Contents 
IEEE Control Systems Magazine welcomes suggestions for books to be reviewed in this column. Please contact either Michael Polis or Zongli Lin, associate editors for book reviews.
In the context of control theory, there does not appear to be a universally accepted definition of a dynamical system, or a classification of such systems, including hybrid dynamical systems (HDSs), switched dynamical systems, and impulsive dynamical systems. Indeed, there does not seem to be universal agreement on whether a switched dynamical system is an HDS or vice versa, an impulsive dynamical system is an HDS or vice versa, or a switched dynamical system is an impulsive dynamical system or vice versa. However, since all of these systems have discontinuous motions (motions that are not continuous with respect to time), there ought to be universal agreement that HDS, switched dynamical systems, and impulsive dynamical systems are discontinuous dynamical systems (DDSs). The book under review addresses Lyapunov stability, dissipativity, and control for a class of DDS that will be defined shortly.
Lyapunov stability and boundedness results for continuous dynamical systems and discrete-time dynamical systems constitute a mature subject, spanning over a century [1], [2]. On the other hand, Lyapunov stability and boundedness results for DDSs have been established more recent-ly. A comprehensive set of Lyapunov stability and boundedness results for DDSs, including converse theo-rems, are given in [3]. For subsequent results, see the tutorial papers [4], [5] and references cited therein. In addition, monographs dealing with specialized topics on switched dynamical systems include [6]–[8]. Since the intersection between these monographs and the book under review is negligible, we will not elaborate further on these works.
About the Book
The authors characterize an HDS as an “interacting countable collection of dynamical systems involving a mixture of contin-uous-time dynamics and discrete events that includes impulsive dynamical systems, hierarchical systems, and switching systems as special cases.” The authors' stated objective is to “develop a unified analysis and control design framework for impulsive and hybrid dynamical systems using a Lyapunov and dissipative systems approach.” We will comment later on the authors' success in achieving their stated goals.
The book is organized into 13 chapters and includes an appendix section. In Chapter I, the authors provide a general discussion of the class of impulsive and hybrid dynamical systems used throughout the book. As pointed out earlier, these terms may mean different things to different people, depending on the context of the problem on hand.
Chapter 2 addresses stability theory for time-invariant and time-varying impulsive systems. The presentation concerns Lyapunov stability results (of an equilibrium) and Lagrange stability results (boundedness of solutions), as well as Krasovskii-LaSalle type invariance results for time-invariant systems. Most of the Lyapunov stability results are based on the original work of Bainov [9], since these results appear to best fit the framework the authors pursue. In addition, the authors address partial stability of impulsive systems.
In the qualitative analysis of general dynamical systems, Lyapunov and Lagrange stability properties of states are of primary interest, while in the qualitative analysis of control systems, input-output properties as well as state properties of the system are addressed. In Chapter 3, the authors extend the notion of dissipative dynamical systems, introduced by Willems [10], [11] for continuous systems, to impulsive dissipative dynamical systems. In doing so, the concepts of system storage function and system supply rate are extended to impulsive dynamical systems. Included in this chapter are Kalman- Yacubovich-Popov type results for characterizing dissipativity for impulsive systems.
In Chapter 4, the results of chapters 2 and 3 are further refined for special classes of impulsive systems, specifically, for nonnegative systems (systems whose state variables are nonnegative) and compartmental systems (systems whose states exchange mass or energy).
Large-scale systems, also called interconnected systems, and in certain contexts, decentralized systems, are frequently analyzed in terms of their lower order and simpler subsystems and in terms of the system interconnecting structure in order to circumvent difficulties encountered when addressing complex systems. In Chapter 5, the authors introduce the notions of vector dissipativity for large-scale nonlinear impulsive dynamical systems. They study the dissipativity properties of these systems in terms of the dissipativity properties of the individual subsystems and the system's interconnecting structure.
Using the results of the preceding chapters, Chapter 6 focuses on the stability of feedback interconnections of dissipative impulsive dynamical systems. Reflecting the authors' research interests, special classes of impulsive systems are addressed, including hybrid controllers for combustion sys-tems, nonlinear impulsive nonnegative dynamical systems, and large-scale impulsive dynamical systems.
In chapters 7 and 8, the authors conduct qualitative analyses of special classes of impulsive hybrid feedback control systems, which again reflects their expertise and research interests, including energy-based control for impulsive Hamiltonian systems and thermodynamic stabilization through energy-dissipative hybrid controllers.
In Chapter 9, ideas from classical optimal control are extended to a hybrid feedback control problem over an infinite horizon involving a hybrid nonlinear nonquadratic performance functional. This functional involves a continuous-time cost (cor-responding to the continuous-time dynamics) and a discrete-time cost to address performance at the resetting instants. In contrast to variational methods used in the literature for optimal control of impulsive and hybrid systems, the authors consider an approach involving feedback controllers that guarantee closed-loop stability by the use of an underlying Lyapunov function. In future work, it might be interesting to consider using the less conservative stability results for DDSs established in [12] or results involving multiple Lyapunov functions [5].
In chapters 10 and 11, the authors extend the optimal feedback control framework of Chapter 9 to include disturbance rejection control and robust control for nonlinear impulsive dynamical systems with bounded exogenous disturbances.
Chapters 12 and 13 employ an input/output description for left-continuous dynamical systems. This system formulation, which is very general, includes as special cases hybrid dynamical systems and impulsive dynamical systems. By their very nature, these system descriptions do not involve specific system structures and may therefore be more difficult to apply. Similarly, as was done in earlier chapters, the authors develop stability and dissipativity results using the state flow of the system. In Chapter 13, Poincare's theorem is adapted to left-continuous dynamical systems to analyze the stability properties of periodic motions in impulsive and hybrid dynamical systems. The exposition in these two chapters is quite a bit shorter than in the preceding chapters, suggesting that more research can be expected in this area.
Concluding Comments
The authors have achieved their objective “to develop a unified analysis and control design framework for impulsive and hybrid dynamical systems using a Lyapunov and dissipative systems approach.” This book fills a void in the area of systems research and is a welcome addition to the literature on hybrid and impulsive systems. Whereas a great part of this work reflects the authors' own research interests and results, neverthe-less, the breadth and depth of the book goes beyond the norm achieved in the literature on the present and related topics.
The book is well organized, well written, and rigorous in the development of the subject on hand. The authors are to be commended for their scholarly contribution on a subject that is still evolving. Their monograph will be of great use to many researchers within the control systems community. The book can also serve as the basis for a graduate course in control systems. Prerequisites for such a course include a background in analysis and linear algebra at the intermediate level as well as the usual first year graduate courses in linear and nonlinear systems.