Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control

by Wassim M. Haddad, VijaySekhar Chellaboina, Sergey G. Nersesov

Overview

This book develops a general analysis and synthesis framework for impulsive and hybrid dynamical systems. Such a framework is imperative for modern complex engineering systems that involve interacting continuous-time and discrete-time dynamics with multiple modes of operation that place stringent demands on controller design and require implementation of increasing complexity—whether advanced high-performance tactical fighter aircraft and space vehicles, variable-cycle gas turbine engines, or air and ground transportation systems.



Impulsive and Hybrid Dynamical Systems goes beyond similar treatments by developing invariant set stability theorems, partial stability, Lagrange stability, boundedness, ultimate boundedness, dissipativity theory, vector dissipativity theory, energy-based hybrid control, optimal control, disturbance rejection control, and robust control for nonlinear impulsive and hybrid dynamical systems. A major contribution to mathematical system theory and control system theory, this book is written from a system-theoretic point of view with the highest standards of exposition and rigor. It is intended for graduate students, researchers, and practitioners of engineering and applied mathematics as well as computer scientists, physicists, and other scientists who seek a fundamental understanding of the rich dynamical behavior of impulsive and hybrid dynamical systems.

Product Details

ISBN-13:	9780691127156
Publisher:	Princeton University Press
Publication date:	07/23/2006
Series:	Princeton Series in Applied Mathematics , #15
Edition description:	New Edition
Pages:	496
Product dimensions:	6.00(w) x 9.25(h) x (d)
Age Range:	18 Years

About the Author

Wassim M. Haddad is Professor of Aerospace Engineering at the Georgia Institute of Technology. VijaySekhar Chellaboina is Associate Professor of Mechanical, Aerospace, and Biomedical Engineering at the University of Tennessee. Sergey G. Nersesov is Assistant Professor of Mechanical Engineering at Villanova University. Haddad, Chellaboina, and Nersesov previously coauthored Thermodynamics: A Dynamical Systems Approach (Princeton).

Read an Excerpt

Impulsive and Hybrid Dynamical Systems

Stability, Dissipativity, and Control

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By Wassim M. Haddad VijaySekhar Chellaboina Sergey G. Nersesov

Princeton University Press

Copyright © 2006 Princeton University Press
All right reserved.
ISBN: 0-691-12715-8

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Chapter One

1.1 Impulsive and Hybrid Dynamical Systems

Modern complex engineering 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 decision-making units performing logical checks that identify system mode operation and specify the continuous-variable subcontroller to be activated. These multi-echelon systems (see Figure1.1) are classified as hybrid systems (see [6, 126, 161] and the numerous references there in) and involve an interacting countable collection of dynamical systems possessing a hierarchical structure characterized by continuous-time dynamics at the lower-level units and logical decision-making units at the higher level of the hierarchy. The lower-level units directly interact with the dynamical system to be controlled, while the logical decision-making, higher-level units receive information from the lower-level units asinputs and provide (possibly discrete) out put commands, which serve to coordinate and reconcile the (sometimes competing) actions of the lower-level 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 higher-level units perform logical checks that determine system mode operation, while the lower-level units execute continuous-variable commands for a given system mode of operation. Due to their multiechelon hierarchical structure, hybrid dynamical systems are capable of simultaneously exhibiting continuous-time dynamics, discrete-time 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], sampled-data systems [71], discrete-event systems [139], intelligent vehicle/highway systems[113], constrained mechanical systems[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 elements-namely, a continuous time 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 time-varying 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 time-dependent 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 state-dependent 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 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 break down 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 in equality 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 bang-bang 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 basic oxygen needs. However, during stress (such as hemorrhage) when perfusion pressure falls, perfusion of certain regions (e.g., brain and heart) takes precedence over perfusion of other regions, and hierarchical controls(overriding controls) shutdown 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], the fundamental theory of impulsive differential equations is developed in the monographs by Bainov, Lakshmikantham, Perestyuk, Samoilenko, and Simeonov [12-14, 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 [12- 14, 93, 148] to develop in variant set stability theorems, partial stability, Lagrange stability, boundedness and ultimate boundedness, dissipativity theory, vector dissipativity theory, energy-based 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 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 time-dependent and state-dependent impulsive dynamical systems. Furthermore, we state and prove a fundamental result on positive limit sets for state-dependent impulsive dynamical systems. Using this result, we generalize the Krasovskii-LaSalle 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 Kalman-Yakubovitch-Popov 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 continuous-time 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 large-scale nonlinear impulsive dynamical systems. In particular, we introduce a generalized definition of dissipativity for large-scale 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 large-scale impulsive system are shown to be determined from the dissipativity properties of the individual impulsive subsystems making up the large-scale 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 input-output energy, these results generalize the positivity and small gain theorems to the case of nonlinear impulsive dynamical systems.

In Chapter 7, we develop a hybrid control framework for impulsive port-controlled Hamiltonian systems. In particular, we obtain constructive sufficient conditions for hybrid feedback stabilization that provide a shaped energy function for the closed-loop system while preserving a hybrid Hamiltonian structure at the closed-loop level. A novel class of energy-based hybrid controllers is proposed in Chapter 8 as a means for achieving enhanced energy dissipation in Euler-Lagrange, port-controlled 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 closed-loop systems described by impulsive differential equations. In addition, we construct hybrid controllers that guarantee that the closed-loop system is consistent with basic thermodynamic principles. In particular, the existence of an entropy function for the closed-loop system is established that satisfies a hybrid Clausius-type in equality. Extensions to hybrid Euler-Lagrange 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 closed-loop impulsive system. Furthermore, the Lyapunov function is shown to be a solution of a steady-state, hybrid Hamilton-Jacobi-Bellman 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 left-continuous flows, that is, left-continuous 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 left-continuous dynamical systems. Specifically, the classical concepts of system storage functions and supply rates are extended toleft-continuousdynamical systemsproviding ageneralized hybrid system energy interpretation in terms of stored energy, dissipated energy over the continuous-time dynamics, and dissipated energy over the resetting events. Finally, the generalized dissipativity notions are used to develop general stability criteria for feedback interconnections of left-continuous dynamical systems. These results generalize the positivity and small gain theorems to the case of left-continuous and hybrid dynamical systems.

Finally, in Chapter 13 we generalize Poincar'e's theorem to dynamical systems possessing left-continuous flows to address the stability of limit cycles and periodic orbits of left-continuous, hybrid, and impulsive dynamical systems. It is shown that the resetting manifold provides a natural hyperplane for defining a Poincar'e return map. In the special case of impulsive dynamical systems, we show that the Poincar'e map replaces an nth-order impulsive dynamical system by an (n-1)th-order discrete-time system for analyzing the stability of periodic orbits.

(Continues...)

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Table of Contents

Preface xiii

Chapter 1. Introduction 1
1.1 Impulsive and Hybrid Dynamical Systems 1
1.2 A Brief Outline of the Monograph 4

Chapter 2. Stability Theory for Nonlinear Impulsive Dynamical Systems 9
2.1 Introduction 9
2.2 Nonlinear Impulsive Dynamical Systems 11
2.3 Stability Theory of Impulsive Dynamical Systems 20
2.4 An Invariance Principle for State-Dependent Impulsive Dynamical Systems 27
2.5 Necessary and Sufficient Conditions for Quasi-Continuous Dependence 32
2.6 Invariant Set Theorems for State-Dependent Impulsive Dynamical Systems 38
2.7 Partial Stability of State-Dependent Impulsive Dynamical Systems 44
2.8 Stability of Time-Dependent Impulsive Dynamical Systems 56
2.9 Lagrange Stability, Boundedness, and Ultimate Boundedness 63
2.10 Stability Theory via Vector Lyapunov Functions 71

Chapter 3. Dissipativity Theory for Nonlinear Impulsive Dynamical Systems 81
3.1 Introduction 81
3.2 Dissipative Impulsive Dynamical Systems: Input-Output and State Properties 84
3.3 Extended Kalman-Yakubovich-Popov Conditions for Impulsive Dynamical Systems 103
3.4 Specialization to Linear Impulsive Dynamical Systems 119

Chapter 4. Impulsive Nonnegative and Compartmental Dynamical Systems 125
4.1 Introduction 125
4.2 Stability Theory for Nonlinear Impulsive Nonnegative Dynamical Systems 126
4.3 Impulsive Compartmental Dynamical Systems 131
4.4 Dissipativity Theory for Impulsive Nonnegative Dynamical Systems 135
4.5 Specialization to Linear Impulsive Dynamical Systems 143

Chapter 5. Vector Dissipativity Theory for Large-Scale Impulsive Dynamical Systems 147
5.1 Introduction 147
5.2 Vector Dissipativity Theory for Large-Scale Impulsive Dynamical Systems 150
5.3 Extended Kalman-Yakubovich-Popov Conditions for Large-Scale Impulsive Dynamical Systems 175
5.4 Specialization to Large-Scale Linear Impulsive Dynamical Systems 186

Chapter 6. Stability and Feedback Interconnections of Dissipative Impulsive Dynamical Systems 191
6.1 Introduction 191
6.2 Stability of Feedback Interconnections of Dissipative Impulsive Dynamical Systems 191
6.3 Hybrid Controllers for Combustion Systems 199
6.4 Feedback Interconnections of Nonlinear Impulsive Nonnegative Dynamical Systems 208
6.5 Stability of Feedback Interconnections of Large-Scale Impulsive Dynamical Systems 214

Chapter 7. Energy-Based Control for Impulsive Port-Controlled Hamiltonian Systems 221
7.1 Introduction 221
7.2 Impulsive Port-Controlled Hamiltonian Systems 222
7.3 Energy-Based Hybrid Feedback Control 227
7.4 Energy-Based Hybrid Dynamic Compensation via the Energy-Casimir Method 233
7.5 Energy-Based Hybrid Control Design 242

Chapter 8. Energy and Entropy-Based Hybrid Stabilization for Nonlinear Dynamical Systems 249
8.1 Introduction 249
8.2 Hybrid Control and Impulsive Dynamical Systems 251
8.3 Hybrid Control Design for Dissipative Dynamical Systems 258
8.4 Lagrangian and Hamiltonian Dynamical Systems 265
8.5 Hybrid Control Design for Euler-Lagrange Systems 267
8.6 Thermodynamic Stabilization 271
8.7 Energy-Dissipating Hybrid Control Design 277
8.8 Energy-Dissipating Hybrid Control for Impulsive Dynamical Systems 300
8.9 Hybrid Control Design for Nonsmooth Euler-Lagrange Systems 308
8.10 Hybrid Control Design for Impact Mechanics 313

Chapter 9. Optimal Control for Impulsive Dynamical Systems 319
9.1 Introduction 319
9.2 Impulsive Optimal Control 319
9.3 Inverse Optimal Control for Nonlinear Affine Impulsive Systems 330
9.4 Nonlinear Hybrid Control with Polynomial and Multilinear Performance Functionals 333
9.5 Gain, Sector, and Disk Margins for Optimal Hybrid Regulators 337
9.6 Inverse Optimal Control for Impulsive Port-Controlled Hamiltonian Systems 345

Chapter 10. Disturbance Rejection Control for Nonlinear Impulsive Dynamical Systems 351
10.1 Introduction 351
10.2 Nonlinear Impulsive Dynamical Systems with Bounded Disturbances 352
10.3 Specialization to Dissipative Impulsive Dynamical Systems with Quadratic Supply Rates 358
10.4 Optimal Controllers for Nonlinear Impulsive Dynamical Systems with Bounded Disturbances 366
10.5 Optimal and Inverse Optimal Nonlinear-Nonquadratic Control for Affine Systems with L2 Disturbances 375

Chapter 11. Robust Control for Nonlinear Uncertain Impulsive Dynamical Systems 385
11.1 Introduction 385
11.2 Robust Stability Analysis of Nonlinear Uncertain Impulsive Dynamical Systems 386
11.3 Optimal Robust Control for Nonlinear Uncertain Impulsive Dynamical Systems 395
11.4 Inverse Optimal Robust Control for Nonlinear Affine Uncertain Impulsive Dynamical Systems 402
11.5 Robust Nonlinear Hybrid Control with Polynomial Performance Functionals 406

Chapter 12. Hybrid Dynamical Systems 411
12.1 Introduction 411
12.2 Left-Continuous Dynamical Systems 412
12.3 Specialization to Hybrid and Impulsive Dynamical Systems 418
12.4 Stability Analysis of Left-Continuous Dynamical Systems 422
12.5 Dissipative Left-Continuous Dynamical Systems: Input-Output
and State Properties 427
12.6 Interconnections of Dissipative Left-Continuous Dynamical Systems 435

Chapter 13. Poincaré Maps and Stability of Periodic Orbits for Hybrid Dynamical Systems 443
13.1 Introduction 443
13.2 Left-Continuous Dynamical Systems with Periodic Solutions 444
13.3 Specialization to Impulsive Dynamical Systems 451
13.4 Limit Cycle Analysis of a Verge and Foliot Clock Escapement 458
13.5 Modeling 459
13.6 Impulsive Differential Equation Model 462
13.7 Characterization of Periodic Orbits 464
13.8 Limit Cycle Analysis of the Clock Escapement Mechanism 468
13.9 Numerical Simulation of an Escapement Mechanism 472

Appendix A. System Functions for the Clock Escapement Mechanism 477

Bibliography 485
Index 501

What People are Saying About This

V. Lakshmikantham

This book is a significant and timely contribution to the field. Interest in the study of hybrid systems has been growing exponentially in recent years, and the investigation of impulsive differential equations has also drawn much attention. In combining these two important areas, Impulsive and Hybrid Dynamical Systems captures the rich behavior of both in a manner applicable to many applied, technical, and real-world problems. It provides all the necessary tools for the benefit of users.
— V. Lakshmikantham, Florida Institute of Technology

From the Publisher

"This book is a significant and timely contribution to the field. Interest in the study of hybrid systems has been growing exponentially in recent years, and the investigation of impulsive differential equations has also drawn much attention. In combining these two important areas, Impulsive and Hybrid Dynamical Systems captures the rich behavior of both in a manner applicable to many applied, technical, and real-world problems. It provides all the necessary tools for the benefit of users."—V. Lakshmikantham, Florida Institute of Technology

"This carefully written book fills a void in the literature on hybrid and impulsive systems. No book in print has the depth and breadth of this one. The authors present their material in a rigorous and mathematically sound manner."—Anthony Michel, University of Notre Dame

Anthony Michel

This carefully written book fills a void in the literature on hybrid and impulsive systems. No book in print has the depth and breadth of this one. The authors present their material in a rigorous and mathematically sound manner.
— Anthony Michel, University of Notre Dame

Editorial Reviews

Wassim Haddad, Winner of the 2014 Pendray Aerospace Literature Award, American Institute of Aeronautics and Astronautics

From the Publisher

This book fills a void in the are of systems research and is a welcome addition to the literature on hybrid and impulsive systems. The book is well organized, well written, and rigorous in the development of their subject on hand. The authors are to be commended for their scholarly contribution on a subject that is still evolving.

IEEE Control Systems Magazine - Anthony N. Michel

With the growing interest in hybrid dynamical systems, the book forms a welcome text dealing with a restricted well-defined class of hybrid systems. Typical subjects that receive attention throughout are set-stability, energy based control, inverse-optimal control, etc. The book may be viewed as a welcome addition in the areas of hybrid systems, containing rigorous and detailed results.

Mathematical Reviews

With the growing interest in hybrid dynamical systems, the book forms a welcome text dealing with a restricted well-defined class of hybrid systems. Typical subjects that receive attention throughout are set-stability, energy based control, inverse-optimal control, etc. The book may be viewed as a welcome addition in the areas of hybrid systems, containing rigorous and detailed results.

"Mathematical Reviews k Nijmeijer

This book fills a void in the are of systems research and is a welcome addition to the literature on hybrid and impulsive systems. The book is well organized, well written, and rigorous in the development of their subject on hand. The authors are to be commended for their scholarly contribution on a subject that is still evolving.
— Anthony N. Michel

IEEE Control Systems Magazine