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1 of 16 SMALL - GAIN THEOREMS of LASALLE TYPE for HYBRID SYSTEMS Daniel Liberzon (Urbana-Champaign) Dragan Nešić (Melbourne) Andy Teel (Santa Barbara) CDC, Maui, Dec 2012

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2 of 16 MODELS of HYBRID SYSTEMS … [ Goebel-Sanfelice-Teel ] Flow: Jumps:

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3 of 16 HYBRID SYSTEMS as FEEDBACK CONNECTIONS continuous discrete Every hybrid system can be thought of in this way But this special decomposition is not always the best one E.g., NCS: network protocol

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4 of 16 HYBRID SYSTEMS as FEEDBACK CONNECTIONS HS1 HS2 Can also consider external signals

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5 of 16 SMALL – GAIN THEOREM Small-gain theorem [ Jiang-Teel-Praly 94 ] gives GAS if: (small-gain condition) Input-to-state stability (ISS) from to [ Sontag 89 ]: ISS from to :

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6 of 16 SUFFICIENT CONDITIONS for ISS This gives strong ISS property [ Cai-Teel 09 ] For : where

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7 of 16 LYAPUNOV – BASED SMALL – GAIN THEOREM on (small-gain condition) is a Lyapunov function for the overall hybrid system Then Pick s.t. on Assume:

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8 of 16 LYAPUNOV – BASED SMALL – GAIN THEOREM Generalizes Lyapunov small-gain constructions for continuous [ Jiang-Mareels-Wang 96 ] and discrete [ Laila-Nešić 02 ] systems decreases along solutions of the hybrid system On the boundary, use Clarke derivative

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9 of 16 LIMITATION on on The strict decrease conditions are often not satisfied off-the-shelf E.g.: Since and we would typically have

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10 of 16 LASALLE THEOREM all nonzero solutions have both flow and jumps Assume: As before, pick and let Then is non-increasing along both flow and jumps and its not constant along any nonzero traj. GAS

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11 of 16 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 ]

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12 of 16 QUANTIZED STATE FEEDBACK QUANTIZER CONTROLLER PLANT Hybrid quantized control: is discrete state – zooming variable

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13 of 16 QUANTIZED STATE FEEDBACK QUANTIZER CONTROLLER PLANT Hybrid quantized control: is discrete state Zoom out to overcome saturation – zooming variable

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14 of 16 QUANTIZED STATE FEEDBACK QUANTIZER CONTROLLER PLANT Hybrid quantized control: is discrete state After the ultimate bound is achieved, recompute partition for smaller region Can recover global asymptotic stability – zooming variable

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15 of 16 SMALL – GAIN ANALYSIS quantization error Zoom in: where ISS from to with gain small-gain condition! ISS from to with some linear gain Can use quadratic Lyapunov functions to compute the gains

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

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