Delayed state feedback and chaos control for time-periodic systems via a symbolic approach |
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| Institution: | 1. Department of Mechanical Engineering, University of Alaska Fairbanks, Fairbanks, AK 99775, USA;2. Department of Mathematical Sciences, University of Alaska Fairbanks, Fairbanks, AK 99775, USA;1. Departamento de Matemática Aplicada II, Universidade de Vigo, 36310 Vigo, Spain;2. Institut für Mathematik, Alpen-Adria Universität Klagenfurt, 9020 Klagenfurt, Austria;1. School of Mathematics and Computational Science in Fuyang Teachers College, Fuyang 236029, PR China;2. School of Mathematical Science, Anhui University, Hefei 230039, PR China;3. Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong;4. Center for Nonlinear Studies, Beijing-Hong Kong-Singapore Joint Center for Nonlinear and Complex Systems, Hong Kong Baptist University, Hong Kong;5. Institute of Computational and Theoretical Studies, Hong Kong Baptist University, Kowloon Tong, Hong Kong;1. Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 4 place Jussieu, F-75005 Paris, France;2. ETH Institut für Theoretische Studien (ETH-ITS), Clausiusstrasse 47, Zürich, Switzerland;3. School of Mathematics, Shandong University, Jinan, Shandong 250100, China;4. Université de Bordeaux, UMR 5251, Institut de Mathématiques de Bordeaux, 351 cours de la Libération, F-33405 Talence, France |
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| Abstract: | This paper presents a symbolic method for a delayed state feedback controller (DSFC) design for linear time-periodic delay (LTPD) systems that are open loop unstable and its extension to incorporate regulation and tracking of nonlinear time-periodic delay (NTPD) systems exhibiting chaos. By using shifted Chebyshev polynomials, the closed loop monodromy matrix of the LTPD system (or the linearized error dynamics of the NTPD system) is obtained symbolically in terms of controller parameters. The symbolic closed loop monodromy matrix, which is a finite dimensional approximation of an infinite dimensional operator, is used in conjunction with the Routh–Hurwitz criterion to design a DSFC to asymptotically stabilize the unstable dynamic system. Two controllers designs are presented. The first design is a constant gain DSFC and the second one is a periodic gain DSFC. The periodic gain DSFC has a larger region of stability in the parameter space than the constant gain DSFC. The asymptotic stability of the LTPD system obtained by the proposed method is illustrated by asymptotically stabilizing an open loop unstable delayed Mathieu equation. Control of a chaotic nonlinear system to any desired periodic orbit is achieved by rendering asymptotic stability to the error dynamics system. To accommodate large initial conditions, an open loop controller is also designed. This open loop controller is used first to control the error trajectories close to zero states and then the DSFC is switched on to achieve asymptotic stability of error states and consequently tracking of the original system states. The methodology is illustrated by two examples. |
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