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

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

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Suppose that there exist such that Main result Theorem 1: 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. Design a multi-estimator Estimation error Control Scheme: For each design a stabilizing controller

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

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then 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) 21 If we have some switching mechanism that provides smallness of w.r.t 2 + A1 smallness of bounded Thm 1 + A2 1 ISS type properties of the switched injected system with respect to

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A3: The plant state is bounded when are bounded. bounded A3 bounded bounded+ 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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