Abstract
We propose convex conditions for stabilization of nonlinear discrete-time systems with time-varying delay in states through a fuzzy Takagi-Sugeno (T-S) modeling. These conditions are developed from a fuzzy Lyapunov-Krasovskii function and they are formulated in terms of linear matrix inequalities (LMIs). The results can be applied to a class of nonlinear systems that can be exactly represented by T-S fuzzy models inside a specific region called the region of validity. As a consequence, we need to provide an estimate of the set of safe initial conditions called the region of attraction such that the closed-loop trajectories starting in this set are assured to remain in the region of validity and to converge asymptotically to the origin. The estimate of the region of attraction is done with the aid of two sets: one dealing with the current state, and the other concerning the delayed states. Then, we can obtain the feedback fuzzy control law depending on the current state, x k , and the maximum delayed state vector, xk−d̅. It is shown that such a control law can locally stabilize the nonlinear discrete-time system at the origin. We also develop convex optimization procedures for the computation of the fuzzy control gains that maximize the estimates of the region of attraction. We present two examples to demonstrate the efficiency of the developed approach and to compare it with other approaches in the literature.
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Recommended by Associate Editor Ohmin Kwon under the direction of Editor Euntai Kim. This work has been supported by the Brazilian Agencies CAPES and CNPq.
Luís F. P. Silva received his B.S. and M.S. degrees in electrical engineering from Federal Center of Technological Educacional, Belo Horizonte, Minas Gerais, Brazil, in 2009 and 2011, respectively, and Ph.D. degree in engineering of automation and systems from Federal University of Santa Catarina, Florianópolis, Santa Catarina, Brazil, in 2016. He is currently a professor in the Department of Mechatronics Engineering at Federal Center of Technological Educacional, Divinópolis, Minas Gerais, Brazil. His current research interests include T-S fuzzy model control, stability analysis and synthesis of control of uncertain linear systems with time-delay in the states and actuators saturated.
Valter J. S. Leite was born in Itaúna (MG, Brazil). He received the Ph.D. degree in Electrical Engineering and in Automatique et Informatique Industrielle from the University of Campinas (Brazil) and from the INSA de Toulouse (France), respectively, in 2005. He has been with CEFET-MG since 1997 and currently he is an Associate Professor in the Department of Mechatronic Engineering at campus Divinópolis (MG, Brazil). Valter is an associate editor of International Journal of Robust and Nonlinear Control, and International Journal of Control, Automation and Electrical Systems, and was an associate editor of Mathematical Problems in Engineering. His main research interests include robust control, delay systems, fuzzy T-S systems and constrained systems.
Eugênio B. Castelan was born in Criciúma (SC, Brazil). He received the Electric Engineering degree, in 1982, and the M.Sc. degree, in 1985, both from UFSC, Brazil, and the Doctoral degree, in 1992, from Paul Sabatier University, France. In 1993, he joined the Department of Automation and Systems at UFSC, Brazil, where he develops his teaching and research activities. In 2003, he spent a year at LAAS du CNRS, France, as an invited researcher in the Group MAC. He was the chair of the Graduate Program on Automation and Systems Engineering at UFSC from 2007 to 2011. He is currently a full professor at UFSC and an associate editor of Journal of the Franklin Institute. His main research interests are on constrained control systems, control theory, fuzzy T-S based control of nonlinear systems, and control applications.
Gang Feng received the Ph.D. degree in Electrical Engineering from the University of Melbourne, Australia. He has been with City University of Hong Kong since 2000 after serving as lecturer/senior lecturer at School of Electrical Engineering, University of New South Wales, Australia, 1992–1999. He is now Chair Professor of Mechatronic Engineering. He has been awarded an Alexander von Humboldt Fellowship, the IEEE Transactions on Fuzzy Systems Outstanding Paper Award, and Changjiang chair professorship from Education Ministry of China. He is listed as a SCI highly cited researcher by Thomoson Reuters. His current research interests include multi-agent systems and control, intelligent systems and control, and networked systems and control. Prof. Feng is an IEEE Fellow, an associate editor of IEEE Trans. Fuzzy Systems and Journal of Systems Science and Complexity, and was an associate editor of IEEE Trans. Automatic Control, IEEE Trans. Systems, Man & Cybernetics, Part C, Mechatronics, and Journal of Control Theory and Applications.
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Silva, L.F.P., Leite, V.J.S., Castelan, E.B. et al. Delay Dependent Local Stabilization Conditions for Time-delay Nonlinear Discrete-time Systems Using Takagi-Sugeno Models. Int. J. Control Autom. Syst. 16, 1435–1447 (2018). https://doi.org/10.1007/s12555-017-0526-z
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DOI: https://doi.org/10.1007/s12555-017-0526-z