Document Type : Research Paper
Author
Department of Mathematics, Thiruvalluvar University, Vellore - 632 106, Tamilnadu, India
Abstract
In this paper, global robust stability of stochastic impulsive recurrent neural networks with time-varying
delays which are represented by the Takagi-Sugeno (T-S) fuzzy models is considered. A novel Linear Matrix Inequality (LMI)-based stability criterion is obtained by using Lyapunov functional theory to guarantee the asymptotic stability of uncertain fuzzy stochastic impulsive recurrent neural networks with time-varying
delays. The results are related to the size of delay and impulses.
Finally, numerical examples and simulations are given to demonstrate the correctness of the theoretical results.
Keywords
\bibitem{SA2} S. Arik, {\it An analysis of exponential stability of
delayed neural networks with time varying delays},
Neural Netw., \textbf{17} (2004), 1027--1031.
\bibitem{SA1} S. Arik, {\it Novel results for global robust stability
of delayed neural networks}, Chaos Solitons Fractals, \textbf{39} (2009), 1604--1614.
\bibitem{Be} R. Belohlavek, {\it Fuzzy logical bidirectional
associative memory}, Information Sciences, \textbf{128}
(2000), 91--103.
\bibitem{BF} S. A. Billings and C. F. Fung, {\it Recurrent radial
basis function networks for adaptive noise cancellation},
Neural Netw., \textbf{8} (1995), 273 -- 290.
\bibitem{BI} J. J. Blake, L. P. Maguire, T. M. McGinnity,
B. Roche and L. J. McDaid, {\it The implementation of fuzzy systems,
neural networks and fuzzy neural networks using FPGAs},
Information Sciences, {\bf 112} (1998), 151--168.
\bibitem{BGFB} B. Boyd, L. Ghoui, E. Feron and V. Balakrishnan,
{\it Linear Matrix Inequalities in System and Control Theory},
philadephia, PA, SIAM, 1994.
\bibitem{CF} Y. Y. Cao and P. M. Frank, {\it Analysis and synthesis of
nonlinear time-delay system via fuzzy control approach},
IEEE Trans Fuzzy Syst., \textbf{8} (2000), 200 –- 211.
\bibitem{GNLC} P. Gahinet, A. Nemirovski, A. Laub and M. Chilali,
{\it LMI control toolbox user's guide}, Massachusetts, The Mathworks,
1995.
\bibitem{GE} Y. Gao and M. J. Er, {\it Modelling, control, and stability
analysis of non-linear systems using generalized fuzzy neural
networks}, Internat. J. Systems Science, \textbf{34} (2003), 427 --
438.
\bibitem{GKW} C. L. Giles, G. M. Kuhn and R. J. Williams, {\it Dynamic
recurrent neural networks: Theory and applications}, IEEE
Trans. Neural Netw., \textbf{5} (1994), 153 -- 156.
\bibitem{ZHG} Z. H. Guan, J. Lam and G. Chen, {\it On impulsive
autoassociative neural networks}, Neural Netw., \textbf{13}
(2000), 63–69.
\bibitem{H} J. Hale and S. M. Verduyn Lunel, {\it Introduction to
Functional Differential Equations}, New York, Springer, 1993.
\bibitem{MH} M. Han, Y. Sun and Y. Fan, {\it An improved fuzzy neural
network based on T–S model}, Expert Syst. Appl.,
\textbf{34} (2008), 2905–-2920.
\bibitem{YYH} Y. Y. Hou, T. L. Liao and Y. Y. Yan, {\it Stability
analysis of Takagi-Sugeno fuzzy cellular neural networks with time
varying delays}, IEEE Trans. Syst. Man Cybrn. Part B,
{\bf 37} \textbf{(3)} (2007), 720--726.
\bibitem{HW} S. Hu and J. Wang, {\it Global asymptotic stability and
global exponential stability of continuous time recurrent neural
networks}, IEEE Trans. Automat. Control, \textbf{47}
(2002), 802 -- 807.
\bibitem{HLM} S. Hu, X. Liao and X. Mao, {\it Stochastic Hopfield
neural networks}, J. Phys. A: Math. Gen., \textbf{9}
(2004), 47 -- 53.
\bibitem{HHL} H. Huang, D. Ho and J. Lam, {\it Stochastic stability
analysis of fuzzy Hopfield neural networks with time-varying
delays}, IEEE Trans. Circ. Syst. II, Exp. Briefs,
\textbf{52} (2005), 251 -- 255.
\bibitem{HC} H. Huang and J. Cao, {\it Exponential stability analysis of
uncertain stochastic neural networks with multiple delays},
Nonlinear Anal: R W A, \textbf{8} (2007), 646--653.
\bibitem{IJFS1} M. Khasheri, M. Bijari and S. R. Hejazi, {\it An extented fuzzy
artificial neural networks model for time series forecasting},
Iranain Journal of Fuzzy Systems, {\bf 8} \textbf{(3)}, (2011), 45--66.
\bibitem{L} V. Lakshmikantham, D. D. Bainov and P. S. Simeonov,
{\it Theory of Impulsive Differential Equations}, Singapore, World
Scientific, 1989.
\bibitem{VL} V. Lakshmikantham and X. Z. Liu, {\it Impulsive hybrid systems and
stability theory}, International Journal of Nonlinear
Differential Equations, \textbf{5}, (1999), 9–17.
\bibitem{LC1} J. Liang and J. Cao, {\it Boundedness and stability for
recurrent neural networks with variable coefficients and
time-varying delays}, Phys. Lett. A, \textbf{318} (2003), 53
-– 64.
\bibitem{LM} X. Liao and X. Mao, {\it Exponential stability and
instability of stochastic neural networks}, Stoch. Anal.
Appl., \textbf{14} (1996), 165 –- 185.
\bibitem{LZ} X. D, Liu and Q. L. Zhang, {\it New approaches to H1
controller designs based on fuzzy observers for T–S fuzzy systems
via LMI}, Automatica, \textbf{39} (2003), 1571 –- 1582.
\bibitem{BL} B. Liu and P. Shi, {\it Delay-range-dependent stability for
fuzzy BAM neural networks with time-varying delays}, Phys.
Lett. A, \textbf{373} (2009), 1830--1838.
\bibitem{IJFS2} J. Liu, Z. Gu, H. Han and S. Hu, {\it T- S fuzzy model - based memory control for discrete - time system
with random input delay}, Iran. J. Fuzzy Syst., \textbf{8(3)} (2011), 67-79.
\bibitem{LC} X. Lou and B. Cui, {\it Robust asymptotic stability of
uncertain fuzzy BAM neural networks with time-varying delays},
Fuzzy sets and systems, \textbf{158} (2007), 2746 -- 56.
\bibitem{XM} X. Mao, N. Koroleva and A. Rodkina, {\it Robust stability
of uncertain stochastic delay differential equations},
Systems Control. lett., \textbf{35} (1998), 325 -– 336.
\bibitem{MS1} M. Syed Ali and P. Balasubramaniam, {\it Robust stability
of uncertian stochastic fuzzy BAM neural networks with time
varying delay}, Phys. Lett. A, \textbf{372} (2008),
5159--5166.
\bibitem{MS2} M. Syed Ali and P. Balasubramaniam, {\it Stability analysis of uncertain fuzzy Hopfield
neural networks with time delays}, Communic.
Nonlin. Numeric. Simul., \textbf{14} (2009), 2776--2783.
\bibitem{MS3} M. Syed Ali and P. Balasubramaniam, {\it Robust stability of uncertain fuzzy Cohen-Grossberg BAM
neural networks with time-varying delays}, Expert Syst.
Appl., \textbf{36} (2009), 10583--10588.
\bibitem{TS} T. Takagi and M. Sugeno, {\it Fuzzy identification of
systems and its applications to modeling and control}, IEEE
Trans. Syst. Man, Cybern., \textbf{15} (1985), 116 -- 132.
\bibitem{TY1} T. Yang, {\it Impulsive control}, IEEE Trans.
Automat. Control, \textbf{44}(5) (1999), 1081-1083.
\bibitem{TY2} T. Yang, {\it Impulsive system and control:theory and
applications}, Huntington, NY, Nova Science Publishers, 2001.
\bibitem{ZP} Y. Zhang and A. H. Pheng, {\it Stability of fuzzy systems
with bounded uncertain delays}, IEEE Trans. Fuzzy Syst.,
\textbf{10} (2002), 92 -– 97.
\bibitem{Z1} H. Zhang, {\it Robust exponential stability of recurrent
neural networks with multiple time varying delays}, IEEE
Trans. Circ. Syst. II, Exp. Briefs, \textbf{54} (2007), 730 -- 734.
\bibitem{YZ1} Y. Zhang and J. T. Sun,{\it Boundedness of the solutions
of impulsive differential systems with time-varying delay},
Appl. Math. Comput., \textbf{154} (2004), 279–288.
\bibitem{YZ2} Y. Zhang and J. T. Sun, {\it Stability of impulsive neural
networks with time delays}, Phys. Lett. A, \textbf{348}
(2005), 44–-50.
\bibitem{QZ} Q. Zhang, X. Wei and J. Xu, {\it Delay-dependent exponential
stability of cellular neural networks with time varying delays},
Chaos Solitons Fractals, \textbf{23} (2005), 1363-1369.
delayed neural networks with time varying delays},
Neural Netw., \textbf{17} (2004), 1027--1031.
\bibitem{SA1} S. Arik, {\it Novel results for global robust stability
of delayed neural networks}, Chaos Solitons Fractals, \textbf{39} (2009), 1604--1614.
\bibitem{Be} R. Belohlavek, {\it Fuzzy logical bidirectional
associative memory}, Information Sciences, \textbf{128}
(2000), 91--103.
\bibitem{BF} S. A. Billings and C. F. Fung, {\it Recurrent radial
basis function networks for adaptive noise cancellation},
Neural Netw., \textbf{8} (1995), 273 -- 290.
\bibitem{BI} J. J. Blake, L. P. Maguire, T. M. McGinnity,
B. Roche and L. J. McDaid, {\it The implementation of fuzzy systems,
neural networks and fuzzy neural networks using FPGAs},
Information Sciences, {\bf 112} (1998), 151--168.
\bibitem{BGFB} B. Boyd, L. Ghoui, E. Feron and V. Balakrishnan,
{\it Linear Matrix Inequalities in System and Control Theory},
philadephia, PA, SIAM, 1994.
\bibitem{CF} Y. Y. Cao and P. M. Frank, {\it Analysis and synthesis of
nonlinear time-delay system via fuzzy control approach},
IEEE Trans Fuzzy Syst., \textbf{8} (2000), 200 –- 211.
\bibitem{GNLC} P. Gahinet, A. Nemirovski, A. Laub and M. Chilali,
{\it LMI control toolbox user's guide}, Massachusetts, The Mathworks,
1995.
\bibitem{GE} Y. Gao and M. J. Er, {\it Modelling, control, and stability
analysis of non-linear systems using generalized fuzzy neural
networks}, Internat. J. Systems Science, \textbf{34} (2003), 427 --
438.
\bibitem{GKW} C. L. Giles, G. M. Kuhn and R. J. Williams, {\it Dynamic
recurrent neural networks: Theory and applications}, IEEE
Trans. Neural Netw., \textbf{5} (1994), 153 -- 156.
\bibitem{ZHG} Z. H. Guan, J. Lam and G. Chen, {\it On impulsive
autoassociative neural networks}, Neural Netw., \textbf{13}
(2000), 63–69.
\bibitem{H} J. Hale and S. M. Verduyn Lunel, {\it Introduction to
Functional Differential Equations}, New York, Springer, 1993.
\bibitem{MH} M. Han, Y. Sun and Y. Fan, {\it An improved fuzzy neural
network based on T–S model}, Expert Syst. Appl.,
\textbf{34} (2008), 2905–-2920.
\bibitem{YYH} Y. Y. Hou, T. L. Liao and Y. Y. Yan, {\it Stability
analysis of Takagi-Sugeno fuzzy cellular neural networks with time
varying delays}, IEEE Trans. Syst. Man Cybrn. Part B,
{\bf 37} \textbf{(3)} (2007), 720--726.
\bibitem{HW} S. Hu and J. Wang, {\it Global asymptotic stability and
global exponential stability of continuous time recurrent neural
networks}, IEEE Trans. Automat. Control, \textbf{47}
(2002), 802 -- 807.
\bibitem{HLM} S. Hu, X. Liao and X. Mao, {\it Stochastic Hopfield
neural networks}, J. Phys. A: Math. Gen., \textbf{9}
(2004), 47 -- 53.
\bibitem{HHL} H. Huang, D. Ho and J. Lam, {\it Stochastic stability
analysis of fuzzy Hopfield neural networks with time-varying
delays}, IEEE Trans. Circ. Syst. II, Exp. Briefs,
\textbf{52} (2005), 251 -- 255.
\bibitem{HC} H. Huang and J. Cao, {\it Exponential stability analysis of
uncertain stochastic neural networks with multiple delays},
Nonlinear Anal: R W A, \textbf{8} (2007), 646--653.
\bibitem{IJFS1} M. Khasheri, M. Bijari and S. R. Hejazi, {\it An extented fuzzy
artificial neural networks model for time series forecasting},
Iranain Journal of Fuzzy Systems, {\bf 8} \textbf{(3)}, (2011), 45--66.
\bibitem{L} V. Lakshmikantham, D. D. Bainov and P. S. Simeonov,
{\it Theory of Impulsive Differential Equations}, Singapore, World
Scientific, 1989.
\bibitem{VL} V. Lakshmikantham and X. Z. Liu, {\it Impulsive hybrid systems and
stability theory}, International Journal of Nonlinear
Differential Equations, \textbf{5}, (1999), 9–17.
\bibitem{LC1} J. Liang and J. Cao, {\it Boundedness and stability for
recurrent neural networks with variable coefficients and
time-varying delays}, Phys. Lett. A, \textbf{318} (2003), 53
-– 64.
\bibitem{LM} X. Liao and X. Mao, {\it Exponential stability and
instability of stochastic neural networks}, Stoch. Anal.
Appl., \textbf{14} (1996), 165 –- 185.
\bibitem{LZ} X. D, Liu and Q. L. Zhang, {\it New approaches to H1
controller designs based on fuzzy observers for T–S fuzzy systems
via LMI}, Automatica, \textbf{39} (2003), 1571 –- 1582.
\bibitem{BL} B. Liu and P. Shi, {\it Delay-range-dependent stability for
fuzzy BAM neural networks with time-varying delays}, Phys.
Lett. A, \textbf{373} (2009), 1830--1838.
\bibitem{IJFS2} J. Liu, Z. Gu, H. Han and S. Hu, {\it T- S fuzzy model - based memory control for discrete - time system
with random input delay}, Iran. J. Fuzzy Syst., \textbf{8(3)} (2011), 67-79.
\bibitem{LC} X. Lou and B. Cui, {\it Robust asymptotic stability of
uncertain fuzzy BAM neural networks with time-varying delays},
Fuzzy sets and systems, \textbf{158} (2007), 2746 -- 56.
\bibitem{XM} X. Mao, N. Koroleva and A. Rodkina, {\it Robust stability
of uncertain stochastic delay differential equations},
Systems Control. lett., \textbf{35} (1998), 325 -– 336.
\bibitem{MS1} M. Syed Ali and P. Balasubramaniam, {\it Robust stability
of uncertian stochastic fuzzy BAM neural networks with time
varying delay}, Phys. Lett. A, \textbf{372} (2008),
5159--5166.
\bibitem{MS2} M. Syed Ali and P. Balasubramaniam, {\it Stability analysis of uncertain fuzzy Hopfield
neural networks with time delays}, Communic.
Nonlin. Numeric. Simul., \textbf{14} (2009), 2776--2783.
\bibitem{MS3} M. Syed Ali and P. Balasubramaniam, {\it Robust stability of uncertain fuzzy Cohen-Grossberg BAM
neural networks with time-varying delays}, Expert Syst.
Appl., \textbf{36} (2009), 10583--10588.
\bibitem{TS} T. Takagi and M. Sugeno, {\it Fuzzy identification of
systems and its applications to modeling and control}, IEEE
Trans. Syst. Man, Cybern., \textbf{15} (1985), 116 -- 132.
\bibitem{TY1} T. Yang, {\it Impulsive control}, IEEE Trans.
Automat. Control, \textbf{44}(5) (1999), 1081-1083.
\bibitem{TY2} T. Yang, {\it Impulsive system and control:theory and
applications}, Huntington, NY, Nova Science Publishers, 2001.
\bibitem{ZP} Y. Zhang and A. H. Pheng, {\it Stability of fuzzy systems
with bounded uncertain delays}, IEEE Trans. Fuzzy Syst.,
\textbf{10} (2002), 92 -– 97.
\bibitem{Z1} H. Zhang, {\it Robust exponential stability of recurrent
neural networks with multiple time varying delays}, IEEE
Trans. Circ. Syst. II, Exp. Briefs, \textbf{54} (2007), 730 -- 734.
\bibitem{YZ1} Y. Zhang and J. T. Sun,{\it Boundedness of the solutions
of impulsive differential systems with time-varying delay},
Appl. Math. Comput., \textbf{154} (2004), 279–288.
\bibitem{YZ2} Y. Zhang and J. T. Sun, {\it Stability of impulsive neural
networks with time delays}, Phys. Lett. A, \textbf{348}
(2005), 44–-50.
\bibitem{QZ} Q. Zhang, X. Wei and J. Xu, {\it Delay-dependent exponential
stability of cellular neural networks with time varying delays},
Chaos Solitons Fractals, \textbf{23} (2005), 1363-1369.
)