Stability analysis of discrete-time recurrent neural networks based on standard neural network models

Original Article

Abstract

In order to conveniently analyze the stability of various discrete-time recurrent neural networks (RNNs), including bidirectional associative memory, Hopfield, cellular neural network, Cohen-Grossberg neural network, and recurrent multiplayer perceptrons, etc., the novel neural network model, named standard neural network model (SNNM) is advanced to describe this class of discrete-time RNNs. The SNNM is the interconnection of a linear dynamic system and a bounded static nonlinear operator. By combining Lyapunov functional with S-Procedure, some useful criteria of global asymptotic stability for the discrete-time SNNMs are derived, whose conditions are formulated as linear matrix inequalities. Most delayed (or non-delayed) RNNs can be transformed into the SNNMs to be stability analyzed in a unified way. Some application examples of the SNNMs to the stability analysis of the discrete-time RNNs shows that the SNNMs make the stability conditions of the RNNs easily verified.

Keywords

Standard neural network model (SNNM) Global asymptotic stability Time-delay system Time-varying system Discrete-time Linear matrix inequality (LMI) Hopfield neural network Bidirectional associative memory Cohen-Grossberg neural network 

References

  1. 1.
    Warwich K (1995) Neural networks: an introduction. Neural network applications in control. In: Irwin GW, Warwick K, Hunt KJ (eds) The Institution of Electrical Engineers, London, pp 1–16Google Scholar
  2. 2.
    Suykens JAK, Vandewalle JPL, De Moor BLR (1996) Artificial neural networks for modeling and control of non-linear systems. Kluwer, BostonGoogle Scholar
  3. 3.
    Medsker LR, Jain LC (eds) (2000) Recurrent neural networks: design and applications, CRC Press LLC, Boca RatonGoogle Scholar
  4. 4.
    Miyoshi S, Yanai HF, Okada M (2004) Associative memory by recurrent neural networks with delay elements. Neural Netw 17(1):55–63. doi:10.1016/S0893-6080(03)00207-7 MATHCrossRefGoogle Scholar
  5. 5.
    Xia Y, Wang J (2004) A recurrent neural network for nonlinear convex optimization subject to nonlinear inequality constraints. IEEE Trans Circuits Syst I Regul Pap 51(7):1385–1394. doi:10.1109/TCSI.2004.830694 CrossRefMathSciNetGoogle Scholar
  6. 6.
    Liao XF, Chen GR, Sanchez EN (2002) LMI-based approach for asymptotically stability analysis of delayed neural networks. IEEE Trans Circ Syst I Fundam Theory Appl 49(7):1033–1039. doi:10.1109/TCSI.2002.800842 CrossRefMathSciNetGoogle Scholar
  7. 7.
    Liao XF, Chen GR, Sanchez EN (2002) Delay-dependent exponential stability analysis of delayed neural networks: an LMI approach. Neural Netw 15(7):855–866. doi:10.1016/S0893-6080(02)00041-2 CrossRefMathSciNetGoogle Scholar
  8. 8.
    Guo S, Huang L (2004) Exponential stability of discrete-time Hopfield neural networks. Comput Math Appl 47(8–9):1249–1256. doi:10.1016/S0898-1221(04)90119-8 MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Liang J, Cao J, Ho DWC (2005) Discrete-time bidirectional associative memory neural networks with variable delays. Phys Lett A 335(2–3):226–234. doi:10.1016/j.physleta.2004.12.026 MATHCrossRefGoogle Scholar
  10. 10.
    Xiong W, Cao J (2005) Global exponential stability of discrete-time Cohen–Grossberg neural networks. Neurocomputing 64:433–446. doi:10.1016/j.neucom.2004.08.004 CrossRefGoogle Scholar
  11. 11.
    Liu MQ (2006) Discrete-time delayed standard neural network model and its application. Sci China Ser F Inf Sci 49(2):137–154. doi:10.1007/s11432-006-0137-4 MATHCrossRefGoogle Scholar
  12. 12.
    Barabanov NE, Prokhorov DV (2002) Stability analysis of discrete-time recurrent neural networks. IEEE Trans Neural Netw 13(2):292–303. doi:10.1109/72.991416 CrossRefGoogle Scholar
  13. 13.
    Zhang SL, Liu MQ (2005) Stability analysis of discrete-time BAM neural networks based on standard neural network models. J Zhejiang Univ Sci 6A(7):689–696. doi:10.1631/jzus.2005.A0689 MATHCrossRefGoogle Scholar
  14. 14.
    Wang L, Xu Z (2006) Sufficient and necessary conditions for global exponential stability of discrete-time recurrent neural networks. IEEE Trans Circuits Syst I Regul Pap 53(6):1373–1380. doi:10.1109/TCSI.2006.874179 CrossRefMathSciNetGoogle Scholar
  15. 15.
    Mak KL, Peng JG, Xu ZB, Yiu KFC (2007) A new stability criterion for discrete-time neural networks: nonlinear spectral radius. Chaos Solitons Fractals 31(2):424–436. doi:10.1016/j.chaos.2005.09.075 MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hu S, Wang J (2002) Global stability of a class of discrete-time recurrent neural networks. IEEE Trans Circuits Syst I Fundam Theory Appl 49(8):1104–1117. doi:10.1109/TCSI.2002.801284 CrossRefMathSciNetGoogle Scholar
  17. 17.
    Liu MQ (2007) Delayed standard neural network models for control systems. IEEE Trans Neural Netw 18(5):1376–1391. doi:10.1109/TNN.2007.894084 CrossRefGoogle Scholar
  18. 18.
    Chandrasekharan PC (1996) Robust control of linear dynamical systems. Academic Press, LondonGoogle Scholar
  19. 19.
    Rios-Patron E (2000) A general framework for the control of nonlinear systems, PhD thesis, University of IllinoisGoogle Scholar
  20. 20.
    Boyd SP, Ghaoui LE, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. SIAM, PhiladelphiaMATHGoogle Scholar
  21. 21.
    Li CD, Liao XF, Zhang R (2004) Global asymptotical stability of multi-delayed interval neural networks: an LMI approach. Phys Lett A 328(6):452–462. doi:10.1016/j.physleta.2004.06.053 MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Xie L, de Souza CE (1992) Robust H control for linear systems with norm-bounded time-varying uncertainty. IEEE Trans Automat Contr 37(8):1188–1191. doi:10.1109/9.151101 CrossRefGoogle Scholar
  23. 23.
    Khargonekar PP, Petersen IR, Zhou K (1990) Robust stabilization of uncertain linear systems: quadratic stability and H control theory. IEEE Trans Automat Contr 35(3):356–361. doi:10.1109/9.50357 MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Bhaya A, Kaszkurewicz E, Kozyakin VS (1996) Existence and stability of a unique equilibrium in continuous-valued discrete-time asynchronous Hopfield neural networks. IEEE Trans Neural Netw 7(3):620–628. doi:10.1109/72.501720 Google Scholar
  25. 25.
    Guo S, Huang L (2004) Periodic oscillation for discrete-time Hopfield neural networks. Phys Lett A 329(3):199–206. doi:10.1016/j.physleta.2004.07.007 MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Hopfield JJ (1984) Neurons with graded response have collective computational properties like those of two-state neurons. Proc Natl Acad Sci USA 81(10):3088–3092. doi:10.1073/pnas.81.10.3088 CrossRefGoogle Scholar
  27. 27.
    Gahinet P, Nemirovski A, Laub AJ, Chilali M (1995) LMI control toolbox- for use with Matlab. The MATH Works, Inc., NatickGoogle Scholar
  28. 28.
    Kosko B (1987) Adaptive bidirectional associative memories. Appl Opt 26(23):4947–4960CrossRefGoogle Scholar
  29. 29.
    Kosko B (1988) Adaptive bi-directional associative memories. IEEE Trans Syst Man Cybern 18(1):49–60. doi:10.1109/21.87054 CrossRefMathSciNetGoogle Scholar
  30. 30.
    Arik S, Tavsanoglu V (2005) Global asymptotic stability analysis of bidirectional associative memory neural networks with constant time delays. Neurocomputing 68:161–176. doi:10.1016/j.neucom.2004.12.002 CrossRefGoogle Scholar
  31. 31.
    Cao J, Liang J, Lam J (2004) Exponential stability of high-order bidirectional associative memory neural networks with time delays. Physica D Nonlinear Phenomena 199(3–4):425–436. doi:10.1016/j.physd.2004.09.012 MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Wang L, Zou X (2004) Capacity of stable periodic solutions in discrete-time bidirectional associative memory neural networks. IEEE Trans Circuits Syst II Express Briefs 51(6):315–319. doi:10.1109/TCSII.2004.829571 CrossRefGoogle Scholar
  33. 33.
    Cohen MA, Grossberg S (1983) Absolute stability of global pattern formation and parallel memory storage by competitive neural network. IEEE Trans Syst Man Cybern 13(5):815–826MATHMathSciNetGoogle Scholar
  34. 34.
    Wang W, Cao J (2006) LMI-based criteria for globally robust stability of delayed Cohen-Grossberg neural networks. IEE Proc Contr Theory Appl 153(4):397–402. doi:10.1049/ip-cta:20050197 CrossRefMathSciNetGoogle Scholar
  35. 35.
    Guo S, Huang L (2006) Stability analysis of Cohen-Grossberg neural networks. IEEE Trans Neural Netw 17(1):106–117. doi:10.1109/TNN.2005.860845 CrossRefGoogle Scholar
  36. 36.
    Chen Y (2006) Global asymptotic stability of delayed Cohen-Grossberg neural network. IEEE Trans Circuits Syst I Regul Pap 53(2):351–357. doi:10.1109/TCSI.2005.856047 CrossRefMathSciNetGoogle Scholar
  37. 37.
    Cao J, Liang J (2004) Boundedness and stability for Cohen-Grossberg neural networks with time-varying delays. J Math Anal Appl 296(2):665–685. doi:10.1016/j.jmaa.2004.04.039 MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2008

Authors and Affiliations

  1. 1.Department of System Science and Engineering, College of Electrical EngineeringZhejiang UniversityHangzhouPeople’s Republic of China

Personalised recommendations