Automatica 64 (2016) 270–277Contents lists available at ScienceDirectAutomaticajournal homepage: www.elsevier.com/locate/automaticaBrief paperGeneralized switching signals for input-to-state stability of switchedsystems ?Atreyee Kundu a,1 , Debasish Chatterjee b , Daniel Liberzon ca Control Systems Technology Group, Department of Mechanical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlandsb Systems & Control Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, Indiac Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USAa r t i c l e i n f oArticle history:Received 6 February 2015Received in revised form10 July 2015Accepted 23 October 2015Keywords:Switched systemsInput-to-state stabilityMultiple-ISS Lyapunov functionsa b s t r a c tThis article dealswith input-to-state stability (ISS)of continuous-time switched nonlinearsystems. Givena family of systems with exogenous inputs such that not all systems in the family are ISS, we characterizea new and general class of switching signals under which the resulting switched system is ISS. Ourstabilizing switching signals allow the number of switches to grow faster than an affine function of thelength of a time interval, unlike in the case of average dwell time switching. We also recast a subclassof average dwell time switching signals in our setting and establish analogs of two representative priorresults.© 2015 Elsevier Ltd. All rights reserved.1. IntroductionA switched system comprises of two components—a family ofsystems and a switching signal. The switching signal selects anactive subsystem at every instant of time, i.e., the system fromthe family that is currently being followed (Liberzon, 2003, Sec-tion 1.1.2). Stability of switched systems is broadly classifiedinto two categories—stability under arbitrary switching (Liberzon,2003, Chapter 2) and stability under constrained switching (Liber-zon, 2003, Chapter 3). In the former category, conditions on thefamily of systems are identified such that the resulting switchedsystem is stable under all admissible switching signals; in the lat-ter category, given a family of systems, conditions on the switch-ing signals are identified such that the resulting switched systemis stable. In this article our focus is on stability of switched systemswith exogenous inputs under constrained switching.Prior study in the direction of stability under constrainedswitching primarily utilizes the concept of slow switching vis-a-vis(average) dwell time switching. Exponential stability of a switchedlinear system under dwell time switching was studied in Morse?The material in this paper was not presented at any conference. This paper wasrecommended for publication in revised form by Associate Editor Graziano Chesiunder the direction of Editor Richard Middleton.E-mail addresses: a.kundu@tue.nl (A. Kundu), dchatter@iitb.ac.in(D. Chatterjee), liberzon@illinois.edu (D. Liberzon).1Tel.: +31 40 247 4850; fax: +31 40 243 4970.(1996). In Xie, Wen, and Li (2001) the authors showed that aswitched nonlinear system is ISS under dwell time switching if allsubsystems are ISS. A class of state-dependent switching signalsobeying dwell time property under which a switched nonlinearsystem is integral input-to-state stable (iISS) was proposed in DePersis,DeSantis,Morse(2003).Thedwelltimerequirementforsta-bilitywasrelaxedtoaveragedwelltimeswitching toswitchedlinearsystems with inputs and switched nonlinear systems without in-puts in Hespanha and Morse (1999). ISS of switched nonlinear sys-tems under average dwell time was studied in Vu, Chatterjee, andLiberzon(2007).ItwasshownthatiftheindividualsubsystemsareISS and their ISS-Lyapunov functions satisfy suitable conditions,then the switched system has the ISS, exponentially-weighted ISS,and exponentially-weighted iISS properties under switching sig-nals obeying sufficiently large average dwell time. Given a familyof systems such that not all systems in the family are ISS, it wasshownintherecentwork(Yang&Liberzon,2014)thatitispossibleto construct a class of hybrid Lyapunov functions to guarantee ISSof the switched system provided that the switching signal neitherswitches too frequently nor activates the non-ISS subsystems fortoo long. In Müller and Liberzon (2012) input/output-to-state sta-bility (IOSS) of switched nonlinear systems with families in whichnot all subsystems are IOSS, was studied. It was shown that theswitched system is IOSS under a class of switching signals obeyingaverage dwell time property and constrained point-wise activationof unstable subsystems.Given a family of systems, possibly containing non-ISS dy-namics, in this article we study ISS of switched systems underhttp://dx.doi.org/10.1016/j.automatica.2015.11.0270005-1098/ © 2015 Elsevier Ltd. All rights reserved.
