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SMALL - GAIN THEOREMS of LASALLE TYPE for HYBRID SYSTEMS

SMALL - GAIN THEOREMS of LASALLE TYPE for HYBRID SYSTEMS. Daniel Liberzon ( Urbana-Champaign ) Dragan Ne šić (Melbourne ) Andy Teel (Santa Barbara). CDC, Maui, Dec 2012. MODELS of HYBRID SYSTEMS. …. Flow:. Jumps:. [ Goebel-Sanfelice-Teel ].

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SMALL - GAIN THEOREMS of LASALLE TYPE for HYBRID SYSTEMS

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  1. SMALL-GAIN THEOREMS of LASALLE TYPE for HYBRID SYSTEMS Daniel Liberzon(Urbana-Champaign) DraganNešić(Melbourne) Andy Teel (Santa Barbara) CDC, Maui, Dec 2012

  2. MODELS of HYBRID SYSTEMS … Flow: Jumps: [Goebel-Sanfelice-Teel]

  3. HYBRID SYSTEMS as FEEDBACK CONNECTIONS E.g., NCS: network protocol continuous discrete Every hybrid system can be thought of in this way But this special decomposition is not always the best one

  4. HYBRID SYSTEMS as FEEDBACK CONNECTIONS HS1 HS2 Can also consider external signals

  5. 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:

  6. SUFFICIENT CONDITIONS for ISS • For : • For : where This gives “strong” ISS property [Cai-Teel ’09]

  7. LYAPUNOV– BASED SMALL–GAIN THEOREM Assume: on on (small-gain condition) Pick s.t. Then is a Lyapunov function for the overall hybrid system

  8. LYAPUNOV– BASED SMALL–GAIN THEOREM decreases along solutions of the hybrid system On the boundary, use Clarke derivative Generalizes Lyapunov small-gain constructions for continuous [Jiang-Mareels-Wang ’96] and discrete [Laila-Nešić ’02] systems

  9. LIMITATION The strict decrease conditions on are often not satisfied off-the-shelf E.g.: on Since and we would typically have

  10. LASALLE THEOREM Assume: As before, pick and let Then is non-increasing along both flow and jumps and it’s not constant along any nonzero traj. GAS • all nonzero solutions have both flow and jumps

  11. 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]

  12. QUANTIZED STATE FEEDBACK – zooming variable PLANT Hybrid quantized control: is discrete state QUANTIZER CONTROLLER

  13. QUANTIZED STATE FEEDBACK – zooming variable PLANT Hybrid quantized control: is discrete state QUANTIZER CONTROLLER Zoom out to overcome saturation

  14. QUANTIZED STATE FEEDBACK – zooming variable PLANT Hybrid quantized control: is discrete state QUANTIZER CONTROLLER After the ultimate bound is achieved, recompute partition for smaller region Can recover global asymptotic stability

  15. SMALL–GAIN ANALYSIS quantization error ISS from to with some linear gain Zoom in: where ISS from to with gain small-gain condition! Can use quadratic Lyapunov functions to compute the gains

  16. CONCLUSIONS • Basic idea: small-gain analysis tools are naturally • applicable to hybrid systems • Main technical results: (weak) Lyapunovfunction • constructions for hybrid system interconnections • Applications: • Quantized feedback control • Networked control systems • Event-triggered control[Tabuada] • Other ???

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