STABILIZING a SWITCHED LINEAR SYSTEM by SAMPLED - DATA QUANTIZED FEEDBACK

1. Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign STABILIZING a SWITCHED LINEAR SYSTEM bySAMPLED-DATA QUANTIZED FEEDBACK Daniel Liberzon 50th CDC-ECC, Orlando, FL, Dec 2011, last talk in the program!

2. PROBLEM FORMULATION Switched system: are (stabilizable) modes is a (finite) index set is a switching signal sampling period Information structure: Sampling:state is measured at times Quantization:each is encoded by an integer from 0 to and sent to the controller, along with ( ) Data rate: Objective:design an encoding & control strategy s.t. based on this limited information about and

3. MOTIVATION • Switching: • ubiquitous in realistic system models • lots of research on stability & stabilization under switching • tools used: common & multiple Lyapunov functions, • slow switching assumptions • Quantization: • coarse sensing (low cost, limited power, hard-to-reach areas) • limited communication (shared network resources, security) • theoretical interest (how much info is needed for a control task) • tools used: Lyapunov analysis, data-rate/MATI bounds Commonality of tools is encouraging Almost no prior work on quantized control of switched systems (except quantized MJLS [Nair et. al. 2003, Dullerud et. al. 2009])

4. NON-SWITCHED CASE Quantized control of a single LTI system: [Baillieul, Brockett-L, Hespanha, Nair-Evans, Petersen-Savkin,Tatikonda] System is stabilized if error reduction factor at growth factor on Crucial step: obtaining a reachable setover-approximation at next sampling instant How to do this for switched systems?

5. SLOW-SWITCHING and DATA-RATE ASSUMPTIONS • dwell time (lower bound on time between switches) • average dwell time (ADT) s.t. number of switches on (sampling period) Implies: switch on each sampling interval Define (usual data-rate bound for individual modes)

6. ENCODING and CONTROL STRATEGY Goal: generate, on the decoder/controller side, a sequence of points and numbers s.t. (always -norm) Let Pick s.t. is Hurwitz Define state estimate on by Define control on by

7. GENERATING STATE BOUNDS Choosing a sequence that grows faster than system dynamics, for some we will have Inductively, assuming we show how to find s.t. Let Case 1 (easy): sampling interval with no switch

8. GENERATING STATE BOUNDS – unknown to the controller Before the switch: as on previous slide, but this is unknown known Instead, pick some and use as center Intermediate bound: Case 2 (harder): sampling interval witha switch (triangle inequality)

9. GENERATING STATE BOUNDS After the switch: on , closed-loop dynamics are , or Auxiliary system in : We know: As before, unknown lift Instead, pick & use as center (known)

10. GENERATING STATE BOUNDS unknown lift Instead, pick & use as center project onto Projecting onto -component and letting we obtain the final bound:

11. STABILITY ANALYSIS: OUTLINE : on This is exp. stable DT system w. input data-rate assumption exp and : satisfies if from to , then • contains a switch If satisfies then exp as same true for • sampling interval with no switch Thus, the overall is exp. stable “cascade” system Lyapunov function ADT Intersample behavior, Lyapunov stability – see paper

12. SIMULATION EXAMPLE (data-rate assumption holds) switch Theoretical lower bound on is about 10 times larger

13. CONCLUSIONS and FUTURE WORK • Contributions: • Studied stabilization of switched systems with quantization • Main step is computing over-approximations of reachable sets • Data-rate bound is the usual one, maximized over modes • Extensions: • Relaxing the slow-switching assumption • Refining reachable set bounds • Allowing state jumps • Challenges: • State-dependent switching (hybrid systems) • Nonlinear dynamics • Output feedback