Stability Analysis for Recurrent Neural Networks with Time-varying Delay

Yuan-Yuan Wu; Yu-Qiang Wu

doi:10.1007/s11633-009-0223-3

August 2009

Article Contents

Yuan-Yuan Wu and Yu-Qiang Wu. Stability Analysis for Recurrent Neural Networks with Time-varying Delay. International Journal of Automation and Computing, vol. 6, no. 3, pp. 223-227, 2009. doi: 10.1007/s11633-009-0223-3

Cite as: Yuan-Yuan Wu and Yu-Qiang Wu. Stability Analysis for Recurrent Neural Networks with Time-varying Delay. International Journal of Automation and Computing, vol. 6, no. 3, pp. 223-227, 2009. doi: 10.1007/s11633-009-0223-3

Stability Analysis for Recurrent Neural Networks with Time-varying Delay

Yuan-Yuan Wu¹,
Yu-Qiang Wu²

1.
School of Automation, Southeast University, Nanjing 210096, PRC;
2.
Research Institute of Automation, Qufu Normal University, Qufu 273165, PRC

Received: 2008-09-22

Fund Project:

supported by National Natural Science Foundation of China (No.60674027)

Abstract

This paper is concerned with the stability analysis for static recurrent neural networks (RNNs) with time-varying delay. By Lyapunov functional method and linear matrix inequality technique, some new delay-dependent conditions are established to ensure the asymptotic stability of the neural network. Expressed in linear matrix inequalities (LMIs), the proposed delay-dependent stability conditions can be checked using the recently developed algorithms. A numerical example is given to show that the obtained conditions can provide less conservative results than some existing ones.
- Static neural networks,
- time-varying delay,
- asymptotical stability,
- delay-dependent,
- linear matrix inequalities (LMIs)

References

[1]	J.Cao,L.Wang.Exponential Stability and Periodic Oscillatory Solution in BAM Networks with Delays.IEEE Transactions on Neural Networks,vol.13,no.2,pp.457-463,2002.
[2]	S.Arik.Global Asymptotic Stability of a Larger Class of Neural Networks with Constant Time Delay.Physics Letters A,vol.311,no.6,pp.504-511,2002.
[3]	T.Chen,L.Rong.Delay-independent Stability Analysis of Cohen-Grossberg Neural Networks.Physics Letters A,vol.317,no.5-6,pp.436-449,2003.
[4]	V.Singh.A Novel Global Robust Stability Criterion for Neural Networks with Delay.Physics Letters A,vol.337,no.4-6,pp.369-373.2005.
[5]	V.Singh.Simplified LMI Condition for Global Asymptotic Stability of Delayed Neural Networks.Chaos,Solitons and Fractals,vol.29,no.2,pp.470-473,2006.
[6]	Y.Liu,Z.Wang,X.Liu.Global Exponential Stability of Generalized Recurrent Neural Networks with Discrete and Distributed Delays.Neural Networks,vol.19,no.5,pp.667-675,2006.
[7]	T.Li,Q.Luo,C.Y.Sun,B.Y.Zhang.Exponential Stability of Recurrent Neural Networks with Time-varying Delay.Nonlinear Analysis:Real World Applications,to be published.
[8]	J.Yu,K.Zhang,S.Fei,T.Li.Simplified Exponential Stability Analysis for Recurrent Neural Networks with Discrete and Distributed Time-varying Delays.Applied Mathematics and Computation,vol.205,no.1,pp.465-474,2008.
[9]	T.Li,S.Fei.Exponential State Estimation for Recurrent Neural Networks with Distributed Delays.Neurocomputing,vol.71,no.1-3,pp.428-438,2007.
[10]	Q.Song,Z.Wang.Neural Networks with Discrete and Distributed Time-varying Delays:A General Stability Analysis.Chaos,Solitons and Fractals,vol.37,no.5,pp.1538-1547,2008.
[11]	Z.Wang,Y.Liu,X.Liu.On Global Asymptotic Stability of Neural Networks with Discrete and Distributed Delays.Physics Letters A,vol.345,no.4-6,pp.299-308,2005.
[12]	J.H.Park,H.J.Cho.A Delay-dependent Asymptotic Stability Criterion of Cellular Neural Networks with Timevarying Discrete and Distributed Delays.Chaos,Solitons and Fractals,vol.33,no.2,pp.436-442,2007.
[13]	Z.Wang,H.Shu,Y.Liu,D.W.C.Ho,X.Liu.Robust Stability Analysis of Generalized Neural Networks with Discrete and Distributed Time Delays.Chaos,Solitons and Fractal,vol.30,no.4,pp.886-896,2006.
[14]	H.J.Cho,J.H.Park.Novel Delay-dependent Robust Stability Criterion of Delayed Cellular Neural Networks.Chaos,Solitons and Fractals,vol.32,no.3,pp.1194-1200,2007.
[15]	R.S.Gau,C.H.Lien,J.G.Hsieh.Global Exponential Stability for Uncertain Cellular Neural Networks with Multiple Time-varying Delays via LMI Approach.Chaos,Solitons and Fractals,vol.32,no.4,pp.1258-1267,2007.
[16]	Y.G.Chen,W.P.Bi.New Robust Exponential Stability Analysis for Uncertain Neural Networks with Time-varying Delay.International Journal of Automation and Computing,vol.5,no.4,pp.395-400,2008.
[17]	Y.He,M.Wu,J.H.She.Delay-dependent Exponential Stability of Delayed Neural Networks with Time-varying Delay.IEEE Transactions on Circuits and Systems Ⅱ:Express Briefs,vol.53,no.7,pp.553-557,2006.
[18]	Q.Zhang,X.Wei,J.Xu.Delay-dependent Exponential Stability Celluar Neural Networks with Time-varying Delays.Chaos,Solitons and Fractals,vol.23,no.4,pp.1363-1369,2005.
[19]	W.H.Chen,X.M.Lu,Z.H.Guan,W.X.Zheng.Delaydependent Exponential Stability of Neural Networks with Variable Delay:An LMI Approach.IEEE Transactions on Circuits and Systems Ⅱ:Express Briefs,vol.53,no.9,pp.837-842,2006.
[20]	C.C.Hua,C.N.Long,X.P.Guan.New Results on Stability Analysis of Neural Networks with Time-varying Delays.Physics Letters A,vol.352,no.4-5,pp.335-340,2006.
[21]	Y.He,Q.G.Wang,M.Wu.LMI-based Stability Criteria for Neural Networks with Multiple Time-varying Delays.Physica D:Nonlinear Phenomena,vol.212,no.1-2,pp.126-136,2005.
[22]	Y.He,G.P.Liu,D.Rees.New Delay-dependent Stability Criteria for Neural Networks with Time-varying Delays.IEEE Transactions on Neural Networks,voh 18,no.1,pp.310-314,2007.
[23]	S.Y.Xu,J.Lam,D.W.C.Ho,Y.Zou.Novel Global Asymptotic Stability Criteria for Delayed Celluar Neural Networks.IEEE Transactions on Circuits and Systems Ⅱ:Express Briefs,vol.52,no.6,pp.349-353,2005.
[24]	T.Li,L.Guo,C.Lin,C.Y.Sun.New Results on Global Asymptotic Stability Analysis for Neural Networks with Time-varying Delays.Nonlinear Analysis:Real World Applications,vol.10,no.1,pp.554-562,2008.
[25]	Y.Chen,Y.Wu.Novel Delay-dependent Stability Criteria of Neural Networks with Time-varying Delay.Neurocomputing,vol.72,no.4-6,pp.1065-1070,2009.
[26]	J.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.
[27]	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.
[28]	X.Y.Lou,B.T.Cui.Delay-dependent Criteria for Robust Stability of Uncertain Switched Hopfield Neural Networks.International Journal of Automation and Computing,vol.4,no.3,pp.304-314,2007.
[29]	Z.B.Xu,H.Qiao,J.Peng,B.Zhang.A Comparative Study on Two Modeling Approaches in Neural Networks.Neural networks,vol.17,no.1,pp.73-85,2004.

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Stability Analysis for Recurrent Neural Networks with Time-varying Delay

1. School of Automation, Southeast University, Nanjing 210096, PRC;

2. Research Institute of Automation, Qufu Normal University, Qufu 273165, PRC

Received: 2008-09-22

Fund Project:

supported by National Natural Science Foundation of China (No.60674027)

Key words:

Abstract: This paper is concerned with the stability analysis for static recurrent neural networks (RNNs) with time-varying delay. By Lyapunov functional method and linear matrix inequality technique, some new delay-dependent conditions are established to ensure the asymptotic stability of the neural network. Expressed in linear matrix inequalities (LMIs), the proposed delay-dependent stability conditions can be checked using the recently developed algorithms. A numerical example is given to show that the obtained conditions can provide less conservative results than some existing ones.

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Reference (29)

[1]	J.Cao,L.Wang.Exponential Stability and Periodic Oscillatory Solution in BAM Networks with Delays.IEEE Transactions on Neural Networks,vol.13,no.2,pp.457-463,2002.
[2]	S.Arik.Global Asymptotic Stability of a Larger Class of Neural Networks with Constant Time Delay.Physics Letters A,vol.311,no.6,pp.504-511,2002.
[3]	T.Chen,L.Rong.Delay-independent Stability Analysis of Cohen-Grossberg Neural Networks.Physics Letters A,vol.317,no.5-6,pp.436-449,2003.
[4]	V.Singh.A Novel Global Robust Stability Criterion for Neural Networks with Delay.Physics Letters A,vol.337,no.4-6,pp.369-373.2005.
[5]	V.Singh.Simplified LMI Condition for Global Asymptotic Stability of Delayed Neural Networks.Chaos,Solitons and Fractals,vol.29,no.2,pp.470-473,2006.
[6]	Y.Liu,Z.Wang,X.Liu.Global Exponential Stability of Generalized Recurrent Neural Networks with Discrete and Distributed Delays.Neural Networks,vol.19,no.5,pp.667-675,2006.
[7]	T.Li,Q.Luo,C.Y.Sun,B.Y.Zhang.Exponential Stability of Recurrent Neural Networks with Time-varying Delay.Nonlinear Analysis:Real World Applications,to be published.
[8]	J.Yu,K.Zhang,S.Fei,T.Li.Simplified Exponential Stability Analysis for Recurrent Neural Networks with Discrete and Distributed Time-varying Delays.Applied Mathematics and Computation,vol.205,no.1,pp.465-474,2008.
[9]	T.Li,S.Fei.Exponential State Estimation for Recurrent Neural Networks with Distributed Delays.Neurocomputing,vol.71,no.1-3,pp.428-438,2007.
[10]	Q.Song,Z.Wang.Neural Networks with Discrete and Distributed Time-varying Delays:A General Stability Analysis.Chaos,Solitons and Fractals,vol.37,no.5,pp.1538-1547,2008.
[11]	Z.Wang,Y.Liu,X.Liu.On Global Asymptotic Stability of Neural Networks with Discrete and Distributed Delays.Physics Letters A,vol.345,no.4-6,pp.299-308,2005.
[12]	J.H.Park,H.J.Cho.A Delay-dependent Asymptotic Stability Criterion of Cellular Neural Networks with Timevarying Discrete and Distributed Delays.Chaos,Solitons and Fractals,vol.33,no.2,pp.436-442,2007.
[13]	Z.Wang,H.Shu,Y.Liu,D.W.C.Ho,X.Liu.Robust Stability Analysis of Generalized Neural Networks with Discrete and Distributed Time Delays.Chaos,Solitons and Fractal,vol.30,no.4,pp.886-896,2006.
[14]	H.J.Cho,J.H.Park.Novel Delay-dependent Robust Stability Criterion of Delayed Cellular Neural Networks.Chaos,Solitons and Fractals,vol.32,no.3,pp.1194-1200,2007.
[15]	R.S.Gau,C.H.Lien,J.G.Hsieh.Global Exponential Stability for Uncertain Cellular Neural Networks with Multiple Time-varying Delays via LMI Approach.Chaos,Solitons and Fractals,vol.32,no.4,pp.1258-1267,2007.
[16]	Y.G.Chen,W.P.Bi.New Robust Exponential Stability Analysis for Uncertain Neural Networks with Time-varying Delay.International Journal of Automation and Computing,vol.5,no.4,pp.395-400,2008.
[17]	Y.He,M.Wu,J.H.She.Delay-dependent Exponential Stability of Delayed Neural Networks with Time-varying Delay.IEEE Transactions on Circuits and Systems Ⅱ:Express Briefs,vol.53,no.7,pp.553-557,2006.
[18]	Q.Zhang,X.Wei,J.Xu.Delay-dependent Exponential Stability Celluar Neural Networks with Time-varying Delays.Chaos,Solitons and Fractals,vol.23,no.4,pp.1363-1369,2005.
[19]	W.H.Chen,X.M.Lu,Z.H.Guan,W.X.Zheng.Delaydependent Exponential Stability of Neural Networks with Variable Delay:An LMI Approach.IEEE Transactions on Circuits and Systems Ⅱ:Express Briefs,vol.53,no.9,pp.837-842,2006.
[20]	C.C.Hua,C.N.Long,X.P.Guan.New Results on Stability Analysis of Neural Networks with Time-varying Delays.Physics Letters A,vol.352,no.4-5,pp.335-340,2006.
[21]	Y.He,Q.G.Wang,M.Wu.LMI-based Stability Criteria for Neural Networks with Multiple Time-varying Delays.Physica D:Nonlinear Phenomena,vol.212,no.1-2,pp.126-136,2005.
[22]	Y.He,G.P.Liu,D.Rees.New Delay-dependent Stability Criteria for Neural Networks with Time-varying Delays.IEEE Transactions on Neural Networks,voh 18,no.1,pp.310-314,2007.
[23]	S.Y.Xu,J.Lam,D.W.C.Ho,Y.Zou.Novel Global Asymptotic Stability Criteria for Delayed Celluar Neural Networks.IEEE Transactions on Circuits and Systems Ⅱ:Express Briefs,vol.52,no.6,pp.349-353,2005.
[24]	T.Li,L.Guo,C.Lin,C.Y.Sun.New Results on Global Asymptotic Stability Analysis for Neural Networks with Time-varying Delays.Nonlinear Analysis:Real World Applications,vol.10,no.1,pp.554-562,2008.
[25]	Y.Chen,Y.Wu.Novel Delay-dependent Stability Criteria of Neural Networks with Time-varying Delay.Neurocomputing,vol.72,no.4-6,pp.1065-1070,2009.
[26]	J.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.
[27]	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.
[28]	X.Y.Lou,B.T.Cui.Delay-dependent Criteria for Robust Stability of Uncertain Switched Hopfield Neural Networks.International Journal of Automation and Computing,vol.4,no.3,pp.304-314,2007.
[29]	Z.B.Xu,H.Qiao,J.Peng,B.Zhang.A Comparative Study on Two Modeling Approaches in Neural Networks.Neural networks,vol.17,no.1,pp.73-85,2004.

Stability Analysis for Recurrent Neural Networks with Time-varying Delay

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