80 likes | 91 Views
STABILITY ANALYSIS of HYBRID SYSTEMS via SMALL-GAIN THEOREMS. Daniel Liberzon Univ. of Illinois at Urbana-Champaign, USA. Dragan Ne šić University of Melbourne, Australia. HSCC ’06. HYBRID SYSTEMS as FEEDBACK CONNECTIONS. See paper for more general setting.
E N D
STABILITY ANALYSIS of HYBRID SYSTEMS via SMALL-GAIN THEOREMS Daniel Liberzon Univ. of Illinois at Urbana-Champaign, USA Dragan Nešić University of Melbourne, Australia HSCC ’06
HYBRID SYSTEMS as FEEDBACK CONNECTIONS See paper for more general setting • Other decompositions possible • Can also have external signals continuous discrete
SMALL–GAIN THEOREM • Input-to-state stability (ISS) from to [Sontag ’89]: • ISS from to : (small-gain condition) Small-gain theorem [Jiang-Teel-Praly ’94] gives GAS if:
SUFFICIENT CONDITIONS for ISS • ISS from to if ISS-Lyapunov function [Sontag ’89]: • ISS from to if: and # of discrete events on is [Hespanha-L-Teel, CDC’05]
LYAPUNOV– BASED SMALL–GAIN THEOREM and # of discrete events on is Hybrid system is GAS if:
SKETCH of PROOF is nonstrictly decreasing along trajectories Trajectories along which is constant? None! GAS follows by LaSalle principle for hybrid systems [Lygeros et al. ’03, Sanfelice-Goebel-Teel ‘05]
APPLICATION: QUANTIZED CONTROL quantization error ISS from to with some linear gain Zoom in: where ISS from to with gain small-gain condition! [Nešić-L, CDC’05]
http://decision.csl.uiuc.edu/ liberzon CONCLUSIONS • Main idea: small-gain analysis tools are naturally • applicable to hybrid systems • Ongoing work: Lyapunov function constructions • for hybrid systems (more on this in the paper) • Applications: • Quantized feedback control • Networked control systems [CDC paper, Nešić-Teel] • Other ???