Switched and impulsive systems. Analysis, design and applications.
type: Item
instance of: scholarly article
title: Switched and impulsive systems. Analysis, design and applications.
zbMATH Open document ID: 1060.93004
author: Zhengguo Li
author: Yeng Chai Soh
author: Changyun Wen
published in: Lecture Notes in Control and Information Sciences
publication date: 2005-03-15T00:00:00Z
review text: 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 str…
reviewed by: A. Akutowicz
Mathematics Subject Classification ID: 93-02
Mathematics Subject Classification ID: 93C30
Mathematics Subject Classification ID: 93B12
Mathematics Subject Classification ID: 34A37
Mathematics Subject Classification ID: 93C65
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 68U35
Mathematics Subject Classification ID: 68U10
Mathematics Subject Classification ID: 94A60
Mathematics Subject Classification ID: 90B22
zbMATH DE Number: 2146306
zbMATH Keywords: scheduling of server systems
zbMATH Keywords: hybrid systems
zbMATH Keywords: stability
zbMATH Keywords: stabilization
zbMATH Keywords: applications
zbMATH Keywords: Lyapunov function
zbMATH Keywords: matrix measures
zbMATH Keywords: switching
zbMATH Keywords: cycles
zbMATH Keywords: synchronization of chaotic systems
zbMATH Keywords: secure communication
zbMATH Keywords: service quality on the internet
zbMATH Keywords: scalable video coding systems
MaRDI profile type: Publication
full work available at URL: b100366
OpenAlex ID: W2499792849
cites work: A unified framework for hybrid control: model and optimal control theory
cites work: Periodicity and chaos from switched flow systems: contrasting examples of discretely controlled continuous systems
cites work: Switching in systems and control
cites work: Hybrid dynamical systems. Controller and sensor switching problems
cites work: Switched linear systems. Control and design.
DOI: 10.1007/B100366
description: scientific article; zbMATH DE number 2146306