Download presentation

Presentation is loading. Please wait.

Published byLarry Hunger Modified over 3 years ago

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 ???

Similar presentations

OK

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

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

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on any one mathematician paul A ppt on bill gates Ppt on carburetor parts Ppt on democracy and equality in indian Ppt on pay per click Ppt on trade of goods and services in international market Ppt on pin photodiode Convert doc file to ppt online Ppt on hyperion financial management Ppt on oxidation and reduction reactions