1. 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

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

3. 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

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

5. 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 :

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

7. 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:

8. 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

9. 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

10. 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

11. 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 ]

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

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

14. 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

15. 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

16. 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 ???