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Impulsive and Hybrid Dynamical Systems

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

Wassim M. Haddad
VijaySekhar Chellaboina
Sergey G. Nersesov
Copyright Date: 2006
Pages: 496
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  • Book Info
    Impulsive and Hybrid Dynamical Systems
    Book Description:

    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 Systemsgoes 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.

    eISBN: 978-1-4008-6524-6
    Subjects: Mathematics, Technology, Physics

Table of Contents

  1. Front Matter
    (pp. i-viii)
  2. Table of Contents
    (pp. ix-xii)
  3. Preface
    (pp. xiii-xvi)
    Wassim M. Haddad, VijaySekhar Chellaboina and Sergey G. Nersesov
  4. Chapter One Introduction
    (pp. 1-8)

    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 multiechelon systems (see Figure 1.1) are classified ashybridsystems (see [6, 126, 161] and the numerous references therein) and involve aninteractingcountable collection of dynamical systems possessing a hierarchical structure characterized...

  5. Chapter Two Stability Theory for Nonlinear Impulsive Dynamical Systems
    (pp. 9-80)

    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 bestableif, 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...

  6. Chapter Three Dissipativity Theory for Nonlinear Impulsive Dynamical Systems
    (pp. 81-124)

    In control engineering, dissipativity theory provides a fundamental framework for the analysis and control design of dynamical systems using an input-output system description based on system-energy-related 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...

  7. Chapter Four Impulsive Nonnegative and Compartmental Dynamical Systems
    (pp. 125-146)

    Nonnegative systems [20, 57, 58, 132] are essential in capturing the phenomenological features of a wide range of dynamical systems involving dynamic states whose values are nonnegative. A subclass of nonnegative dynamical systems are compartmental systems [4, 25, 47, 50, 83, 84, 114, 115, 125, 149]. These systems are derived from mass and energy balance considerations and are comprised of homogeneous interconnected macroscopic subsystems or compartments which exchange variable quantities of material via intercompartmental flow laws. Since biological and physiological systems have numerous input-output properties related to conservation, dissipation, and transport of mass and energy, nonnegative and compartmental systems are...

  8. Chapter Five Vector Dissipativity Theory for Large-Scale Impulsive Dynamical Systems
    (pp. 147-190)

    Modern complex dynamical systems¹ are highly interconnected and mutually interdependent, both physically and through a multitude of information and communication network constraints. The sheer size (i.e., dimensionality) and complexity of these large-scale dynamical systems often necessitate a hierarchical decentralized architecture for analyzing and controlling these systems. Specifically, in the analysis and control-system design of complex large-scale dynamical systems it is often desirable to treat the overall system as a collection of interconnected subsystems. The behavior of the composite (i.e., large-scale) system can then be predicted from the behaviors of the individual subsystems and their interconnections. The need for decentralized analysis...

  9. Chapter Six Stability and Feedback Interconnections of Dissipative Impulsive Dynamical Systems
    (pp. 191-220)

    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 input-output energy, these results generalize the positivity and small-gain theorems [53, 74, 142, 165, 171] to the case of nonlinear...

  10. Chapter Seven Energy-Based Control for Impulsive Port-Controlled Hamiltonian Systems
    (pp. 221-248)

    In a recent series of papers [136–138] a passivity-based control framework for port-controlled Hamiltonian systems is established. Specifically, the authors in [136–138] develop a controller design methodology that achieves stabilization via system passivation. In particular, the interconnection and damping matrix functions of the port-controlled Hamiltonian system are shaped so that the physical (Hamiltonian) system structure is preserved at the closed-loop level, and the closed-loop 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 closed-loop level, the passivity-based controller...

  11. Chapter Eight Energy and Entropy-Based Hybrid Stabilization for Nonlinear Dynamical Systems
    (pp. 249-318)

    Energy is a concept that underlies our understanding of all physical phenomena and is a measure of the ability of a dynamical system to produce changes (motion) in its own system state as well as changes in the system states of its surroundings. In control engineering, dissipativity theory [165], which encompasses passivity theory, provides a fundamental framework for the analysis and control design of dynamical systems using an input-output system description based on system-energy-related considerations [107, 134, 160]. The notion of energy here refers to abstract energy notions for which a physical system energy interpretation is not necessary. As noted...

  12. Chapter Nine Optimal Control for Impulsive Dynamical Systems
    (pp. 319-350)

    In this chapter, we consider a hybrid feedback optimal control problem over an infinite horizon involving a hybrid nonlinear-nonquadratic performance functional. The performance functional involves a continuous-time cost for addressing performance of the continuous-time system dynamics and a discrete-time cost for addressing performance at the resetting instants. Furthermore, the hybrid cost functional can be evaluated in closed form as long as the nonlinear-nonquadratic 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. This Lyapunov function is shown to be a solution of a steady-state, hybrid...

  13. Chapter Ten Disturbance Rejection Control for Nonlinear Impulsive Dynamical Systems
    (pp. 351-384)

    In this chapter, we develop an optimality-based theory for disturbance rejection of nonlinear impulsive dynamical systems with bounded energy exogenous disturbances. The key motivation for developing an optimal and inverse optimal nonlinear hybrid control theory that additionally guarantees disturbance rejection is that it provides a class of candidate disturbance rejection hybrid controllers parameterized by the hybrid cost functional that is minimized. In order to address the optimality-based disturbance rejection nonlinear hybrid control problem we extend the nonlinear-nonquadratic, hybrid controller analysis and synthesis framework presented in Chapter 9. Specifically, using nonlinear hybrid dissipativity theory developed in Chapter 3, with appropriate storage...

  14. Chapter Eleven Robust Control for Nonlinear Uncertain Impulsive Dynamical Systems
    (pp. 385-410)

    Although the theory of impulsive dynamical systems has received considerable attention in the literature [12, 14, 39, 61, 62, 79, 93, 148], robust analysis and control design techniques for uncertain nonlinear impulsive dynamical systems remain relatively undeveloped. In this chapter, we extend the analysis and control design framework for nonlinear impulsive dynamical systems developed in Chapters 2 and 9 to address robustness considerations for impulsive dynamical systems. In particular, we build on the results of Chapter 9 to develop an optimality-based framework for addressing the problem of nonlinear-nonquadratic optimal hybrid control foruncertainnonlinear impulsive dynamical systems with structured parametric...

  15. Chapter Twelve Hybrid Dynamical Systems
    (pp. 411-442)

    As discussed in Chapter 1, hybrid dynamical systems involve an interacting countable collection of dynamical systems possessing a mixture of continuous-time dynamics and discrete-time dynamics that include impulsive dynamical systems, hierarchical systems, and switching systems as special cases. Even though numerous results focusing on specific forms of hybrid systems have been developed in the literature (see for example [51] and the numerous references therein), the development of a general model for hybrid dynamical systems has received little attention in the literature. Notable exceptions include [30, 121, 169]. In particular, the authors in [121, 169] introduce a general (undisturbed) hybrid dynamical...

  16. Chapter Thirteen Poincaré Maps and Stability of Periodic Orbits for Hybrid Dynamical Systems
    (pp. 443-476)

    In Chapter 12 a unified dynamical systems framework for a general class of systems possessing left-continuous flows, that is, left-continuous dynamical systems, was developed. Stability results of left-continuous dynamical systems are also considered in Chapter 12. The extension of the Krasovskii-LaSalle 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 left-continuous dynamical system is a disconnected set in then-dimensional state space making the construction of a Lyapunov-like function...

  17. Appendix A. System Functions for the Clock Escapement Mechanism
    (pp. 477-484)
  18. Bibliography
    (pp. 485-500)
  19. Index
    (pp. 501-505)