ISS of Switched Systems and Application to Adaptive Control Linh Vu, Debasish Chatterjee, Daniel Liberzon Coordinated Science Laboratory, University of Illinois, Urbana-Champaign, U.S.A CDC 2005

Switched Nonlinear Systems where is a switching signal and is some index set where ISS: -ISS: -iISS: LTI systems: Hespanha-Morse. “Stability of switched systems with average dwell-time”. (‘99)

Slow switching Average dwell-time switching where is the number of switches in is a chatter bound is the average dwell-time Hespanha-Morse. “Stability of switched systems with average dwell-time”. (‘99)

ü ý þ Main result Theorem 1: Suppose that there exist such that Û ISS ISS if Then the switched system is -ISS if -iISS if ISS ® AS : Hespanha-Morse. (‘99)

Application: Supervisory Control A parameterized plant with unknown parameters Goal: stabilize the plant when and guarantee bounded when is bounded. Control Scheme: For each design a stabilizing controller Design a multi-estimator Estimation error

Controllers + Multi-estimator = Injected systems fixed Plant P Controller Multi- estimator C E Controllers + Multi-estimator = Injected systems since

ü ý þ + + A1: (criterion for the multi-estimator design) A2: The injected systems satisfy ISS hypotheses of Theorem 1 with respect to (criteria for controllers + multi-estimator design) 2 1 If we have some switching mechanism that provides smallness of w.r.t ü ý þ Þ bounded Thm 1 + A2 1 ISS type properties of the switched injected system with respect to then 2 + A1 smallness of

ü ý þ A3: The plant state is bounded when are bounded. A3 Þ + Scaled-independent hysteresis switching logic gives: Hespanha-Morse. (‘99) 1 for some chosen 2 (smallness of w.r.t )

Theorem 2: Suppose that the plant satisfies A3 the controllers and the multi-estimator satisfy A1 and A2. Then under the scaled-independent hysteresis switching logic, all continuous states of the closed-loop systems are bounded for arbitrary initial conditions and bounded disturbance. Further, if then can make

Example: Controllers: Multi-estimator:

Conclusion ISS type properties of switched system under average dwell-time switching Application to supervisory control of nonlinear uncertain systems in the presence of disturbances Future work: Relax the requirement of the existence of a constant for nonlinear systems Include unmodeled dynamics (non-exact matching) Use other slow switching mechanisms