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STABILITY ANALYSIS of HYBRID SYSTEMS via SMALL-GAIN THEOREMS

Explore stability analysis of hybrid systems using small-gain theorems. Learn about input-to-state stability (ISS), Lyapunov functions, and gas properties. Discover applications in quantized control and networked control systems.

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STABILITY ANALYSIS of HYBRID SYSTEMS via SMALL-GAIN THEOREMS

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  1. 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

  2. HYBRID SYSTEMS as FEEDBACK CONNECTIONS See paper for more general setting • Other decompositions possible • Can also have external signals continuous discrete

  3. 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:

  4. 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]

  5. LYAPUNOV– BASED SMALL–GAIN THEOREM and # of discrete events on is Hybrid system is GAS if:

  6. 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]

  7. 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]

  8. 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 ???

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