Switched and impulsive systems. Analysis, design and applications.
This text is an introduction to hybrid systems, their stability and stabilization, and a presentation of various applications. Hybrid systems are called here ``switched and impulsive systems''. Such systems evolve on \(Q\times\mathbb{R}^n\) where \(Q\) is a finite set. The dynamics of an ODE is coupled to the dynamics of an automaton, and once the continuous component reaches a certain set \(S\), the state of the automaton switches and a new ODE is activated. This is why the systems under consideration are called ``switched''. They are also called ``impulsive'' because the initial condition of the ODE after switching differs in general from the continuous state before switching. Such systems can be controlled by ``switched and impulsive'' controls which take the form \(X(t^+_k)=X(t^-_k)+U(t_k,X)\), and \(U(t_k,X)\) is a ``switching control'' of the form \(U(t_k,X)=K_{m(t_k)}(X)\) with \(K_j:\mathbb{R}^n\to\mathbb{R}^n\) and \(m(t_k)= I(X(t_k))\) with \(I:\mathbb{R}^n\to Q\). Here the controller is not only determined by \(I\) and \(K_j\) but also by the \(\{t_k\}_{k\in\mathbb{N}}\), and \(X\) is the state of the continuous variable. Obviously, such controls are open-loop and not the most general ones one could construct. For uncontrolled hybrid systems, a notion of Lyapunov stability is introduced and sufficient conditions for Lyapunov stability (existence of a Lyapunov function) are proposed. In particular, the Lyapunov function should decrease when evaluated at suitable times between switches. Since the condition is quite restrictive, the authors consider a weaker case with multiple Lyapunov functions associated to each ODE between switches, which are however supposed linear (this case is more restrictive concerning the dynamics). Another criterion using matrix measures is obtained; the proof uses a global Lyapunov function. Another case investigates situations where all the continuous dynamics are unstable but stability is obtained due to the switching. Other strong assumptions involving cycles are needed here. (Let us remark that example 3.5.1, page 59, is artificial and is constructed for the need of the cause; the authors adjust parameters so that a non-generic condition is satisfied). Some criteria for stabilization are obtained too. The controllers have to identify the continuous dynamics in action from state knowledge. For each continuous dynamics, a stabilizing controller is designed. The switching law for the controller should mimic the switching law for the system, which is not realizable in practice. These issues are reconsidered in the nonlinear case in another chapter. The last part of the book is devoted to applications: synchronization of chaotic systems, chaos-based secure communication, scheduling of switched server systems, service quality on the internet, switched scalable video coding systems. Finally, the authors consider open problems. There are 218 references, and an index which could be improved. The book is of interest as a study on stability issues for hybrid systems and for original applications.
scientific article; zbMATH DE number 2146306