~ + ~=c(02 sinO  OcosO),
o= c~ cos 0 + u,
(8.118) (8.119)
where ~ is the normalized cart position and u represents the nondimensionalized control torque. In the normalized equations (8.118) and (8.119), the symbol () represents differentiation with respect to the normalized time r and the parameter c represents the coupling between the translational and rotational motions and is defined by me (8.120) c ~ ;:;=:====:~==========c= ..)(1 + me2 )(M + m)
286 0.45
CHAPTER 8
V
0.4
0.35
0.3
is: 0.25
e
"E w
0.2
0.15
0.1
0.05
2
3
4
5 Time
6
7
8
9
10
Figure 8.10 Closedloop entropy versus time.
/
M
Figure 8.11 Rotational/translational proofmass actuator.
Since the plant energy function (8.114) is not positive definite in ~4, we first design a control law u = ko(} + u, where ko > 0, with associated positive definite normalized plant energy function given by
287
ENERGY AND ENTROPYBASED HYBRID STABILIZATION
Description Cart mass Arm mass Arm eccentricity Arm inertia Spring stiffness Coupling parameter
Parameter M m e
I k c:
Value 1.3608 0.096 0.0592 0.0002175 186.3 0.200
Units kg kg m kgm 2 N/m 
Table 8.1 Problem data for the RTAC [36].
To design a statedependent hybrid controller for (8.118) and (8.119), . 1'2 1 2 . l' let ne = 1, Ve(~e, ~e,(j) = 2me~e + 2ke(~e 0) ,.ce(~e, ~e, 0) = 2me~;!ke(~e  0)2, Yq = 0, and fJ(Yq) = Yq, where me> 0 and ke > O. Then the statedependent hybrid controller has the form (8.122) (8.123) (8.124) with the resetting set (8.66) taking the form
To show that Al holds, we show that upon reaching a nonequilibrium point x(r) ~ [~(r),e(r),O(r),e(r),~e(r),ee(r)lT t/. Z that is in the closure of Z, the continuoustime dynamics x = fe(x) remove x(r) from Z, and thus necessarily move the trajectory a finite distance away from Z. If x(r) t/. Z is an equilibrium point, then x(s) t/. Z, s 2 r, which is also consistent with AI. The closure of Z is given by (8.126) Furthermore, the points x* satisfying [0*  ~~, e~lT = 0 have the form
x* ~ [~ ~ 0 that is, ~e = 0 and
e0
0] T ,
(8.127)
ee = O. It follows that x* t/. Z, although x* E Z.
CHAPTER 8
288
To show that the continuoustime dynamics x = fe(x) remove x* from Z, note that d .. . (8.128) dr Vs(~,~, 0, 0) = keO(~e  0) and d2 .... dr2 Vs( ~,~, 0,0) = keO(~e  0) d 33 Vs(~, dr d44 Vs(~, dr
.,.
+ keO(~e 
0),
(8.129)
e, 0, iJ) = keO(3) (~e  0) + 2keB(ee  iJ) + keiJ(ee  B), (8.130) e, 0, iJ) = ke
o(4)
(~e  0) + 3keO(3) (ee  iJ) + 3keB( ee  B)
+keiJ(~~3)  0(3)),
(8.131)
where g(n)(r) ~ d~~~). Since d2 Vs(~,~,"0,1 dr2 0) x=x. = keO'2 ,
(8.132)
it follows that if iJ =1= 0, then the continuoustime dynamics x = fe(x) remove x* from Z. If iJ = 0, then it follows from (8.129)(8.131) that d2
.. 1
=0,
(8.133)
d3 .. 1 . d r 3 Vs(~,~, 0, 0) x=x*,9=O
= 0,
(8.134)
d4 .. 1 . dr 4 Vs(~,~,O,O) x=x',9=O
= 3keq..2 ,
(8.135)
d r 2Vs(~,~,0,O)
.
x=x',9=O
where in the evaluation of (8.134) and (8.135) we use the fact that if ~e = 0 and = 0, then = 0, which follows immediately from the continuoustime dynamics. Since if iJ = and B=1= 0, then the lowestorder nonzero time derivative of Vs(~, 0, iJ) is negative, it follows that the continuoustime dynamics remove x* from Z. However, if iJ = and B= 0, then it follows from the continuoustime dynamics that x* is necessarily an equilibrium point, in which case the trajectory never again enters Z. Therefore, we can conclude that Al is indeed valid for this system. Also, since fd(X + fd(X)) = 0, it follows from (8.125) that if x E Z, then x + fd(X) rt z, and thus A2 holds. For thermodynamic stabilization, the output y is modified as y = [~, 0, iJ]T and the resetting set (8.70) is given by
ee
ee
e,
°
e,
Z = { (~,
e, 0, iJ, ~e, ee) E
]R6 :
keiJ(~e  O)[Vs(~, e, 0, iJ)  Vc(~e, ee, 0)]
°
289
ENERGY AND ENTROPYBASED HYBRID STABILIZATION
(8.136)
Furthermore, the entropy function S(E) is given by
To illustrate the behavior of the closedloop impulsive dynamical system, let me = 0.2, ke = 1, and ko = 1 with initial conditions ~(o) = 1, ~(o) = 0, 0(0) = 0, 0(0) = 0, ~e(O) = 0, and ~e(O) = O. For thermodynamic stabilization, the initial conditions are given by ~(O) = 0.6, ~(O) = 0, 0(0) = 0, 0(0) = 0, ~e(O) = 0.8, and ~e(O) = o. For this system, the transversality condition is sufficiently complex that we have been unable to show it analytically. This condition was verified numerically, and hence, Assumption 8.1 holds. Figures 8.12 and 8.13 show the translational position of the cart and the angular position of the rotational proof mass versus time. Figure 8.14 shows the control torque versus time. Note that the compensator states are the only states that reset. Furthermore, the control torque versus time is discontinuous at the resetting times. A comparison of the plant energy, control energy, and total energy is shown in Figure 8.15. Figures 8.168.19 show analogous representations for the thermodynamically stabilizing compensator. Finally, Figure 8.20 shows the closedloop system entropy versus time. Note that the entropy of the closedloop system strictly increases between resetting events. Our final example considers the design of a hybrid controller for the combustion system we considered in Section 6.3. Recall that this model is given by
XI(t) =alxl(t) + 0IX2(t)  ,B(XI(t)X3(t) + X2(t)X4(t)) + UI(t), X1(0) = XlO, t ~ 0, (8.138) X2(t) = OIXI(t) + alx2(t) + ,B(X2(t)X3(t)  XI(t)X4(t)) + U2(t), X2(0) = X2Q, (8.139) X3(t) = a2x3(t) + 02X4(t) + ,B(x~(t)  x~(t)) + U3(t), X3(0) = X3Q, (8.140)
X4(t) = 02X3(t) + a2x4(t)
+ 2,BxI(t)X2(t) + U4(t),
X4(0) = X4Q, (8.141)
where x ~ [Xl, X2, X3, X4jT E ]R4 is the plant state, U ~ [UI, U2, U3, u4jT E ]R4 is the control input, i = 1, ... ,4, aI, a2 E ]R represent growth/ decay constants, 01, O2 E ]R represent frequency shift constants, ,B = ((y + 1)/8,),)WI, where,), denotes the ratio of specific heats, WI is
290
CHAPTER 8
frequency of the fundamental mode, and Ui, i = 1, ... ,4, are control input signals. For the data parameters 0:1 = 5, 0:2 = 55, 01 = 4, O2 = 32, "( = 1.4, WI = 1, and x(O) = [1, 1, 1, 1jT, the openloop (Ui(t) == 0, i = 1,2,3,4) dynamics (8.138)(8.141) result in a limit cycle instability. In addition, with the plant energy defined by Vp(x) £ !(x~+x~+x~+x~), (8.138)(8.141) is dissipative with respect to the supply rate i),Ty, where i), £ [Ul + O:IXl, U2 + 0:1X2, U3, u4jT and y£x. Next, consider the reducedorder dynamic compensator given by (8.19)(8.21) with fcc(x c, y) = Acxc + BcY, rJ(Y) = 0, hcc(xc, y) = BJ XC, where Xc £ [xcI, x c2jT E JR2,
Ac
= [ ~1
~],
Bc
=
[~ ~ ~ ~],
(8.142)
and controller energy given by Vc(xc) = !xJ Xc. Furthermore, the resetting set (8.27) is given by Z = {(x,x c): xJBcx = 0, Xc f= O}. To illustrate the behavior of the closedloop impulsive dynamical system, we choose the initial condition xc(O) = [0, ojT. For this system a straightforward, but lengthy, calculation shows that Al and A2 hold. However, the transversality condition is sufficiently complex that we have been unable to show it analytically. This condition was verified numerically and Assumption 8.1 appears to hold. Figure 8.21 shows the state trajectories of the plant versus time, while Figure 8.22 shows the state trajectories of the compensator versus time. Figure 8.23 shows the control inputs Ul and U2 versus time. Note that the compensator states are the only states that reset. Furthermore, the control force versus time is discontinuous at the resetting times. A comparison of the plant energy, controller energy, and total energy is shown in Figure 8.24. Note that the proposed energybased hybrid controller achieves finitetime stabilization. Next, we consider the case where 0:1 = 0 and 0:2 = 0, that is, there is no decay or growth in the system. The other system parameters remain as before. In this case, the system is loss less with respect to the supply rate U T y. For this problem we consider an entropybased hybrid dynamic compensator given by (8.19)(8.21) with fcc(xc,y) = Acxc + BcY, rJ(Y) = 0, hcc(xc,y) = BJx c, where Xc £ [Xc1, Xc2,Xc3, Xc4]T E JR4,
30
o o o
f ~],
(8.143)
291
ENERGY AND ENTROPYBASED HYBRID STABILIZATION 0.02 r,___~,,r,____,_____,___._____,
0.015
0.01
Ic:
0.005
~
~
1ii
o
~
"iii
:ii
F
0.005
0.01
0.015
0.02 0L    . . L . . .     ' . 2  3L    . L 4 L S SL    . L 7 L a'g'..J10 Time (s)
Figure 8.12 Translational position of the cart versus time.
and controller energy given by Vc(x c) = ~xJ Xc. Furthermore, the entropy function S(E) is given by S(E) = loge[l + Vp(x)] + loge[l + Vc(x c)], and the resetting set (8.70) is given by
Z = {(x, xc) : x; Bcx[Vc(xc)  Vp(x)] = 0, Xc
=1=
o} .
To illustrate the behavior of the closedloop impulsive dynamical system, we choose initial condition xc(O) = [0,0,0, ojT. Straightforward calculations show that Assumptions A1 and A2 hold. However, the transversality condition is sufficiently complex that we have been unable to show it analytically. This was verified numerically, and hence, Assumption 8.1 appears to hold. Figure 8.25 shows the state trajectories of the plant versus time, while Figure 8.26 shows the state trajectories of the compensator versus time. Figure 8.27 shows the control input versus time. Note that the compensator states are the only states that reset. Furthermore, the control force versus time is discontinuous at the resetting times. A comparison of the plant energy, controller energy, and total energy is shown in Figure 8.28. Finally, Figure 8.29 shows the closedloop system entropy versus time. Note that the entropy of the closedloop system strictly increases between resetting events.
292
CHAPTER 8
0.5 0.4 0.3 0.2
~c: ~ II> ~
:e
0.1 0
'3
g> 0.1
0.02 "c: w
\
\"
T
0.015
\
'I
'oo 71 (Xi) > 0, where {xd~1 rj. Z is such that limi>oo Xi = Xo and limi> 00 71 (Xi) exists, then limi> 00 71 (Xi) = 71(XO).
Proof. The proof is similar to the proof of Proposition 8.2 and, hence, is omitted. 0 Next, we present a hybrid controller design framework for lossless impulsive dynamical systems. Specifically, we consider impulsive dynamical systems yp of the form given by (8.144)(8.146) where u(·) satisfies sufficient regularity conditions such that (8.144) has a unique solution between the resetting times. Furthermore, we consider hybrid resetting dynamic controllers Yc of the form
xc(t) = fcc(xc(t), y(t)), ~xc(t) = 17(y(t))  xc(t), ycc(t) = hcc(xc(t), y(t)), Ydc(t) = hdc(xc(t), y(t)),
xc(O) = XcO, (xc(t), y(t)) rj. Zc, (8.157) (8.158) (xc(t), y(t)) E Zc, (8.159) (8.160)
where xc(t) E 'Dc ~ ~nc, 'Dc is an open set with 0 E 'Dc, y(t) E ~l, Ycc(t) E ]R.mc, Ydc(t) E ]R.md, fcc: 'Dc x]R.I t ]R.nc is smooth on 'Dc and satisfies fcc (0, 0) = 0, 17 : ~l t'Dc is continuous and satisfies 17(0) = 0, hcc : 'Dc x ]RI t ]Rmc is continuous and satisfies hcc(O,O) = 0, and hdc : 'Dc x ]RI t ]Rmd is continuous. We assume that yp is lossless with respect to the hybrid supply rate (sc (u c, y), Sd ( Ud, y) ), and hence, there exists a continuous, nonnegativedefinite storage function V';, : 'Dp t ]R such that V';,(O) = 0 and
V';,(xp(t)) = V';,(xp(to))
L
+
+
it
Sc(Uc(O"), y(O"))dO"
to
Sd(Ud(tk), y(tk)),
t ~ to,
(8.161)
kEZ[t,to)
for all to, t ~ 0, where xp(t), t ~ to, is the solution to (8.144) and (8.145) with (u c , Ud) E Uc x Ud. Equivalently, it follows from Theorem 3.2 that over the interval t E (tk' tk+1], (8.161) can be written as
V';,(xp(i))  V';,(xp(t)) =
1£
sc(uc(O"),y(O"))dO",
tk < t::; i::; tk+l, k E
Z+,
(8.162)
303
ENERGY AND ENTROPYBASED HYBRID STABILIZATION
Vs(Xp(tk)
+ fdp(Xp(tk), Ud(tk))) 
Vs(Xp(tk)) = Sd(Ud(tk), y(tk)), k E
Z+. (8.163)
In addition, we assume that the nonlinear impulsive dynamical system Qp is completely reachable and zerostate observable, and there exist
functions Kc : ]Rl t ]Rmc and Kd : ]Rl t ]Rmd such that Kc(O) = 0, Kd(O) = 0, Sc(Kc(Y), y) < 0, y i= 0, and Sd(Kd(Y), y) < 0, Y i= 0, so that all storage functions Vs(x p), xp E 'Dp , of Qp are positive definite. Finally, we assume that Vs (.) is continuously differentiable. Next, consider the negative feedback interconnection of Qp and Qc given by Y = Ucc and (u c, Ud) = (Ycc, Ydc). In this case, the closedloop system Q is given by
x(t) = fc(x(t)), ~x(t) = fd(X(t)),
X(O) = Xo, x(t) E Z,
X(t) (j. Z,
t
2 0,
(8.164) (8.165)
where t 2 0, x(t) ~ [xJ(t),xr(t)jT, Z ~ Zl U Z2, Zl ~ {x E 'D : (xp, hcc(xc, hp(xp))) E Zp}, Z2 ~ {x E 'D : (xc, hp(xp)) E Zc},
fc(x) ~ [ fcp(x p, hcc(xc, hp(xp))) ] fcc(x c, hp(xp)) , fd(X) ~ [ fdp(X p , hdc(Xc, hp(xP)))XZl (x) ] . (17(h p(xp))  XC )Xz2 (x)
(8.166)
Assume that there exists an infinitely differentiable function Vc : 'Dc x l I · such that Vc(x c, y) 2 0, Xc E 'Dc, y E ]R, Vc(x c, y) = 0 If and only if Xc = 17(Y), and
]R t]R+
Vc(Xc(t), y(t)) = scc(ucc(t), Ycc(t)),
(xc(t), y(t)) (j. Zc,
°
t 2 0, (8.167)
where Sec : ]Rl X ]Rmc t ]R is such that scc(O,O) = and is locally integrable for all inputoutput pairs satisfying (8.157)(8.160). As in Section 8.3, we associate with the plant a positivedefinite, continuously differentiable function Vp(xp) ~ Vs(x p), which we will refer to as the plant energy. Furthermore, we associate with the controller a nonnegativedefinite, infinitelydifferentiable function Vc(x c, y) called the controller emulated energy. Finally, we associate with the closedloop system the function (8.168)
called the total energy.
304
CHAPTER 8
Next, we construct the resetting set for
Yc in the following form
Z2 = {(xp, xc) E 'Dp x 'Dc : Llc Vc(xc, hp(xp)) and Vc(x c, hp(xp)) > O}
= {(xp, xc) E'Dp
=0
'Dc : scc(hp(xp), hcc(xc, hp(xp))) and Vc(x c , hp(xp)) > O}. X
=0 (8.169)
The resetting set Z2 is thus defined to be the set of all points in the closedloop state space that correspond to decreasing controller emulated energy. By resetting the controller states, the plant energy can never increase after the first resetting event. Furthermore, if the closedloop system total energy is conserved between resetting events, then a decrease in plant energy is accompanied by a corresponding increase in emulated energy. Hence, this approach allows the plant energy to flow to the controller, where it increases the emulated energy but does not allow the emulated energy to flow back to the plant after the first resetting event. For practical implementation, knowledge of Xc and y is sufficient to determine whether or not the closedloop state vector is in the set Z2. The next theorem gives sufficient conditions for asymptotic stability of the closedloop system y using statedependent hybrid controllers. Theorem 8.7 Consider the closedloop impulsive dynamical system y given by (8.164) and (8.165) with the resetting set Z2 given by (8. 169}. Assume that 'Dei C 'D is a compact positively invariant set o
with respect to Y such that 0 E 'Dei, assume that if Xo E Zl then Xo + fd(xo) E Zl \Z1, and if Xo E Zl \Zl, then fdp(xpo, hdc(XcO, hp(xpo))) = 0, where Zl = {x E 'D : Xp(x) = O} with an infinitely differentiable function Xp (.), and assume that YP is loss less with respect to the hybrid supply rate (sc(u c , y), Sd(Ud, y)) and with a positivedefinite, continuously differentiable storage function Vp(xp), Xp E 'Dp. In addition, assume there exists a smooth function Vc : 'Dc x lRl t lR+ such that Vc(xc,y) 2:: 0, Xc E 'Dc, y E lRl, Vc(xc,Y) = 0 if and only if Xc = 1J(y), and (8.167) holds. Furthermore, assume that every Xo E Z is transversal to (8.164) with Xc (x) = Llc Vc(x c, hp(xp)), and sc(Uc,y)
+ scc(ucc,Ycc) = 0, Sd(Ud, y) < 0,
rt z,
x E Zl,
(8.170) (8.171)
Ucc = hp(xp), Uc = Ycc = hcc(xc, hp(xp)), and Yd = hdc(Xc, hp(xp)). Then the zero solution x(t) == 0 to the closedloop system y is asymptotically stable. In addition, the total energy function V(x) of y given by (8.168) is strictly decreasing
where y Ydc
=
=
x
305
ENERGY AND ENTROPYBASED HYBRID STABILIZATION
across resetting events. Finally, if 'Dp = ]Rnp , 'Dc = ]Rnc, and V (.) is radially unbounded, then the zero solution x(t) == 0 to Q is globally asymptotically stable. Proof. First, note that since Vc(xc,y) 20, Xc
E
'Dc, y
E
]Rl, it
follows that Z = ZI U {(xp, xc) E'Dp X 'Dc : Lic Vc(x c, hp(xp)) = 0 and Vc(x c, hp(xp)) 2 O} = ZI U {(xp, xc) E'Dp
X
'Dc :
Xc(x) = O},
(8.172)
where Xc(x) = Llc Vc(x c, hp(xp)). Next, we show that if the transversality condition (8.155) and (8.156) holds, then Assumptions AI, A2, and 8.1 hold, and, for every Xo E 'Dci, there exists 7 2 0 such that X(7) E Z. Note that if Xo E Z\Z, that is, Xp(x(O)) = 0 or Vc(xc(O), hp(x(O))) = 0 and Llc Vc(xc(O), hp(xp(O))) = 0, it follows from the transversality condition that there exists 8 > 0 such that for all t E (0,8], Xp(x(t)) =1= 0 and Llc Vc(xc(t), hp(xp(t))) =1= O. Hence, since Vc(xc(t), hp(xp(t))) = Vc(xc(O), hp(xp(O))) +tLlc Vc(Xc(7), hp(Xp (7))) for some 7 E (O,t] and Vc(xc,y) 20, Xc E 'Dc, y E ]Rl, it follows that Vc(xc(t), hp(xp(t))) > 0, t E (0,8], which implies that Al is satisfied. Furthermore, if x E Z then, since Vc(x c, y) = 0 if and only if Xc = 'f}(y), it follows from (8.165) that x + fd(x) E Z2\Z2, and hence, X+ fd(x) E Z\Z. Hence, A2 holds. Assumption 8.1 and the fact that for every Xo ¢ Z, Xo =1= 0, there exists 7 > 0 such that X(7) E Z follow as in the proof of Theorem 8.3 with Proposition 8.3 invoked in the place of Proposition 8.2. To show that the zero solution x(t) == 0 to Q is asymptotically stable, consider the Lyapunov function candidate corresponding to the total energy function V(x) given by (8.168). Since Qp is lossless with respect to the hybrid supply rate (sc(u c, y), Sd(Ud, y)) and (8.167) and (8.170) hold, it follows that
V(x(t)) = sc(uc(t), y(t))
+ scc(ucc(t), ycc(t)) = 0,
x(t) ¢ Z. (8.173)
Furthermore, it follows from (8.163), (8.166), and (8.169) that ~V(X(tk))
= Vp(xp(tt))  Vp(Xp(tk)) +Vc(xc(tt), hp(xp(tt)))  Vc(Xc(tk), hp(Xp(tk))) = Sd(Ud(tk), y(tk))XZ1 (X(tk)) +[Vc('f}(hp(Xp(tk))) , hp(Xp(tk)))
306
CHAPTER 8
 Vc(Xc(tk), hp(Xp(tk)) )]XZ2(X(tk)) = Sd(Ud(tk), y(tk))xZl (X(tk))  Vc(Xc(tk), hp(Xp(tk)) )XZ2 (X(tk)) < 0, X(tk) E Z, k E Z+.
(8.174)
Thus, it follows from Theorem 8.2 that the zero solution x(t) == 0 to 9 is asymptotically stable. Finally, if Vp = ~np, Vc = ~nc, and V(·) is radially unbounded, then global asymptotic stability is immediate.
o
It is important to note that Theorem 8.7 also holds for the case where (8.171) is replaced by Sd(Ud,y) ::; 0, x E Zl. In this case, it can be shown using similar arguments as in the proof of Theorem 8.3 that for every Xo f$ Z, Xo i= 0, there exists r > 0 such that x(r) E Z2. Finally, we specialize the hybrid controller design framework just presented to impulsive portcontrolled Hamiltonian systems. Specifically, consider the statedependent impulsive portcontrolled Hamiltonian system given by
Xp(t) = Jcp(xp(t))
(~:: (Xp(t))) T + Gp(xp(t))uc(t),
xp(O) = xpo, ~xp(t)
= Jdp(Xp(t)) ( a1l ax: (xp(t))
y(t)=GJ(xp(t))
(xp(t), uc(t)) f$ Zp,
)T + Gp(xp(t))Ud(t),
(8.175)
(xp(t),uc(t))
(8.176)
E
Zp,
(~~:(xp(t))) T,
(8.177)
~ 0, xp(t) E Vp ~ ~np, Vp is an open set with ~m, Ud(t) E ~m, y(t) E ~m, 1lp : Vp ~ ~ is an
0 E V p, infinitely differentiable Hamiltonian function for the system (8.175)(8.177), Jcp : Vp ~ ~npxnp is such that Jcp(xp) = Jc~(xp), xp E V p,
where t
uc(t) E
Jcp (xp) ( ~~: (xp) )T, xp E V p, is Lipschitz continuous on V p, G p : Vp ~ ~npxm, Jdp : Vp ~ ~npxnp is such that Jdp(Xp) = JJ;(x p), xp E V p,
Jdp(Xp)
(a;;: (xp))T, xp
Zp ~ Zxp x ZU c C Vp 1lp (.) is such that
X
E V p, is continuous on V p, and
~m is the resetting set. Furthermore, assume
'lip (x p + Jdp (xp) (':: (xp)
r
+ Gp (xp)'" )
307
ENERGY AND ENTROPYBASED HYBRID STABILIZATION
Finally, we assume that 1tp(O) = 0 and 1tp(xp) > 0 for all xp i 0 and xp E Dp. Next, consider the fixedorder, energybased hybrid controller
xc(t) = Jcc(xc(t))
(~~: (xc(t))) T + Gcc(xc(t))y(t),
xc(O) = XcO, b.xc(t) = Xc(t), (Xc(t), y(t)) Uc(t) =
(Xc(t), y(t)) E Zc,
G~(xc(t)) (~~: (xc(t)))
01t Ud(t) = G~(xp(t)) ( ox: (Xp(t)) where t 2: 0, xc(t) E Dc
~ ]Rnc,
tt Zc,
(8.179) (8.180)
T ,
(8.181)
)T ,
(8.182)
Dc is an open set with 0 E Dc,
b.xc(t) ~ xc(t+)  xc(t), 1tc : Dc + ]R is an infinitely differentiable Hamiltonian function for (8.179), Jcc : Dc + ]Rncxnc is such that Jcc(x c) = Jc~(xc), Xc E Dc, Jcc(xc)(~~; (xc))T, Xc E Dc, is Lipschitz continuous on Dc, G cc : Dc + ]Rnc xm , and the resetting set Zc C Dp x Dc given by
Finally, we assume that 1tc(O) = 0 and 1t c(x c) > 0 for all Xc
Xc
E
i 0 and
Dc·
Note that 1tp(x p), xp E Dp, is the plant energy and 1t c(x c), Xc E Dc, is the controller emulated energy. Furthermore, the closedloop system energy is given by 1t(xp, xc) ~ 1tp(x p) + 1tc(x c). The resetting set Z is given by Z ~ Zl U Z2, where
Zj
~ { (xp, xc) E Dp XDc , (xp, ~G;';(xc) (':: (xc)) T) E Zp } , (8.184)
Z2
~ {(Xp,xc) E Dp x Dc , (xc> G~(xp) ('::(Xp)r) E
ze} (8.185)
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CHAPTER 8
Here, we assume that Zl = {(xp,x c) E Dp x Dc : XI(xp,xc) = O}. Furthermore, if (xp,x c) E Zl then xp + Jdp(XP)(~~p(xp))Tp T
Gp(xp)G p (xp)(
ax: (xp))
OJl
(a;;: (xp))T 
Jdp (xp) sume that
T

E Zl \Zl, and if (xp,
xc)

E Zl \Zl then
Gp(xp)GJ(xp)(~~: (xp))T = O. Finally, we as
Next, note that the total energy function H(xp, xc) along the trajectories of the closedloop dynamics (8.175)(8.185) satisfies d dt H(xp(t), xc(t)) = 0,
(xp(t), xc(t)) ¢ Z,
(8.187)
b.H(Xp(tk), Xc(tk)) =  {}{}Hp (Xp(tk) )Gp(Xp(tk))G~ (Xp(tk)) xp
{}H ( {}X: (Xp(tk))
)T
XZl (Xp(tk), Xc(tk))
Hc(Xc(tk) )XZ2 (Xp(tk), Xc(tk)), (Xp(tk), Xc(tk)) E Z, k E Z+.
(8.188)
Here, we assume that every (xpo, XcO) E Z is transversal to the closedloop dynamical system given by (8.175)(8.185) with Xp(xp, xc) = Xl (xp, xc) and Xc(xp, xc) = Hc(xc). Furthermore, we assume Dei C Dp x Dc is a compact positively invariant set with respect to the
Jt
o
closedloop dynamical system (8.175)(8.185), such that 0 E Dei. In this case, it follows from Theorem 8.7, with V';;(xp) = Hp(xp), Vc(xc, y) = Hc(xc), sc(uc, y) = uJy, Sd(Ud, y) = uJy, and Sec (ucc , Ycc) = uJcYcc, that that the zero solution (xp(t), xc(t)) == (0,0) to the closedloop system (8.175)(8.185) is asymptotically stable.
8.9 Hybrid Control Design for Nonsmooth EulerLagrange Systems
In this section, we present a hybrid feedback control framework for nonsmooth EulerLagrange dynamical systems. Consider the governing equations of motion of an npdegreeoffreedom dynamical system
ENERGY AND ENTROPYBASED HYBRID STABILIZATION
309
given by the hybrid EulerLagrange equation
:t [~~
(q(t), q(t))] T q(O) = qQ,
_
[~~ (q(t), q(t))] T = Uc(t), q(O) = qQ,
] _ [ P(q(t))  q(t) ] [ D.q(t) D.q(t) Q(q(t))  q(t) ,
(q(t), q(t)) ~ Zp, (8.189)
(q(t), q(t)) E Zp,
(8.190)
with outputs (8.191) where t 2: 0, q E JR.11. p represents the generalized system positions, q E lRnp represents the generalized system velocities, £ : lRnp x lRnp  t lR denotes the system Lagrangian given by £(q, q) = T(q, q)  U(q), where T : lRnp x lRnp  t lR is the system kinetic energy and U : lRnp  t lR is the system potential energy, U c E lRnp is the vector of generalized control forces acting on the system, Zp c lRnp x lRnp is the resetting set such that the closure of Zp is given by Zp ~ {(q, q) : H(q, q) = O},
(8.192)
where H : lRnp x lRnp  t lR is an infinitely differentiable function. Here, P : lRnp  t lRnp and Q : lRnp  t lRnp are continuous functions such that if (q, q) E Zp, then (P(q), Q(q)) E Zp \Zp, and if (q, q) E Zp \Zp, then (P(q), Q(q)) = (q, q), T(P(q), Q(q)) + U(P(q)) < T(q, q) + U(q), (q, q) E Zp, and hI : lRnp  t lRh and h2 : lRnp  t lRl  h are continuously differentiable functions such that hl(O) = 0, h2 (0) = 0, and hl(q) ¢. 0. We assume that the system kinetic energy is such that T(q, q) = ~qT[~(q,q)jT, T(q, 0) = 0, and T(q,q) > 0, q =1= 0, q E lRnp . Furthermore, let 1t : lRnp x lRnp  t lR denote the Legendre transformation of the Lagrangian function £(q, q) with respect to the generalized velocity q defined by 1t(q,p) ~ qTp  £(q,q), where p denotes the vector of generalized momenta given by (8.193) where the map from the generalized velocities q to the generalized momenta p is assumed to be bijective. Now, if 1t(q,p) is lower bounded, then we can always shift 1t(q,p) so that, with a minor abuse of notation, 1t(q,p) 2: 0, (q,p) E lRnp x lRnp. In this case, using (8.189) and
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CHAPTER 8
the fact that
d[f'( .)] ac( .). ac( .) .. dtt..q,q = aq q,qq+ a£l q,qq,
(8.194)
Jt
it follows that 1l (q, p) = u; £l, (q, £l) ¢ Zp. We also assume that the system potential energy U(·) is such that U(O) = 0 and U(q) > 0, q i= 0, q E Vq ~ ]Rnp , which implies that 1l(q,p) = T(q,£l)+U(q) > 0, (q, £l) i= 0, (q, £l) E Vq x ]Ritp. Next, consider the energybased hybrid controller
:t [~~:(qe(t)'£le(t),yq(t))]T qe(O) = qeO,

£le(O) = £leo,
[~~:(qe(t)'£le(t),yq(t))]T = 0, (qe(t), £le(t), y(t)) ¢ Ze,
] = [ 'f/(Yq(t))  qe(t) ] [ boqe(t) bo£le(t) £le(t) ' ue(t) =
(() () ()) qe t ,£le t ,Y t
E
(8.195) ( ) Ze, 8.196
[a~e (qe(t), £le(t), yq(t))] T ,
(8.197)
where t ~ 0, qe E ]Rite represents virtual controller positions, qe E ]Rite represents virtual controller velocities, Yq ~ hl(q), Ce : ]Rite X ]Rite X ]Rh t ]R denotes the controller Lagrangian given by Ce(qe, £le, Yq) ~ Te(qe, £le)  Ue(qe, Yq), where Te : ]Rne x ]Rile t ]R is the controller kinetic energy, Ue : ]Rite X ]Rh t ]R is the controller potential energy, 'f/(') is a continuously differentiable function such that 'f/(O) = 0, Ze C ]Rite X ]Rite X ]Rl is the resetting set, boqe(t) ~ qe(t+)  qe(t), and bo£le(t) ~ £le(t+)  £le(t). We assume that the controller kinetic energy Te(qe, £le) is such that Te(qe, £le) = ~£lJ[Wc(qe, £le)jT, with Te(qe, 0) = 0 and Te(qe,£le) > 0, £le i= 0, £le E ]Rite. Furthermore, we assume that Ue('f/(Yq), Yq) = 0 and Ue(qe, Yq) > 0 for qe i= 'f/(Yq), qe E Vqe ~ ]Rite. As in Section 8.5, note that Vp(q, £l) ~ T(q, £l) + U(q) is the plant energy and Vc(qe, £le, Yq) ~ Te(qe, £le) + Ue(qe, Yq) is the controller emulated energy. Furthermore, V(q, £l, qe, £le) ~ Vp(q, £l) + Vc(qe, £le, Yq) is the total energy of the closedloop system. It is important to note that the Lagrangian dynamical system (8.189) is not lossless with outputs Yq or y. Next, we study the behavior of the total energy function V(q, £l, qe, £le) along the trajectories of the closedloop system dynamics. For the closedloop system, we define our resetting set as Z ~ Zl U Z2, where Zl ~ {(q,£l,qe,£le) : (q,£l) E Zp} and Z2 ~ {(q, £l, qe, £le) : (qe, £le, y) E Ze}. Note that
:t
Vp(q, £l) =
:t
1l(q,p) = u'[£l,
(q, £l, qe, £le) ¢ Z.
(8.198)
311
ENERGY AND ENTROPYBASED HYBRID STABILIZATION
Jt
To obtain an expression for Vc(qc,(Zc, Yq) when (q, q, qc, qc) ¢ Z, define the controller Hamiltonian by (8.199) where the virtual controller momentum Pc is given by Pc( qc, qc, Yq)
=
[~~: (qc, qc, yq)] T.
Then 'Hc(qc, qc,Pc, Yq) = Tc(qc, qc)+Uc(qc, Yq). Now, it follows from (8.195) and the structure of Tc(qc, qc) that, for t E
(tk' tk+1], 0= :t lPc(qc(t) , qc(t), yq(t))]T qc(t) 
~~: (qc(t) , qc(t), Yq(t))qc(t)
= :t [p;(qc(t),qc(t),Yq(t))qc(t)]  p;(qc(t),qc(t),Yq(t))iic(t)
+ ~~c (qc(t), qc(t), Yq (t))ijc (t) + 8;c (qc(t), qc(t), Yq(t))q(t) uq
u~
 :t.cc(qc(t) , qc(t), Yq(t))
= :tlP;(qc(t), qc(t), Yq(t))qc(t)  .cc(qc(t), qc(t), Yq(t))]
+ 8~c (qc(t), qc(t), Yq(t))q(t) = :t 'Hc(qc(t), qc(t),Pc(t), Yq(t)) + 8~c (qc(t), qc(t), Yq(t))q(t) = :t Vc(qc(t) , qc(t), Yq(t)) + 8~c (qc(t), qc(t), Yq(t))q(t), (q(t), q(t), qc(t), qc(t)) ¢ Z.
(8.200)
Hence,
:t V( q(t), q(t), qc(t), qc(t)) =
u; (t)q(t) 
=0,
8~c (qc(t), qc(t), Yq(t) )q(t)
(q(t),q(t),qc(t),qc(t)) ¢ tk < t ::; tk+ 1,
z, (8.201)
which implies that the total energy of the closedloop system between resetting events is conserved. The total energy difference across resetting events is given by ~V(q(tk)' q(tk), qc(tk), qc(tk)) = Vp(q(tt), q(tt))  Vp(q(tk), q(tk))
+Tc(qc(tt), qc(tt)) +Uc(qc(tt), Yq(tk))
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CHAPTER 8
 Ve(qe(tk), qe(tk), Yq(tk)) = [Vp(P(q(tk)), Q(q(tk)))  Vp(q(tk), q(tk))] ·XZ1 (q(tk), q(tk), qe(tk), qe(tk))  Vc(qe(tk), qe(tk), Yq(tk)) ·XZ2 (q(tk), q(tk), qe(tk), qe(tk)) < 0, (q(tk), q(tk), qe(tk), qe(tk)) k E
Z+,
E
Z,
(8.202)
which implies that the resetting law (8.196) ensures the total energy decrease across resetting events. Here, we concentrate on an energy dissipating statedependent resetting controller that affects a oneway energy transfer between the plant and the controller. Specifically, consider the closedloop system (8.189)(8.197), where Ze is defined by
Ze
~ { (q, q, qe, qe) :
:t
Vc(qe, qe, Yq) = 0 and Vc(qe, qe, Yq) >
o} . (8.203)
Since Yq = hl(q) and
:tVe(qe,qe,Yq)
= 
[88~e(qe,qe,yq)] q= [8~e(qe,yq)] q, (qe, qe, y) rf Ze,
(8.204)
it follows that (8.203) can be equivalently rewritten as
Ze = {(q, q, qe, qe) :
[8~e (qe, h1 (q))] q =
and Vc(qe,qe,h 1 (q)) >
o}.
0 (8.205)
Once again, for practical implementation, knowledge of qe, qe, and Y is often sufficient to determine whether or not the closedloop state vector is in the set Ze. The next theorem gives sufficient conditions for stabilization of nonsmooth EulerLagrange dynamical systems using statedependent hybrid controllers. For this result define the closedloop system states x ~ [qT,qT,q;!"qJjT.
ENERGY AND ENTROPYBASED HYBRID STABILIZATION
313
Theorem 8.8 Consider the closedloop dynamical system 9 given by (8.189}(8.197), with the resetting set Ze given by (8.203). Assume that Dei C Dq x JRn p x Dqe x JRne is a compact positively ino
variant set with respect to 9 such that 0 E Dei. Furthermore, assume that the transversality condition (8.155) and (8.156) holds with Xp(x) = H(q,q) and Xe(x) = JtVc(qe,qe,Yq). Then the zero solution x(t) == 0 to 9 is asymptotically stable. In addition, the total energy function V (x) of 9 is strictly decreasing across resetting events. Finally, if Dq = JRnp, Dqe = JRne, and the total energy function V(x) is radially unbounded, then the zero solution x(t) == 0 to 9 is globally asymptotically stable. The proof is similar to the proof of Theorem 8.7 with Vp(x p) = Vp(q, q), Vc(x e, y) = Vc(qe, qe, yq), y = U ee = x p, U e = Yee = ~, se(ue,y) = uJp(y), Sd(Ud,y) = 0, Vp(P(q),Q(q)) Vp(q, q) < 0, (q, q) E Zp, See (u ee , Yee) = y'£cp(ue ), where p(y) = Proof.
p ([
~])
= q, T/(Y) replaced by [T/(6 q )
],
and noting that (8.201)
and (8.202) hold.
0
8.10 Hybrid Control Design for Impact Mechanics
In this section, we apply the energydissipating hybrid controller synthesis framework presented in Section 8.9 to the constrained inverted pendulum shown in Figure 8.30, where m = 1 kg and L = 1 m. In the case where IB(t)1 < Be ::; ~, the system is governed by the dynamic equation of motion
O(t)  gsinB(t) = ue(t),
B(O) = Bo,
e(O) = eo,
t::::: 0, (8.206)
where g denotes the gravitational acceleration and U e () is a (thruster) control force. At the instant of collision with the vertical constraint IB(t)1 = Be, the system resets according to the resetting law (8.207)
where e E [0, 1) is the coefficient of restitution. Defining q = Band q = we can rewrite the continuoustime dynamics (8.206) and resetting dynamics (8.207) in Lagrangian form (8.189) and (8.190) with £(q, q) = !q2 _ g cos q, P(q) = q, Q(q) = eq, and Zp = {(q, q) E JR2 : q = Be, q > O} U {(q, q) E JR2 : q = Be, q < O}.
e,
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CHAPTER 8
I
~/ 1
/ / / /
/
Figure 8.30 Constrained inverted pendulum.
Next, to stabilize the equilibrium point (qe,qe) the hybrid dynamic compensator
qe(t)
+ keqe(t) = keq(t),
qe(O) = qeD,
=
(0,0), consider
(q(t), q(t), qe(t), qe(t)) ¢ Ze, t ~ 0,
 qe(t) ] ' [ Llqe(t) Llqe(t) ] = [ q(t)qe(t)
(8.208)
(() () () ()) t E Z e, (8 .2 09) qt,qt,qet,qe
ue(t) = kpq + ke(qe(t)  q(t)),
(8.210)
where kp > 9 and ke > 0, with the resetting set (8.203) taking the form
To illustrate the behavior of the closedloop impulsive dynamical system, let Be = ~, 9 = 9.8, e = 0.5, kp = 9.9, and ke = 2 with initial conditions q(O) = 0, q(O) = 1, qe(O) = 0, and qe(O) = O. For this system a straightforward, but lengthy, calculation shows that Assumptions AI, A2, and 8.1 hold. Figure 8.31 shows the phase portrait of the closedloop impulsive dynamical system with Xl = q and X2 = q. Figure 8.32 shows the controlled plant position and velocity states versus time, while Figure 8.33 shows the controller position and velocity versus time. Figure 8.34 shows the control force versus time. Note that for this example the plant velocity and the controller velocity are the only states that reset. Furthermore, in this
315
ENERGY AND ENTROPYBASED HYBRID STABILIZATION
0.8 0.6
0.4
0.2
"N 0
0.2
0.4
0.6
0.8 0.3
0.2
0.1
o
0.1
x,
0.2
0.3
0.4
0.5
0.6
Figure 8.31 Phase portrait of the constraint inverted pendulum.
case, the control force is continuous since the plant position and the controller position are continuous functions of time.
316
CHAPTER 8
0.4
0
10
5
15
20
25
30
35
40
25
30
35
40
Time
1
1\
0.5
~
rJ
0 0.5 1
0

5
10
15
20 Time
Figure 8.32 Plant position and velocity versus time.
s
l:r x
0.1 0.2 0.3 0.4
0
5
10
15
20
25
30
35
40
25
30
35
40
Time
"g 0.4
0.2
eN u
x
\~
0 0.2 0.4
~
~
0.6 0.8
0
5
10
15
20 Time
Figure 8.33 Controller position and velocity versus time.
317
ENERGY AND ENTROPYBASED HYBRID STABILIZATION
3r.......~
2
o 1
3
4
5 6 _7L~LL~LL~
o
5
10
15
20 Time
25
30
Figure 8.34 Control signal versus time.
35
40
Chapter Nine Optimal Control for Impulsive Dynamical Systems
9.1 Introduction
In this chapter, we consider a hybrid feedback optimal control problem over an infinite horizon involving a hybrid nonlinearnonquadratic performance functional. The performance functional involves a continuoustime cost for addressing performance of the continuoustime system dynamics and a discretetime cost for addressing performance at the resetting instants. Furthermore, the hybrid cost functional can be evaluated in closed form as long as the nonlinearnonquadratic cost functional considered is related in a specific way to an underlying Lyapunov function that guarantees asymptotic stability of the nonlinear closedloop impulsive system. This Lyapunov function is shown to be a solution of a steadystate, hybrid HamiltonJacobiBellman equation, and hence, guarantees both optimality and stability of the feedback controlled impulsive dynamical system. The overall framework provides the foundation for extending linearquadratic feedback control methods to nonlinear impulsive dynamical systems. We note that the optimal control framework for impulsive dynamical systems developed herein is quite different from the quasivariational inequality methods for impulsive and hybrid control developed in the literature (e.g., [1618,30]). Specifically, quasivariational methods do not guarantee asymptotic stability via Lyapunov functions and do not necessarily yield feedback controllers. In contrast, the proposed approach provides hybrid feedback controllers guaranteeing closedloop stability via an underlying Lyapunov function.
9.2 Impulsive Optimal Control
In this section, we consider an optimal control problem for nonlinear impulsive dynamical systems involving a notion of optimality with respect to a hybrid nonlinearnonquadratic performance functional. Specifically, we consider the following impulsive optimal control problem.
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CHAPTER 9
Impulsive Optimal Control Problem. Consider the nonlinear im
pulsive controlled system given by
x(t) = Fc(x(t), uc(t), t), x(to) = xo, X(tf) = Xf, uc(t) E Uc, (9.1) (t, x(t)) ¢ Sx, ~x(t) = Fd(X(t), Ud(t), t), Ud(t) E Ud, (t, x(t)) E Sx, (9.2)
°
where t 2: 0, x(t) E V ~ ]Rn is the state vector, V is an open set with E V, (uc(t), Ud(tk)) E Uc x Ud ~ ]Rmc X ]Rmd, t E [to, tf], k E Z[to,tf)' is the hybrid control input, x(to) = Xo is given, X(tf) = Xf is fixed, Fe : V X Uc x ]R t ]Rn is Lipschitz continuous and satisfies Fe (0, 0, t) = for every t E [to, tf], Fd : Sx x Ud t ]Rn is continuous and satisfies Fd(O, 0, t) = for every t E [to, trj, and Sx c [0,00) x ]Rn. Then determine the control inputs (uc(t), Ud(tk)) E Uc x Ud, t E [to, tf], k E Z[to,tf)' such that the hybrid performance functional
°
°
J(xo, ucO, Ud('), to) =
itf to
+
Lc(x(t), uc(t), t)dt
L
Ld(X(tk), Ud(tk), tk)
(9.3)
kEZ[to,tf)
is minimized over all admissible control inputs (u c 0, Ud (.)) E Uc X Ud, where Lc : V x Uc x ]R t ]R and Ld : Sx X Ud t ]R are given. Next, we present a hybrid version of Bellman's principle of optimality which provides necessary and sufficient conditions, with a given hybrid control (uc(t), Ud(tk)) E Uc x Ud, t 2: to, k E Z[t,to), for minimizing the performance functional (9.3). Lemma 9.1 Let (u~ 0, uci 0) E Uc X Ud be an optimal hybrid control that generates the trajectory x(t), t E [to, tf], with x(to) = xo. Then the trajectory x (.) from (to, xo) to (tf' Xf) is optimal if and only if for all t', t" E [to, tf], the portion of the trajectory x(·) going from
(t',x(t')) to (t",x(t")) optimizes the same cost functional over [t',t"], where x( t') = Xl is a point on the optimal trajectory generated by (u~
0, uci (.)).
Proof. Let u~ (.) E Uc and uci (.) E Ud solve the Impulsive Optimal Control Problem and let x(t), t E [to, tf], be the solution to (9.1) and (9.2) generated by u~() and uci(')' Next, ad absurdum, suppose there exist t' 2: to, t" ::; tf, and Uc(t), t E [t', t"], Ud (tk), k E Z[tl ,til), such that
OPTIMAL CONTROL FOR IMPULSIVE DYNAMICAL SYSTEMS
321
where x(t) is a solution of (9.1) and (9.2) for all t E tt', til] with uc(t) = uc(t), Ud(tk) = Ud(tk), x(t /) = X(t'), and x(t") = x(t"). Now, define u~(t),
uco(t) ~ { uc(t), u~ (t),
[to, t/], tt', til], [til, tf],
Then,
=
1 t'
to
+
L
+
1 t'
t'
to
+ +
Lc(x(t), uc(t), t)dt
L
1 +
Ld(X(tk), Ud(tk), tk)
til
+
0 and "Id > 0, and consider the nonlinear
impulsive dynamical system (10.1) and (10.2) with performance functional (10.18). Assume there exist functions P1Wd : V t lR1xdd and P2Wd : V t Ndd, and a continuously differentiable function V : V t lR such that
V(O) =0, V(x) >0, x E V, x =I 0, V'(x)fe(x) < 0, x t/. Zx, x =I 0, V(x + fd(x))  V(x) SO, x E Zx, V(x + fd(x) + J1d(X)Wd) = V(x + fd(x)) + P1Wd(X)Wd +WJp2Wd (X)Wd, x E Zx,
(10.86) (10.87) (10.88) (10.89) Wd E Wd,
(10.90)
"Id ( f3d Idd  P2Wd x)
> 0,
x E Zx,
(10.91)
Le(x)
+ V'(x)fe(x) + 4f3e VI(x)Jle(X)J~(x)V'T(x) = 0,
Ld(X)
+ V(x + fd(x))
"Ie
+~PIWd(X)(;: Idd 
(10.92)
 V(x) P2Wd (x))lPl'v d(x)
= 0,
x E Zx.
(10.93)
Then there exists a neighborhood Vo ~ V of the origin such that if Xo E Vo, then the zero solution x(t) == 0 of the undisturbed (i.e.,
365
DISTURBANCE REJECTION CONTROL
(We(t), Wd(tk) == (0,0))) system (10.1) and (10.2) is asymptotically stable. If, in addition, re(x) ~ 0, x ¢ Zx and rd(X) ~ 0, x E Zx, then J(xo) ~ J(xo) where J(xo)
~
1
00
[Le(x(t))
= V(xo),
L
+ re(x(t))]dt +
(10.94)
[Ld(X(tk))
+ r d(X(tk))],
kEZ[O,oo)
(10.95) (10.96) (10.97)
and where x(t), t ~ 0, is a solution to (10.1) and (10.2) with (we(t), Wd(tk)) == (0,0). Furthermore, if Xo = 0, then the solution x(t), t ~ 0, to (10.1) and (10.2) satisfies V(x(T)) ~ 'Y,
'Y
= 'Ye + 'Yd, T
~ 0,
we() E We/3c' Wd () E Wd/3d' (10.98)
Finally, if 1)
= jRn and V(x) ~
00
°
as
"xii
~ 00,
(10.99)
then the zero solution x(t) == to the undisturbed system (10.1) and (10.2) is globally asymptotically stable. Proof. The proofs for local and global asymptotic stability, as well as the performance bound (10.94) are identical to the proofs of local and global asymptotic stability given in Theorem 10.1 and performance bound (10.28). Next, with (Se(Ze, we), Sd(Zd, Wd)) = ('ftw; We, jtWJWd), and re(x) and rd(X) given by (10.96) and (10.97), respectively, it follows from Proposition 10.1 that
V(x(T))
~ ;:
lT
w[(t)we(t)dt+ ;:
L
we () E We/3c' which yields (10.98).
WJ(tk)Wd(tk),
T
~ 0,
kEZ[O,oo)
Wd() E Wd/3d' (10.100)
o
366
CHAPTER 10
10.4 Optimal Controllers for Nonlinear Impulsive Dynamical Systems with Bounded Disturbances
In this section, we consider a hybrid control problem involving a notion of optimality with respect to an auxiliary cost which guarantees a bound on the worstcase value of a nonlinearnonquadratic hybrid cost functional over a prescribed set of bounded exogenous disturbances. The optimal hybrid feedback controllers are derived as a direct consequence of Theorem 10.1 and provide a generalization of the hybrid HamiltonJacobiBellman conditions for time invariant, infinitehorizon problems considered in Chapter 9. In particular, we develop nonlinear hybrid feedback controllers for nonlinear impulsive statedependent dynamical systems with bounded energy disturbances that additionally minimize a nonlinearnonquadratic hybrid cost functional. To address the optimal hybrid control problem let V c ~n be an open set with 0 E V. Furthermore, let We ~ ~dc and Wd ~ ~dd, and let Se : ~Pc X ~dc + ~ and Sd : ~Pd X ~dd + ~ be given functions. Consider the controlled nonlinear impulsive dynamical system
x(t) = Fe(x(t), ue(t)) + J 1e (x(t))w e(t),
x(O) = XO, x(t) ¢ Zx, (10.101) we(t) EWe, x(t) E Zx, Wd(t) E Wd, (10.102)
with performance variables
Ze(t) = he(x(t), ue(t)) + he(x(t))we(t), Zd(t) = hd(X(t), Ud(t))
+ hd(X(t))Wd(t),
x(t) ¢ Zx,
we(t) EWe, (10.103)
x(t)
E
Zx,
Wd(t)
E Wd,
(10.104) where Fe : ~n
X ~mc + ~n
satisfies Fe (0,0) = 0, J 1e
: ~n + ~nxdc,
he : ~n X ~mc + ~Pc satisfies he(O,O) = 0, he: ~n + ~PcXdc, hd: ~n X ~md + ~Pd, hd : ~n + ~Pdxdd, and the hybrid control (Ue(),Ud(')) is restricted Fd : ~n X ~md + ~n,
J 1d
: ~n + ~nxdd,
to the class of admissible controls consisting of measurable functions such that (ue(t), Ud(tk)) E Ue X Ud for all t 2: 0 and k E Z[O,oo), where the control constraint sets Ue and Ud are given with (0,0) E Ue X Ud. Given a hybrid control law (cPe (.), cPd (.)) and a hybrid feedback controllaw (ue(t),Ud(t)) = (cPe(x(t)),cPd(X(t))), the closedloop system shown in Figure 10.1 has the form
367
DISTURBANCE REJECTION CONTROL
Figure 10.1 Feedback interconnection of 9 and 9c.
x(t) = Fe(x(t), 00 with (we(t), Wd(tk)) == (0, (10.109)
On.
Theorem 10.3 Consider the nonlinear controlled impulsive dynam
368
CHAPTER 10
ical system (10.101)(10.104) with hybrid performance functional J(xo, ueO, Ud(')) =
1
00
L
Le(x(t), ue(t))dt +
Ld(X(tk), Ud(tk)),
kEZ[O,oo)
(10.110)
where (ueO, Ud(')) is an admissible hybrid control. Assume there exist functions r e : V X Ue t JR, r d : V X Ud t JR, PI Wd : V X Ud t JR1xdd, P2Wd : V X Ud t Ndd, a continuously differentiable function V : V t JR, and a hybrid control law ¢e : V t Ue and ¢d : V t Ud such that
V(O) =0, V(x) >0, ¢e(O) =0, V'(x)Fe(x, ¢e(x)) < 0, V'(X)Jle(X)W e :::; se(ze, we)
(10.111)
x E V, X ¢ Zx,
x
=1=
x
0,
=1=
0,
+ Le(x, ¢e(x)) + fe(x, ¢e(x)),
(10.112) (10.113) (10.114) X
¢ Zx,
We EWe, Ze E JRPc, (10.115) (10.116) V(x + Fd(X, ¢d(X)))  V(x) :::; 0, x E Zx, P1Wd (x, ¢d(X))Wd + P2Wd (x, ¢d(X))Wd :::; Sd(Zd, Wd) +Ld(X, ¢d(X)) + r d(X, ¢d(X)), x E Zx, Wd E Wd, Zd E JRPd
wJ
(10.117)
V(X + Fd(x, Ud) + Jld(X)Wd) = V(x + Fd(X, Ud)) +P1wd (x, Ud)Wd + P2Wd (x, Ud)Wd, X E Zx, Ud E Ud, Wd E Wd, He(x, ¢e(x)) = 0, X ¢ Zx, Hd(X, ¢d(X)) = 0, x E Zx, He(x, u e) 2': 0, X ¢ Zx, Ue E Ue, Hd(X, Ud) 2': 0, x E Zx, Ud E Ud,
wJ
(10.118) (10.119) (10.120) (10.121) (10.122)
where He(x, u e) ~ Le(x, ue) + r e(x, u e) + V' (x )Fe(x, u e), (10.123) Hd(X, Ud) ~ Ld(X, Ud) + r d(X, Ud) + V(x + Fd(X, Ud))  V(x). (10.124)
Then, with the hybrid feedback control (UeO,Ud(')) = (¢e(x(')),¢d (x(·))), there exists a neighborhood Vo ~ V of the origin such that if Xo E Vo and (we(t), Wd(tk)) == (0,0), the zero solution x(t) of the closedloop system (10.105) and (10.106) is asymptotically stable. If,
"= °
369
DISTURBANCE REJECTION CONTROL
in addition, rc(x,¢c(x)) then
~
0, x ¢ Zx, and rd(X,¢d(X)) ~ 0, x E Zx,
J(xo, ¢c(x(,)), ¢d(X('))) ::; :r(xo, ¢c(x(,)), ¢d(X('))) = V(xo), (10.125) where :r(xo, uc(), Ud(')) ~ faoo [Lc(x(t), uc(t))
L
+
+ rc(x(t), uc(t))]dt
[Ld(X(tk), Ud(tk))
+ r d(X(tk), Ud(tk))],
kEZro,oo)
(10.126)
and where (Uc(),Ud(')) is admissible and x(t), t ~ 0, is a solution to (10.101) and (10.102) with (wc(t), Wd(tk)) == (0,0). In addition, if Xo E V o then the hybrid feedback control (u c(), Ud(')) = (¢c(x(,)), ¢d(X('))) minimizes :r(xo, uc(), Ud(')) in the sense that :r(xo, ¢c(x(,)), ¢d(X('))) =
min
(uc(' ),Ud (·))E C(xo)
:r(xo, u c(), Ud('))' (10.127)
Furthermore, the solution x(t), t fies the dissipativity constraint
~
faT sc(zc(t), wc(t))dt + L
0, to (10.105) and (10.106) satis
Sd(Zd(tk), Wd(tk))
+ V(xo)
~ 0,
kEZ[O,T)
T ~ 0,
Finally, if V =
jRn,
Uc =
jRmc,
V(x)
t
00
W c ()
Ud = as
E
jRmd,
Ilxll
£2,
Wd() E £2.
(10.128)
and t
00,
(10.129)
then the zero solution x(t) == 0 of the undisturbed closedloop system (10.105) and (10.106) is globally asymptotically stable. Proof. Local and global asymptotic stability is a direct consequence of (10.111)(10.114) and (10.116) by applying Theorem 10.1 to the closedloop system (10.105) and (10.106). Furthermore, using (10.119) and (10.120), the performance bound (10.125) is a restatement of (10.28) as applied to the closedloop system. Next, let (u c(), Ud(')) E C(xo) and let x(t), t ~ 0, be the solution to (10.101) and (10.102) with (wc(t),Wd(tk)) == (0,0). Then (10.127) follows from Theorem 9.3 with Lc(x, u c) and Ld(X, Ud) replaced by Lc(x, uc) +
370
CHAPTER 10
rc(x, uc) and Ld(X, Ud)+r d(X, Ud), respectively, and J(xo, ucO, Ud(')) replaced by .J(xo,ucO, UdO). Finally, using (10.115), (10.117), and (10.118), condition (10.128) is a restatement of (10.30) as applied to the closedloop system. 0 Next, we specialize Theorem 10.3 to linear impulsive dynamical systems with bounded energy disturbances. Specifically, we consider the case in which Fc(x, u c) = Acx + Bcuc, Jlc(x) = Dc, hc(x, u c) = ElcX + E2cUc, and Jzc(x) = E ooc , where Ac E jRnxn, Bc E jRnxmc, Dc E jRnxdc, Elc E jRPc xn , E2c E jRPcxmc, and Eooc E jRPcxdc, and
Fd(X, Ud) = AdX
and J2d(X) =
+ BdUd,
Eood,
where
J1d(X) = Dd, hd(x, Ud) = E1dX + E2dUd, Ad E jRnxn, Bd E jRnxmd, Dd E jRnxdd ,
and Eood E jRPdxdd. First, we consider the case where (sc(zc, Wc), Sd(Zd, Wd)) = b;w;wc z; ZC, 'YJWJWd  zJ Zd), where 'Yc > 0 and 'Yd > 0 are given. For the following result assume Eooc = 0, R12c ~ ElcE2c = 0, Eood = 0, t:. T t:. T t:. and R12d = E 1d E 2d = 0, and define RIc = ElcEl c > 0, R2c = T t:. 1 T t:. T t:. T E 2c E2c > 0, Sc = BcR2c Bc , RId = EldEld > 0, R 2d = E 2d E 2d > 0, E 1d E jRPd xn , E 2d E jRPdxmd,
R2ad ~ R 2d + BJ PBd + B;f/DdbJ1dd  DJ PDd)1 DJ PBd, Pad ~ BJpAd+BJPDdbJlddDdPDd)IDJpAd, for arbitrary P E jRnxn
when the indicated inverse exists. Corollary 10.3 Consider the linear impulsive controlled dynamical system
WcO
E £2,
(10.130)
WdO
E
£2,
(10.131)
Zc(t) = ElcX(t) + E 2c u c(t), x(t) (j. Zx, Zd(t) = EldX(t) + E2dUd(t) , x(t) E Zx,
(10.132) (10.133)
with hybrid performance functional
J(xo, uJ), Ud(')) =
10 +
00
[x T (t)RlcX(t)
L
+ u;(t)R2cUc(t)]dt
[XT(tk)RldX(tk)
+ uJ(tk)R2dUd(tk)],
kEZ[O,oo)
(10.134)
371
DISTURBANCE REJECTION CONTROL
where (U c(), Ud ()) is admissible. Assume there exists a positivedefinite matrix P E jRnxn such that 0= xT(A; P
+ PAc + RIc + "'1;2 PDcD; P 
PScP)x,
x ¢ Zx, (10.135)
o< 'Y~Idd 
DJ P Dd, (10.136) O=xT(AJPAd  P+RId +AJPDd(')'~Idd  DJPDd)IDJPAd P:!'dR2a~Pad)X,
x
E
(10.137)
Zx.
Then, with the hybrid feedback control law Uc= 0,
(10.168) (10.169)
x
E 1),
x
=1=
0,
(10.170)
V'(x)[fe(x)  ~Ge(X)R2a~(x)(Lie(x) + G~(x)V'T(x) + 2J;(x)he(x))] +re(x, 0, "Id > 0, (u e(), Ud(')) is admissible, and x(t), t ~ 0, is a solution to (i0.161) and (10.162) with (W e() , Wd(')) == (0,0). Then the zero solution x(t) == 0 of the undisturbed (i.e., ((We(),Wd(')) == (0,0))) closedloop system x(t) = fe(x(t))
+ Ge(x(t))d(X)]T (R2d(X) + P2(X) + r dudud (X))[Ud  ¢>d(X)], x E Zx,
(11.114)
where R2c(X) > 0, X tj. Zx, and R2d(X) + P2(x) + rdUdUd(x) > 0, x E Zx, conditions (11.61) and (11.62) hold. The result now follows as a direct consequence of Theorem 11.2. 0 Note that (11.93)(11.98) imply
V(x) ~ V' (x)[fco (x)
+ tlfc(x) + Gc(x)¢>c(x)] < 0,
406
CHAPTER 11
x ¢ Zx,
D.V(x) ~ V(x
X
I 0,
D.!eO E ~e, (11.115)
+ !dO(X) + D.!d(X) + Gd(X) 0, and let k E Z+. Since, by assumption, TkO is continuous, it follows that for sufficiently small 01 > 0, Tk(X) and Tk+l(X),X E B81(xo), where B81(xo) denotes the open ball centered at Xo with radius 01, are well defined and finite. Hence, it follows from Axiom vii)' that s(t,O,·,O),t E (Tk('), Tk+10], is uniformly bounded on B81 (xo). Now, since 9 is continuous between resetting events it follows that for e > and k E Z+ there exists J = J(e, k) > Osuch that if It  t'l < J, then
°
Ils(t,O,x,O)  s(t',O,x,O)11
and k E Z+, I.k(A, xo) = infxEB>.(xo) Tk(X) and Tk(A, xo) =
°
E
B81(XO), t,t'
E
417
HYBRID DYNAMICAL SYSTEMS
Tk(X) are well defined and lim.x>o ~k(A, xo) = lim.x>o Tk (A, xo) = Tk(XO)' (Note that ~k(A, xo) ~ Tk(X) ~ Tk(A, xo), for all x E B.x(xo).) Hence, there exists 8' = 8'(6) > such that Tk(8',xo)~(8',xo) < 8 and Tk+1(8',XO)  ~k+1(8',xo) < 6. Next, let TJ> be SUPXEB.\(xo)
°
°
such that
Tk 8 ,Xo  ~ 8 ,xo) < Tk(XO)  ~k(8 ,xo) + TJ < 8, Tk+1 (8', xo)  ~k+1 (8', xo) < TJ + Tk+1 (8', xo)  Tk+l(XO) < (
' )
( '
,
Then, it follows from Axiom vii)' that there exists 8" such that
+ Ilxo  yll < 8, + TJ,Tk+1(XO)  TJ), and t'
Now, if It  t'l
(Tk(XO)
E
6.
(12.3)
= 8" (E, TJ, k) >
°
E
3' y E BtSlI(xo), (Tk(XO) + TJ, Tk+1(XO)  TJ).
Ils(t,O,xo,O)  s(t,O,y,O)11
such that Tk(6",xo)  Tk(XO) < 'fl and Tk+ 1(xo)  1:k+1 (6" , xo) < 'fl. Note that the above inequalities guarantee that if t = t' E (Tk(XO) + 'fl,Tk+l(XO)  'fl), then t E (Tk(XO), Tk+1(XO)] and t' E (Tk(Y), Tk+l(Y)]' Y E B811 (XO). Furthermore, letting 6k = 6k(c,'fl,xo,k) = min{6',6"}, it follows from the joint continuity of 9 that for t = t' E (Tk(XO) + 'fl,Tk+1(XO)  'fl), Ils(t,O,xo,O)  s(t,O,y,O)11 < c, Y E B8k (XO)' Similarly, we can obtain 6kl = 6kl (c, 'fl, Xo, k) > such that an analogous inequality can be constructed for all Y E B8k_l (xo) and t E (Tkl(XO) + 'fl, Tk(XO)  'fl). Recursively repeating this procedure for m = k  2, ... ,1, and choosing 6 = 6(c, 'fl, Xo, k) = 6(c, 'fl, Xo, T) = min{61, ... ,6k}, it follows that Ils(t,O,xo,O)  s(t,O,y,O)11 < c, y E B8(XO), t E [0, T], and It  TI(Xo)1 > 'fl, l = 1, ... , k, which implies that
°
°
9 is a stationary leftcontinuous dynamical system satisfying Axiom vii)'.
0
12.3 Specialization to Hybrid and Impulsive Dynamical Systems
In this section, we show that hybrid dynamical systems [30, 169J and impulsive dynamical systems [12, 14,79,93, 148] are a specialization of leftcontinuous dynamical systems. We start our presentation by considering a definition of a controlled hybrid dynamical system that includes the definition given in [30] as a special case. For this definition let Q ~ Z+, where Z+ denotes the set of nonnegative integers. Definition 12.4 A hybrid dynamical system 9H is the 13tuple (V, Q, U,U,Y,Y,q,x,se,jd,S,he,h d ), where q: [0,00) x [0,00) x V x Q x U + Q, x : [0,00) x [0,00) x V x Q x U + V, Se = {Seq}qEQ, Seq: [0,00) x [0,00) x V xU + V, S = {Sq}qEQ' Sq C [O,oo)x V xU, fd = {fdq}qEQ, fdq : Sq + V x Q, he : V x Ue + Y e, and hd : V X Ud + Yd are such that the following axioms hold: i) For every q E Q, to E [0,00), and u E U, Seq(·,to, ',u) is jointly continuous on [to, 00) x V. ii) Foreveryq E Q, to E [0,00), Xo E V, andu EU, Seq(to,to,xo,u)
= Xo·
iii) For every q E Q, to E [0,00), and Xo E V, Seq(t,tO,xO,Ul)
=
419
HYBRID DYNAMICAL SYSTEMS
Scq(t,tO,xO,U2) for all t ul(r) = u2(r), r E [to,t].
E [to,oo)
and Ul,U2
E
U satisfying
ivY For every q E Q, to, tl, t2 E [0,00), to ::::; tl ::::; t2, Xo E u E U, Scq(t2' to, Xo, u) = Scq(t2' tl, Scq(tl' to, xu, u), u).
1),
and
v) For every qo E Q, to E [0,00), Xo E 1), and u E U, q(.) and x(·) are such that q(t,to,xO,qO,u) = qo and x(t,to,xo,qo,u) = Scqo(t,to,xo,u), for all to ::::; t::::; tl, where tl ~ min{t ~ to : (t, ScqO (t, to, xu, u), u(t)) ¢ Sqo} exists. Furthermore, [x T (tt, to, xu, qo, u), qT(tt, to, xu, qo, u)]T = fdqo(tl, x(h), U(tl)) + [xT(tl, to,xO,qO,u), qT(tl,tO,xO,qO,u)]T and for (Xl,ql) ~ (x(tt,to,xo, qO,u),q(tt, to,xO,qO,u)), q(.) andx(·) are such thatq(t,to,xO,qO, u) = ql and x(t,to,xo,qo,u) = Scql(t,tl,Xl,U), for all tl < t::::; t2, where t2 ~ min{t > tl: (t,Scql(t,tl,xl,U), u(t)) ¢ SqJ exists, and so on. vi) There exists y E Y such that y(t) = (hc(x(t,to,xO,qO,u),uc(t)), hd(x(t, to, xo,qo, U),Ud(t))) for all Xo E 1), u E U, to E [0,00), and t E [to, 00). It follows from Definition 12.4 that hybrid dynamical systems involve switchings between a countable collection of continuous dynamical systems. To ensure that the switchings or resetting times are well defined and distinct we make the following additional assumptions:
AI. If (t,x(t,to,xO,qO,u),u(t)) E Sq\Sq, where Sq denotes the closure of the set Sq, then there exists c > 0 such that, for all 0< 8 < c,
Scq(t + 8, t, x(t, to, xu, qo, u), u) ¢ Sq. A2. If (tk, X(tk' to, xu, qo, u), U(tk)) E {)SqnSq, where {)Sq denotes the boundary of the set Sq, then there exists c > 0 such that, for all 0< 8 < c,
Scq(tk
+ 8, tk, x(tt, to, xu, qo, u), u) ¢ Sq, q E
Q.
Assumption Al ensures that if a trajectory reaches the closure of Sq at a point that does not belong to Sq, then the trajectory must be directed away from Sq, that is, a trajectory cannot enter Sq through a point that belongs to the closure of Sq but not to Sq. Equivalently,
420
CHAPTER 12
Al implies that a trajectory can only reach Sq through a point belonging to both Sq and its boundary. Furthermore, A2 ensures that when a trajectory intersects the boundary of a resetting set Sq, it instantaneously exits Sq and the continuoustime dynamics becomes the active element of the hybrid dynamical system. Since a continuous trajectory starting outside Sq and intersecting the interior of Sq must first intersect the boundary of Sq, it follows that no trajectory can reach the interior of Sq. To show that gH is a leftcontinuous dynamical system, let s : [0,00) x [0, 00) x (1) x Q) xU  t 1)x Q be such that s(to, to, (xo, qo), u) = (xo, qo), and for every k = 1,2, ... , s(t, to, (xo, qo), u) = (seqk_l (t, tkl, Xkl, u), qkl), s(tt, to, (xo, qo), u)
= fdqkl (tk' Xk, U(tk)) + [xI, qIlT.
tkl < t ~ tk, (12.6)
(12.7)
Note that s satisfies Axioms i)v) of Definition 12.1 so that the controlled hybrid dynamical system gH generates a leftcontinuous dynamical system on 1)x Q given by the octuple (1) x Q,U, U, y, Y, s, he, hd). Since the resetting events Te = {tl' t2, ... } can be a function of time t, the system state x(t, to, xo, qo, u), and the system input u, hybrid dynamical systems can involve system jumps at variable times, and hence, in general are timevarying leftcontinuous dynamical systems. In the case where the resetting events are defined by a prescribed sequence of times which are independent of the system trajectories and system inputs, that is, Sq = Tq x 1) xU, where Tq C [0,00) and q E Q is a closed discrete set, we refer to gH as a timedependent hybrid dynamical system. Alternatively, in the case where the resetting events are defined by the manifold Sq = [0,00) x Sxq x U, where Sxq C 1), q E Q, that is, Sq is independent of time and the inputs, we refer to gH as a statedependent hybrid dynamical system. More generally, if the resetting events are defined by the manifold Sq = ([0,00) x Sxq xU) U ([0,00) x 1) x Suq), where Suq C U, q E Q, we refer to gH as an input/statedependent hybrid dynamical system. Note that if {seq }qEQ are continuous trajectories such that Axiom vi) in Definition 12.2 holds, then state and input/statedependent hybrid dynamical systems are stationary leftcontinuous dynamical systems. As in the case of impulsive dynamical systems, the analysis of hybrid dynamical systems can be quite involved. In particular, such systems can exhibit Zenoness and beating, as well as confluence. Even though Al and A2 allow for the possibility of confluence and Zeno solutions, A2 precludes the possibility of beating. In the case of stationary state
421
HYBRID DYNAMICAL SYSTEMS
dependent hybrid dynamical systems several interesting observations can be made regarding quasicontinuous dependence and Zeno solutions. Specifically, if the first resetting time is continuous with respect to all initial conditions and all system solutions are nonZeno, then the hybrid dynamical system can be shown to satisfy Axiom vii). Furthermore, if the second resetting time is continuous with respect to all initial conditions on the resetting surfaces Sq, q E Q, and all convergent solutions starting from D\ UqEQ Sq are Zeno, then all the trajectories approach the set UqEQSq \Sq as t ~ 00. The proofs of these facts follow as in the proofs of Propositions 2.1 and 2.3. The notion of a controlled hybrid dynamical system OR given by Definition 12.4 generalizes all the existing notions of dynamical systems wherein the state space has a fixed dimension. For example, if Q = {q} and S = 0, then OR denotes a continuoustime dynamical system with a continuous flow [165]. Alternatively, if Q = {q}, S = Sq, and Seq denotes the solution to an ordinary differential equation (12.8) where Xq(t) E D, t ::::: to, and feq : [0,00) x D x Ue ~ ]Rn, then the hybrid dynamical system OR is characterized by the impulsive differential equation
x(t) = feq(t, x(t), ue(t)), ~x(t) = fdq(t, x(t), Ud (t)),
x(to) = xo, (t, x(t), ue(t)) ¢ Sq, (12.9) (t, x(t), ue(t)) ESq. (12.10)
More generally, if Q is a (finitely or infinitely) countable set and {seq }qEQ denote the solutions to a set of ordinary differential equations, then OR can be represented by a set of coupled ordinary differential equations and difference equations or, equivalently, a set of impulsive differential equations with discontinuous vector fields. Specifically, for every q E Q, let Seq denote the solution to the ordinary differential equation (12.11) where xq(t) ED, t ::::: to, and feq : [0,00) x D x Ue ~ ]Rn. In this case, the hybrid dynamical system OR is characterized by the impulsive differential equation
°
[ x(to) ] = [ Xo ] , [ x(t) ] = [ feq(t)(t,x(t),ue(t)) ] q(t) 'q(to) qo (t, x(t), ue(t)) ¢ Sq(t) , (12.12)
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] [ ~x(t) ~q(t) = fdq(t) (t, x(t), Ud (t)),
(t, X(t), Uc(t))
E
Sq(t). (12.13)
Finally, note that if ~x(t) = 0 in (12.13), then (12.12) specializes to the case of switched hybrid systems involving continuous flows but discontinuous vector fields [29,101,140]' that is, a Filippov dynamical system. We close this section by noting that several of the classical hybrid dynamical system models developed in the literature [7,11,31,130, 157, 167J are a special case of the impulsive dynamical system (12.12) and (12.13). Specifically, the Witsenhausen model [167], the Tavernini model [157J, the NerodeKohn model [130], and the AntsaklisStiverLemmon model [7J are special cases of an autonomous version of (12.12) and (12.13) with ~x(t) == 0, uc(t) == 0, and Ud(t) == o. Hence, these models belong to the class of switched hybrid system models with continuous flows and discontinuous vector fields. Alternatively, the BackGuckenheimerMyers model [l1J is a special case of an autonomous version of (12.12) and (12.13) with uc(t) == 0 and Ud(t) == O. Finally, the Brockett models [31J are a special case of an autonomous version of (12.12) and (12.13) with ~x(t) == O. For a further discussion of these models the interested reader is referred to [30J. 12.4 Stability Analysis of LeftContinuous Dynamical Systems
In this section, we present uniform Lyapunov, uniform asymptotic, and uniform exponential stability results for leftcontinuous dynamical systems. Furthermore, for strong leftcontinuous dynamical systems we present an invariant set stability theorem that generalizes the KrasovskiiLaSalle invariance principle to systems with leftcontinuous flows. For the statement of the following result we define .
6..
1
V(t,s(t,to,xo,u)) = hm tT [V(t,s(t,to,xo,u)) T>tV(r,s(r,to,xo,u))J,
(12.14)
for a given continuous function V : [to, 00) x V + [0,00), whenever the limit on the righthand side exists. Note that V(t,s(t,to,xo,u)) is leftcontinuous on [to, 00), and is continuous everywhere on [to, 00) except on the discrete set TC. Furthermore, we assume that the origin is an equilibrium point of the undisturbed leftcontinuous dynamical system g.
423
HYBRID DYNAMICAL SYSTEMS
Theorem 12.1 Suppose there exist a continuous function V : [0, (0) x'O + [0, (0) and class K functions o{) and {3(.) satisfying
a(llxll) :S V(t, x) :S {3(llxll),
xED, t E [to, (0), (12.15) V(t,s(t,to,xo,O)):S V(t,S(T,to,XO,O)), t ~ T ~ to. (12.16)
°
Then the equilibrium point x = of the undisturbed leftcontinuous dynamical system 9 is uniformly Lyapunov stable. If, in addition, for every Xo E '0, V(·) is such that V(s(t, to, Xo, 0)), t E T, exists and V(t,s(t,to,xo,O)):S 'Y(lls(t,to,xo, 0)11),
t E T,
(12.17)
°
where 'Y : [0, (0) + [0, (0) is a class K function, then the equilibrium point x = of the undisturbed leftcontinuous dynamical system 9 is uniformly asymptotically stable. Alternatively, if there exist scalars e:, &, iJ > 0, and p ~ 1 such that
&llxll P :S V(t, x) :S iJllxllP ,
XED, t E [0,(0), (12.18) (12.19) V(t, s(t, to, xo, 0)) :S e:V(t, s(t, to, xo, 0)), t E T,
°
then the equilibrium point x = of the undisturbed leftcontinuous dynamical system 9 is uniformly exponentially stable. Finally, if '0 = ]R.n and a(·) is a class Koo function, then (12.17) implies (respectively, (12.18) and (12.19) imply) that the equilibrium point x = of the undisturbed leftcontinuous dynamical system 9 is globally uniformly asymptotically (respectively, exponentially) stable.
°
Proof. i) It follows from (12.16) that V(t, s(t, to, Xo, 0)), t ~ to, is a nonincreasing function of time. Moreover, for all t E (to, 00 ), V(t+O",s(t+O",to,xo,O)):S V(tO",s(tO",to,xo,O)),
°
for every sufficiently small 0" > 0. [0, (0) x '0, letting 0" + yields
Since V(·,·) is continuous on
V(t+,s(t+,to,xo,O)):S V(t,s(t,to,xo,O)),
°
(12.20)
t E [to, (0).
(12.21)
Next, let e: > be such that Bo:(O) ~ {x E ]R.n: Ilxll < e:} C '0, define 'rJ ~ a(e:), and define '01'/ ~ {x E Bo:(O) : there exists t E [0,(0) such that V(t, x) < 'rJ}. Now, since V(t, s(t, to, Xo, 0)) is a nonincreasing function of time, '01'/ x [0, (0) is a positive invariant set with respect to the leftcontinuous dynamical system g. Next, let 8 = 8(e:) > be such that {3(8) = a(e:). Hence, it follows from (12.15)
°
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that for all (xo, to) E B8(0) x [0,00), a(lls(t, to, xo, 0)11) ~ V(t, s(t, to, xo, 0)) ~ V(to, xo)
< ,6(8) = a(e:), t 2: to, (12.22)
and hence, s(t, to, xo, 0) E Bo:(O), t 2: to, establishing uniform Lyapunov stability of the equilibrium point x = 0 of g. ii) Uniform Lyapunov stability follows from i). Next, let e: > 0 and 8 = 8(e:) > 0 be such that for every Xo E B8(0), s(t, to, xo, 0) E Bo:(O) , t 2: to, for all to E [0,00) (the existence of such a (8,e:) pair follows from uniform Lyapunov stability), and assume that (12.17) holds. Since by (12.16) V(t,s(t,to,xo,O)) is a nonincreasing function of time and, since V (', .) is bounded from below, it follows from the BolzanoWeierstass theorem [146] that there exists L 2: 0 such that limt+oo V(t, s(t, to, xo, 0)) = L. Now, suppose, ad absurdum, for some Xo E B8(0) and to E [0,00), L> O. Since V(·,·) is continuous and V(to, 0) = 0 for all to E [0,00) it follows that V L ~ {x E Bo:(O) : V(t, x) ~ L for all t E [O,oon is nonempty and s(t, to, xo, 0) ¢ VL, t 2: to. Thus, as in the proof of i), there exists 8 > 0 such that B8(0) c VL. Hence, it follows from (12.21) and (12.17) that for any Xo E B8(0) and t 2: to,
V(t, s(t, to, xo, 0)) = V(to, xo) + rt V(T, S(T, to, XO, O))dT
+
L
lto
[V(tt, s(tt,to,xo,O))
iEZ[to,t)
 V(ti' S(ti' to, XO, 0))]
~V(to,xo) 
rt "Y(lls(T,to,xo,O)ll)dT
ltD
~ V(to, xo)  "Y(8)t,
(12.23)
where Z[to,t) ~ {k E Z+: to < tk ~ t} and [0, oo)\T = {tl, t2," .}. Letting t 2: V(t~~8)L, it follows that V(t, x(t)) ~ L, which is a contradiction. Hence, L = 0, and, since Xo E B8(0) and to E [0,00) was chosen arbitrarily, it follows that V(t, s(t, to, xo, 0)) t 0 as t t 00 for all Xo E B8(0) and to E [0,00). Now, since V(t,s(t,to,xo,O)) 2: a(lls(t, to, xo, 0) II) 2: 0 it follows that a(lls(t, to, xo, 0) II) t 0 or, equivalently, s(t, to, xo, 0) t 0, t t 00, establishing uniform asymptotic stability of the equilibrium point x = 0 of g. iii) Let e: > 0 and rJ ~ a(e:) be given as in the proof of i). Now, (12.19) implies that V(t,s(t,to,xo,O)) ~ 0, t E T, Xo E V, to E
425
HYBRID DYNAMICAL SYSTEMS
[0,(0), and hence, using (12.21), it follows that V(t,s(t,to,xo,O)) is a nonincreasing function of time and V", x [0, (0) C V x [0, (0) is a positive invariant set with respect to g. It follows from (12.19) that for all (xo,to) E V", x [0,(0) with tk E [0,(0)\7, k E Z+,
V(t,s(t,to,xo,O))::; eV(t,s(t,to,xo,O)),
to ::; t ::; tl,
(12.24)
which implies that
V(t,s(t,to,xo,O))::; V(to,xo)ec(tt o),
to ::; t ::; tl'
(12.25)
Similarly,
V(t,s(t,to,xo,O))::; eV(t,s(t,to,xo,O)),
tl < t ::; t2,
(12.26)
which, using (12.21) and (12.25), yields
V(t, s(t, to, Xo, 0)) ::; V( tt, s( tt, to, Xo, 0) )eC(tt 1 ) ::; V(tl, S(tl, to, Xo, 0))ec(tt1 ) < _ V(t 0, x 0 )eC(tltO)ec(ttl)
= V(to, xo)ec(tto), tl < t ::; t2' (12.27) Recursively repeating the above arguments for tk < t ::; tk+1, k 3,4, ... , it follows that
V(t, s(t, to, Xo, 0)) ::; V(to, xo)ec(tto),
t ~ to.
Now, it follows from (12.18) and (12.28) that for all t
~
=
(12.28)
to
alls(t, to, Xo, 0) liP ::; V(t, s(t, to, Xo, 0)) < _ V(t 0, x 0 )ec(tto) ::; ,allxoliPec(tto),
(12.29)
and hence,
t
~
to,
establishing exponential stability of the equilibrium point x The global results follow using standard arguments.
(12.30)
= 0 of g. 0
Next, we present a generalization of the KrasovskiiLaSalle invariance principle to strong leftcontinuous dynamical systems. This result is predicated on the generalized positive limit set theorem (The
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CHAPTER 12
orem 2.2) for systems with leftcontinuous flows satisfying the quasicontinuous dependence property given in Axiom vii). For the statement of the next result define the ,),level set V 1(,),) ~ {x E Vc: V(x)
= ')'},
where,), E JR, Vc ~ V, and V : Vc + JR is a continuous function, and let My denote the largest invariant set (with respect to the strong leftcontinuous dynamical system Q) contained in VI (')'). Theorem 12.2 Let s(t, 0, Xo, 0), t 2": 0, denote a trajectory of the undisturbed strong leftcontinuous dynamical system 9 and let Vc c V be a compact positively invariant set with respect to g. Assume there exists a continuous function V : Vc + JR such that V(s(t, 0, xo, 0)) ::::; V(S(7, 0, XO, 0)), 7 ::::; t, for all Xo E VC. If Xo E Vc, then
°::;
°
o
s(t, 0, xo, 0) + M ~ UYEIR My as t + 00. If, in addition, E Vc, V(O) = 0, V(x) > 0, x E Vc, x =1= 0, and for every Xo E Vc there exists an unbounded infinite sequence {7n}~=1 such that V(S(7n+1' 0, XO, 0)) < V(S(7n ,0,XO,0)), n = 1,2, ... , then the origin is an asymptotically stable equilibrium point of the undisturbed strong leftcontinuous dynamical system g.
Proof. Since V(·) is continuous on the compact set Vc, there exists (3 E JR such that V(x) 2": (3, x E Vc. Hence, since V(s(t, 0, xo, 0)), t 2": 0, is nonincreasing, ')'xo ~ limt~oo V(s(t, 0, Xo, 0)), Xo E Vc, exists. Now, for every y E w(xo) there exists an increasing unbounded sequence {tn}~=o with to = 0, such that s(tn' 0, Xo, 0) + y as n + 00, and, since V(·) is continuous, it follows that V(y) = V(limn~oos(tn,O,xo,O)) = limntoo V(s(tn,O,xo,O)) = ')'xo' Hence, y E V 1(')'xo) for all y E w(xo), or, equivalently, w(xo) ~ VI (')'xo)' Now, since Vc is compact and positively invariant, it follows that s(t, 0, Xo, 0), t 2": 0, is bounded for all Xo E Vc, and hence, it follows from Theorem 2.2 that w(xo) is a nonempty, compact invariant set. Thus, w(xo) is a subset of the largest invariant set contained in VI (')'xo) , that is, w(xo) ~ Myxo' Hence, for all Xo E Vc, w(xo) ~ M. Since s(t, 0, xo, 0) + w(xo) as t + 00, it follows that s(t, 0, Xo, 0) + M as t + 00. Finally, if V(O) = 0, V(x) > 0, x E Vc, x =1= 0, and for every Xo E Vc there exists an unbounded sequence {7n}~=I' with 71 = 0, such that V(S(7n+l' 0, Xo, 0)) < V(S(7n , 0, Xo, 0)), n = 1,2, ... , then V(s(t,O,xo,O)) is a nonincreasing function of time, and hence, there exists ')'xo 2": such that limt~oo V(s(t,O,xo,O)) = ')'xo' Now, suppose
°
HYBRID DYNAMICAL SYSTEMS
427
ad absurdum, "(xo > O. Since Ve is a compact positively invariant set with respect to g, it follows that s(t, 0, xo, 0) is bounded for all Xo EVe, and hence, it follows from Theorem 2.2 that w(xo) is a nonempty, compact invariant set. Thus, w(xo) ~ M'Yxo. Since for every Xo EVe, V(S(Tn+l' 0, XO, 0)) < V(S(Tn, 0, Xo, 0)), n = 1,2, ... , it follows that M'Yxo is not an invariant set which is a contradiction. Hence, "(xo = 0 and, since V(·) is continuous and V(O) = 0, it follows that M contains no invariant set other than the set {O}, and hence, the origin is an asymptotically stable equilibrium point of the undisturbed strong leftcontinuous dynamical system g. 0
12.5 Dissipative LeftContinuous Dynamical Systems: InputOutput and State Properties
In this section, we extend the notion of dissipative dynamical systems to develop the concept of dissipativity for leftcontinuous dynamical systems. The presentation here very closely parallels the presentation given in Section 3.2, and hence, the comments are brief and many of the proofs are omitted. We begin by considering the leftcontinuous dynamical system 9 with input U = (u e , Ud) and output Y = (Ye,Yd). Recall that a function (se(ue,Ye),Sd(Ud,Yd)), where Se : Ue x Yc + lR and Sd : Ud x Yd + lR are such that se(O,O) = 0 and Sd(O,O) = 0, is a hybrid supply rate 3 if sc(uc,Yc) is locally integrable, that is, for all inputoutput pairs ue(t) E Ue, Ye(t) E Yc,
J/
se(·,·) satisfies ISe(ue(s), Ye(s))1 ds < 00, t, i 2: O. Note that since all inputoutput pairs Ud(tk) E Ud, Yd(tk) E Yd, are defined for the resetting events tk ETc, Sd(·,·) satisfies LkEZ [t,t). ISd(Ud(tk), Yd(tk))1 < 00, where k E Z[t,i) ~ {k : t ~ tk < i}. For the remainder of this chapter, we use the notation s(t, to, Xo, u), t 2: to, and x(t), t 2: to, interchangeably to denote the trajectory of 9 with initial time to, initial condition Xo, and input u.
Definition 12.5 A leftcontinuous dynamical system 9 is dissipative with respect to the hybrid supply rate (se, Sd) if the dissipation in
3More generally, the hybrid supply rate (Se, Sd) can be an explicit function of time, that is, Se : [0,00) X Ue X Y e > ffi. and Sd : [0,00) X Ud X Yd > R
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CHAPTER 12
equality
o~
iT
L
+
Se(ue(t), Ye(t))dt
to
Sd(Ud(tk), Yd(tk)),
T
~ to,
kEZ[to,T)
(12.31) is satisfied for all T ~ to and all (u e0, Ud (. )) E Ue x Ud with Xo = O. A leftcontinuous dynamical system g is exponentially dissipative with respect to the hybrid supply rate (se, Sd) if there exists a constant c > 0, such that the dissipation inequality (12.31) is satisfied, with se(ue(t), Ye(t)) replaced by ectse(Ue(t) , Ye(t)) and Sd(Ud(tk), Yd(tk)) replaced by ectksd(Ud(tk), Yd(tk)), for all T ~ to and (ueO, Ud(')) E Ue x Ud with Xo = O. A leftcontinuous dynamical system is lossless with respect to the hybrid supply rate (se, Sd) if g dissipative with respect to the hybrid supply rate (se, Sd) and the dissipation inequality {12.31} is satisfied as an equality for all T ~ to and (u e(. ), Ud (.)) E Ue x Ud with Xo = s(T, to, xo, u) = O. Next, define the available storage Va(to, xo) of the leftcontinuous dynamical system g by
Va (to, Xo)
~
inf
(UcO,Ud('», T?,to
+
L
[iT to
Se (u e(t), Ye (t) )dt
Sd(Ud(tk), Yd(tk))]'
(12.32)
kEZ[to,T)
Note that Va (to , xo) ~ 0 for all (to, xo) E lR x V since Va(to, xo) is the supremum over a set of numbers containing the zero element (T = to). It follows from (12.32) that the available storage of a leftcontinuous dynamical system g is the maximum amount of generalized stored energy which can be extracted from g at any time T. Furthermore, define the available exponential storage of the leftcontinuous dynamical system g by
Va(to, xo)
~+
inf
(UC('),Ud('»' T?,to
L kEZ[to,T)
where c > O.
[iT to
ectse(ue(t), Ye(t))dt
ectkSd(Ud(tk), Yd(tk))] ,
(12.33)
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HYBRID DYNAMICAL SYSTEMS
Definition 12.6 Consider the leftcontinuous dynamical system 9 with input u = (u e , Ud), output y = (Ye, Yd), and hybrid supply rate (se, Sd). A continuous nonnegativedefinite function Vs : JR x V t JR satisfying Vs(t,O) = 0, t E JR, and Vs(T, x(T)) ::; Vs(to, xo)
+
L
+
T se(ue(t), Ye(t))dt r lto (12.34)
Sd(Ud(tk), Yd(tk)),
where x(T) = s(T, to, xo, u), T ~ to, and (ue(t), Ud(tk)) E Ue X Ud, is called a storage function for g. A continuous nonnegativedefinite function Vs : JR x V t JR satisfying Vs(t, 0) = 0, t E JR, and eCTVs(T, x(T)) ::; ect°Vs(to, xo)
+
L
+
T ct r e se( ue(t), Ye(t) )dt lto
ectksd(Ud(tk), Yd(tk)),
(12.35)
kEZ[to,T)
where
E:
> 0, is called an exponential storage function for g.
Note that for every to E [0,00), Xo E V, and u E U, Vs(t, s(t, to, xo, u)) is leftcontinuous on [to, 00), and is continuous everywhere on [to, 00) except on Te.
Definition 12.7 A leftcontinuous dynamical system 9 is completely reachable if for all (to, xo) E JR xV, there exist a finite time ti ::; to, square integrable input ue(t) defined on [til to], and input Ud(tk) defined on k E Z[tj,to)' such that s(to, ti, 0, u) = Xo· Theorem 12.3 Consider the leftcontinuous dynamical system 9 with input u = (u e , Ud) and output Y = (Ye, Yd), and assume that 9 is completely reachable. Then 9 is dissipative (respectively, exponentially dissipative) with respect to the hybrid supply rate (se, Sd) if and only if the available system storage Va(to,xo) given by (12.32) (respectively, the available exponential system storage Va(to, xo) given by (12.33)) is finite for all to E JR and Xo E V and Va(t, 0) = 0, t E lR. Moreover, if Va(t,O) = 0, t E JR, and Va(to, xo) is finite for all to E JR and Xo E V, then Va(t,x), (t,x) E JR x V, is a storage function (respectively, exponential storage function) for g. Finally, all storage functions (respectively, exponential storage functions) Vs (t, x), (t, x) E JR xV, for 9 satisfy 0::; Va(t,x) ::; Vs(t,x),
(t,x) E JR x V.
(12.36)
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CHAPTER 12
Proof. The proof is identical to that of the impulsive dynamical system case given in Theorem 3.1 and, hence, is omitted. 0 The following corollary is immediate from Theorem 12.3. Corollary 12.1 Consider the leftcontinuous dynamical system g and assume that g is completely reachable. Then g is dissipative (respectively, exponentially dissipative) with respect to the hybrid supply rate (se, Sd) if and only if there exists a continuous storage function (respectively, exponential storage function) Vs(t,x), (t,x) E lRxD, satisfying (12.34) (respectively, {12.35}}. The next result gives necessary and sufficient conditions for dissipativity and exponential dissipativity over an interval t E (tk' tk+l] involving the consecutive resetting times tk, tk+1 E Te. Theorem 12.4 g is dissipative with respect to the hybrid supply rate (se, Sd) if and only if there exists a continuous, nonnegativedefinite function Vs : lR x D ~ lR such that, for all k E Z+, Vs(i, s(i, to, xo, u))  Vs(t, s(t, to, xo, u)) ::;
it
Se(Ue(S), Ye(s))ds,
tk < t ::; i ::; tk+1,
(12.37)
Vs(tt, s(tt, to, xo, u))  Vs(tkl S(tk' to, xo, u)) ::; Sd(Ud(tk), Yd(tk)). (12.38) Furthermore, g is exponentially dissipative with respect to the hybrid supply rate (se, Sd) if and only if there exist a continuous, nonnegativedefinite function Vs : lR x D ~ lR and a scalar E: > 0 such that ectVs(i, s(i, to, xo, u))  ectVs(t, s(t, to, xo, u))
: ; it
ecSSe(Ue(S),Ye(s))ds,
tk < t::; i::; tk+1, (12.39)
Vs(tt, s(tt, to, Xo, u))  Vs(tk, S(tkl to, xo, u)) ::; Sd(Ud(tk), Yd(tk)). (12.40) Finally, g is lossless with respect to the hybrid supply rate (se, Sd) if and only if there exists a continuous, nonnegativedefinite function Vs : lR x D ~ lR such that (12.37) and (12.38) are satisfied as equalities.
Proof. Let k E Z+ and suppose g is dissipative with respect to the hybrid supply rate (se, Sd)' Then, there exists a continuous
431
HYBRID DYNAMICAL SYSTEMS
nonnegativedefinite function Vs : lR x V  t lR such that (12.34) holds. Now, since for tk < t ::; i ::; tk+1, Z[t,i) = 0, (12.37) is immediate. Next, note that
Vs(tt, s(tt, to, xo, u))  Vs(tk, S(tk' to, xo, u))
: ; r se(ue(s), Ye(s))ds + Sd(Ud(tk), Yd(tk)), t+
ltk
k
which, since Z[tk,tt)
= {k},
(12.41)
implies (12.38).
Conversely, suppose (12.37) and (12.38) hold, let i 2 t 2 0, and let Z[t,i) = {i,i + 1, ... ,j}. (Note that if Z[t,i) = 0 the converse is a direct consequence of (12.34).) In this case, it follows from (12.37) and (12.38) that
Vs(i, x(i))  Vs(t, x(t)) = Vs(i, x(i)) Vs(tj,x(tj))
+ Vs(tj,x(tj)) 
Vs(tj_l,X(tj_l))
+Vs(tj_l,X(tj_l))  ...  Vs(tt,x(tt)) Vs(t,x(t))
: ; Ii
Se(Ue(S), Ye(s))ds +
L
+ Vs(tt,x(tt))
Sd(Ud(tk), Yd(tk)),
kEZ[t,i)
which implies that 9 is dissipative with respect to the hybrid supply rate (se, Sd). Finally, similar constructions show that 9 is exponentially dissipative (respectively, lossless) with respect to the hybrid supply rate (se, Sd) if and only if (12.39) and (12.40) are satisfied (respectively, 0 (12.37) and (12.38) are satisfied as equalities). If in Theorem 12.4 Vs(' s(·, to, xo, u)) exists almost everywhere on [to, 00) except the discrete set Te, then an equivalent statement for
dissipativeness of the leftcontinuous dynamical system 9 with respect to the hybrid supply rate (se, Sd) is
Vs(t, s(t, to, xo, u))::; se(ue(t), Ye(t)), tk < t ::; tk+l, (12.42) ~Vs(tk' s(tk, to, xo, u))::; Sd(Ud(tk), Yd(tk)), k E Z+, (12.43) where ~Vs(tk,S(tk,to,xo, u)) ~ Vs(tt,s(tt,to,xo,u))  Vs(tk,S(tk, to, Xo, u)), k E Z+, denotes the difference of the storage function Vs(t,x) at the times tk, k E Z+, of the leftcontinuous dynamical
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system g. Furthermore, an equivalent statement for exponential dissipativeness of the leftcontinuous dynamical system 9 with respect to the hybrid supply rate (sc, Sd) is given by Vs(t, s(t, to, xo, u))
+ EVs(t, s(t, to, xo, u)):::; sc(uc(t), yc(t)), tk
0, and (12.43). The following theorem provides sufficient conditions for guaranteeing that all storage functions (respectively, exponential storage functions) of a given dissipative (respectively, exponentially dissipative) leftcontinuous dynamical system are positive definite. For this result we need the following definition.
Definition 12.8 A leftcontinuous dynamical system 9 with input u = (u c , Ud) and output y = (Yc, Yd) is zerostate observable if (u c (t), Ud(tk)) == (0,0) and (Yc(t), Yd(tk)) == (0,0) implies s(t, to, xo, u) == o. Furthermore, a leftcontinuous dynamical system 9 is minimal if it is zerostate observable and completely reachable. Theorem 12.5 Consider the leftcontinuous dynamical system 9 and assume that 9 is completely reachable and zerostate observable. Furthermore, assume that 9 is dissipative (respectively, exponentially dissipative) with respect to the hybrid supply rate (sc, Sd) and there exist functions /'l,c : Yc  t Uc and /'l,d : Yd  t Ud such that /'l,c(O) = 0, /'l,d(O) = 0, sc(/'l,c(Yc), Yc) < 0, Yc i= 0, and Sd(/'l,d(Yd), Yd) < 0, Yd i= 0. Then all the storage functions (respectively, exponential storage functions) Vs(t, x), (t, x) E lR x V, for 9 are positive definite, that is, Vs(,O) = 0 and Vs(t,x) > 0, (t,x) E lR x V, x i= O. Proof. The proof is identical to that of the impulsive dynamical system case given in Theorem 3.3 and, hence, is omitted. 0 Next, we introduce the concept of required supply of a leftcontinuous dynamical system g. Specifically, define the required supply Vr(to, xo) of the leftcontinuous dynamical system 9 by Vr(to, xo)
~ (uc(·),Ud(·)), inf [ (to sc(uc(t), yc(t))dt T~to JT +
L kEZ[T,tO)
Sd(Ud(tk), Yd(tk))] ,
(12.45)
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HYBRID DYNAMICAL SYSTEMS
where T S to and u E U are such that s(to, T, 0, u) = Xo. Note that Vr(to,O) = 0, to E R It follows from (12.45) that the required supply of a leftcontinuous dynamical system is the minimum amount of generalized energy which can be delivered to the leftcontinuous dynamical system in order to transfer it from a zero initial state to a given state Xo. Similarly, define the required exponential supply of the leftcontinuous dynamical system g by Vr(to, xo)
~ (uc(·),Ud()), inf [ (to ectse(ue(t) , Ye(t))dt T90 iT +
L
ectksd(Ud(tk), Yd(tk))] '
(12.46)
kEZ[T,tO)
where c > 0, and T S to and u E U are such that s(to, T, 0, u) = Xo. Next, using the notion of required supply, we show that all storage functions for a leftcontinuous dynamical system are bounded from above by the required supply and bounded from below by the available storage.
Theorem 12.6 Consider the leftcontinuous dynamical system g and assume that g is completely reachable. Then g is dissipative (respectively, exponentially dissipative) with respect to the hybrid supply rate (Se,Sd) if and only ifO S Vr(t,x) < 00, t E~, x E 'O. Moreover, if Vr(t, x) is finite and nonnegative for all (to,xo) E ~x'O, then Vr(t, x), (t, x) E ~ X V, is a storage function (respectively, exponential storage function) for g. Finally, all storage functions (respectively, exponential storage functions) Vs (t, x), (t, x) E ~ xV, for g satisfy
Os Va(t, x) S Vs(t, x)
S Vr(t, x) < 00,
(t, x) E ~ x 'O.
(12.47)
Proof. The proof is identical to that of the impulsive dynamical system case given in Theorem 3.4 and, hence, is omitted. 0 Theorem 12.7 Consider the leftcontinuous dynamical system g and assume g is completely reachable to and from the origin. Then g is loss less with respect to the hybrid supply rate (se, Sd) if and only if there exists a continuous storage function Vs(t,x), (t,x) E ~ x V, satisfying (12.34) as an equality. Furthermore, if g is lossless with respect to the hybrid supply rate (se, Sd), then Va(t, x) = Vr(t, x), and hence, the storage function Vs(t,x), (t,x) E ~ x V, is unique and is
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given by Vs(to, xo)
=
IT+ Se(ue(t), Ye(t))dt  L to
=
[t;_
Sd(Ud(tk), Yd(tk))
kEZ[tO,T+)
Se(Ue(t), Ye(t))dt
+
L
Sd(Ud(tk), Yd(tk))
kEZ[T_,tO)
(12.48) with x(to) = xo, Xo E V, and (ueO, Ud(')) E Ue x Ud, for any T+ and T_ > to such that s( T_, to, xo, u) = 0 and s(T+, to, xo, u)
> to = O.
Proof. The proof is identical to that of the impulsive dynamical system case given in Theorem 3.5 and, hence, is omitted. 0 Finally, we provide two definitions of leftcontinuous dynamical systems which are dissipative (respectively, exponentially dissipative) with respect to hybrid supply rates of a specific form. Definition 12.9 A leftcontinuous dynamical system 9 with input u = (u e , Ud), output Y = (Ye, Yd), me = le, and md = ld is passive (respectively, exponentially passive) if 9 is dissipative (respectively, exponentially dissipative) with respect to the hybrid supply rate (se(ue,Ye), Sd(Ud,Yd)) = (2u~Ye,2uJYd). Definition 12.10 A leftcontinuous dynamical system 9 with input U = (u e , Ud) and output Y = (Ye, Yd) is nonexpansive (respectively, exponentially nonexpansive) if 9 is dissipative (respectively, exponentially dissipative) with respect to the hybrid supply rate (se(ue,Ye), Sd(Ud,Yd)) = (·;;u~ue  yJ'Ye,'I'JUJUd  yJYd), where'l'e and 'I'd > 0 are given. In light of the above definitions, the following result is immediate. Proposition 12.3 Consider the leftcontinuous dynamical system 9 with input U = (Ue,Ud), output Y = (Ye,Yd), storage function Vs(', .), and hybrid supply rate (se, Sd)' Then the following statements hold: i) If 9 is dissipative, se(O, Ye) ~ 0, Ye E Yc, Sd(O, Yd) ~ 0, Yd E Yd , and Vs(',·) satisfies (12.15), then the equilibrium point x = 0 of the undisturbed system 9 is Lyapunov stable.
HYBRID DYNAMICAL SYSTEMS
435
ii) If Q is exponentially dissipative, sc(O, Yc) ~ 0, Yc E Yc, Sd(O, Yd) ~ 0, Yd E Yd , and Vs(',·) satisfies (12.15), then the equilibrium point x = 0 of the undisturbed system Q is asymptotically stable. If, in addition, Vs (', .) satisfies (12.18), then the equilibrium point x = 0 of the undisturbed system Q is exponentially stable. iii) If Q is passive (respectively, nonexpansive), then the equilibrium point x = 0 of the undisturbed system Q is Lyapunov stable.
iv) If Q is exponentially passive (respectively, exponentially nonexpansive) and Vs(',·) satisfies (12.15), then the equilibrium point x = 0 of the undisturbed system Q is asymptotically stable. If, in addition, Vs(',·) satisfies (12.18), then the equilibrium point x = 0 of the undisturbed system Q is exponentially stable. v) If Q is a strong leftcontinuous dynamical system, zerostate observable, and non expansive, then the equilibrium point x = 0 of the undisturbed system Q is asymptotically stable. Proof. The result is a direct consequence of Theorems 12.4, 12.1, and 12.2 using standard arguments. 0
12.6 Interconnections of Dissipative LeftContinuous
Dynamical Systems In this section, we consider interconnections of dissipative leftcontinuous dynamical systems. Specifically, consider a finite collection of leftcontinuous dynamical systems Qa = (Va, Ua , Ua , Ya, Ya , Sa, hcm hda), where a spans over a finite index set A,4 and consider the spaces l1, [;, and Y. Here, the elements of Ua and Ya are internal inputs and outputs, respectively, while the elements of l1 and Yare external inputs and outputs, respectively. Next, we introduce an interconnection function I : [; X IIaEAYa  t Y X IIaEAUa , where IIaEA denotes the Cartesian set product. Figure 12.1 illustrates the concept of a finite collection of leftcontinuous dynamical subsystems Qa interconnected through the interconnection constraint I yielding an interconnected system g = IIaEAQa/I. The following definition provides wellposedness conditions for the interconnected system g to qualify as a leftcontinuous dynamical system.
y,
4More generally, count ably infinite sets with an appropriate measure on A can also be considered.
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Yl
=
(Yel, Ydl)
Ul
=
(Ucl' Udl)
9
~h
I Yn
=
(Ycn, Ydn)
Un
=
(Ucn , Udn)
gn
Figure 12.1 System interconnection
9 = II"EAg" \'I.
Definition 12.11 The leftcontinuous dynamical system 9 = (IIaEA 'Va, U, U, j), Y, IIaEASa, hc, hd) is an interconnection of the leftcontinuous dynamical systems ga = (Va, Ua , Ua , Ya, Y a , Sa, hca, h da ), a E A, through the interconnection constraint I if for every Xa E 'Va, U E and t E Lto,,?O), there exist unique ma'l!..s 'l/Ja ~ [0, 00) x IIaEA'Va x U  t Ua , (hc,hd): [0,00) X IIaEA'Va x U  t Y, and Sa: [0,00) x [0,00) X IIaEA'Va xU  t 'Va, such that ua(t) = 'l/Ja(t, sa(t, to, Xa , u), u) and IIaEA Sa satisfies Axioms i )iv ).
q,
A straightforward but key property of a leftcontinuous interconnected dynamical system is that if the component subsystems are dissipative and the interconnection constraint does not introduce any new supply or dissipation, then the interconnected system is dissipative. Hence, the following result is immediate. For this result let Bc : Uc x t tJR and Bd : Ud x Yd  t JR be given.
Proposition 12.4 Let ga, a E A, be a finite collection of leftcontinuous dissipative dynamical systems with hybrid supply rates Sa = (sca(ucw Yca), Sda(Udw Yda)), where Sca : Uca x Yea  t JR and Sda : Uda X Yda  t JR, and storage functions Vsa(', .). Furthermore, let the interconnection constraint I: UxIIaEAYa  t YXIIaEAUa be such that Bc = I:aEA Sca and Bd = I:aEA Sda' Then the interconnected system 9 = IIaEAga/I is dissipative with respect to the hybrid supply rate
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HYBRID DYNAMICAL SYSTEMS
(Se, Sd) = (2:aEA sea> 2:aEA Sda) and has a storage function Vs(',·) 2:aEA Vsa(', .).
=
Proof. The result is a direct consequence of Theorem 12.4 by summing both sides of the inequalities
Vsa(i, xa(i))  Vsa(t, Xa(t)):S
1£
Sea(Uea(S), Yea(s))ds,
tk < t :S i:s tk+1, Vsa(tt, xa(tt))  Vsa(tk, Xa(tk)) :S Sda( Ud(tk), Yd(tk)), for k E Z+, over a E A, and using the assumptions and Sd = 2:aEA Sda'
Se
=
(12.49) (12.50) 2:aEA sea 0
The following corollary is a direct consequence of Proposition 12.4.
Corollary 12.2 Consider the leftcontinuous dynamical systems Ql and Q2 with inputoutput pairs (Uel, Udl; YCl' Ydl) and (U e2, Ud2, ; Ye2, Yd2), respectively. Then the following statements hold: i) If Ql and Q2 are passive, then the parallel interconnection of Ql and Q2 is passive. ii) If Ql and Q2 are passive, then the negative feedback interconnection of Ql and Q2 is passive. iii) IfQl and Q2 are nonexpansive with gains (reb 'Ydl) and (re2, 'Yd2) , respectively, then the cascade interconnection of Ql and Q2 is nonexpansive with gain (rel'Ye2, 'Ydl'Yd2).
Proof. The result is a direct consequence of Proposition 12.4 by noting the interconnection constraints for cascade, parallel, and feedback interconnections are given by (Ue,Ud) = (Uel,Udl), (U e2,Ud2) = (Yel,Ydl), (Ye,Yd) = (Ye2,Yd2); (Ue,Ud) = (Uel,Udl) = (Ue2,Ud2), (Ye,Yd) = (Yel +Ye2,Ydl +Yd2); and (Ue,Ud) = (Uel +Ye2,Udl +Yd2), (Ye,Yd) = (Yel,Ydl) = (U e2,Ud2), respectively. Now, the result is immediate by noting that the above interconnection constraints satisfy the required constraints on Be and Sd in Proposition 12.4. 0 Next, we consider stability of feedback interconnections of dissipative leftcontinuous dynamical systems. Specifically, using the notion of dissipative and exponentially dissipative leftcontinuous dynamical systems, with appropriate storage functions and hybrid supply rates,
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g
+ Figure 12.2 Feedback interconnection of 9 and 9c.
we construct Lyapunov functions for interconnected leftcontinuous dynamical systems by appropriately combining storage functions for each subsystem. Here, for simplicity of exposition, we restrict our attention to stationary leftcontinuous dynamical systems. Furthermore, we assume that for the dynamical system g, V = jRn, Ue = jRme, Ud = jRmd, Yc = jRlc, and Yd = jRld. We begin by considering the negative feedback interconnection of the stationary leftcontinuous dynamical system g with the stationary leftcontinuous feedback system ge given by the octuple (jRnc, U, jRmce X A I I AD. AD. jRmdc , y, lR ce x lR dc, Se, hee, hde), where U = Uee X Ude, Y = Yee X Yde, Se : [0, (0) x [0, (0) x jRnc X U * jRne, hee : jRnc X jRmcc * jRlcc, and hde : jRne X jRmdc * jRlde. We refer to se(t, 0, XeO, '11), t 2 0, as the trajectory of ge corresponding to an initial condition XeO E jRnc and input '11 = (u ee , Ude) E U, where Uee E Uee and Ude E Ude' Furthermore, for ge, let 7;;e denote the set of resetting times and let Tc denote the complement of 7;;e, that is, [to, oo)\7;;e. Note that with the feedback interconnection given by Figure 12.2, (u ee , Ude) = (Ye, Yd) and (Yee, Yde) = (u e, Ud). For the ensuing results, we assume that the negative feedback interconnection of g and ge is well posed, that is, the feedback interconnection generates an undisturbed stationary leftcontinuous dynamical system on jRn X jRnc with trajectory s(t, 0, (xo,xeo),O) ~ (s(t,O,XO,u),se(t,O,xeO,Y) and initial condition (xo, XeD) E jRn x jRne. Theorem 12.8 Consider the feedback system consisting of the sta
tionary leftcontinuous dynamical systems g and ge with inputoutput pairs (Ue,Ud;Ye,Yd) and (Uee,Ude;Yee,Yde), respectively, and with (u ee , Ude) = (Ye, Yd) and (Yee, Yde) = (u e, Ud). Assume g and ge are zerostate observable and dissipative with respect to the hybrid supply
439
HYBRID DYNAMICAL SYSTEMS
rates (sc(uc,Yc), Sd(Ud, Yd)) and (scc(u cc , Ycc), Sdc(Udc, Ydc)), and with continuous positive definite, radially unbounded storage functions lis (.) and liscO, respectively, such that lIs(O) = 0 and lIsc(O) = O. Furthermore, assume there exists a scalar 0" > 0 such that Sc (Uc, Yc) + O"scc(ucc , ycc) ::; 0 and Sd(Ud, Yd) + O"Sdc(Udc, Ydc) ::; O. Then the following statements hold:
i) The negative feedback interconnection of (I and (lc is Lyapunov stable. ii) If (I and (lc are exponentially dissipative with respect to the
hybrid supply rates (sc(u c, Yc), Sd(Ud, Yd)) and (Scc (u cc , Ycc), Sdc (Udc, Ydc)), respectively, and lis (.) and lIsc 0 are such that there exist constants a, a c, (3, (3c > 0 such that
allxl1 2 ::; lis (x) ::; (3llxI1 2, X E ]Rn, a c Ilxc 112 ::; lIsc(x c) ::; (3c Ilxc 11 2, Xc E ]Rnc,
(12.51) (12.52)
then the negative feedback interconnection of (I and (lc is globally exponentially stable. Proof. Let TC ~ TC u ~c, T ~ [to,oo)\T C, and tk ETc, k = 1,2, .... i) Consider the Lyapunov function candidate V(x, xc) = lis (x ) + O"lIsc(xc). Now, the corresponding Lyapunov left derivative of V(x, xc) along the state trajectories (x(t), xc(t)) = (s(t, 0, xo, u), sc(t, 0, XcO, V)), t E (tk, tk+l], is given by V(x(t), xc(t)) = "Vs(x(t)) + O""Vsc(xc(t)) ::; sc(Uc(t), Yc(t)) + O"Scc(ucc (t) , Ycc(t)) ::; 0, t E T,
(12.53)
and the Lyapunov difference of V(x, xc) at the resetting times tk ETc, k E Z+, is given by ~ V(X(tk), xc(h))
= ~lIs(X(tk)) + O"~lIsc(Xc(tk)) ::; Sd(Ud(tk), Yd(tk)) ::; O.
+ O"Sdc(Udc(tk), Ydc(tk)) (12.54)
Now, Lyapunov stability of the negative feedback interconnection of (I and (lc follows as a direct consequence of Theorem 12.1.
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ii) If 9 and ge are exponentially dissipative and (12.51) and (12.52) hold, then it follows that
V(x(t), xe(t)) = Vs(x(t)) + O"Vse(xe(t)) :s: ceVs(x(t))  O"Cee Vse(xe(t)) + se(ue(t), Ye(t)) +O"see(uee(t), Yee(t)) :s:  min{ce, cee} V(x(t), xe(t)), tk < t :s: tk+1, (12.55) and ~V(X(tk),Xe(tk))' tk ETc, k E Z+, satisfies (12.54). Now, Theorem 12.1 implies that the negative feedback interconnection of 9 and ge is globally exponentially stable. D The next result presents Lyapunov, asymptotic, and exponential stability of dissipative feedback systems with quadratic hybrid supply rates. Theorem 12.9 Let Qe E §lc, Se E jRlcxmc, Re E §mc, Qd E §ld, Sd E jR1dxmd, Rd E §md, Qee E §lcc, Sec E jRlccxmcc, Ree E §mcc, Qde E §ldc, Sde E jR1dcxmdc, and Rde E §mdc. Consider the closedloop system consisting of the leftcontinuous dynamical systems 9 and ge, and assume 9 and ge are zerostate observable. Furthermore, assume 9 is dissipative with respect to the quadratic hybrid supply rate (se(ue,Ye),Sd(Ud,Yd)) = (y'[QeYe+2y'[Seue+u'[Reue, yJQdYd+ 2yJ SdUd + uJ RdUd) and has a radially unbounded storage function Vs (.), and ge is dissipative with respect to the quadratic hybrid supply rate (sec (uee , Yee) , Sde(Ude, Yde)) = (yJeQeeYee + 2yJeSeeuee + uJeReeuee , yJeQdeYde + 2yJeSdeUde + UJeRdeUde) and has a radially unbounded storage function VseO. Finally, assume there exists a scalar 0" > 0 such that (12.56) (12.57)
Then the following statements hold:
i) The negative feedback interconnection of 9 and ge is Lyapunov stable. ii) If 9 and ge are exponentially dissipative with respect to the hybrid supply rates (se(Ue, Ye), Sd(Ud, Yd)) and (see( Uee ' Yee) , Sde( Ude, Yde)), respectively, and there exist constants 0:, O:e, f3, f3e > 0
441
HYBRID DYNAMICAL SYSTEMS
such that (12.51) and (12.52) hold, then the negative feedback interconnection of 9 and 9c is globally exponentially stable. iii) If Qc < 0, Qd < 0, and 9 and 9c are strong leftcontinuous dynamical systems, then the negative feedback interconnection of 9 and 9c is globally asymptotically stable.
Proof. Statements i) andii) are a direct consequence of Theorem 12.8 by noting
= [ :c: ] T Qc [ :ccc ],
(12.58)
Sd(Ud, Yd) + aSdc(Udc, Ydc) = [ :d: ] T Qd [ :d: ],
(12.59)
sc(uc, Yc) + ascc (U cc , Ycc)
°
and hence, sc(uc, Yc)+ascc(ucc , Ycc) ~ and Sd(Ud, Yd)+asdc(Udc, Ydc) ~ 0. To show iii) consider the Lyapunov function candidate V(x, xc) = V';,(x) + aV';,c(xc). Noting that U cc = Yc and Ycc = U c it follows that the corresponding Lyapunov left derivative of V(x, xc) along the trajectories (x(t), xc(t)) = (s(t, 0, XQ, u), sc(t, 0, XcQ, y)) satisfies
V(x(t), xc(t)) = "Vs(x(t)) + aVsc(xc(t)) ~ sc(uc(t), Yc(t)) + ascc(ucc(t), Ycc(t)) = yJ(t)QcYc(t) + 2yJ(t)Scuc(t) + uJ(t)Rcuc(t) +a[y~(t)QccYcc(t) + 2y~(t)Sccucc(t) + uJc (t)Rccucc (t)] _ [ Yc(t) Ycc(t) ~
0,
tk
]T Q
0, it follows that (se(ue,Ye), Sd(Ud,Yd)) = (2keyJYe,  2kdYdYd) < (0,0), (Ye,Yd) =1= (0,0). Alternatively, if a leftcontinuous dynamical system g is dissipative (respectively, exponentially dissipative) with respect to a hybrid supply rate (se(ue,Ye),Sd(Ud,Yd)) = ("/;uJue  yJYe,I'~UdUd yJYd) , where I'e,/'d > 0, then, with (/'i;e(Ye),/'i;d(Yd)) = (0,0), it follows that (se(ue,Ye), Sd(Ud,Yd)) = (yJYe,yJYd) < (0,0), (Ye,Yd) =1= (0,0). Hence, if g is zerostate observable it follows from Theorem 12.5 that all storage functions of g are positive definite. Corollary 12.3 Consider the closedloop system consisting of the stationary leftcontinuous dynamical systems g and ge, and assume g and ge are zerostate observable. Then the following statements hold:
i) If g and ge are exponentially passive with storage functions Vs (.) and Vse(·), respectively, such that (12.51) and (12.52) hold, then the negative feedback interconnection of g and ge is exponentially stable.
°
ii) If g and ge are exponentially nonexpansive with gains I'e, I'd > and I'ee,/'de > 0, and storage functions Vs() and Vse(·), respectively, such that (12.51) and (12.52) hold and I'el'ee :::; 1 and I'dl'de :::; 1, then the negative feedback interconnection of g and ge is exponentially stable. Proof. The proof is a direct consequence of Theorem 12.9. Specifically, statement i) follows from Theorem 12.9 with Qe = 0, Qd = 0, Qee = 0, Qde = 0, Se = Imc ' Sd = I md , Sec = I mcc ' Sde = I mdc ' Re = 0, Rd = 0, Ree = 0, and Rde = 0. Statement ii) follows from Theorem 12.9 with Qe = Ilc' Qd = lId' Qee = Ilcc' Qde = Ildc ' Se = 0, Sd = 0, Sec = 0, Sde = 0, and Re = I';Imc ' Rd = I'~Imd' Ree = I';elmcc, and Rde = I'~elmdc· 0
Chapter Thirteen Poincare Maps and Stability of Periodic Orbits for Hybrid Dynamical Systems
13.1 Introduction
In Chapter 12 a unified dynamical systems framework for a general class of systems possessing leftcontinuous flows, that is, leftcontinuous dynamical systems, was developed. Stability results of leftcontinuous dynamical systems are also considered in Chapter 12. The extension of the KrasovskiiLaSalle invariant set theorem to hybrid and impulsive dynamical systems presented in Chapter 12 provides a powerful tool in analyzing the stability properties of periodic orbits and limit cycles of dynamical systems with impulse effects. However, the periodic orbit of a leftcontinuous dynamical system is a disconnected set in the ndimensional state space making the construction of a Lyapunovlike function satisfying the invariance principle a difficult task for highorder nonlinear systems. In such cases, it becomes necessary to seek alternative tools to study the stability of periodic orbits of hybrid and impulsive dynamical systems, especially if the trajectory of the system can be relatively easily integrated. In this chapter, we generalize Poincare's theorem to leftcontinuous dynamical systems, and hence, to hybrid and impulsive dynamical systems. Specifically, we develop necessary and sufficient conditions for stability of periodic orbits based on the stability properties of a fixed point of a discretetime dynamical system constructed from a Poincare return map. As opposed to dynamical systems possessing continuous flows requiring the construction of a hyperplane that is transversal to a candidate periodic trajectory necessary for defining the return map, the resetting set, which provides a criterion for determining when the states of the leftcontinuous dynamical system are to be reset, provides a natural candidate for the transversal surface on which the Poincare map of a leftcontinuous dynamical system can be defined. Hence, the Poincare return map is defined by a subset of the resetting set that induces a discretetime mapping from this subset onto the resetting set. This mapping traces the leftcontinuous tra
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jectory of the leftcontinuous dynamical system from a point on the resetting set to its next corresponding intersection with the resetting set. In the case of impulsive dynamical systems possessing sufficiently smooth resetting manifolds, we show the Poincare return map can be used to establish a relationship between the stability properties of an impulsive dynamical system with periodic solutions and the stability properties of an equilibrium point of an (n  l)thorder discretetime system. These results are used to analyze the periodic orbits for a verge and folio clock escapement mechanism [145J which exhibits impulsive dynamics. 13.2 LeftContinuous Dynamical Systems with Periodic Solutions
In this section, we generalize Poincare's theorem to leftcontinuous dynamical systems. We begin by specializing Definition 12.3 to undisturbed systems (i.e., u(t) == 0). For this definition V ~ ]Rn and Txo ~ [0,(0), Xo E V, is a dense subset of the semiinfinite interval [0,(0) such that [0,(0)\Txo is (finitely or infinitely) countable. Definition 13.1 A strong leftcontinuous dynamical system on V is the triple (V, [O,oo),s), where s: [0,(0) X V  t V is such that the
following axioms hold:
i} (Leftcontinuity): s(·, xo) is leftcontinuous in t, that is, limr>tS(7, xo) = s(t, xo) for all Xo E V and t E (0, (0).
ii} (Consistency): s(O, xo)
= Xo for all Xo
iii} (Semigroup property): S(7, s(t, xo)) and t, 7 E [0, (0).
E V.
= S(t+7, xo) for all Xo
EV
iv} (Quasicontinuous dependence): For every Xo E V, there exists Txo ~ [0, (0) such that [0, oo)\Txo is countable and for every c > and t E Tx o' there exists 8(c, Xo, t) > such that if Ilxo yll < 8(c, xo, t), Y E V, then Ils(t, xo)  s(t, y)11 < c.
°
°
As in Chapter 12, we denote the strong leftcontinuous dynamical system (V, [0,(0), s) by g. Furthermore, we refer to s(t, xo), t ~ 0, as the trajectory of g corresponding to Xo E V, and for a given trajectory s(t, xo), t ~ 0, we refer to Xo E V as an initial condition of g. The trajectory s(t, xo), t ~ 0, of g is bounded if there exists "( > Osuch that IIs(t, xo)11 < ,,(, t ~ 0. The next proposition is a specialization of Proposition 12.1 to the case where g is undisturbed (i.e., u(t) == 0).
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POINCARE MAPS FOR HYBRID DYNAMICAL SYSTEMS
Proposition 13.1 Let the triple (V, [0, (0), s), where s : [0, (0) xV V, be such that Axioms i)iii) hold and
t
°
iv}' For every Xo E V, c, 'Tj > 0, and T E Tx o' there exists 8(c, 'Tj, Xo, T) > such that if Ilxo  yll < 8(c, 'Tj, Xo, T), y E V, then for every t E Txo n [0, T] such that It  TI > 'Tj, for all T E [0, T]\Txo' Ils(t, xo)  s(t, y) II < c. Furthermore, if t E Txo is an accumulation point of [0, oo)\Txo' then s(t,·) is continuous at Xo. IfAxioms i)  iv)' hold, then 9 is a strong leftcontinuous dynamical system. Henceforth, we refer to a strong leftcontinuous dynamical system as a leftcontinuous dynamical system wherein Axiom iv)' holds in place of Axiom iv). This minor abuse in terminology considerably simplifies the ensuing presentation. Furthermore, we assume that Txo in Definition 13.1 is given by Txo ~ {t E [0,(0): s(t,xo) = s(t+,xon so that [O,oo)\Txo corresponds to the (countable) set of resetting times, where the trajectory s(·, xo) is discontinuous. Next, define
Zxo ~ {x E ~n : there exists t E [0, oo)\Txo such that x =s([O,oo)\Txo,xo),
°
/':,.

= s(t, xon (13.1) /':,.
and Z = UxoEVZxo' Furthermore, let Ti(XO), i E Z+, where TO(XO) = and Tl(XO) < T2(XO) < "', denote the resetting times, that is, {Tl(XO),T2(XO), ... } = [0, (0)\'1;0' Next, we present a key assumption on the resetting times Ti (.), i E Z+.
Assumption 13.1 For every i E Z+, TiO is continuous and for every Xo E V, there exists c( xo) > 0 such that Ti+1 (xo)  Ti (xo) 2:: c(xo), i E Z+. The next result is a restatement of Proposition 12.2 and shows that 9 is a strong leftcontinuous dynamical system if and only if the trajectory of 9 is jointly continuous between resetting events, that is, for every c > 0 and k E Z+ there exists 8 = 8(c, k) > 0 such that if It  t'l + Ilxo  yll < 8, then Ils(t,xo)  s(t',y)11 < c, where Xo,y E V, t E (Tk(XO), Tk+1(XO)], and t' E (Tk(y), Tk+1(Y)]·
Proposition 13.2 Consider the dynamical system 9 satisfying Axioms i)iii) and Assumption 13.1. Then 9 is a strong leftcontinuous dynamical system if and only if the trajectory s(t, xo), t 2:: 0, of 9 is jointly continuous between resetting events.
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Definition 13.2 A solution s(t, xo) of g is periodic if there exists a finite time T > 0, known as the period, such that s(t+T, xo) = s(t, xo) for all t ~ o. A set 0 C V is a periodic orbit of g if 0 = {x E V : x = s(t, xo), 0 S; t S; T} for some periodic solution s(t, xo) of g. Note that the set Zxo is identical for all Xo E O. Furthermore, if for every Xo E ]Rn there exists c(xo) such that Ti+1(XO)  Ti(XO) ~ c(xo), i E Z+, then it follows that Zxo contains a finite number of (isolated) points. Finally, for every Xo E 0 it follows that THN(XO) = Ti(XO)+T, i = 2, 3, ... , where N denotes the number of points in Zxo. Next, to extend Poincare's theorem to hybrid dynamical systems let Z c Z be such that 0 n Z is a singleton. Note that the existence of such a Z is guaranteed since all the points in 0 n Z are isolated. Now, we define the Poincare return map P : Z t Z by P(x) ~ S(TN+1(X),X),
x E
Z.
(13.2)
Note that if p EOn Z, then S(TN+1(p),p) = p. Furthermore, if Assumption 13.1 holds then TN+1 () is continuous, and hence it follows that P(·) is well defined. Next, define the discretetime system given by z(k
+ 1) = P(z(k)),
k E
Z+,
z(O) E
Z.
(13.3)
It is easy to see that p is a fixed point of (13.3).
For notational convenience define the set 8 xo .7] ~ {t E Txo: It  TI > 'T}, T E [0, (0) \ Tx o} denoting the set of all nonresetting times that are at least a distance 'T} away from the resetting times. Next, we introduce the notions of Lyapunov and asymptotic stability of a periodic orbit for the leftcontinuous dynamical system g.
Definition 13.3 A periodic orbit 0 of g is Lyapunov stable if for all c > 0 there exists 8 = 8(c) > 0 such that if dist(xo, 0) < 8, then dist(s(t, xo), 0) < c, t ~ O. A periodic orbit 0 of g is asymptotically stable if it is Lyapunov stable and there exists 8 > 0 such that if dist(xo, 0) < 8, then dist(s(t, xo), 0) t 0 as t t 00. The following key lemma is needed for the main stability result of this section. Lemma 13.1 Consider the strong leftcontinuous dynamical system E Z generates the periodic orbit 0 ~ {x E V : x = s(t,p),O S; t S; T}, where s(t,p), t ~ 0, is the periodic solution with the period T = TN +1 (p) . Then the following statements hold:
g. Assume the point p
POINCARE MAPS FOR HYBRID DYNAMICAL SYSTEMS
447
i) The periodic orbit 0 is Lyapunov stable if and only if for every c > 0 and for every Po E 0 there exists 6' = 6' (c, po) > 0 such that if Xo E Bc5 ,(po), then dist(s(t, xo), 0) < c, t ~ O. ii) The periodic orbit 0 is asymptotically stable if and only if it is Lyapunov stable and for every Po E 0 there exists 6' = 6' (po) > 0 such that if Xo E Bc5' (po), then dist(s(t, xo), 0)  t 0 as t  t 00. Proof. i) Necessity is immediate. To show sufficiency, assume that for every c > 0 and for every po E 0 there exists 6' = 6' (c, po) > 0 such that if Xo E Bc5'(PO), then dist(s(t,xo),O) < c, t ~ O. Here, we assume that 6' = 6' (c, po) > 0 is the largest value such that the above distance inequality holds. Next, let 6 = 6(c) = infpoEo 6'(c,po) and suppose, ad absurdum, that 6 = O. In this case, there exists a sequence {POk}~l E 0 such that limk+oo 6'(c,POk) = O. Since {POd~l is a bounded sequence, it follows from the BalzanoWeierstrass theorem [146] that there exists a convergent subsequence {qOd~l E {POk}~l such that limk+oo qOk = q and limk+oo 6' (c, qOk) = O. Note, that since 0 is closed and {qOk}k=l E 0, it follows that q E 0, and hence, 6'(c,q) > O. Thus, it follows that there exists ij E {qOk}~l such that, for sufficiently smallp, > 0, Bc5'(c,ij)+/l(ij) c Bc5'(c,q)(q). Now, since, for every Po E 0, 6' = 6' (c, po) > 0 is assumed to be the largest value such that dist( s(t, xo), 0) < c, t ~ 0, for all Xo E Bc5'(PO), it follows that there exists x~ E Bc5'(c,ij)+/l(ij) and t' ~ 0 such that dist(s(t', x~), 0) > c. However, since Bc5'(c,ij)+/l(ij) C Bc5'(c,q)(q), then for x~ E Bc5'(c,ij)+/l(ij) it follows that dist(s(t, x~), 0) < c for all t ~ 0, which is a contradiction. Hence, for every c > 0 there exists 6 = 6(c) > 0 such that for every PO E 0 and Xo E Bc5 (po) it follows that dist(s(t, xo), 0) < c, t ~ O. Next, given Xo ED such that dist(xo,O) = infpoEo Ilxo  poll < 6, it follows that there exists a point p* E 0 such that dist(xo, 0) ~ IIxo  p*II < 6, which implies that Xo E Bc5(P*), and hence, dist(s(t, xo), 0) < c, t ~ 0, establishing Lyapunov stability. ii) The proof is analogous to i) and, hence, is omitted. D The following theorem generalizes Poincare's theorem to strong leftcontinuous dynamical systems by establishing a relationship between the stability properties of the periodic orbit 0 and the stability properties of an equilibrium point of the discretetime system (13.3).
Theorem 13.1 Consider the strong leftcontinuous dynamical system 9 with the Poincare return map defined by {13.2}. Assume that Assumption 13.1 holds and the point P E Z generates the periodic orbit
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o
~ {X E V : X = s(t,p), 0::; t ::; T}, where s(t,p), t ~ 0, is the periodic solution with the period T = 7N+1(P) such that S(TN+1(P),P) = p. Then the following statements hold:
i} p EOn Z is a Lyapunov stable fixed point of (13.3) if and only if the periodic orbit 0 of 9 generated by p is Lyapunov stable. ii} p EOn Z is an asymptotically stable fixed point of (13.3) if and only if the periodic orbit 0 of 9 generated by p is asymptotically stable.
°
Proof. i) To show necessity, let E > and note that the set Zp = {x E V : x = s(Tz(p),p) = Pl, 1 = 1, ... ,N} contains N points, where p ~ Pl. Furthermore, let p+ = limT~O S(T,p) and let t > 0. It follows from joint continuity of solutions of 9 that there exists 8 = 8(p, t) such that if Ilx~ p+1I + Itt'l < 8, then Ils(t, x~)  s(t',p+)11 < t, where t E (O,Tl(X~)] and t' E (0, Tl(P+)]. Next, as shown in the proof of Proposit ion 12.2, it follows that lim>.>o I:1 (>\, p+) = lim>.>o 71 (.A, p+) = T1 (p+). Hence, choosing 5' = 5' (p, t) > such that 5' < and
°
4,
4
71W,P+)  I:1W,P+) + fJ, < where fJ, is a sufficiently small constant, it follows from the joint continuity property that, since Ilx~ p+11 + It  T1(P+)1 < 8, Ils(t, x~)  S(T1(p+),p+)11 < t,
x~ E BOI(P+),
t E [I:1W,P+)  fJ"T1(X~)]. (13.4)
°
Next, let fJ > be such that fJ < T1(P+)  I:1W,P+) + fJ,. Now, it follows from the strong quasicontinuous dependence property iv)' that there exists 5" = 5" (p, t) > such that
°
Ils(t,p+)  s(t,x~)11
there exists 51 = 51(8) = 51 (t) such that Ilx+ p+11 < 8 for all x E BOI (p)nZ, where x+ = limT~o S(T,X). Hence, it follows that for t > 0, there exists 51 = 51 (t) such that dist(s(t, x~), 0)
< t,
x~ E
BOI (p)
n Z,
t E [0, T2(X~)].
(13.7)
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POINCARE MAPS FOR HYBRID DYNAMICAL SYSTEMS
Similarly, for every point in Zp there exists a neighborhood such that an analogous condition to (13.7) holds. Specifically, for c > 0 and PN E Zp there exists 8N = 8N(c) < c such that dist(s(t, x~), 0) < c for all x~ E 88N(PN) n Z, t E [0, T2(X~)l. Analogously, for PN1 E Zp, there exists 8N 1 = 8N1 (8N) = 8N 1(c) such that dist(s(t, x~), 0) < 8N < c for all x~ E 88N_1 (PN1) n Z, t E [0, T2(X~)l. Recursively repeating this procedure and using the semigroup property iii), it follows that for c > 0 there exists 81 = 81 (c) > 0 such that dist(s(t,x~), 0)
< c,
x~ E 881 (p)
n z,
t E [O,TN+1(X~)J.
(13.8)
Next, it follows from Lyapunov stability of the fixed point P E Zp of the discretetime dynamical system (13.3) that, for 81 > 0, there exists 8~ = 8~ (81 ) > 0 such that Ilz(k + 1)  pI! = I!P(z(k))  pI! < 81 for all z(O) E 8 811 (p) n Z. Hence, using (13.8) and the semigroup property iii), it follows that dist(s(t, x~), 0)
< c,
x~ E
8 811 (p)
n Z,
t ~ O.
(13.9)
Using similar arguments as above, for every PO E 0 there exists 8 8(c,po) such that
=
(13.10) where m is the number of resettings required for s(t, xo), t ~ 0, to reach 8 811 (p) n Z. Finally, it follows from (13.10), (13.9), and the semigroup property iii) that dist(s(t, xo), 0) < c,
Xo E 88(PO),
t ~ 0,
(13.11)
which, using Lemma 13.1, proves Lyapunov stability of the periodic orbit O. Next, we show sufficiency. Assume that 0 is a Lyapunov stable periodic orbit. Furthermore, choose c > 0 and let t E (0, c] be such that there does not exist a point of Zp in 8€(p) other than P E Zp. Note that t > 0 exists since Zp is a finite set. Now, using the fact that g is leftcontinuous, it follows that for sufficiently small 8 > 0 there exists J = J(8) such that 8 ::; J < t and dist(x,O)
> 8,
x E 8€(p)\88(p)
n Z.
(13.12)
Here, we let J > 0 be the smallest value such that (13.12) holds. Note that in this case lim 8(8)
8>0
= o.
(13.13)
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Now, it follows from Assumption 13.1 and the joint continuity of solutions of 9 that for t > 0 there exists 6' (t) > 0 such that (13.14) Hence, using (13.13), we can choose 8 = 8(t) > 0 such that J(8) ~ 6'(t). Next, it follows from the Lyapunov stability of 0 that for 8 = 8(t) > 0 there exists 6 = 6(8) = 6(t) > 0 such that if Xo == z(O) E Bo(p) n Z, then dist(s(t, xo), 0) < 8, t 2: O. Now, using (13.12) and (13.14), it follows that
Ilz(k + 1)  pil
= IIP(z(k)) 
pil = Ils(TN+1(z(k)), z(k))  pil 0 such that if dist(x~, 0) < 6, then for every c > 0 there exists T = T(c, x~) > 0 such that dist(s(t, x~), 0) < c for all t > T. Next, using similar arguments as in i), for any c > 0 there exists 8 = 8(c) > 0 such that
dist(s(t, x~), 0) < c,
x~ E B8(p)
n Z, t E
[0, TN+1 (x~)l.
(13.16)
Now, it follows from the asymptotic stability of p that for every BOI(p) n Z there exists K = K(8,x~) = K(c,x~) E Z+ such that
Ils(T(N+1)'k(X~), x~)  pil < 8,
k > K.
x~ E
(13.17)
Choose 1 > K and let T = T(c, x~) = T(N+l)'I(X~). Then, it follows from (13.16) and (13.17) that for a given c > 0 there exists T = T(c, x~) > 0 such that dist(s(t, x~), 0)
< c,
and hence, dist( s( t, xb), 0)
+
x~ E BOI(p) n Z,
t > T,
0 as t
xb E BOI (p) n Z.
+ 00
for all
(13.18)
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POINCARE MAPS FOR HYBRID DYNAMICAL SYSTEMS
Finally, using similar arguments as in i) it can be shown that for every Po E 0 there exists fJ = fJ(po) > 0 such that Ils(Tm(XO),xo)PI! < fJ' for all Xo E B5(PO) , where m is the number of resettings required for s(t, xo), t ~ 0, to reach B5'(P) n Z. This argument along with (13.18), the semigroup property iii), and ii) of Lemma 13.1 implies asymptotic stability of the periodic orbit O. Finally, we show sufficiency. Assume that 0 is an asymptotically stable periodic orbit of g. Hence, P E Z is a Lyapunov stable fixed point of (13.3) and there exists fJ > 0 such that if Xo E Bp(fJ), then for every sequence {tdk::o such that tk t 00 as k t 00,
dist(s(tk, xo), 0)
t
0,
k
t
(13.19)
00.
Next, choose 8 E (O,fJ] such that there are no points of Zp in B8(p) other than P E Z. Once again, 8 > 0 exists since Zp is a finite set. Since P E Z is a Lyapunov stable fixed point of (13.3) it follows that for 8 > 0 there exists 8 = 8(8) > 0 such that if z(O) == Xo E Bj(p) n i, then z(k+ 1) = P(z(k)) = S(T(N+1).(k+1) (xo), xo) E B8(p)nZ, k E Z+. Next, choose a sequence {tdk::o = {T(N+1).k(XO)}k'=o' Xo E Bj(p) n Z, and note that T(N+1).k(XO) t 00 as k t 00. Hence, it follows from (13.19) that dist(z(k + 1),0)
= dist(P(z(k)), 0) = dist(s(T(N+1).(k+1) (xo), xo), 0) t
0,
k
t
00,
Xo
E
Bj(p)
n Z.
(13.20)
Since P E Zp is the only point of 0 in B8(p) n Z, (13.20) implies that dist(z(k + l),p) t 0 as k t 00 for all z(O) == Xo E Bj(P) n Z, which establishes asymptotic stability of the fixed point P E Zp of (13.3). 0
13.3 Specialization to Impulsive Dynamical Systems
In this section, we specialize Poincare's theorem for strong leftcontinuous dynamical systems to statedependent impulsive dynamical systems. Recall that a statedependent impulsive dynamical system 9 has the form
±(t) = fc(x(t)), ~x(t) = fd(X(t)),
x(O) = Xo, x(t) x(t) E Zx,
f/. Zx,
(13.21) (13.22)
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where t ~ 0, x(t) E V ~ ]Rn, V is an open set, fe : V t ]Rn, fd : Zx t ]Rn is continuous, and Zx c V is the resetting set. Here, we assume that the continuoustime dynamics feO are such that the solution to (13.21) is jointly continuous in t and Xo between resetting events. A sufficient condition ensuring this is Lipschitz continuity of feO. As in Section 13.2, for a particular trajectory x(t) we denote the resetting times of (13.21) and (13.22) by 7k(XO), that is, the kth instant of time at which x(t) intersects Zx. Thus, the trajectory of the system (13.21) and (13.22) from the initial condition x(O) = Xo E V is given by 'Ij.;(t,xo) for 0 < t ::; 71(XO), where 'Ij.;(t,xo) denotes the solution to the continuoustime dynamics (13.21). If and when the trajectory reaches a state Xl ~ X(71(XO)) satisfying Xl E Zx, then the state is instantaneously transferred to xt ~ Xl + fd(XI) according to the resetting law (13.22). The solution x(t), 71 (xo) < t ::; 72(XO), is then given by 'Ij.;(t  71(XO),xt), and so on for all Xo E V. Note that the solution x(t) of (13.21) and (13.22) is leftcontinuous, that is, it is continuous everywhere except at the resetting time 7k(XO), and
Xk ~X(7k(XO)) = lim X(7k(XO)  c), e>O+
xt ~X(7k(XO))
+ fd(X(7k(XO)),
(13.23) (13.24)
for k = 1,2, .... Here, we assume that assumptions A1 and A2 of Chapter 2 hold, that is: Al. If x(t) E Zx \Zx, then there exists c > 0 such that, for all o < & < c, X (t + &) ¢ Zx. A2. If X E Zx, then X + fd(X) ¢ Zx. Note that it follows from the definition of 7kO that 7l(X) > 0, X ¢ Zx, and 71(X) = 0, X E Zx. Furthermore, since for every X E Zx, x + fd(X) ¢ Zx, it follows that 72(X) = 7l(X) + 71 (x + fd(X)) > O.
Finally, note that it follows from A1 and A2 that the resetting times
7k(XO) are well defined and distinct. Recall that since not every bounded solution of an impulsive dy
namical system over a forward time interval can be extended to infinity due to Zeno solutions, we assume that feO and fdO are such that existence and uniqueness of solutions for (13.21) and (13.22) are satisfied in forward time. In this section we assume that fe (.) and fdO are such that 7k(XO) t 00 as k t 00 for all Xo E V. In light of the above, note that the solution to (13.21) and (13.22) with initial condition Xo E V denoted by s(t, xo), t ~ 0, is i) leftcontinuous,
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POINCARE MAPS FOR HYBRID DYNAMICAL SYSTEMS
that is, limT+ts(T,xo) = s(t,xo) for all Xo ED and t E (0,00); ii) consistent, that is, s(O, xo) = Xo, for all Xo E '0; and iii) satisfies the semigroup property, that is, S(T,S(t,Xo)) = S(t+T,Xo) for all Xo ED and t,T E [0,00). To see this, note that s(O,xo) = Xo for all Xo ED and
s(t,xo) = {
1jJ(t, xo), 0::; t ::; T1 (xo), 1jJ(t  Tk(XO), S(Tk(XO), xo) + fd(s(Tk(xo) , xo))), Tk(XO) < t ::; Tk+1(XO), 1jJ(t  T(XO), S(T(XO), xo)), t 2:: T(XO), (13.25)
where T(XO) ~ sUPk>O Tk(XO), which implies that s(·, xo) is leftcontinuous. Furthermore, uniqueness of solutions implies that s(t, xo) satisfies the semigroup property S(T,S(t,Xo)) = S(t+T,Xo) for all Xo ED and
t,T E [0,00).
Next, we present two key assumptions on the structure of the resetting set Zx. Specifically, we assume that the resetting set Zx is such that the following assumptions hold: A3. There exists a continuously differentiable function X : '0 + lR. such that the resetting set Zx = {x E '0 : X (x) = O}; moreover, X'(x) =I 0, x E Zx. A4. a~;x) fe(x) =1= 0, x
E
Zx.
It follows from A3 that the resetting set Zx is an embedded submanifold [81], while A4 assures that the solution of 9 is not tangent to the resetting set Zx. The following proposition shows that under Assumptions A3 and A4, the resetting times TkO are continuous at Xo E '0 for all k E Z+.
Proposition 13.3 Consider the nonlinear statedependent impulsive dynamical system 9 given by {13.21} and {13.22}. Assume that A3 and A4 hold. Then TkO is continuous at Xo ED, where 0 < Tk(XO) < 00, for all k E Z+. Proof. First, it follows from Proposition 2.2 that T1 (.) is continuous at Xo E '0, where 0 < T1(XO) < 00. Since feO is such that the solutions to (13.21) are jointly continuous in t and Xo, it follows that 1jJ(.,.) is continuous in both its arguments. Furthermore, note that
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1P(7t(X),X) = S(Tl(X),X),X E V. Next, it follows from the definition of Tk(X) that for every x E V and k E {1, 2, ... }, Tk(X)
= Tkj(X) + Tj[S(Tk_j(X), x) + fd(s(Tkj(x), x))],
where TO(X) ~ O. (13.26) that T2(X) continuous on V. 3,4, ... , it follows k E Z+.
j
= 1, ... , k,
(13.26) Hence, since fdO is continuous, it follows from = Tl(X) + Tl[S(Tl(X),X) + fd(s(Tl(x),x))] is also By recursively repeating this procedure for k = that Tk (x) is a continuous function on V for all 0
Since feO and fdO are such that the Axioms i)iii) hold for the statedependent impulsive dynamical system g, and 9 is jointly continuous between resetting events, then, with Assumptions A3 and A4 satisfied, it follows from Propositions 13.2 and 13.3 that the statedependent impulsive dynamical system 9 is a strong leftcontinuous dynamical system. Hence, the following corollary to Theorem 13.1 is immediate.
Corollary 13.1 Consider the impulsive dynamical system 9 given by (13.21) and (13.22) with the Poincare return map defined by (13.2). Assume that A3 and A4 hold, and the point p E Zx generates the periodic orbit 0 ~ {x E V: x = s(t,p), 0 ~ t ~ T}, where s(t,p), t 2': 0, is the periodic solution with the period T = TN +1 (p) such that S(TN+1(P),p) = p. Then the following statements hold: i} p EOn Zx is a Lyapunov stable fixed point of {13. 3} if and only
if the periodic orbit 0 of 9 generated by p is Lyapunov stable.
onzx is an asymptotically stable fixed point of (13.3) if and only if the periodic orbit 0 of 9 generated by p is asymptotically stable.
ii} P E
Corollary 13.1 gives necessary and sufficient conditions for Lyapunov and asymptotic stability of a periodic orbit of the statedependent impulsive dynamical system 9 based on the stability properties of a fixed point of the ndimensional discretetime dynamical system involving the Poincare map (13.2). Next, as is the case of the classical Poincare theorem, we present a specialization of Corollary 13.1 that allows us to analyze the stability of periodic orbits by replacing the nthorder impulsive dynamical system by an (n  1)thorder discretetime dynamical system.
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POINCARE MAPS FOR HYBRID DYNAMICAL SYSTEMS
To present this result assume, without loss of generality, that aaX(x) Xn =1= O,x E Zx, where x = [Xl, ... ,xnF. Then, it follows from the implicit function theorem [89] that Xn = g(XI, ... , Xnl), where g(.) is a continuously differentiable function at Xr £ [Xl, ... ,Xn_I]T such that [xi,g(xr)F E Zx. Note that in this case P : Zx > Zx in (13.3) is given by P(x) £ [PI (X), ... , Pn(x)F, where (13.27) Hence, we can reduce the ndimensional discretetime system (13.3) to the (n  l)dimensional discretetime system given by
zr(k + 1) where Zr E lRn 
l ,
= Pr(zr(k)),
k E Z+,
[z;(.),g(zr())]T E Zx, and [
PI(Xng(Xr))
PnI(X~,g(Xr))
1
(13.28)
(13.29)
.
Note that it follows from (13.27) and (13.29) that p £ [Pi,g(Pr)F E Zx is a fixed point of (13.3) if and only if Pr is a fixed point of (13.28).
Corollary 13.2 Consider the impulsive dynamical system g given by {13.21} and {13.22} with the Poincare return map defined by {13.2}. ax(x) Assume that A3 and A4 hold, a=1= 0, X E Zx, and the point P E Zx Xn A
generates the periodic orbit 0 ~ {x E V: X = s(t,p), 0 :S t :S T}, where s(t,p), t ~ 0, is the periodic solution with the period T = TN+1(p) such that S(TN+1(p),p) = p. Then the following statements hold: i} For p = [Pi,g(Pr)F EOn ix, Pr is a Lyapunov stable fixed point of {13.28} if and only if the periodic orbit 0 is Lyapunov stable. ii} For P = [Pi,g(Pr)]T EOn ix, Pr is an asymptotically stable fixed point of {13.28} if and only if the periodic orbit 0 is asymptotically stable.
Proof. i) To show necessity, assume that Pr is a Lyapunov stable fixed point of (13.28) and let E > O. Then it follows from the continuity of g(.) that there exists 5' = 5'(E) > 0 such that (13.30)
456
CHAPTER 13
°
Choosing 8' < ~, it follows from the Lyapunov stability of Pr that for 8' > there exists 8 = 8(c) < 8' such that
where z(k) = [zr(k)T,g(zr(k))]T E Zx,k E Z+, satisfies (13.3). If z(O) E B8(P) n ix, that is,
Ilz(O)  pil = II[zr(O?,g(Zr(O))]T  [p;,g(Pr)]TII ::::; II[Zr(O)T,O]T  [P;,O]TII +11[0, ... ,0,g(zr(0))]T  [0, ... ,0,g(Pr)]TII (13.32)
such that if z(O) E B8(p) n ix, then Ilz(k + 1)  pil < c, which establishes Lyapunov stability of P for the discretetime system (13.3). Now, Lyapunov stability of the periodic orbit 0 follows as a direct consequence of Theorem 13.1. Next, to show sufficiency, assume that the periodic orbit 0 is Lyapunov stable. In this case, it follows from Theorem 13.1 that P = [Pi, g(Pr)]T E onix is a Lyapunov stable fixed point of (13.3). Hence, for every c > there exists 8' = 8' (c) > such that
°
Ilz(k + 1)  pil = IIP(z(k))  pil < c,
°
Z+,
B8 (P) nix. (13.34) Now, it follows from the continuity of g(.) that for 6' > there exists 8 = 8(c) > such that
°
k
E
z(O)
E
°
1
(13.35)
457
POINCARE MAPS FOR HYBRID DYNAMICAL SYSTEMS
Choosing 6 < ~ and zr(O) E BtS(Pr), it follows from (13.35) that
Ilz(O)  pil = II[zr(O?,g(zr(O))]T  [p;,g(Pr)]TII
0 such that t
0,
k
t
00,
zr(O)
E
Ilz(k + 1)  pil = IIP(z(k))  pil = II[P;(zr(k)),Pn(zr(k),g(zr(k)))]T  [p;,g(Pr)]TII t 0, k t 00, z(O) E BtSl(p) nix. (13.38) Using similar arguments as in i), there exists 6> 0 such that if zr(O) E BtS(Pr), then z(O) E BtSl(p) nix. Thus, it follows from (13.38) that
Ilzr(k + 1)  Prll = IIPr(zr(k))  Prll
zr(O) E BtS(Pr), (13.39) which establishes asymptotic stability of Pr for (13.28). 0 t
0,
k
t
00,
458
CHAPTER 13
13.4 Limit Cycle Analysis of a Verge and Foliot Clock Escapement
In the remainder of this chapter, we use impulsive differential equations and Poincare maps to model the dynamics of a verge and foliot clock escapement mechanism, determine conditions under which the dynamical system possesses a stable limit cycle, and analyze the period and amplitude of the oscillations of this limit cycle. Although clocks are one of the most important instruments in science and technology, it is not widely appreciated that feedback control has been essential to the development of accurate timekeeping. As described in [120], feedback control played a role in the operation of ancient water clocks in the form of regulated valves. Alternative timekeeping devices, such as sundials, hourglasses, and burning candles, were developed as well, although each of these had disadvantages. Mechanical clocks were developed in the 12th century to keep both time and the calendar, including the prediction of astronomical events [49,97]. Although early mechanical clocks were expensive, large, and not especially accurate (they were often set using sundials), this technology for timekeeping had inherent advantages of accuracy and reliability as mechanical technology improved. The crucial component of a mechanical clock is the escapement, which is a device for producing precisely regulated motion. The earliest escapement is the weightdriven verge and foliot escapement, which dates from the late 13th century. The feedback nature of the verge and foliot escapement is discussed in [102], which points out that this mechanism is a work of "pure genius." 1 The authors of [102] have performed an important service in identifying this device as a contribution of automatic control technology. It is interesting to note that the verge and foliot escapement was 1 It is important to note here that far more complex and ingenious differential geared mechanisms were inspired, developed, and built by the ancient Greeks over twelve centuries before the development of the first mechanical clock. These technological marvels included Heron's automata and, arguably the greatest fundamental mechanical invention of all time, the Antikythera mechanism. The Antikythera mechanism, most likely inspired by Archimedes, was built around 76 B.C. and was a device for calculating the motions of the stars and planets, as well as for keeping time and calendar. This first analog computer involving a complex array of meshing gears was a quintessential hybrid system that unequivocally shows the singular sophistication, capabilities, and imagination of the ancient Greeks. And as in the case of the origins of much of modern science and mathematics, it shows that modern engineering can also be traced back to the great cosmic theorists of ancient Greece.
POINCARE MAPS FOR HYBRID DYNAMICAL SYSTEMS
459
the only mechanical escapement known from the time of its inception until the middle of the 17th century. In 1657 Huygens modified the verge and foliot escapement by replacing the foliot with a pendulum swinging in a vertical plane and the crown gear mounted horizontally. However, the basic paddle/gear teeth interaction remained the same. The next escapement innovation was the invention of the anchor or recoil escapement by Hooke in 1651 in which a pendulumdriven lever arm alternately engages gear teeth in the same plane. Subsequent developments invoking additional refinements include the deadbeat escapement of Graham and the grasshopper escapement of Harrison. The latter device played a crucial role when the British government sought novel technologies for determining longitude at sea [156J. For details on these and other escapements, see [22,48,72, 141J. Since escapements produce oscillations from stored energy, they can be analyzed as selfoscillating dynamical systems. For details, see [5J. The present development considers only the verge and foliot escapement, which consists of a pair of rotating rigid bodies which interact through collisions. These collisions constitute feedback action which give rise to a limit cycle. This limit cycle provides the crown gear with a constant average angular velocity that determines the clock speed for accurate timekeeping. The verge and foliot is analyzed in [102J under elastic and inelastic conditions. For the latter case expressions were obtained for the period of the limit cycle and for the crown gear angular velocity at certain points in time. Because of the presence of collisions, a hybrid continuousdiscrete model was used to account for instantaneous changes in velocity.
13.5 Modeling
The verge and foliot escapement mechanism shown in Figure 13.1 consists of two rigid bodies rotating on bearings. For simplicity we assume that these bearings are frictionless. The crown gear has teeth spaced equally around its perimeter. The verge and foliot, which henceforth will be referred to as the verge, has two paddles that engage the teeth of the crown gear through alternating collisions. We ignore sliding of the paddles along the crown gear teeth, which may occur in practice. For the orientation shown in Figure 13.1, there is an upper paddle and a lower paddle. Collisions involving the upper paddle impart a positive torque impulse to the verge, while those involving the lower paddle impart a
460
CHAPTER 13
Figure 13.1 Verge and foliot escapement mechanism. The angular velocities
of the crown gear and verge are Be and Bv, respectively, with the sign convention shown. There is a constant torque T applied to the crown gear with positive direction shown.
negative torque impulse to the verge. Each collision imparts a negative torque impulse which acts to retard the motion of the crown gear. The mechanism is driven by a constant torque applied to the crown gear. This torque is usually provided by a mass hanging from a rope which is wound around the shaft. The verge spins freely at all times except at the instant a collision takes place. Energy is assumed to leave the system only through the collisions. The amount of energy lost during each collision is a function of the system geometry as well as the coefficient of restitution e realized in the collision. The crown gear and verge have inertias Ie and I y , contact radii re and r y, and angular velocities Be and By, respectively. The velocities immediately before and after a collision are denoted by the subscripts o and 1, respectively, as in Oeo and 0C1' The motion of the crown gear and verge is governed by the differential equations
..
1
()e(t) = IT e
..
()y(t) =

re .. 1 F(()e(t), ()y(t), ()e(t), ()y(t)),
(13.40)
e
{ + ~: F(()e(t), ()y(t), Oe(t), Oy(t)) ,
upper,
i,;F(()e(t), ()y(t), Oe(t), Oy(t)),
lower,
(13.41)
where the first expression in (13.41) applies to collisions between the crown gear and the upper paddle, and the second expression applies to collisions between the crown gear and the lower paddle. The function
461
POINCARE MAPS FOR HYBRID DYNAMICAL SYSTEMS
file +
..
Be
1
Ie s2
Te
F
iCollision Function
Tv fvBv
..
Bv
1
Iv s2
Figure 13.2 System block diagram showing the interconnection of the crown
gear and verge rigid bodies through the collision block.
F((}c(t), (}v(t), Oe(t), Ov(t)) is the collision force, which is zero when the crown gear and verge are not in contact and is impulsive at the instant of impact. The collision force function F acts equally and oppositely on the crown gear and verge. Defining
(J'~
+1,
{
1,
upper, lower,
(13.42)
(13.41) can be written in the form (13.43) A system diagram is shown in Figure 13.2. To determine the collision force function, we integrate (13.40) and (13.43) across a collision event to obtain .
.
(}q  (}eo
=
. ( 1 It+~t re It+~t hm rds  1 F(s)ds
. . hm. (rv It+~t F(s) ds ). (J'I
(}Vl  (}vo
=
~t>O ~t>O
Ie t~t v
t~t
e
t~t
)
,(13.44) (13.45)
Eliminating the integrated collision force from (13.44) and (13.45) yields (13.46)
462
CHAPTER 13
which is an expression of conservation of linear momentum at the instant of a collision. Expression (13.46) can be rewritten as (13.47) where Me ~
:§. and Tc
Mv ~ ~ are the effective crown gear mass and Tv
6..
6..
effective verge mass, respectively, and Vc = Te(}e and Vv = CITv(}v are the tangential velocities of the crown gear and the verge, respectively. The coefficient of restitution e relates the linear velocities of the crown gear and the verge before and after the collision according to (13.48) which accounts for the loss of kinetic energy in a collision. Solving (13.46) and (13.48) yields
+ e) Mv(1 + e) (M M ) Vco + CI (M M ) Vvo , Te v + e Te v + e
. b,.(}e = .
Mv(1
b,.(}v=CI
+ e) (M M) Vco Tv v+ e Me(1
+ e) (M M) VvO, Tv v+ e Me(1
(13.49)
where (13.50) are the impulsive changes in angular velocity when a collision occurs. These quantities depend on the geometry as well as the velocities immediately before the collision. The integral of the impulsive force function over a collision event is
l
tl
to
F( ) d = MeMv (1 + e) (V, _ sSM M cO v
+
e
1"T
)
VvO,
(13.51)
where to is a time slightly before the collision and tl is a time slightly after the collision. 13.6 Impulsive Differential Equation Model
In this section, we rewrite the equations of motion of the escapement mechanism in the form of an impulsive differential equation. To describe the dynamics of the verge and foliot escapement mechanism as an impulsive differential equation, define the state (13.52)
463
POINCARE MAPS FOR HYBRID DYNAMICAL SYSTEMS
where Xl is the position of the crown gear, that is, the counterclockwise angle swept by the line connecting the center of the crown gear and the zeroth tooth from the 12 o'clock position; X2 is the position of the verge, that is, the deviation of the mean line of the angular offset between two paddles from the vertical plane perpendicular to the plane of the crown gear; X3 is the angular velocity of the crown gear; and X4 is the angular velocity of the verge. Between collisions the state satisfies
x(t) =
[H ~ !1 J 1 x(t)
°°°°
+[
while the resetting function is given by
T,
(13.53)
°
00
fd(X) =
[
where
°° °°
+ e) r (Iv + ls..) e ~ r~ ~(1
Ge
~ _v.:,_ _.,
(13.54)
(13.55)
,
The resetting set is (13.56) where, for m Z~r:J:er
= {x
= 0, ... ,n,
: re
sin(xI  mae)
reX3  rvx4
= rv tan(x2 + a v /2),
> 0,
(m 1/2)ae + 2p7f ~ p E {0,1,2, ... }}, z~o~er = {X: re sin (mae 
Xl)
+ rvX4 > 0, (m  1/2)ae + (2p 
Xl
~ (m
+ 1/2)ae + 2p7f, (13.57)
= rv tan( X2 + a v /2),
reX3
p E {a, 1,2, ... }},
1)7f ~
Xl
~ (m
+ 1/2)ae + (2p 
1)7f, (13.58)
464
CHAPTER 13
where eYe is the angle between neighboring teeth on the crown gear, eY v is the angular offset of the paddles about the vertical axis, m is the index of the crown gear tooth involved in the collision, and p is the number of full rotations of the crown gear. The crown gear teeth are numbered from 0 to n clockwise, or opposite the direction of increasing Oe, beginning at Oe = O. There must be an odd number of crown gear teeth for the mechanism to function correctly, and thus n is even. 13.7 Characterization of Periodic Orbits
In this section we characterize periodic orbits of the clock escapement mechanism, which henceforth we denote by g. First we integrate the continuoustime dynamics (13.53) to obtain .
T
Oe2 = Oeo
+ Oq tlt + 2Ie tlt
OY2 = OyO
+ By! tlt,
2
(13.59)
,
(13.60)
where Oe2 and OY2 are evaluated immediately before the next collision and tlt is the elapsed time between two successive collisions. For an initial collision involving the upper paddle we have (13.61)
The index m' of the crown gear tooth involved in the subsequent lower collision is given by m'
= m + 1f/eYe + 1/2,
(13.62)
so that the condition (13.63)
must be satisfied at the lower collision. (13.60) into (13.63) yields resin (Oeo + Bqtlt+
Substituting (13.59) and
2~e tlt 2  (m+~) eYe)
= ry tan ( Oyo  By! tlt +
~Y)
(13.64)
A smallangle approximation of (13.61) and (13.64) implies (13.65)
465
POINCARE MAPS FOR HYBRID DYNAMICAL SYSTEMS
(13.66) Subtracting (13.65) from (13.66) yields (13.67) Analogous expressions hold for collisions involving the lower paddle. Solving for b..t yields
b..t
=
d +
vid2 + (re
Q
e  40Tv Ovo)re T / Ie
reT/Ie
'
(13.68)
where d = [(2 + e) Me  eMvl rJ}eo + u [(2 + e)Mv  eMcJ rvBvo . (13.69)
Mv+Me
Mv+Me
Furthermore, the kinetic energy b..T lost in a collision is given by
b..T
MvMe
= 2 (Mv +
2
.
.
2
Me) (e  1) (rv Ovo  ure Oeo) .
(13.70)
Next, we specify conditions that characterize a periodic orbit in the
(Be, Bv) plane. The first condition
(13.71)
requires the verge to reverse direction at every collision. This condition ensures that the absolute value of the verge speed is constant with time. On the other hand, the crown gear will lose speed with every collision and then gain speed between collisions. Thus, the second condition .
.
OC1 = Oeo 
T
b..t
T
(13.72)
requires the crown gear speed to be the same before each collision. The third condition (13.73) requires the range of motion of the verge between collisions to be centered at Ovo = O. This condition keeps the motion of the verge from wandering out of the range of angles within which the mechanism
466
CHAPTER 13
ey
1
O~,~~/. /
/
0.6
.
/
I1.l
~ 0.2
::>
0.4 0.6
fly
/
3
0 0 is small, it follows that in order to avoid the singularity ±~ we need to make sure that a v + E =I= ~ which can be achieved by assuming a v 1, then p(j(_~v ,b, c)) > 1, which implies that the fixed point (y,b, c) of (13.82) is unstable. 0 Next, it follows from the uniqueness of solutions of 9 and the fact that the initial conditions (X~,X2,X3,X4) and (XI,X2,X3,X4), where Xl = x~ + 27r, give rise to identical solutions for g, that the point Xo = (0,  ~v , b, c) is a fixed point for the discretetime system capturing the state of 9 immediately before every (np+ 1)th upper paddle collision for p = 0,1,2, .... Note that whenever an upper paddle collision occurs, the position of the crown gear is completely defined by the position of the verge, and the relation between them results from the collision condition, that is, Xl = !I (X2), where !I : ~ + ~ is defined by (13.57). Thus, the aforementioned fourdimensional system has the form
(13.86)
where
!I ()
is given by
(13.87)
Next, we identify the periodic orbit generated by the point Xo = (O,y,b,c). For any point on this orbit with (X3,X4) = (z,c),z E (a, b], it follows that z = a + tt z , where t z is the time spanned for the crown gear to restore its velocity from the value of a to z. Thus, this point can be characterized by X Xz
=
[
10
+ a(za) 1 + (za)2 1 + c(za) 1 7
_
C
27
C
1
Uv
2
7
Z C
C
,
(13.88)
471
POINCARE MAPS FOR HYBRID DYNAMICAL SYSTEMS
where XlO = lac, I = 0,1,2, ... ,n  1. Similarly, every point on the orbit with (X3, X4) = (z, c), Z E (a, bj, is characterized by X'10 I
Xz
=
[
+ a(za) 1 + (za)2 1 7 O:v _
2
C
27
c(za) I 7
Z
C
C
1 ,
(13.89)
c where x~o
=
2ltl a c , I
= 0,1,2, ... ,n 
1. Since the initial conditions (X~,X2,X3,X4) and (Xl,X2,X3,X4), where Xl = x~ + 27f, give rise to identical solutions for 9, it follows that 0 ~ {y E ~4 : y = x z } U {y E . ~4 : y = x~} is the periodic orbit of 9. The expressions given by (13.88) and (13.89) imply that points Xo = (XlO,  ~v ,b, c) E Zx or Xo = (x~o, ~v, b, c) E Zx generate O. Next, we show that 0 is asymptotically stable. For this result let 1) be a sufficiently small neighborhood of 0 for which the state of 9 is defined.
Theorem 13.2 Consider the impulsive dynamical system 9. Then the following statements hold: i) If p(J( T' b, c)) < 1, then the periodic orbit 0 of9 generated by Xo = (XlO,  ~v ,b, c) E Zx or Xo = (x~o, T' b, c) E Zx is asymptotically stable. ii) If p( J ( ~v , b, c)) > 1, then the periodic orbit 0 of 9 generated by Xo = (XlO' _~v, b, c) E Zx or Xo = (x~o, ~v, b, c) E Zx is unstable. Proof. First, we show that the assumptions of Corollary 13.2 hold for 9. To see that Assumption A3 holds, note that for Z~~er given by (13.57) with a smallangle approximation, X'(x) = [rc, rv , 0, OJ =1= d ax(x) I 0 ,X E Zupper O, X E Z xupper m ,an ~ r x m · Fu r thermore, £or zlower xm given by (13.58) with a smallangle approximation, X'(x) = [rc, rv , 0, OJ I 0 I 0 r , x E zlower x m , and ax(x) aX! r , x E zlower x m . Note , that in both cases XO is a continuously differentiable function by Assumption 13.2. To see that Assumption A4 holds, note that a~~x) fc(x) = rcx3  rvx4 > upper ax(x) Jc f ( ) t O, x E Z xm ,and ax X  rcX3 + rv X4 > 0 ,X E zlower xm. Nex, to show i) assume that p(J(~v,b,c)) < 1. Then it follows from Proposition 13.4 that the fixed point ( ~v , b, c) of (13.82) is asymptotically stable, and hence, by Corollary 13.2, the periodic orbit 0 of 9 is asymptotically stable. Finally, the proof to ii) follows analogously.
o
472
CHAPTER 13
The condition p( J(  o;,v ,b, c)) < 1 guarantees (local) asymptotic stability of the escapement mechanism. Alternatively, it follows from physical considerations that for each choice of clock parameters, if the value of the coefficient of restitution e is sufficiently close to 1, then the escapement mechanism dissipates less energy during a collision event than it gains from the rotational torque between collisions. This leads to instability of the mechanism. However, the Jacobian matrix J is sufficiently complex that we have been unable to show analytically the explicit dependence of the spectral radius of J on the parameter e.
13.9 Numerical Simulation of an Escapement Mechanism
In this section we numerically integrate the equations of motion (13.53) (13.58) to illustrate convergence of the trajectories to a stable limit cycle. We choose the parameters T = 1 N·m, e = 0.05, Ic = 10 kg·m 2 , Iv = 0.15 kg· m 2 , rc = 1 m, rv = 0.3 m, and O:c = 24 deg. For these parameters it follows from (13.75)(13.79) that the periodic orbit has an average crown gear velocity of 0.257 rad/sec, a crown gear velocity of 0.297 rad/sec prior to collisions, a crown gear velocity of 0.216 rad/sec after collisions, a verge speed of 0.813 rad/sec, and a period of 1.63 sec. Furthermore, the eigenvalues of the Jacobian matrix (13.81) are Al = 0.7191, A2 = 0.2072, and A3 = 0.0149, which implies that the fixed point ( o;,v ,b, c) of (13.82) is asymptotically stable, and hence, by Theorem 13.2 the periodic orbit of the escapement mechanism is asymptotically stable. An initial verge position of eva = 0 is chosen for all simulations. We assume that the verge and the crown gear are in contact at the start of the simulation, which determines the crown gear's initial position. For a collection of four initial conditions, Figure 13.5 shows the trajectories of the system in terms of the verge and crown gear velocities Ov and Oc. For each choice of initial conditions it can be seen that the trajectory approaches a periodic orbit, which is discontinuous due to the impulsive nature of the collisions. Numerical computation of the amplitude and period of this orbit from the simulation data yields 0.257 rad/sec and 1.63 sec, respectively, which agrees with the values given by (13.75) and (13.79). The kinetic energy time histories of the verge, crown gear, and total system are shown in Figure 13.6 for the system considered in Figure 13.5. It can be seen that the verge kinetic energy converges, whereas the crown gear and total system kinetic energies converge to periodic
473
POINCARE MAPS FOR HYBRID DYNAMICAL SYSTEMS 3 2
oqf~ ..... U
0
1 0
~
~1
1
'2 = 0.1775, and >'3 = 0.0046. Since the fixed point ( ~v ,b, c) of the discretetime system (13.82) is unstable, it follows from Theorem 13.2 that the periodic orbit of the escapement mechanism is also unstable. Figure 13.8 shows the nonconverging velocity phase portrait of the system. Finally, Figure 13.9 shows p(J(  ~v ,b, c)) versus the coefficient of restitution e for several values of the torque T.
474
CHAPTER 13
~ 0.06~_~~~
~O.05
&l 0.04 ~
~ 0.03 ~
:2 0.02 ~
~O.01
~
OL_~
o
s
_ _ L_ _
~
_ _~_~_ _~_ __ L_ _~_~
8
10
Time, sec
12
14
16
18
Z
~O.6 C 0.5
f;iI
~ 0.4 ~
~ 03
..
~ 0.2
"~ O~~~ C 0.1
o
_ _ L_ _~_ _~_ _L_~_ _ L_ _~_~
0
8
10
Time, sec
12
14
16
18
~ 06,,,,___,,,,,,
~O.5
&l 04 ~
u
~
~
0.3
:2 0.2
'"
~O.1 O~~~
o
_ __ L _ __ L_ _
~_
8
_L_~
10
Time, sec
_ _ L_ _~_~
12
14
16
18
Figure 13.6 Verge, crown gear, and total kinetic energy time histories starting
from rest.
475
POINCARE MAPS FOR HYBRID DYNAMICAL SYSTEMS
e = 0.1
0.3 !.)
' 0.25
€
o 03
:>
0.2
til
'CD
60
20
40L__ ;'(x2(k)) = ¥C(rea e  4rvx2(k)). Next, it follows from (A.3) that x~'(k) and x1'(k) are the velocities of the crown gear and the verge, respectively, immediately after the lower paddle collision, and hence, the positions of the crown gear and the verge before the successive upper paddle collision are given by
x 1 (k + 1) = x"'(k) + x"'(k)6.t 1 3 2 + ~6.t2 21 2, x2(k + 1) = x~'(k)
+ x1' (k)6.t2'
e
where x1'(k) and x~'(k) are positions of the crown gear and the verge, respectively, immediately after the lower paddle collision. Using a similar procedure as outlined above, the conditions for the lower and upper paddle collisions, respectively, are given by
resin (x1'(k)
+x~'(k)6.t2 + 2~e 6.t~ 
(m+ l)ae)
= rv tan (x~'(k) +x1'(k)6.t2 + ~v),
(A.19)
which can be approximated by
re (x1'(k)  mac  ~e) = rv ( x~'(k)
re (xr'(k)
+ x~'(k)6.t2 + 2~e 6.t~ 
+ ~v) ,
(A.20)
(m + l)ae)
= rv (x~'(k) +x1'(k)6.t2 + ~v).
(A.21)
481
SYSTEM FUNCTIONS FOR THE CLOCK ESCAPEMENT MECHANISM
Subtracting (A.21) from (A.20) gives
reT A 2 ( ",() ",( )) reae ", 2Ie ut 2 + rex3 k  rvx4 k ilt2  2  2rvx2 (k) = 0, so that
ilt = (rex~'(k)  rvxX'(k))
reT
2
Ic (rex~'(k)
 rvxX'(k))2
+ rJ; (reae + 4rvx~'(k))
+~~~
!£.! Ie
(A.22)
From (A.l)(A.3) and (A.17) it follows that
x~'(k) =
(( reGe
+ 1)2 
rvGereGv + a ( Ge + :e) ) x3(k)
+ (rvGe( reGe + 1) + r;GvGe  rvGe + {3 (Ge + +( Ge +
r~) )
~ h/(ax3(k) + (3x4(k))2 +
A. re Similarly, from (A.l)(A.3) and (A.17) it follows that
x4(k) (A.23)
x~'(k) = (r;GeGv  reGv + (rvGv + l)reGv  aGv)x3(k)
+(reGvrvGe + (rvGv + 1)2  (3G v )x4(k) GvV(ax3(k) + (3x4(k))2 + A.
(A.24)
Now, using (A.23) and (A.24) we obtain
rex~'(k)  rvxX'(k) = ')'x3(k)  oX4(k)  vV(ax3(k)
+ (3x4(k))2 + A, (A.25)
where
')' = (re(reGe + 1)2  2r;rvGcGv + rcG;r; +a(1 + rvGv  reGc)), 0= ( rvr;G~ + 2r;reGvGc  rv( rvGv + 1)2 +(3(1 + rvGv  rcGe)), v = (1 + rvGv  reGe). Next, using x~'(k) that
(A.26) (A.27) (A.28)
= x2(k) +x~(k)iltl' (A.l), and (A.17), it follows
x~'(k) =x2(k) + r:T (rcGvx3(k) 
rv Gvx4(k)
+ x4(k))
Ie
·(ax3(k) + (3x4(k) + V(ax3(k) + (3x4(k))2 + A).
(A.29)
482
APPENDIX A
Thus, (A.22) can be rewritten as
+
vJ(ax3(k) + !3x4(k))2 +,\
reT
T;
+ r:T
((yX3(k) + 8X4(k) + VJ(ax3(k) + !3x4(k))2 + ,\)2
Ie
+ 4rv(rcGvX3(k) 
rvGvx4(k)
+ x4(k))
. (ax3(k) + !3x4(k)
+ J(ax3(k) + !3x4(k))2 +,\) + J.L)"2 ,
1
(A.30) where J.L ~ J.L(x2(k)) = ¥C(rca c + 4rvx2(k)). Finally, it follows from (A.5) using (A.17) and (A.30) that
[~~~~:+ g1 ~~~~j 1 x4(k
+
1)
=A [
x4(k)
Jr§(x3(k), x4(k))g(X2(k), x3(k), x4(k)) [ (Gc + ;}( aX3(k) + !3x4(k) + J(ax3(k) + !3x4(k))2 +,\) Gv(ax3(k) + !3x4(k) + J(ax3(k) + !3x4(k))2 +,\)
*[ + ;e
j( x 2(k), x3(k), x4(k))f(x2(k), x3(k), x4(k))
(yx3(k)
+[
+ 8X4(k) + vJ(ax3(k) + !3x4(k))2 +,\)
:J(X2(k)'~3(k)'X4(k»
1
o
1'
(A.31)
where
§(x3(k), x4(k)) = rcGvx3(k)  rvGvx4(k) g(x2(k), x3(k), x4(k)) = aX3(k) + !3x4(k)
+ x4(k),
+J(ax3(k) + !3x4(k))2 +'\, j(x2(k),X3(k),X4(k)) =~x3(k) + (x4(k) GvV ax3("k):+!3x4:(k""""))=2+,\ , r7 (
f(x2(k), x3(k), x4(k)) = ,x3(k)
1
+ 8X4(k),
483
SYSTEM FUNCTIONS FOR THE CLOCK ESCAPEMENT MECHANISM
+lIJ(ax3(k) + (3x4(k»2 + A + j(x2(k), x3(k), x4(k», j(x2(k), x3(k), x4(k)) = (bX3(k)
+ oX4(k)
+lIJ(ax3(k) + (3x4(k»)2 + A)2 +4rvg(X3(k), x4(k»g(X2(k), x3(k), x4(k» 1
+11)2, f. = r;GvGc  rcGv  aGv + (rvGv + 1)rcGv, (= rcrvGcGv  (3Gv + (rvGv + 1)2. Now, using (A.31) we can characterize the functions h(X2(k), x3(k), x4(k», h(X2(k), x3(k), x4(k», and !4(X2(k), x3(k), x4(k» appearing
in (13.80); namely,
h(X2(k), x3(k), x4(k» = o'n x2(k)
/T
+ 0'12X3(k) + 0'13X4(k)
+ !£..!... g(x3(k), x4(k»g(X2(k), x3(k), x4(k» Ie
1 
+ !£. !(x2(k), x3(k), x4(k» Ie
1(X2(k), x3(k), x4(k», h(X2(k), x3(k), x4(k» = 0'21X2(k) + 0'22X3(k) + 0'23 X4(k) +WJ(ax3(k) + (3x4(k»2 + A
(A.32)
+ !(x2(k), x3(k), x4(k», rC = 0'31X2(k) + 0'32X3(k) + 0'33X4(k) GvJ(aX3(k) + (3X4(k»2 + A,
(A.33)
1
!4(X2(k), x3(k), x4(k» where w matrix
=
.!!... Tc
Gc
A given by
+ 1Tc
A
(A.34)
and o'iJ· denotes the (i,j) component of the
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Index
adiabatically isolated system, 276 admissible stabilizing controllers, 398 asymptotically stable, 21, 56 asymptotically stable equilibrium point, 9 asymptotically stable matrix, 151 asymptotically stable periodic orbit, 446 asymptotically stable with respect to Xl, 46 asymptotically stable with respect to Xl uniformly in X20, 46 available exponential storage, 89 available exponential storage of a leftcontinuous dynamical system, 428 available storage, 88 available storage of a leftcontinuous dynamical system, 428 beating, 2, 14, 420 Bellman's principle of optimality, 320 bilinear matrix inequalities, 335 bounded, 70 bounded trajectory, 29 bounded with respect to Xl uniformly in X2, 64 boundedness, 63 cascade interconnection, 437 class K function, 48 class Koo function, 48 closed system, 134 combustion control, 199 combustion systems, 199 compartmental systems, 125 completely reachable leftcontinuous dynamical system, 429 completely reachable system, 90 confluence, 2, 14, 420 connected system, 272 connective stability, 148 conservation of momentum, 224 continuoustime dynamics, 12
controller kinetic energy, 267 controller Lagrangian, 267 controller potential energy, 267 damping injection, 250 deadlock, 2 derived cost functional, 331 determinism axiom, 413 disconnected system, 152, 272 disk margin guarantees, 337 dissipation matrix, 152 dissipation rate function, 259 dissipative dynamical system, 81 dissipative dynamical systems, 250 dissipative leftcontinuous dynamical system, 427 dissipative nonnegative system, 136 dissipative system, 88 dynamical invariants, 236 emulated energy, 251, 260, 268 energy shaping, 249 energybased hybrid control, 251, 267 energyCasimir functions, 236 entropy function, 273 equilibrium point, 12, 414 essentially nonnegative function, 127 essentially nonnegative matrix, 72, 127 EulerLagrange equations, 266 EulerLagrange system, 265 exponential storage function, 90 exponential storage function of a leftcontinuous dynamical system, 429 exponentially dissipative leftcontinuous dynamical system, 428 exponentially dissipative nonnegative system, 136 exponentially dissipative system, 88 exponentially nonaccumulative system, 142 exponentially nonexpansive leftcontinuous dynamical system,
502
INDEX
434 exponentially nonexpansive system, 112 exponentially passive leftcontinuous dynamical system, 434 exponentially passive system, 112 exponentially stable, 21 exponentially stable with respect to Xl uniformly in X20, 47 exponentially vector dissipative system, 151, 176
hybrid HamiltonJacobiBellman conditions, 395 hybrid HamiltonJacobiBellman equation, 319, 328 hybrid performance functional, 319, 320 hybrid return difference condition, 341 hybrid sector margin, 339 hybrid supply rate, 87 hybrid systems, 1 hybrid vector inequality, 149
feedback interconnection, 437
impact dynamics, 224 impact function, 224 impulsive differential equations, 2 impulsive dynamical system, 2, 11 impulsive optimal control problem, 320 impulsive portcontrolled Hamiltonian systems, 221 inflowclosed system, 134 inputoutput nonnegative system, 135 input / statedependent hybrid dynamical system, 420 interconnected leftcontinuous dynamical system, 436 interconnection function, 435 invariance principle, 30 invariant set, 29 invariant set theorem, 38 inverse optimal control problem, 330 isolated dynamical system, 414
generalized energy balance equation, 179 generalized energy conservation equation, 180 generalized momenta, 266 generalized positions, 265 generalized velocities, 265 given cost functional, 331 globally asymptotically stable, 21, 57 globally asymptotically stable with respect to Xl, 46 globally asymptotically stable with respect to Xl uniformly in X20, 46 globally bounded, 70 globally bounded with respect to Xl uniformly in X2, 64 globally exponentially stable, 21 globally exponentially stable with respect to Xl uniformly in X20, 47 globally ultimately bounded, 70 globally ultimately bounded with respect to Xl uniformly in X2, 64 globally uniformly asymptotically stable, 57 globally uniformly bounded, 67 globally uniformly exponentially stable, 57 globally uniformly ultimately bounded, 68 HamiltonJacobiIsaacs equation, 352 Hamiltonian equations of motion, 266 Hamiltonian function, 223, 266 hybrid disk margin, 339 hybrid disturbance rejection problem, 351 hybrid dynamical system, 411, 418 hybrid EulerLagrange system, 309 hybrid feedback control, 326 hybrid gain margin, 339
jointly continuous between resetting events, 416 Kamke condition, 72 Lagrange stability, 63 Lagrange stable, 67, 70 Lagrange stable with respect to Xl, 64 Lagrangian function, 265 Lagrangian systems, 250 largescale dynamical systems, 147 leftcontinuous dynamical system, 411, 412 leftcontinuous trajectory, 13 Legendre transformation, 265 Lie derivative, 255 Lienard system, 277 livelock, 2 lossless leftcontinuous dynamical system, 428 lossless nonnegative system, 136 lossless system, 88
INDEX
lowersemicontinuous function, 35 Lyapunov, Alexandr Mikhailovich, 9 Lyapunov stable, 21, 56 Lyapunov stable equilibrium point, 9 Lyapunov stable periodic orbit, 446 Lyapunov stable set, 32 Lyapunov stable with respect to Xl, 46 Lyapunov stable with respect to Xl uniformly in X20, 46 Lyapunov's direct method, 9 Mmatrix, 126 meaningful performance criterion, 337 mechanical clock, 458 minimal,96 minimal leftcontinuous dynamical system, 432 multilinear cost functional, 333, 335
503 portcontrolled Hamiltonian system, 134, 250, 263 positive limit point, 28 positive limit set, 28 positive orbit, 28 positively invariant set, 29 positivity theorem, 191 practical stability, 63 quasicontinuous dependence, 415 quasicontinuous dependence property, 27 quasimonotone increasing functions, 71 quasivariational method, 319
wlimit set, 28 optimal control, 319 optimal hybrid feedback control, 326 orbit, 28 outflowclosed system, 134
Rayleigh dissipation function, 265 required exponential supply, 98 required exponential supply of a leftcontinuous dynamical system, 433 required supply, 97 required supply of a leftcontinuous dynamical system, 432 resetting law, 12 resetting set, 12 resetting times, 12 robust analysis, 385 robust control, 385 robust inverse optimal hybrid control problem, 402 robust optimal control problem, 395 robust optimal hybrid control, 386 robust performance, 385 robust stability, 385 robustified hybrid HamiltonJ acobiBellman equation, 402 rotationalj translational proofmass actuator (RTAC), 280
parallel interconnection, 437 partial stability, 44 partial stabilization, 45 passive leftcontinuous dynamical system, 434 passive system, 112 performance criterion, 337 period, 446 periodic orbit, 446 periodic solution, 446 plant energy, 260, 268 Poincare return map, 443, 446 Poincare's theorem, 443 polynomial cost functional, 333, 406
second law of thermodynamics, 272 sector margin guarantees, 337 semistable matrix, 151 shaped dissipation function, 229 shaped Hamiltonian function, 229, 240 shaped interconnection function, 229 smallgain theorem, 191 solution of an impulsive system, 12 stability margins, 337 stability of feedback largescale systems, 214 stability of feedback systems, 191 stability of nonnegative feedback systems, 208
negative limit point, 28 negative limit set, 28 negative orbit, 28 negatively invariant set, 29 net energy flow function, 271 nonaccumulative system, 142 noncontinuability of solutions, 2 non expansive leftcontinuous dynamical system, 434 nonexpansive system, 112 nonnegative dynamical system, 136 nonnegative function, 127 nonnegative systems, 125 nonsingular Mmatrix, 126
504 stable equilibrium point, 9 statedependent differential equations, 2 statedependent hybrid dynamical system, 420 statedependent impulsive dynamical systems, 18 stationary leftcontinuous dynamical system, 414 storage function, 81, 90 storage function of a leftcontinuous dynamical system, 429 strong leftcontinuous dynamical system, 415 strongly zerostate observable, 96 structured parametric uncertainty, 385 supply rate, 81, 427 system kinetic energy, 224, 265 system Lagrangian, 265 system potential energy, 224, 265 thermoacoustic instabilities, 200 thermodynamically consistent hybrid control, 271 thermodynamically stabilizing compensator, 276 timedependent differential equations, 2 timedependent hybrid dynamical system, 420 timedependent impulsive dynamical systems, 18 total energy, 260, 268 trajectory, 28 ultimate boundedness, 63 ultimately bounded, 70 ultimately bounded with respect to Xl uniformly in X2, 64 undisturbed system, 414 uniformly asymptotically stable, 57, 414
INDEX
uniformly bounded, 67 uniformly exponentially stable, 57, 414 uniformly Lyapunov stable, 56, 414 uniformly ultimately bounded, 67 unstable, 21 uppersemicontinuous function, 35 vector available storage, 152 vector comparison principle, 71 vector differential inequalities, 71 vector dissipative system, 151, 176 vector exponentially nonexpansive system, 181 vector exponentially passive system, 180 vector hybrid dissipation inequality, 152, 176 vector hybrid dissipativity, 149 vector hybrid supply rate, 149, 151 vector lossless system, 152, 176 vector Lyapunov function, 71, 148 vector nonexpansive system, 181 vector passive system, 180 vector required supply, 162 vector storage function, 149, 152, 176 verge and foliot escapement, 458 virtual controller momentum, 268 virtual controller positions, 267 virtual controller velocities, 267 weak quasicontinuous dependence property, 253 Zmatrix, 126 Zeno solutions, 2, 14, 420 Zeno time, 18 zerostate observable, 96 zerostate observable leftcontinuous dynamical system, 432 zeroth law of thermodynamics, 272