New Stability Criteria for Recurrent Neural Networks with a Time-varying Delay

Hong-Bing Zeng; Shen-Ping Xiao; Bin Liu

doi:10.1007/s11633-010-0564-y

February 2011

Article Contents

Hong-Bing Zeng, Shen-Ping Xiao and Bin Liu. New Stability Criteria for Recurrent Neural Networks with a Time-varying Delay. International Journal of Automation and Computing, vol. 8, no. 1, pp. 128-133, 2011. doi: 10.1007/s11633-010-0564-y

Cite as: Hong-Bing Zeng, Shen-Ping Xiao and Bin Liu. New Stability Criteria for Recurrent Neural Networks with a Time-varying Delay. International Journal of Automation and Computing, vol. 8, no. 1, pp. 128-133, 2011. doi: 10.1007/s11633-010-0564-y

New Stability Criteria for Recurrent Neural Networks with a Time-varying Delay

1.
School of Electrical and Information Engineering, Hunan University of Technology, Zhuzhou 412008, PRC;
2.
School of Information Science and Engineering, Central South University, Changsha 410083, PRC 3School of Engineering, Australian National University, ACT 0200, Australia

Author Biography:

Shen-Ping Xiao received the B.Sc.degree in engineering from Northeastern Univer-sity,Shenyang,PRC in 1988,and Ph.D.
Corresponding author: Hong-Bing Zeng received the B.Sc.degree in electrical engineering from Tianjin University of Technology and Education, Tianjin,PRC in 2003,and M.Sc.

Received: 2009-12-09

Fund Project:

supported by National Natural Science Foundation of China (No.60874025);Natural Science Foundation of Hunan Province of China (No.10JJ6098)

Abstract

This paper deals with the stability of static recurrent neural networks (RNNs) with a time-varying delay. An augmented Lyapunov-Krasovskii functional is employed, in which some useful terms are included. Furthermore, the relationship among the timevarying delay, its upper bound and their difierence, is taken into account, and novel bounding techniques for 1(t) are employed. As a result, without ignoring any useful term in the derivative of the Lyapunov-Krasovskii functional, the resulting delay-dependent criteria show less conservative than the existing ones. Finally, a numerical example is given to demonstrate the effectiveness of the proposed methods.
- Stability,
- recurrent neural networks (RNNs),
- time-varying delay,
- delay-dependent,
- augmented Lyapunov-Krasovskii functional

References

[1]	P.J.Angeline,G.M.Saunders,J.B.Pollack.An evo- lutionary algorithm that constructs recurrent neural net- works.IEEE Transactions on Neural Networks,vol.5,no.1, pp.54-65,1994.
[2]	C.C.Ku,K.Y.Lee.Diagonal recurrent neural networks for dynamic systems control.IEEE Transactions on Neural Networks,vol.6,no.1,pp.144-156,1995.
[3]	A.N.Michel,D.Liu.Qualitative Analysis and Synthesis of Recurrent Neural Networks,New York,USA:Marcel Dekker,2002.
[4]	Y.R.Liu,Z.D.Wang,X.H.Liu.Global exponential stability of generalized recurrent neural networks with dis- crete and distributed delays.Neural Networks,vol.19,no.5, pp.667-675,2006.
[5]	T.Li,Q.Luo,C.Y.Sun,B.Y.Zhang.Exponential stability of recurrent neural networks with time-varying discrete and distributed delays.Nonlinear Analysis:Real World Appli- cations,vol.10,no.4,pp.2581-2589,2009.
[6]	J.D.Cao,J.Wang.Global asymptotic and robust stability of recurrent neural networks with time delays.IEEE Trans- actions on Circuits Systems I:Regular Papers,vol.52,no.2, pp.417-426,2005.
[7]	C.G.Li,X.F.Liao.Robust stability and robust periodicity of delayed recurrent neural networks with noise disturbance. IEEE Transactions on Circuits Systems I:Regular Papers, vol.53,no.10,pp.2265-2273,2006.
[8]	X.F.Liao,K.W.Wong.Robust stability of interval bidi- rectional associative memory neural network with time de- lays.IEEE Transactions on Systems,Man,and Cybernet- ics,Part B:Cybernetics,vol.34,no.2,pp.1142-1154,2004.
[9]	Y.R.Liu,Z.D.Wang,A.Serranob,X.H.Liu.Discrete- time recurrent neural networks with time-varying delays: Exponential stability analysis.Physics Letters A,vol.362, no.5-6,pp.480-488,2007.
[10]	Z.G.Zeng,J.Wang.Improved conditions for global ex- ponential stability of recurrent neural networks with time- varying delays.IEEE Transactions on Neural Networks, vol.17,no.3,pp.623-635,2006.
[11]	Z.G.Zeng,J.Wang,X.X.Liao.Global exponential stabil- ity of a general class of recurrent neural networks with time- varying delays.IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications,vol.50,no.10, pp.1353-1358,2003.
[12]	M.Wu,F.Liu,P.Shi,Y.He,R.Yokoyama.Improved free- weighting matrix approach for stability analysis of discrete- time recurrent neural networks with time-varying delay. IEEE Transactions on Circuits and Systems II:Express Briefs,vol.55,no.7,pp.690-694,2008.
[13]	J.Sun,G.P.Liu,J.Chen,D.Rees.Improved stability criteria for neural networks with time-varying delay.Physics Letters A,vol.373,no.3,pp.342-348,2009.
[14]	O.M.Kwon,J.H.Park.Improved delay-dependent stabil- ity criterion for neural networks with time-varying delays. Physics Letters A,vol.373,no.5,pp.529-535,2009.
[15]	Y.Y.Wu,T.Li,Y.Q.Wu.Improved exponential stabil- ity criteria for recurrent neural networks with time-varying discrete and distributed delays.International Journal of Au- tomation and Computing,vol.7,no.2,pp.199-204,2010.
[16]	G.D.Zong,J.Liu.New delay-dependent global asymptotic stability condition for Hopfield neural networks with time- varying delays.International Journal of Automation and Computing,vol.6,no.4,pp.415-419,2009.
[17]	Z.B.Xu,H.Qiao,J.G.Peng,B.Zhang.A comparative study on two modeling approaches in neural networks.Neu- ral Networks,vol.17,no.1,pp.73-85,2004.
[18]	J.L.Liang,J.D.Cao.A based-on LMI stability criterion for delayed recurrent neural networks.Chaos,Solitons and Fractals,vol.28,no.1,pp.154-160,2006.
[19]	H.Y.Shao.Delay-dependent stability for recurrent neural networks with time-varying delays.IEEE Transactions on Neural networks,vol.19,no.9,pp.1647-1651,2008.
[20]	Y.He,Q.G.Wang,C.Lin,M.Wu.Augmented Lyapunov functional and delay-dependent stability criteria for neu- tral systems.International Journal of Robust and Nonlinear Control,vol.15,no.18,pp.923-933,2005.
[21]	X.M.Zhang,Q.L.Han.New Lyapunov-Krasovskii func- tionals for global asymptotic stability criteria of delayed neural networks.IEEE Transactions on Neural networks, vol.20,no.3,pp.533-539,2009.

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通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

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New Stability Criteria for Recurrent Neural Networks with a Time-varying Delay

1. School of Electrical and Information Engineering, Hunan University of Technology, Zhuzhou 412008, PRC;

2. School of Information Science and Engineering, Central South University, Changsha 410083, PRC 3School of Engineering, Australian National University, ACT 0200, Australia

Corresponding author: Hong-Bing Zeng received the B.Sc.degree in electrical engineering from Tianjin University of Technology and Education, Tianjin,PRC in 2003,and M.Sc.

Received: 2009-12-09

Fund Project:

supported by National Natural Science Foundation of China (No.60874025);Natural Science Foundation of Hunan Province of China (No.10JJ6098)

Key words:

Stability /
recurrent neural networks (RNNs) /
time-varying delay /
delay-dependent /
augmented Lyapunov-Krasovskii functional

Abstract: This paper deals with the stability of static recurrent neural networks (RNNs) with a time-varying delay. An augmented Lyapunov-Krasovskii functional is employed, in which some useful terms are included. Furthermore, the relationship among the timevarying delay, its upper bound and their difierence, is taken into account, and novel bounding techniques for 1(t) are employed. As a result, without ignoring any useful term in the derivative of the Lyapunov-Krasovskii functional, the resulting delay-dependent criteria show less conservative than the existing ones. Finally, a numerical example is given to demonstrate the effectiveness of the proposed methods.

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Reference (21)

[1]	P.J.Angeline,G.M.Saunders,J.B.Pollack.An evo- lutionary algorithm that constructs recurrent neural net- works.IEEE Transactions on Neural Networks,vol.5,no.1, pp.54-65,1994.
[2]	C.C.Ku,K.Y.Lee.Diagonal recurrent neural networks for dynamic systems control.IEEE Transactions on Neural Networks,vol.6,no.1,pp.144-156,1995.
[3]	A.N.Michel,D.Liu.Qualitative Analysis and Synthesis of Recurrent Neural Networks,New York,USA:Marcel Dekker,2002.
[4]	Y.R.Liu,Z.D.Wang,X.H.Liu.Global exponential stability of generalized recurrent neural networks with dis- crete and distributed delays.Neural Networks,vol.19,no.5, pp.667-675,2006.
[5]	T.Li,Q.Luo,C.Y.Sun,B.Y.Zhang.Exponential stability of recurrent neural networks with time-varying discrete and distributed delays.Nonlinear Analysis:Real World Appli- cations,vol.10,no.4,pp.2581-2589,2009.
[6]	J.D.Cao,J.Wang.Global asymptotic and robust stability of recurrent neural networks with time delays.IEEE Trans- actions on Circuits Systems I:Regular Papers,vol.52,no.2, pp.417-426,2005.
[7]	C.G.Li,X.F.Liao.Robust stability and robust periodicity of delayed recurrent neural networks with noise disturbance. IEEE Transactions on Circuits Systems I:Regular Papers, vol.53,no.10,pp.2265-2273,2006.
[8]	X.F.Liao,K.W.Wong.Robust stability of interval bidi- rectional associative memory neural network with time de- lays.IEEE Transactions on Systems,Man,and Cybernet- ics,Part B:Cybernetics,vol.34,no.2,pp.1142-1154,2004.
[9]	Y.R.Liu,Z.D.Wang,A.Serranob,X.H.Liu.Discrete- time recurrent neural networks with time-varying delays: Exponential stability analysis.Physics Letters A,vol.362, no.5-6,pp.480-488,2007.
[10]	Z.G.Zeng,J.Wang.Improved conditions for global ex- ponential stability of recurrent neural networks with time- varying delays.IEEE Transactions on Neural Networks, vol.17,no.3,pp.623-635,2006.
[11]	Z.G.Zeng,J.Wang,X.X.Liao.Global exponential stabil- ity of a general class of recurrent neural networks with time- varying delays.IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications,vol.50,no.10, pp.1353-1358,2003.
[12]	M.Wu,F.Liu,P.Shi,Y.He,R.Yokoyama.Improved free- weighting matrix approach for stability analysis of discrete- time recurrent neural networks with time-varying delay. IEEE Transactions on Circuits and Systems II:Express Briefs,vol.55,no.7,pp.690-694,2008.
[13]	J.Sun,G.P.Liu,J.Chen,D.Rees.Improved stability criteria for neural networks with time-varying delay.Physics Letters A,vol.373,no.3,pp.342-348,2009.
[14]	O.M.Kwon,J.H.Park.Improved delay-dependent stabil- ity criterion for neural networks with time-varying delays. Physics Letters A,vol.373,no.5,pp.529-535,2009.
[15]	Y.Y.Wu,T.Li,Y.Q.Wu.Improved exponential stabil- ity criteria for recurrent neural networks with time-varying discrete and distributed delays.International Journal of Au- tomation and Computing,vol.7,no.2,pp.199-204,2010.
[16]	G.D.Zong,J.Liu.New delay-dependent global asymptotic stability condition for Hopfield neural networks with time- varying delays.International Journal of Automation and Computing,vol.6,no.4,pp.415-419,2009.
[17]	Z.B.Xu,H.Qiao,J.G.Peng,B.Zhang.A comparative study on two modeling approaches in neural networks.Neu- ral Networks,vol.17,no.1,pp.73-85,2004.
[18]	J.L.Liang,J.D.Cao.A based-on LMI stability criterion for delayed recurrent neural networks.Chaos,Solitons and Fractals,vol.28,no.1,pp.154-160,2006.
[19]	H.Y.Shao.Delay-dependent stability for recurrent neural networks with time-varying delays.IEEE Transactions on Neural networks,vol.19,no.9,pp.1647-1651,2008.
[20]	Y.He,Q.G.Wang,C.Lin,M.Wu.Augmented Lyapunov functional and delay-dependent stability criteria for neu- tral systems.International Journal of Robust and Nonlinear Control,vol.15,no.18,pp.923-933,2005.
[21]	X.M.Zhang,Q.L.Han.New Lyapunov-Krasovskii func- tionals for global asymptotic stability criteria of delayed neural networks.IEEE Transactions on Neural networks, vol.20,no.3,pp.533-539,2009.

New Stability Criteria for Recurrent Neural Networks with a Time-varying Delay

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