Stability of Switched Dynamical Systems

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The objective of control is to in .uence the behavior of dynamical systems.Achiev-ing fast and accurate control under di .erent environmental conditions,even whileassuring stability and robustness,is the aim of all control system design.Whenenvironmental changes are signi .cant (e.g.faults o subsystem failures)the designmay equire the use of multiple controllers,each of which is appropriate in a speci .ccontext.By a switching system we mean a hybrid dynamical system with a decisionmechanism which orchestrates the switching between di .erent controllers.With greatadvances in computer technology,and the concomitant increase in the complexity ofdynamical systems,switching system are becoming ubiquitous.The analysis andsynthesis of such systems is consequently assuming major importance.Stability is the primary equirement of any control system.Once this is assured,optimization of the performance of the overall system can be undertaken within aframework of stability.Following this,modi .cations in the control laws can be un-dertaken to make the system robust under di .erent perturbations.This is the basicphilosophy used in conventional control system design as well as in this proposal.Even though research on switching systems has been in progress for many years andnumerous contributions have been made to the .eld,basic questions still remain unan-swered.For example the stability of many classes of linear,nonlinear,and adaptiveswitching systems still remains to be answered,and very little progress has been madein shaping the transient response resulting from the switching action.Both of theseimportant issues are included in the proposal.The proposal consists of three parts.Since stability is undoubtedly the most impor-tant unresolved issue,the .rst part of the proposal is devoted to it.Switching betweengeneral linear time-invariant systems and positive linear systems is treated in this con-text.Special attention is given to systems with diagonal Lyapunov functions in viewof their importance to switching in decentralized systems.Also treated are piecewiselinear systems which are practically important and theoretically tractable.Followingthis,in the second part of the proposal,the performance that can be achieved byswitching is considered.The third and last part of the proposal is devoted to switch-ing in adaptive systems.Intellectual Merit: The PI believes that the research proposed addresses funda-mental questions whose solutions are essential for a better understanding of switchingsystems.This in turn is vital for the design of future control systems.Broader Impacts: The esearch proposed will enhance collaboration between thePI and other international esearchers (particularly Dr.Robert Shorten at the Hamil-ton Institute,Ireland and Dr.Jens Kalkuhl of Daimler-Chrysler Research),lead toincreased interaction with industry,and most important of all,provide training toboth graduate and undergraduate students on advanced control techniques.1