Stability and Control of Large-Scale Dynamical Systems: A Vector Dissipative Systems Approach (Princeton Series in Applied Mathematics) - Hardcover

Haddad, Wassim M.; Nersesov, Sergey G.

Hardcover

ISBN 10: 0691153469 ISBN 13: 9780691153469

Publisher: Princeton University Press, 2011

Synopsis

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.

About this title

About the Author

Wassim M. Haddad is a professor in the School of Aerospace Engineering and chair of the Flight Mechanics and Control Discipline at Georgia Institute of Technology. Sergey G. Nersesov is an associate professor in the Department of Mechanical Engineering at Villanova University.

From the Back Cover

"This solid book provides a unique perspective and detailed account of the theory and applications of vector Lyapunov functions. I am confident it will be well received by researchers and graduate students working in control theory and large-scale systems."--Mehran Mesbahi, University of Washington

"With admirable and sound scholarship, and great scientific maturity, this book gives a methodical development of vector Lyapunov functions and vector control Lyapunov functions for the design of distributed control systems. It is a welcome addition to the literature and develops a rigorous foundation for applications in an important emerging new area."--Frank L. Lewis, University of Texas, Arlington

From the Inside Flap

"This solid book provides a unique perspective and detailed account of the theory and applications of vector Lyapunov functions. I am confident it will be well received by researchers and graduate students working in control theory and large-scale systems."--Mehran Mesbahi, University of Washington

"With admirable and sound scholarship, and great scientific maturity, this book gives a methodical development of vector Lyapunov functions and vector control Lyapunov functions for the design of distributed control systems. It is a welcome addition to the literature and develops a rigorous foundation for applications in an important emerging new area."--Frank L. Lewis, University of Texas, Arlington

Excerpt. © Reprinted by permission. All rights reserved.

Stability and Control of Large-Scale Dynamical Systems

A Vector Dissipative Systems ApproachBy Wassim M. Haddad Sergey G. Nersesov

PRINCETON UNIVERSITY PRESS

Copyright © 2011 Princeton University Press
All right reserved.
ISBN: 978-0-691-15346-9

Contents

Preface...............................................................................................................xiiiChapter 1. Introduction...............................................................................................1Chapter 2. Stability Theory via Vector Lyapunov Functions.............................................................9Chapter 3. Large-Scale Continuous-Time Interconnected Dynamical Systems...............................................45Chapter 4. Thermodynamic Modeling of Large-Scale Interconnected Systems...............................................75Chapter 5. Control of Large-Scale Dynamical Systems via Vector Lyapunov Functions.....................................93Chapter 6. Finite-Time Stabilization of Large-Scale Systems via Control Vector Lyapunov Functions.....................107Chapter 7. Coordination Control for Multiagent Interconnected Systems.................................................127Chapter 8. Large-Scale Discrete-Time Interconnected Dynamical Systems.................................................153Chapter 9. Thermodynamic Modeling for Discrete-Time Large-Scale Dynamical Systems.....................................181Chapter 10. Large-Scale Impulsive Dynamical Systems...................................................................211Chapter 11. Control Vector Lyapunov Functions for Large-Scale Impulsive Systems.......................................271Chapter 12. Finite-Time Stabilization of Large-Scale Impulsive Dynamical Systems......................................289Chapter 13. Hybrid Decentralized Maximum Entropy Control for Large-Scale Systems......................................305Chapter 14. Conclusion................................................................................................351Bibliography..........................................................................................................353Index.................................................................................................................367

Chapter One

Introduction

1.1 Large-Scale Interconnected Dynamical Systems

Modern complex dynamical systems 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 decentralized analysis and control design of large-scale systems is a direct consequence of the physical size and complexity of the dynamical model. In particular, computational complexity may be too large for model analysis while severe constraints on communication links between system sensors, actuators, and processors may render centralized control architectures impractical. Moreover, even when communication constraints do not exist, decentralized processing may be more economical.

In an attempt to approximate high-dimensional dynamics of large-scale structural (oscillatory) systems with a low-dimensional diffusive (non-oscillatory) dynamical model, structural dynamicists have developed thermodynamic energy flow models using stochastic energy flow techniques. In particular, statistical energy analysis (SEA) predicated on averaging system states over the statistics of the uncertain system parameters have been extensively developed for mechanical and acoustic vibration problems. Thermodynamic models are derived from large-scale dynamical systems of discrete subsystems involving stored energy flow among subsystems based on the assumption of weak subsystem coupling or identical subsystems. However, the ability of SEA to predict the dynamic behavior of a complex large-scale dynamical system in terms of pairwise subsystem interactions is severely limited by the coupling strength of the remaining subsystems on the subsystem pair. Hence, it is not surprising that SEA energy flow predictions for large-scale systems with strong coupling can be erroneous.

Alternatively, a deterministic thermodynamically motivated energy flow modeling for large-scale structural systems is addressed in. This approach exploits energy flow models in terms of thermodynamic energy (i.e., ability to dissipate heat) as opposed to stored energy and is not limited to weak subsystem coupling. Finally, a stochastic energy flow compartmental model (i.e., a model characterized by conservation laws) predicated on averaging system states over the statistics of stochastic system exogenous disturbances is developed in. The basic result demonstrates how compartmental models arise from second-moment analysis of state space systems under the assumption of weak coupling. Even though these results can be potentially applicable to large-scale dynamical systems with weak coupling, such connections are not explored in.

An alternative approach to analyzing large-scale dynamical systems was introduced by the pioneering work of Šiljak and involves the notion of connective stability. In particular, the large-scale dynamical system is decomposed into a collection of subsystems with local dynamics and uncertain interactions. Then, each subsystem is considered independently so that the stability of each subsystem is combined with the interconnection constraints to obtain a vector Lyapunov function for the composite large-scale dynamical system, guaranteeing connective stability for the overall system.

Vector Lyapunov functions were first introduced by Bellman and Matrosov and further developed by Lakshmikantham et al., with exploiting their utility for analyzing large-scale systems. Extensions of vector Lyapunov function theory that include matrix-valued Lyapunov functions for stability analysis of large-scale dynamical systems appear in the monographs by Martynyuk. The use of vector Lyapunov functions in large-scale system analysis offers a very flexible framework for stability analysis since each component of the vector Lyapunov function can satisfy less rigid requirements as compared to a single scalar Lyapunov function. Weakening the hypothesis on the Lyapunov function enlarges the class of Lyapunov functions that can be used for analyzing the stability of large-scale dynamical systems. In particular, each component of a vector Lyapunov function need not be positive definite with a negative or even negative-semidefinite derivative. The time derivative of the vector Lyapunov function need only satisfy an element-by-element vector inequality involving a vector field of a certain comparison system. Moreover, in large-scale systems several Lyapunov functions arise naturally from the stability properties of each subsystem. An alternative approach to vector Lyapunov functions for analyzing large-scale dynamical systems is an input-output approach, wherein stability criteria are derived by assuming that each subsystem is either finite gain, passive, or conic.

In more recent research, Šiljak developed new and original concepts for modeling and control of large-scale complex systems by addressing system dimensionality, uncertainty, and information structure constraints. In particular, the formulation in develops control law synthesis architectures using decentralized information structure constraints while addressing multiple controllers for reliable stabilization, decentralized optimization, and hierarchical and overlapping decompositions. In addition, decomposition schemes for large-scale systems involving system inputs and outputs as well as dynamic graphs defined on a linear space as one-parameter groups of invariant transformations of the graph space are developed in.

Graph theoretic concepts have also been used in stability analysis and decentralized stabilization of large-scale interconnected systems. In particular, graph theory is a powerful tool in investigating structural properties and capturing connectivity properties of large-scale systems. Specifically, a directed graph can be constructed to capture subsystem interconnections wherein the subsystems are represented as nodes and energy, matter, or information flow is represented by edges or arcs. A related approach to graph theory for modeling large-scale systems is bond-graph modeling, wherein connections between a pair of subsystems are captured by a bond and energy, matter, or information is exchanged between subsystems along connections. More recently, a major contribution to the analysis and design of interconnected systems is given in. This work builds on the work of bond graphs by developing a modeling behavioral methodology wherein a system is viewed as an interconnection of interacting subsystems modeled by tearing, zooming, and linking.

In light of the fact that energy flow modeling arises naturally in large-scale dynamical systems and vector Lyapunov functions provide a powerful stability analysis framework for these systems, it seems natural that dissipativity theory on the subsystem level, can play a key role in unifying these analysis methods. Specifically, dissipativity theory provides a fundamental framework for the analysis and design of control systems using an input, state, and output description based on system energy related considerations. The dissipation hypothesis on dynamical systems results in a fundamental constraint on their dynamic behavior wherein a dissipative dynamical system can deliver to its surroundings only a fraction of its energy and can store only a fraction of the work done to it. Such conservation laws are prevalent in large-scale dynamical systems such as aerospace systems, power systems, network systems, structural systems, and thermodynamic systems.

Since these systems have numerous input, state, and output properties related to conservation, dissipation, and transport of energy, extending dissipativity theory to capture conservation and dissipation notions on the subsystem level would provide a natural energy flow model for large-scale dynamical systems. Aggregating the dissipativity properties of each of the subsystems by appropriate storage functions and supply rates would allow us to study the dissipativity properties of the composite large-scale system using vector storage functions and vector supply rates. Furthermore, since vector Lyapunov functions can be viewed as generalizations of composite energy functions for all of the subsystems, a generalized notion of dissipativity, namely, vector dissipativity, with appropriate vector storage functions and vector supply rates, can be used to construct vector Lyapunov functions for nonlinear feedback large-scale systems by appropriately combining vector storage functions for the forward and feedback large-scale systems. Finally, as in classical dynamical system theory, vector dissipativity theory can play a fundamental role in addressing robustness, disturbance rejection, stability of feedback interconnections, and optimality for large-scale dynamical systems.

The design and implementation of control law architectures for large-scale interconnected dynamical systems is a nontrivial control engineering task involving considerations of weight, size, power, cost, location, type, specifications, and reliability, among other design considerations. All these issues are directly related to the properties of the large-scale system to be controlled and the system performance specifications. For conceptual and practical reasons, the control processor architectures in systems composed of interconnected subsystems are typically distributed or decentralized in nature. Distributed control refers to a control architecture wherein the control is distributed via multiple computational units that are interconnected through information and communication networks, whereas decentralized control refers to a control architecture wherein local decisions are based only on local information. In a decentralized control scheme, the large-scale interconnected dynamical system is controlled by multiple processors operating independently, with each processor receiving a subset of the available subsystem measurements and updating a subset of the subsystem actuators. Although decentralized controllers are more complicated to design than distributed controllers, their implementation offers several advantages. For example, physical system limitations may render it uneconomical or impossible to feed back certain measurement signals to particular actuators.

Since implementation constraints, cost, and reliability considerations often require decentralized controller architectures for controlling large-scale systems, decentralized control has received considerable attention in the literature. A straightforward decentralized control design technique is that of sequential optimization, wherein a sequential centralized subcontroller design procedure is applied to an augmented closed-loop plant composed of the actual plant and the remaining subcontrollers. Clearly, a key difficulty with decentralized control predicated on sequential optimization is that of dimensionality. An alternative approach to sequential optimization for decentralized control is based on subsystem decomposition with centralized design procedures applied to the individual subsystems of the large-scale system. Decomposition techniques exploit subsystem interconnection data and in many cases, such as in the presence of very high system dimensionality, are absolutely essential for designing decentralized controllers.

1.2 A Brief Outline of the Monograph

The main objective of this monograph is to develop a general stability analysis and control design framework for nonlinear large-scale interconnected dynamical systems, with an emphasis on vector Lyapunov function methods and vector dissipativity theory. The main contents of the monograph are as follows. In Chapter 2, we establish notation and definitions and develop stability theory for large-scale dynamical systems. Specifically, stability theorems via vector Lyapunov functions are developed for continuous-time and discrete-time nonlinear dynamical systems. In addition, we extend the theory of vector Lyapunov functions by constructing a generalized comparison system whose vector field can be a function of the comparison system states as well as the nonlinear dynamical system states. Furthermore, we present a generalized convergence result which, in the case of a scalar comparison system, specializes to the classical Krasovskii-LaSalle invariant set theorem.

In Chapter 3, we extend the notion of dissipative dynamical systems to develop an energy flow modeling framework for large-scale dynamical systems based on vector dissipativity notions. Specifically, using vector storage functions and vector supply rates, dissipativity properties of a composite large-scale system are shown to be determined from the dissipativity properties of the subsystems and their interconnections. Furthermore, extended Kalman-Yakubovich-Popov conditions, in terms of the subsystem dynamics and interconnection constraints, characterizing vector dissipativeness via vector system storage functions, are derived. In addition, these results are used to develop feedback interconnection stability results for large-scale nonlinear dynamical systems using vector Lyapunov functions. Specialization of these results to passive and nonexpansive large-scale dynamical systems is also provided.

In Chapter 4, we develop connections between thermodynamics and large-scale dynamical systems. Specifically, using compartmental dynamical system theory, we develop energy flow models possessing energy conservation and energy equipartition principles for large-scale dynamical systems. Next, we give a deterministic definition of entropy for a large-scale dynamical system that is consistent with the classical definition of entropy and show that it satisfies a Clausius-type inequality leading to the law of non-conservation of entropy. Furthermore, we introduce a new and dual notion to entropy, namely, ectropy, as a measure of the tendency of a dynamical system to do useful work and grow more organized, and show that conservation of energy in an isolated thermodynamic large-scale system necessarily leads to nonconservation of ectropy and entropy. In addition, using the system ectropy as a Lyapunov function candidate, we show that our large-scale thermodynamic energy flow model has convergent trajectories to Lyapunov stable equilibria determined by the system initial subsystem energies.

In Chapter 5, we introduce the notion of a control vector Lyapunov function as a generalization of control Lyapunov functions, and show that asymptotic stabilizability of a nonlinear dynamical system is equivalent to the existence of a control vector Lyapunov function. Moreover, using control vector Lyapunov functions, we construct a universal decentralized feedback control law for a decentralized nonlinear dynamical system that possesses guaranteed gain and sector margins in each decentralized input channel. Furthermore, we establish connections between the notion of vector dissipativity developed in Chapter 3 and optimality of the proposed decentralized feedback control law. The proposed control framework is then used to construct decentralized controllers for large-scale nonlinear systems with robustness guarantees against full modeling uncertainty. In Chapter 6, we extend the results of Chapter 5 to develop a general framework for finite-time stability analysis based on vector Lyapunov functions. Specifically, we construct a vector comparison system whose solution is finite-time stable and relate this finite-time stability property to the stability properties of a nonlinear dynamical system using a vector comparison principle. Furthermore, we design a universal decentralized finite-time stabilizer for large-scale dynamical systems that is robust against full modeling uncertainty.

Next, using the results of Chapter 5, in Chapter 7 we develop a stability and control design framework for time-varying and time-invariant sets of nonlinear dynamical systems. We then apply this framework to the problem of coordination control for multiagent interconnected systems. Specifically, by characterizing a moving formation of vehicles as a time-varying set in the state space, a distributed control design framework for multivehicle coordinated motion is developed by designing stabilizing controllers for time-varying sets of nonlinear dynamical systems. In Chapters 8 and 9, we present discrete-time extensions of vector dissipativity theory and system thermodynamic connections of large-scale systems developed in Chapters 3 and 4, respectively.

(Continues...)

Excerpted from Stability and Control of Large-Scale Dynamical Systemsby Wassim M. Haddad Sergey G. Nersesov Copyright © 2011 by Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

Edition details

• Publisher: Princeton University Press
• Publication date: 2011
• Language: English
• ISBN 10: 0691153469
• ISBN 13: 9780691153469
• Binding: Hardcover
• Edition number: 1
• Number of pages: 384