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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 Hinfinity optimal control is achieved for delayed recurrent neural networks with noise uncertainty. Our objective is to build globally stabilizing control laws to accomplish the inputtostate stability together with the optimality for delayed recurrent neural networks, and to attenuate noises to a predefined level with stability margins. The formulation of Hinfinity control is developed by using Lyapunov technique and solving a HamiltonJacobiIsaacs (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 Hinfinity optimal control, noise attenuation, inputtostate stability, Lyapunov technique, HamiltonJacobiIsaacs (HJI) equation.
Cite this paper
Ziqian Liu,
Nonlinear Hinfinity Control of Delayed Recurrent Neural Networks Influenced by Uncertain Noise, SCIREA Journal of Information Science and Systems Science. Vol.
1
, No.
1
,
2016
, pp.
1

24
.
References
[ 1 ]  Z. Tu, J. Jian, K. Wang, Global exponential stability in Lagrange sense for recurrent neural networks with both timevarying 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 discretetime recurrent neural networks with timevarying 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 timevarying discrete and distributed delays, Nonlinear Analysis: Real World Applications. 10 (2009) 2581–2589. 
[ 5 ]  Y. Guo, New results on inputtostate 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 continuoustime recurrent neural networks, IEEE Transactions on Systems, Man, and Cybernetics – Part B: Cybernetics. 39 (2009) 763–772. 
[ 13 ]  H. Shao, Delaydependent 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 timevarying 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 LMIbased 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 CohenGrossberg 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 timedelayed 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, Delaydependent control for discretetime uncertain recurrent neural networks with interval timevarying 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 timevarying delay, Neural Processing Letters. 26 (2007) 101–119. 
[ 38 ]  S. Das, O. Olurotimi, Noisy recurrent neural networks: the continuoustime 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 timevarying 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, SpringerVerlag, 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 Highorder Neural Networks, SpringerVerlag, New York, 2000. 