Control vector Lyapunov functions for large-scale impulsive dynamical systems |
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| Institution: | 1. Department of Mechanical Engineering, Villanova University, Villanova, PA 19085-1681, United States;2. School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, United States;1. Department of Mechanical Engineering, Eindhoven University of Technology, The Netherlands;2. Université de Lorraine, CRAN, UMR 7039, France;3. CNRS, CRAN, UMR 7039, France;4. Department of Electrical and Electronic Engineering, The University of Melbourne, Australia;1. Tianjin Key Laboratory for Control Theory & Applications in Complicated System, Tianjin University of Technology, Tianjin 300384, China;2. School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore;3. Temasek Laboratories, National University of Singapore, Singapore 117508, Singapore;4. School of Computer Science, Nanyang Technological University, Singapore 639798, Singapore;5. Energy Research Institute, Nanyang Technological University, Singapore 639798, Singapore;1. State Key Laboratory for Turbulence and Complex Systems, Department of Advanced Manufacturing and Robotics, BIC-ESAT, College of Engineering, Peking University, Beijing 100871, China;2. Chongqing Key Laboratory of Nonlinear Circuits and Intelligent Information Processing, College of Electronic and Information Engineering, Southwest University, Chongqing 400715, China;1. GREYC, UMR 6072 CNRS, Université de Caen, ENSICAEN, 6 Bd Maréchal Juin, 14050 Caen Cedex, France;2. Unité de Recherche CONPRI, ENIG Gabès, Rue Omar Ibn El Khattab, 6029 Gabès, Tunisie |
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| Abstract: | Vector Lyapunov theory has been developed to weaken the hypothesis of standard Lyapunov theory in order to enlarge the class of Lyapunov functions that can be used for analyzing system stability. In this paper, we provide generalizations to the recent extensions of vector Lyapunov theory for continuous-time systems to address stability and control design of impulsive dynamical systems via vector Lyapunov functions. Specifically, we provide a generalized comparison principle involving hybrid comparison dynamics that are dependent on the comparison system states as well as the nonlinear impulsive dynamical system states. Furthermore, we develop stability results for impulsive dynamical systems that involve vector Lyapunov functions and hybrid comparison inequalities. Based on these results, we show that partial stability for state-dependent impulsive dynamical systems can be addressed via vector Lyapunov functions. Furthermore, we extend the recently developed notion of control vector Lyapunov functions to impulsive dynamical systems. Using control vector Lyapunov functions, we construct a universal hybrid decentralized feedback stabilizer for a decentralized affine in the control nonlinear impulsive dynamical system that possesses guaranteed gain and sector margins in each decentralized input channel. These results are then used to develop hybrid decentralized controllers for large-scale impulsive dynamical systems with robustness guarantees against full modeling and input uncertainty. |
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