1. SMALL-GAIN APPROACH to STABILITY ANALYSIS of HYBRID SYSTEMS CDC ’05 Dragan Nešić University of Melbourne, Australia Daniel Liberzon Univ. of Illinois at Urbana-Champaign, USA

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

3. SMALL – GAIN THEOREM Small-gain theorem [Jiang-Teel-Praly ’94] gives GAS if: Input-to-state stability (ISS) from to [ Sontag ’89 ] : ISS from to : (small-gain condition)

4. SUFFICIENT CONDITIONS for ISS [ Hespanha-L-Teel, TuC16.5 ] # of discrete events on is ISS from to if: and ISS from to if ISS-Lyapunov function [ Sontag ’89 ] :

5. LYAPUNOV – BASED SMALL – GAIN THEOREM Hybrid system is GAS if [L-Nešić, HSCC ’06]: and # of discrete events on is

6. SKETCH of PROOF [ Jiang-Mareels-Wang ’96 ] is nonstrictly decreasing along trajectories Trajectories along which is constant?None! GAS follows by LaSalle for hybrid systems [ Lygeros et al. ’03 ]

7. APPLICATION: QUANTIZED CONTROL quantization error Zoom in: where ISS from to with gain small-gain condition! External disturbances: [L-Nešić, ThC14.6] ISS from to with some linear gain

8. CONCLUSIONS Main idea: small-gain analysis tools are naturally applicable to hybrid systems Ongoing work: Lyapunov function constructions for hybrid systems Applications: Quantized feedback control Networked control systems [ CDC paper, Nešić-Teel ] Other ??? 