Stability and Control of Large-Scale Dynamical Systems

Stability and Control of Large-Scale Dynamical Systems: A Vector Dissipative Systems Approach

Wassim M. Haddad
Sergey G. Nersesov
Copyright Date: 2011
Pages: 392
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  • Book Info
    Stability and Control of Large-Scale Dynamical Systems
    Book Description:

    Modern complex large-scale dynamical systems exist in virtually every aspect of science and engineering, and are associated with a wide variety of physical, technological, environmental, and social phenomena, including aerospace, power, communications, and network systems, to name just a few. This book develops a general stability analysis and control design framework for nonlinear large-scale interconnected dynamical systems, and presents the most complete treatment on vector Lyapunov function methods, vector dissipativity theory, and decentralized control architectures.

    Large-scale dynamical systems are strongly interconnected and consist of interacting subsystems exchanging matter, energy, or information with the environment. The sheer size, or dimensionality, of these systems necessitates decentralized analysis and control system synthesis methods for their analysis and design. Written in a theorem-proof format with examples to illustrate new concepts, this book addresses continuous-time, discrete-time, and hybrid large-scale systems. It develops finite-time stability and finite-time decentralized stabilization, thermodynamic modeling, maximum entropy control, and energy-based decentralized control.

    This book will interest applied mathematicians, dynamical systems theorists, control theorists, and engineers, and anyone seeking a fundamental and comprehensive understanding of large-scale interconnected dynamical systems and control.

    eISBN: 978-1-4008-4266-7
    Subjects: Mathematics

Table of Contents

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

    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 necessitates 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 aggregate or composite (i.e., large-scale) system can then be predicted from the behaviors of the individual subsystems and their interconnections. The need for...

  5. Chapter Two Stability Theory via Vector Lyapunov Functions
    (pp. 9-44)

    In this chapter, we introduce the notion of vector Lyapunov functions for stability analysis of nonlinear dynamical systems. The use of vector Lyapunov functions in dynamical system theory offers a flexible framework for stability analysis because 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 the analysis of large-scale...

  6. Chapter Three Large-Scale Continuous-Time Interconnected Dynamical Systems
    (pp. 45-74)

    In this chapter, we develop vector dissipativity notions for large-scale nonlinear dynamical systems, a notion not previously considered in the literature. In particular, we introduce a generalized definition of dissipativity for large-scale nonlinear dynamical systems in terms of avector dissipation inequalityinvolving a vector supply rate, a vector storage function, and an essentially nonnegative, semistable dissipation matrix. The vector dissipation inequality reflects the fact that some of the supplied generalized energy of the large-scale system is stored, and some is dissipated. The dissipated generalized energy is nonnegative and is given by the difference of what is supplied and what...

  7. Chapter Four Thermodynamic Modeling of Large-Scale Interconnected Systems
    (pp. 75-92)

    In this chapter, we use vector dissipativity theory to provide connections between large-scale dynamical systems and thermodynamics. Specifically, using a large-scale dynamical systems theory prospective for thermodynamics, we show that vector dissipativity notions lead to a precise formulation of the equivalence between dissipated energy (heat) and work in a large-scale dynamical system. Next, we give a deterministic definition of entropy for a large-scale dynamical system that is consistent with the classical thermodynamic definition of entropy and show that it satisfies a Clausius-type inequality leading to the law of entropy nonconservation. Furthermore, we introduce a dual notion to entropy, namely,ectropy,...

  8. Chapter Five Control of Large-Scale Dynamical Systems via Vector Lyapunov Functions
    (pp. 93-106)

    One of the most basic issues in system theory is the stability of dynamical systems. The most complete contribution to the stability analysis of nonlinear dynamical systems is due to Lyapunov [128]. Lyapunov’s results, along with the Krasovskii-LaSalle invariance principle [116, 120, 121], provide a powerful framework for analyzing the stability of nonlinear dynamical systems. Lyapunov methods have also been used by control system designers to obtain stabilizing feedback controllers for nonlinear systems. In particular, for smooth feedback, Lyapunov-based methods were inspired by Jurdjevic and Quinn [105] who give sufficient conditions for smooth stabilization based on the ability of constructing...

  9. Chapter Six Finite-Time Stabilization of Large-Scale Systems via Control Vector Lyapunov Functions
    (pp. 107-126)

    The notions of asymptotic and exponential stability in dynamical systems theory imply convergence of the system trajectories to an equilibrium state over the infinite horizon. In many applications, however, it is desirable that a dynamical system possesses the property that trajectories that converge to a Lyapunov stable equilibrium state must do so in finite time rather than merely asymptotically. Most of the existing control techniques in the literature ensure that the closed-loop system dynamics of a controlled system are Lipschitz continuous, which implies uniqueness of system solutions in forward and backward times. Hence, convergence to an equilibrium state is achieved...

  10. Chapter Seven Coordination Control for Multiagent Interconnected Systems
    (pp. 127-152)

    Modern complex multiagent dynamical systems are highly interconnected and mutually interdependent, both physically and through a multitude of information and communication networks. Distributed decision-making for coordination of networks of dynamic agents involving information flow can be naturally captured by graph-theoretic notions. These dynamical network systems cover a very broad spectrum of applications, including cooperative control of unmanned air vehicles (UAVs), autonomous underwater vehicles (AUVs), distributed sensor networks, air and ground transportation systems, swarms of air and space vehicle formations, and congestion control in communication networks, to cite but a few examples. Hence, it is not surprising that a considerable research...

  11. Chapter Eight Large-Scale Discrete-Time Interconnected Dynamical Systems
    (pp. 153-180)

    Since most physical processes evolve naturally in continuous-time, it is not surprising that the bulk of large-scale dynamical system theory has been developed for continuous-time systems. Nevertheless, it is the overwhelming trend to implement controllers digitally. Hence, in this chapter we extend the notions of dissipativity theory [70,170,171] to developvector dissipativitynotions for large-scale nonlinear discrete-time dynamical systems. In particular, we introduce a generalized definition of dissipativity for large-scale nonlinear discrete-time dynamical systems in terms of avector dissipation inequalityinvolving avector supply rate, a vector storage function, and a nonnegative, semistable dissipation matrix. Generalized notions of vector...

  12. Chapter Nine Thermodynamic Modeling for Discrete-Time Large-Scale Dynamical Systems
    (pp. 181-210)

    Thermodynamic principles have been repeatedly used in continuous-time dynamical system theory as well as information theory for developing models that capture the exchange of nonnegative quantities (e.g., mass and energy) between coupled subsystems [21,27,30,69,147,170,179]. In particular, conservation laws (e.g., mass and energy) are used to capture the exchange of material between coupled macroscopic subsystems known as compartments. Each compartment is assumed to be kinetically homogeneous, that is, any material entering the compartment is instantaneously mixed with the material in the compartment. These models are known ascompartmentalmodels and are widespread in engineering systems as well as biological and ecological...

  13. Chapter Ten Large-Scale Impulsive Dynamical Systems
    (pp. 211-270)

    The complexity of modern controlled large-scale 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. As discussed in Chapter 1, such systems typically possess a multiechelon hierarchical hybrid decentralized control architecture characterized by continuous-time dynamics at the lower levels of the hierarchy and discrete-time dynamics at the higher levels of the hierarchy. The lower-level units directly interact with the dynamical system to be controlled, while the higher-level units receive information...

  14. Chapter Eleven Control Vector Lyapunov Functions for Large-Scale Impulsive Systems
    (pp. 271-288)

    The mathematical descriptions of many hybrid dynamical systems can be characterized by impulsive differential equations [11,13,82,94,117,155]. As shown in Chapter 10, 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...

  15. Chapter Twelve Finite-Time Stabilization of Large-Scale Impulsive Dynamical Systems
    (pp. 289-304)

    As noted in Chapter 6, finite-time stability implies Lyapunov stability and convergence of system trajectories to an equilibrium state in finite time, and hence, is a stronger notion than asymptotic stability. For continuous-time dynamical systems, finite-time stability implies non-Lipschitzian dynamics [24,86] giving rise to non-uniqueness of solutions in reverse time. Uniqueness of solutions in forward time, however, can be preserved in the case of finite-time convergence. Sufficient conditions that ensure uniqueness of solutions in forward time in the absence of Lipschitz continuity are given in [1, 58, 108, 176]. Finite-time convergence to a Lyapunov stable equilibrium for continuous-time systems, that...

  16. Chapter Thirteen Hybrid Decentralized Maximum Entropy Control for Large-Scale Systems
    (pp. 305-350)

    In this chapter, we develop a novel energy- and entropy-based hybrid decentralized control framework for vector lossless and vector dissipative largescale dynamical systems based on subsystem decomposition. The notion of energy here refers to abstract energy notions for which a physical system energy interpretation is not necessary. These dynamical systems cover a broad spectrum of applications, including mechanical systems, fluid systems, electromechanical systems, electrical systems, combustion systems, structural vibration systems, biological systems, physiological systems, power systems, telecommunications systems, and economic systems, to cite but a few examples. The concept of an energy-based hybrid decentralized controller can be viewed as a...

  17. Chapter Fourteen Conclusion
    (pp. 351-352)

    In this monograph, we have developed a stability analysis and control design framework for large-scale complex dynamical systems. These systems are composed of interconnected subsystems and include air traffic control systems, power and energy grid systems, manufacturing and processing systems, aerospace and transportation systems, communication and information networks, integrative biological systems, biological neural networks, biomolecular and biochemical systems, nervous systems, immune systems, environmental and ecological systems, molecular, quantum, and nanoscale systems, particulate and chemical reaction systems, economic and financial systems, cellular systems, metabolic systems, planetary ecosystems (e.g., Gaia), and galaxies, to name but a few examples. The relationships between the...

  18. Bibliography
    (pp. 353-366)
  19. Index
    (pp. 367-371)
  20. Back Matter
    (pp. 372-372)