Robust stability of fuzzy Markov type Cohen-Grossberg neural networks by delay decomposition approach

Document Type: Research Paper

Authors

1 Department of Social Sciences, Tamil Nadu Agricultural University, Coim- batore - 641 003, Tamilnadu, India

2 Department of Mathematics, Gandhigram Rural Institute - Deemed University, Gandhigram - 624 302, Tamilnadu, India

3 Department of Computer Science, Government Arts College, Melur, Madurai - 625 106, Tamilnadu, India

Abstract

In this paper, we investigate the delay-dependent robust stability of fuzzy Cohen-Grossberg neural networks with Markovian jumping parameter and mixed time varying delays by delay decomposition method. A new Lyapunov-Krasovskii functional (LKF) is constructed by nonuniformly dividing discrete delay interval into multiple subinterval, and choosing proper functionals with different weighting matrices corresponding to different subintervals in the LKFs. A new delay-dependent stability condition is derived with Markovian jumping parameters by T-S fuzzy model. Based on the linear matrix inequality (LMI) technique, maximum admissible upper bound (MAUB) for the discrete and distributed delays are calculated by the LMI Toolbox in MATLAB. Numerical examples are given to illustrate the effectiveness of the proposed method.

Keywords


bibitem{AO} S. Arik and Z. Orman, textit{Global stability analysis of Cohen-Grossberg neural networks with time-varying delays}, Phys. Lett. A, textbf{341} (2005), 410-421.

bibitem{YYC} Y. Y. Cao and P. M. Frank, textit{Stability analysis and synthesis of nonlinear time-delay systems via linear Takagi-Sugeno fuzzy models}, Fuzzy Sets and Systems, textbf{124} (2001), 213-229.

bibitem{CG} M. A. Cohen and S. Grossberg, textit{Absolute stability of global pattern formation and parallel memory storage by competitive neural networks}, IEEE Trans. Syst. Man Cybern., textbf{13} (1983), 815-826.

bibitem{GAO} M. Gao, B. Cui and X. Lou, textit{Robust exponential stability of Markovian jumping neural networks with time-varying delay}, Int. J. Neural Syst., textbf{18(3)} (2008), 207-218.

bibitem{QHAN} Q. L. Han, textit{A new delay-dependent stability criterion for linear neutral systems with norm-bounded uncertainties in all system matrices}, Int. J. Syst. Sci., textbf{36} (2005), 469-475.

bibitem{HWG} J. Hu, Z. Wang and H. Gao, textit{A delay-fractioning approach to robust sliding mode control for discrete-time stochastic systems with randomly occurring nonlinearities}, IMA J. Mathematical Control Inf, textbf{28} (2011), 345-363.

bibitem{HWGS} J. Hu, Z. Wang, H. Gao and L. K. Stergioulas, textit{Robust sliding mode control for discrete stochastic systems with mixed time-delays, randomly occurring uncertainties and randomly occurring nonlinearities}, IEEE Trans. Industrial Electronics, textbf{59} (2012), 3008-3015.

bibitem{HWNS} J. Hu, Z. Wang, Y. Niu and L. K. Stergioulas, textit{$H_infty$ sliding mode observer design for a class of nonlinear discrete time-delay systems: a delay-fractioning approach}, Int. J. Robust Nonlinear Control, textbf{22} (2012), 1806-1826.

bibitem{JZW} C. Ji, H. G. Zhang and Y. Wei, textit{LMI approach for global robust stability of Cohen-Grossberg neural networks with multiple delays}, Neurocomputing, textbf{71} (2008), 475-485.

bibitem{HLI} H. Li, B. Chen, Q. Zhou and W. Qian,  textit{Robust stability for uncertain delayed fuzzy Hopfield neural networks with Markovian jumping parameters}, IEEE Trans. Syst. Man. Cybern. B, textbf{39(1)} (2009), 94-102.

bibitem{TLIS1} T. Li and S. M. Fei, textit{Stability analysis of Cohen-Grossberg neural networks with time-varying and distributed delays}, Neurocomputing, textbf{71(4-6)} (2008), 1069-1081.

bibitem{LSF} T. Li, A. G. Song and S. M. Fei, textit{Robust stability of stochastic Cohen-Grossberg neural networks with mixed time-varying delays}, Neurocomputing, textbf{73} (2009), 542-551.

bibitem{MAR} M. Mariton, textit{Jump linear systems in Automatic control}, New York: Marcel-Dekker, 1990.

bibitem{PPKL2} J. H. Park, C. H. Park, O. M. Kwon and S. M. Lee, textit{A new stability criterion for bidirectional associative memory neural networks of neutral-type}, Appl. Math. Comput., textbf{199} (2008), 716-722.

bibitem{PPKL1} J. H. Park, C. H. Park, O. M. Kwon and S. M. Lee, textit{Simplified stability criteria for fuzzy Markovian jumping Hopfield neural networks of neutral type with interval time-varying delays}, Expert Syst. with Appl., textbf{39} (2012), 5625-5633.

bibitem{LBR} L. B. Rong, textit{LMI-based criteria for robust stability of Cohen-Grossberg neural networks with delays}, Phys. Lett. A, textbf{339} (2005), 63-73.

bibitem{SY} L. Sheng and H. Yang, textit{Robust stability of uncertain Markovian jumping Cohen-Grossberg neural networks with mixed time-varying delays}, Chaos Solitons Fractals, textbf{42} (2009), 2120-2128.

bibitem{QKS} Q. K. Song and J. D. Cao, textit{Stability analysis of Cohen-Grossberg neural networks with both time-varying and continuously distributed delays}, J. Comput. Appl. Math., textbf{197(1)} (2006), 188-203.

bibitem{WWS} W. W. Su and Y. M. Chen, textit{Global robust stability criteria of stochastic Cohen-Grossberg neural networks with discrete and distributed delays}, Commun. Nonlinear Sci. Numerical Simulat., textbf{14(2)} (2009), 520-528.

bibitem{SAL} M. Syed Ali and P. Balasubramaniam, textit{Robust stability of uncertain fuzzy Cohen-Grossberg BAM neural networks with time-varying delays}, Expert Syst. Appl., textbf{36} (2009), 10583-10588.

bibitem{TTS} T. Takagi and M. Sugeno, textit{Fuzzy identification of systems and its application to modeling and control}, IEEE Trans. Syst. Man. Cybern., textbf{15} (1985), 116-132.

bibitem{WLL} Z. Wang, Y. Liu and X. Liu, textit{Exponential stabilization of a class of stochastic system with Markovian jump parameters and mode-dependent mixed time-delays}, IEEE Trans. Automatic Control, textbf{55} (2010), 1656-1662.

bibitem{WLWL} Z. Wang, Y. Liu, G. Wei and X. Liu, textit{A note on control of a class of discrete-time stochastic systems with distributed delays and nonlinear disturbances}, Automatica, textbf{46} (2010), 543-548.

bibitem{WWL} Y. Wang, Z. Wang and J. Liang, textit{On robust stability of stochastic genetic regulatory networks with time-delays: a delay fractioning approach}, IEEE Trans. Syst. Man. Cybern. B, textbf{40} (2010), 729-740.

bibitem{HNC} H. N. Wu and K. Y. Cai, textit{Mode-independent robust stabilization for uncertain Markovian jump nonlinear systems via fuzzy control}, IEEE Trans. Syst. Man. Cybern. B textbf{36} (2006), 509-519.

bibitem{WPSC2} Z. G. Wu, J. H. Park, H. Su and J. Chu, textit{New results on exponential passivity of neural networks with time-varying delays}, Nonlinear Anal. B: Real World Appl., textbf{13} (2012), 1593-1599.

bibitem{WPSC1} Z. G. Wu, J. H. Park, H. Su and J. Chu, textit{Passivity analysis of Markov jump neural networks with mixed time-delays and piecewise-constant transition rates}, Nonlinear Anal. B: Real World Appl., textbf{13} (2012), 2423-2431.

bibitem{XIE} L. Xie, textit{Output feedback $H_{infty}$ control of systems with parameter uncertainty}, Int. J. Control, textbf{63} (1996), 741-750.

bibitem{XX} W. Xiong and B. Xu, textit{Some criteria for robust stability of Cohen-Grossberg neural networks with delays}, Chaos Soliton Fractal, textbf{36(5)} (2008), 1357-1365.

bibitem{YC} K. Yuan and J. Cao, textit{An analysis of global asymptotic stability of delayed Cohen-Grossberg neural networks via nonsmooth analysis}, IEEE Trans. Circuits Syst. I, Reg. Papers, textbf{52} (2005), 1854-1861.

bibitem{ZW} H. Zhang and Y. Wang, textit{Stability analysis of Markovian jumping stochastic Cohen-Grossberg neural networks with mixed time delays}, IEEE Trans. Neural Networks, textbf{19} (2008), 366-370.

bibitem{ZRZ} J. Zhang, D. Ren and W. Zhang, textit{Global exponential stability of fuzzy Cohen-Grossberg neural networks with variable delays and distributed delays}, Lecture notes in Computer Science, (2007), 66-74.

bibitem{ZXM} X. M. Zhang and Q. L. Han, textit{New Lyapunov-Krasovskii functionals for global asymptotic stability of delayed neural networks}, IEEE Trans. Neural Netw., textbf{20(3)} (2009), 533-539.