Nonlinear H-infinity Control of Delayed Recurrent Neural Networks Influenced by Uncertain Noise

Volume 1, Issue 1, October 2016     |     PP. 1-24      |     PDF (584 K)    |     Pub. Date: October 16, 2016
DOI:    449 Downloads     8153 Views  

Author(s)

Ziqian Liu, Department of Engineering, State University of New York Maritime College, 6 Pennyfield Avenue, Throggs Neck, NY 10465, USA

Abstract
This paper presents a theoretical design of how a nonlinear H-infinity optimal control is achieved for delayed recurrent neural networks with noise uncertainty. Our objective is to build globally stabilizing control laws to accomplish the input-to-state stability together with the optimality for delayed recurrent neural networks, and to attenuate noises to a predefined level with stability margins. The formulation of H-infinity control is developed by using Lyapunov technique and solving a Hamilton-Jacobi-Isaacs (HJI) equation indirectly. To illustrate the analytical results, three numerical examples are given to demonstrate the effectiveness of the proposed approach.

Keywords
Delayed recurrent neural networks, nonlinear H-infinity optimal control, noise attenuation, input-to-state stability, Lyapunov technique, Hamilton-Jacobi-Isaacs (HJI) equation.

Cite this paper
Ziqian Liu, Nonlinear H-infinity Control of Delayed Recurrent Neural Networks Influenced by Uncertain Noise , SCIREA Journal of Information Science and Systems Science. Volume 1, Issue 1, October 2016 | PP. 1-24.

References

[ 1 ] Z. Tu, J. Jian, K. Wang, Global exponential stability in Lagrange sense for recurrent neural networks with both time-varying delays and general activation functions via LMI approach, Nonlinear Analysis: Real World Applications. 12 (2011) 2174–2182.
[ 2 ] X. Li, X. Fu, P. Balasubramaniam, R. Rakkiyappan, Existence, uniqueness and stability analysis of recurrent neural networks with time delay in the leakage term under impulsive perturbations, Nonlinear Analysis: Real World Applications. 11 (2010) 4092–4108.
[ 3 ] J. Yu, K. Zhang, S. Fei, Exponential stability criteria for discrete-time recurrent neural networks with time-varying delay, Nonlinear Analysis: Real World Applications. 11 (2010) 207–216.
[ 4 ] T. Li, Q. Luo, C. Sun, B. Zhang, Exponential stability of recurrent neural networks with time-varying discrete and distributed delays, Nonlinear Analysis: Real World Applications. 10 (2009) 2581–2589.
[ 5 ] Y. Guo, New results on input-to-state convergence for recurrent neural networks with variable inputs, Nonlinear Analysis: Real World Applications. 9 (2008) 1558–1566.
[ 6 ] B. Liu, L. Huang, Positive almost periodic solutions for recurrent neural networks, Nonlinear Analysis: Real World Applications. 9 (2008) 830–841.
[ 7 ] X. Liao, Q. Luo, Z. Zeng, Y. Guo, Global exponential stability in Lagrange sense for recurrent neural networks with time delays, Nonlinear Analysis: Real World Applications. 9 (2008) 1535–1557.
[ 8 ] J. Liang, J. Cao, Global output convergence of recurrent neural networks with distributed delays, Nonlinear Analysis: Real World Applications. 8 (2007) 187–197.
[ 9 ] Q. Song, J. Cao, Z. Zhao, Periodic solutions and its exponential stability of reaction–diffusion recurrent neural networks with continuously distributed delays, Nonlinear Analysis: Real World Applications. 7 (2006) 65–80.
[ 10 ] M. S. Mahmoud, Y. Xia, Improved exponential stability analysis for delayed recurrent neural networks, Journal of the Franklin Institute. 348 (2011) 201–211.
[ 11 ] M. U. Akhmet, D. Arugaslan, E. Yilmaz, Stability analysis of recurrent neural networks with piecewise constant argument of generalized type, Neural Networks. 23 (2010) 805–811.
[ 12 ] L. Wang, R. Zhang, Z. Xu, J. Peng, Some characterizations of global exponential stability of a generic class of continuous-time recurrent neural networks, IEEE Transactions on Systems, Man, and Cybernetics – Part B: Cybernetics. 39 (2009) 763–772.
[ 13 ] H. Shao, Delay-dependent approaches to globally exponential stability for recurrent neural networks, IEEE Transactions on Circuits and Systems II: Express Briefs. 55 (2008) 591–595.
[ 14 ] H. Zhang, Z. Wang, D. Liu, Robust exponential stability of recurrent neural networks with multiple time-varying delays, IEEE Transactions on Circuits and Systems II: Express Briefs. 54 (2007) 730–734.
[ 15 ] S. Arik, Global asymptotic stability analysis of bidirectional associative memory neural networks with time delays, IEEE Transactions on Neural Networks. 16 (2005) 580–586.
[ 16 ] Z. Wang, Y. Liu, L. Yu, X. Liu, Exponential stability of delayed recurrent neural networks with Markovian jumping parameters, Physics Letters A. 356 (2006) 346–352.
[ 17 ] J. Cao, D. Huang, Y. Qu, Global robust stability of delayed recurrent neural networks, Chaos, Solitons and Fractals. 23 (2005) 221–229.
[ 18 ] V. Singh, A generalized LMI-based approach to the global asymptotic stability of delayed cellular neural networks, IEEE Transactions on Neural Networks. 15 (2004) 223–225.
[ 19 ] J. Lian, Z. Feng, P. Shi, Observer design for switched recurrent neural networks: an average dwell time approach, IEEE Transactions on Neural Networks. 22 (2011) 1547–1556.
[ 20 ] J. Lian, K. Zhang, Exponential stability analysis for switched Cohen-Grossberg neural networks with average dwell time, Nonlinear Dynamics. 63 (2011) 331–343.
[ 21 ] X Su, Z Li, Y Feng, L Wu, New global exponential stability criteria for interval delayed neural networks, Proceedings of the Institution of Mechanical Engineers - Part I: Journal of Systems and Control Engineering. 225 (2011) 125–136.
[ 22 ] X. Dong, Y. Zhao, Y. Xu, Z. Zhang, P. Shi, Design of PSO fuzzy neural network control for ball and plate system, International Journal of Innovative Computing, Information and Control. 7 (2011) 7091–7103.
[ 23 ] C. K. Ahn, M. K. Song, Filtering for time-delayed switched Hopfield neural networks, International Journal of Innovative Computing, Information and Control. 7 (2011) 1831–1843.
[ 24 ] I. Ahmad, A. Abdullah, A. Alghamdi, Investigating supervised neural networks to intrusion detection, ICIC Express Letters. 4 (2010) 2133–2138.
[ 25 ] Y. E. Shao, An integrated neural networks and SPC approach to identify the starting time of a process disturbance, ICIC Express Letters. 3 (2009) 319–324.
[ 26 ] R. Yang, Z. Zhang, P. Shi, Exponential stability on stochastic neural networks with discrete interval and distributed delays, IEEE Transactions on Neural Networks. 21 (2010) 169–175.
[ 27 ] L. Wan, Q. Zhou, Attractor and ultimate boundedness for stochastic cellular neural networks with delays, Nonlinear Analysis: Real World Applications. 12 (2011) 2561–2566.
[ 28 ] R. Sakthivel, R. Samidurai, S. M. Anthoni, Asymptotic stability of stochastic delayed recurrent neural networks with impulsive effects, Journal of Optimization Theory and Applications. 147 (2010) 583–596.
[ 29 ] Y. Lv, W. Lv, J. Sun, Convergence dynamics of stochastic reaction–diffusion recurrent neural networks with continuously distributed delays, Nonlinear Analysis: Real World Applications. 9 (2008) 1590–1606.
[ 30 ] Z. Liu, H. Schurz, N. Ansari, and Q. Wang, Theoretic design of differential minimax controllers for stochastic cellular neural networks, Neural Networks, 26 (2012) 110–117.
[ 31 ] S. K. Nguang, P. Shi, Fuzzy output feedback control design for nonlinear systems: an LMI approach, IEEE Transactions on Fuzzy Systems. 11 (2003) 331–340.
[ 32 ] S. K. Nguang, P. Shi, Nonlinear filtering of sampled data systems, Automatica. 36 (2000) 303–310.
[ 33 ] L. Wu, X. Su, P. Shi, Mixed approach to fault detection of discrete linear repetitive processes, Journal of the Franklin Institute. 348 (2011) 393–414.
[ 34 ] K. Ezal, Z. Pan, P. V. Kokotovic, Locally optimal and robust backstepping design, IEEE Transactions on Automatic Control. 45 (2000) 260–271.
[ 35 ] T. Basar, P. Bernhard, -Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach, 2nd Edition, Birkhauser, Boston, MA, 1995.
[ 36 ] C. Lu, W. Shyr, K. Yao, C. Liao, C. Huang, Delay-dependent control for discrete-time uncertain recurrent neural networks with interval time-varying delay, International Journal of Innovative Computing, Information and Control. 5 (2009) 3483–3493.
[ 37 ] W. Yu, J. Cao, Robust control of uncertain stochastic recurrent neural networks with time-varying delay, Neural Processing Letters. 26 (2007) 101–119.
[ 38 ] S. Das, O. Olurotimi, Noisy recurrent neural networks: the continuous-time case, IEEE Transactions on Neural Networks. 9 (1998) 913–936.
[ 39 ] J. Cao, J. Wang, Global asymptotic stability of a general class of recurrent neural networks with time-varying delays, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications. 50 (2003) 34–44.
[ 40 ] M. Krstic, H. Deng, Stabilization of Nonlinear Uncertain Systems, Springer-Verlag, New York, 1998.
[ 41 ] J. Primbs, V. Nevistic, J. Doyle, Nonlinear optimal control: a control Lyapunov function and receding horizon perspective, Asian Journal of Control. 1 (1999) 14–24.
[ 42 ] A. Teel, Asymptotic convergence from Stability, IEEE Transactions on Automatic Control. 44 (1999) 2169–2170.
[ 43 ] R. Freeman, P. Kokotovic, Robust Control of Nonlinear Systems, Birkhauser, Boston, MA, 1996.
[ 44 ] E. Todorov, Optimal control theory, in: K. Doya et al (Eds), Bayesian Brain: Probabilistic Approaches to Neural Coding, MIT Press, Massachusetts, 2006, pp. 269–298.
[ 45 ] G. Rovitahkis and M. Christodoulou, Adaptive Control with Recurrent High-order Neural Networks, Springer-Verlag, New York, 2000.