Analysis for a class of singularly perturbed hybrid systems via averaging |
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| Authors: | Wei Wang Andrew R. Teel Dragan Nešić |
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| Affiliation: | 1. Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, 3010, Victoria, Australia;2. Electrical and Computer Engineering Department, University of California, Santa Barbara, CA 93106-9560, USA;1. Department of Electrical Engineering, Faculty of Science and Technology, Tokyo University of Science, Yamazaki 2641, Noda, Japan;2. Interdisciplinary Faculty of Science and Engineering, Shimane University, 1060 Nishikawatsu, Matsue, Japan;3. Ritsumeikan Global Innovation Research Organization (R-GIRO), Ritsumeikan University, 1-1-1 Noji-higashi, Kusatsu, Shiga, Japan;1. School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China;2. Center for Systems and Control, College of Engineering, Peking University, Beijing 100871, China;1. Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL, CNRS UMR 9189), Université Lille 1 / École Centrale de Lille, 59650 Villeneuve d’Ascq, France;2. Non-A, INRIA Lille-Nord Europe, France;1. LAGIS (UMR-CNRS 8219), Ecole Centrale de Lille, BP 48, 59651 Villeneuve D’Ascq, France;2. Xlim (UMR-CNRS 7252), Département SIC, Université de Poitiers, Bât. SP2MI, Bvd Marie et Pierre Curie, BP 30179, 86962 Futuroscope Chasseneuil Cedex, France;3. Non-A project at INRIA Lille, Nord Europe, Parc Scientifique de la Haute Borne, 40 avenue Halley, Bt. A Park Plaza, 59650 Villeneuve dAscq, France;1. College of Energy and Electrical Engineering, Hohai University, Nanjing 210098, PR China;2. Tianjin University, Tianjin 300072, PR China |
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| Abstract: | A class of singularly perturbed hybrid dynamical systems is analyzed. The fast states are restricted to a compact set a priori. The continuous-time boundary layer dynamics produce solutions that are assumed to generate a well-defined average vector field for the slow dynamics. This average, the projection of the jump map in the direction of the slow states, and flow and jump sets from the original dynamics define the reduced, or average, hybrid dynamical system. Assumptions about the average system lead to conclusions about the original, higher-dimensional system. For example, forward pre-completeness for the average system leads to a result on closeness of solutions between the original and average system on compact time domains. In addition, global asymptotic stability for the average system implies semiglobal, practical asymptotic stability for the original system. We give examples to illustrate the averaging concept and to relate it to classical singular perturbation results as well as to other singular perturbation results that have appeared recently for hybrid systems. We also use an example to show that our results can be used as an analysis tool to design hybrid feedbacks for continuous-time plants implemented by fast but continuous actuators. |
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