# (p.xiii) Preface

# (p.xiii) Preface

Modern complex large-scale dynamical systems arise in virtually every aspect of science and engineering and are associated with a wide variety of physical, technological, environmental, and social phenomena. Such systems include large-scale aerospace systems, power systems, communications systems, network systems, transportation systems, large-scale manufacturing systems, integrative biological systems, economic systems, ecological systems, and process control systems. These systems are strongly interconnected and consist of interacting subsystems exchanging matter, energy, or information with the environment. In addition, the subsystem interactions often exhibit remarkably complex system behaviors. Complexity here refers to the quality of a system wherein interacting subsystems form multiechelon hierarchical evolving structures exhibiting emergent system properties.

The sheer size, or dimensionality, of large-scale dynamical systems necessitates decentralized analysis and control system synthesis methods for their analysis and control design. Specifically, in analyzing complex large-scale interconnected dynamical systems it is often desirable to treat the overall system as a collection of interacting subsystems. The behavior and properties of the aggregate large-scale system can then be deduced from the behaviors of the individual subsystems and their interconnections. Often the need for such an analysis framework arises from computational complexity and computer throughput constraints. In addition, for controller design the physical size and complexity of large-scale systems impose severe constraints on the communication links among system sensors, processors, and actuators, which can render centralized control architectures impractical. This problem leads to consideration of decentralized controller architectures involving multiple sensor-processor-actuator subcontrollers without real-time intercommunication. The design and implementation of decentralized controllers is a nontrivial task involving control-system architecture determination and actuator-sensor assignments for a particular subsystem, as well as processor software design for each subcontroller of a given architecture.

In this monograph, we develop a unified stability analysis and control design framework for nonlinear large-scale interconnected dynamical systems based on vector Lyapunov function methods and vector dissipativity theory. The use of vector Lyapunov functions in dynamical system theory offers a very flexible framework for stability analysis since each component of the vector Lyapunov function can satisfy less rigid requirements as compared to (p.xiv) a single scalar Lyapunov function. Moreover, in the analysis of large-scale interconnected nonlinear dynamical systems, several Lyapunov functions arise naturally from the stability properties of each individual subsystem. In addition, since large-scale dynamical systems have numerous input, state, and output properties related to conservation, dissipation, and transport of energy, matter, or information, extending classical dissipativity theory to capture conservation and dissipation notions on the subsystem level provides 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 allows us to study the dissipativity properties of the composite large-scale system using the newly developed notions of vector storage functions and vector supply rates. The monograph is written from a system-theoretic point of view and can be viewed as a contribution to dynamical system and control system theory.

After a brief introduction to large-scale interconnected dynamical systems in Chapter 1, fundamental stability theory for nonlinear dynamical systems using vector Lyapunov functions is developed in Chapter 2. In Chapter 3, we extend classical dissipativity theory to vector dissipativity for addressing large-scale systems using vector storage functions and vector supply rates. Chapter 4 develops connections between thermodynamics and large-scale dynamical systems. A detailed treatment of control design for large-scale systems using control vector Lyapunov functions is given in Chapter 5, whereas extensions of these results for addressing finite-time stability and stabilization are given in Chapter 6. Next, in Chapter 7 we develop a stability and control design framework for coordination control of multiagent interconnected systems. Chapters 8 and 9 present discrete-time extensions of vector dissipativity theory and system thermodynamic connections of large-scale systems, respectively. A detailed treatment of stability analysis and vector dissipativity for large-scale impulsive dynamical systems is given in Chapter 10. Chapters 11 and 12 provide extensions of finite-time stabilization and stabilization of large-scale impulsive dynamical systems. In Chapter 13, a novel class of fixed-order, energy- and entropy-based hybrid decentralized controllers is developed for large-scale dynamical systems. Finally, in Chapter 14 we present conclusions.

The first author would like to thank Dennis S. Bernstein and David C. Hyland for their valuable discussions on large-scale vibrational systems over the years. The first author would also like to thank Paul Katinas for several insightful and enlightening discussions on the statements quoted in ancient Greek on page vii. In some parts of the monograph we have relied on work we have done jointly with Jevon M. Avis, VijaySekhar Chellaboina, Qing Hui, and Rungun Nathan; 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, and the Department of Mechanical Engineering of Villanova University, Villanova,

(p.xv) Pennsylvania, between January 2004 and February 2011. The research support provided by the Air Force Office of Scientific Research and the Office of Naval Research 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, June 2011, *Wassim M. Haddad*

Villanova, Pennsylvania, June 2011, *Sergey G. Nersesov*
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