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**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

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**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)

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**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)

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**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)

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**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

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**Controllers + Multi-estimator = Injected systems**

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

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**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

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**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 )

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**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

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Example: Controllers: Multi-estimator:

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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

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