Local stability conditions for discrete-time cascade locally recurrent neural networks

Patan, K.

International Journal of Applied Mathematics and Computer Science, Tome 20 (2010) no. 1, pp. 23-34.

Résumé

The paper deals with a specific kind of discrete-time recurrent neural network designed with dynamic neuron models. Dynamics are reproduced within each single neuron, hence the network considered is a locally recurrent globally feedforward. A crucial problem with neural networks of the dynamic type is stability as well as stabilization in learning problems. The paper formulates local stability conditions for the analysed class of neural networks using Lyapunov's first method. Moreover, a stabilization problem is defined and solved as a constrained optimization task. In order to tackle this problem, a gradient projection method is adopted. The efficiency and usefulness of the proposed approach are justified by using a number of experiments.

Détail

Keywords: locally recurrent neural network, stability, stabilization, learning, constrained optimization
Mots-clés : sieć lokalnie rekurencyjna, stabilność, stabilizacja, uczenie się, optymalizacja ograniczona

Bibliographie

[1] Back, A. D. and Tsoi, A. C. (1991). FIR and IIR synapses, A new neural network architecture for time series modelling, Neural Computation 3(3): 375-385.

[2] Campolucci, P. and Piazza, F. (2000). Intrinsic stability-control method for recursive filters and neural networks, IEEE Transactions on Circuit and Systems-II: Analog and Digital Signal Processing 47(8): 797-802.

[3] Cannas, B., Cincotti, S., Marchesi, M. and Pilo, F. (2001). Learnig of Chua's circuit attractors by locally recurrent neural networks, Chaos Solitons Fractals 12(11): 2109-2115.

[4] Cao, J., Yuan, K. and Li, H. (2006). Global asymptotical stability of recurrent neural networks with multiple discrete delays and distributed delays, IEEE Transactions on Neural Networks 17(6): 1646-1651.

[5] Ensari, T. and Arik, S. (2005). Global stability analysis of neural networks with multiple time varying delays, IEEE Transactions on Automatic Control 50(11): 1781-1785.

[6] Fasconi, P., Gori, M. and Soda, G. (1992). Local feedback multilayered networks, Neural Computation 4(1): 120-130.

[7] Forti, M., Nistri, P. and Papini, D. (2005). Global exponential stability and global convergence in finite time of delayed neural networks with infinite gain, IEEE Transactions on Neural Networks 16(6): 1449-1463.

[8] Gori, M., Bengio, Y. and Mori, R. D. (1989). BPS: A learning algorithm for capturing the dynamic nature of speech, International Joint Conference on Neural Networks, Washington DC, USA, Vol. II, pp. 417-423.

[9] Gupta, M. M., Jin, L. and Homma, N. (2003). Static and Dynamic Neural Networks. From Fundamentals to Advanced Theory, John Wiley Sons, Hoboken, NJ.

[10] Gupta, M. M. and Rao, D. H. (1993). Dynamic neural units with application to the control of unknown nonlinear systems, Journal of Intelligent and Fuzzy Systems 1(1): 73-92.

[11] Marcu, T., Mirea, L. and Frank, P. M. (1999). Development of dynamical neural networks with application to observer based fault detection and isolation, International Journal of Applied Mathematics and Computer Science 9(3): 547-570.

[12] Patan, K. (2007). Stability analysis and the stabilization of a class of discrete-time dynamic neural network, IEEE Transactions on Neural Networks 18(3): 660-673.

[13] Patan, K. (2008a). Aproximation of state-space trajectories by locally recurrent globally feed-forward neural networks, Neural Networks 21(1): 59-64.

[14] Patan, K. (2008b). Artificial Neural Networks for the Modelling and Fault Diagnosis of Technical Processes, Lecture Notes in Control and Information Sciences, Vol. 377, Springer-Verlag, Berlin.

[15] Patan, K. (2008c). Stability criteria for three-layer locally recurrent networks, Proceedings of the 17th IFAC World Congress on Automatic Control, Seoul, Korea, (on CDROM).

[16] Patan, K. and Parisini, T. (2005). Identification of neural dynamic models for fault detection and isolation: The case of a real sugar evaporation process, Journal of Process Control 15(1): 67-79.

[17] Patan, K., Witczak, M. and Korbicz, J. (2008). Towards robustness in neural network based fault diagnosis, International Journal of Applied Mathematics and Computer Science 18(4): 443-454, DOI: 10.2478/v10006-008-0039-2.

[18] Tsoi, A. C. and Back, A. D. (1994). Locally recurrent globally feedforward networks: A critical review of architectures, IEEE Transactions on Neural Networks 5(2): 229-239.

[19] Xiang-Qun, L. and Zhang, H. Y. (2000). Fault detection and diagnosis of permanent-magnet DC motor based on parameter estimation and neural network, IEEE Transactions on Industrial Electronics 47(5): 1021-1030.

[20] Zhang, J., Morris, A. J. and Martin, E. B. (1998). Long term prediction models based on mixed order locally recurrent neural networks, Computers Chemical Engineering 22(7-8): 1051-1063.