Abstract
In this paper, we are concerned with the robust H∞ problem for a class of discrete-time neural networks with uncertainties. Under a weak assumption on the activation functional, some novel summation inequality techniques and using a new Lyapunov-Krasovskii (L-K) functional, a delay-dependent condition guaranteeing the robust asymptotically stability of the concerned neural networks is obtained in terms of a Linear Matrix Inequality(LMI). It is shown that this stability condition is less conservative than some previous ones in the literature. The controller gains can be derived by solving a set of LMIs. Finally, two numerical examples result are given to illustrate the effectiveness of the developed theoretical results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Sakthivel, K. Mathiyalagan, and S. Marshal Anthoni, “Robust H ∞ control for uncertain discrete-time stochastic neural networks with time-varying delays,” IET Control Theory Appl., vol. 6, pp. 1220–1228, July 2012.
A. Arunkumar, R. Sakthivel, K. Mathiyalagan, and S. Marshal Anthoni, “Robust stability criteria for discrete-time switched neural networks with various activation functions,” Appl. Math. Comput., vol. 218, pp. 10803–10816, July 2012.
L. J. Banu, P. Balasubramaniam, and K. Ratnavel, “Robust stability analysis for discrete-time uncertain neural networks with leakage time-varying delay,” Neurocomputing, vol. 151, pp. 808–816, March 2015.
X. G. Liu, F. X. Wang, and Y. J. Shu, “A novel summation inequality for stability analysis of discrete-time neural networks,” J. Comput. Appl. Math., vol. 304, pp. 160–171, October 2016.
L. Jin, Y. Hen, and W. Wu, “Improved delay-dependent stability analysis of discrete-time neural networks with time-varying delay,” J. Franklin Inst., vol. 354, no. 4, pp. 1922–1936, March 2016.
M. Luo, S. Zhong, R. Wang, and W. Kang, “Robust stability analysis for discrete-time stochastic neural networks systems with time-varying delays,” Appl. Math. Comput., vol.209, no. 2, pp. 305–313, March 2009.
T. Zhang, “Comment on delay-dependent robust H ∞ filtering for uncertain discrete-time singular systems with interval time-varying delay,” Automatica, vol. 53, pp. 291–292, March 2015.
D. Liu, L. Wang, Y. Pan, and H. Ma, “Mean square exponential stability for discrete-time stochastic fuzzy neural networks with mixed time-varying delay,” Neurocomputing, vol. 171, pp. 1622–1628, January 2016.
J. Chen, I.T. Wu, and C.H. Lien, “Robust exponential stability for uncertain discrete-time switched systems with interval time-varying delay through a switching signal,” J. Appl. Rrh. Technol., vol. 12, no. 6, pp. 1187–1197, December 2014.
Y. Shan, S. Zhong, J. Cui, L. Hou, and Y. Li, “Improved criteria of delay-dependent stability for discrete-time neural networks with leakage delay,” Neurocomputing, vol. 226, pp. 409–419, November 2017.
K. Ramakrishnan and G. Ray, “Robust stability criteria for a class of uncertain discrete-time systems with time-varying delay,” Appl. Math. Model., vol. 37 no. 3, pp. 1468–1479, February 2013.
B. Yeon and H. Ahn, “Stability analysis of spatiall inter-connected discrete-time systems with random delays and structured uncertainties,” J. Franklin Inst., vol. 350, no. 7, pp. 1719–1738, September 2013.
G. Chesi and R. H. Middleton, “Robust stability and performance analysis of 2D mixed continuous-discrete-time systems with uncertainty,” Automatica, vol. 67, pp. 233–243, May 2016.
L. J. Banu and P. Balasubramaniam, “Admissibility analysis for discrete-time singular systems with randomly occurring uncertainties via delay-divisioning approach,” ISA Trans., vol. 59, pp. 354–362, November 2015.
L. Jarina Banu, and P. Balasubramaniam, “Robust stability analysis for discrete-time neural networks with timevarying leakage delays and random parameter uncertainties,” Neurocomputing, vol. 179, no. 29, pp. 134–126, February 2016.
Y. Li and G. H. Yang, “Robust adaptive fuzzy control of a class of uncertain switched nonlinear systems with mismatched uncertainties,” Inf. Sci., vol. 339, no. 20, pp. 290–309, April 2016.
M. Hashemi, J. Askari, and J. Ghaisar, “Adaptive decentralised dynamic surface control for non-linear large-scale systems against actuator failures,” IET Control Theory Appl. vol. 10, no. 1, pp. 44–57, January 2016.
H. Liu, Y. Pan, S. Li, and Y. Chen, “Adaptive fuzzy backstepping control of fractional-order nonlinear systems,” IEEE Trans. Systems, Man Cybern. Syst. vol. 47 no.8 pp. 2209–2217 August 2017.
Y. Pan and H. Yu, “Composite learning from adaptive dynamic surface control,” IEEE Transa. Automat. Cont. vol. 61 no. 9 pp. 2603–2609 September 2016.
H. Liu, Y. Pan, S. Li, and Y. Chen, “Synchronization for fractional-order neural networks with full/under-actuation using fractional-order sliding mode control,” Int. J. Mach. Learn. & Cyber., 2017.
T. Fujinami, Y. Saito, M. Morishita, Y. Koike, and K. Tanida, “A hybrid mass damper system controlled by H ∞ control theory for reducing bending-torsion vibration of an actual building,” Earthq. Eng. Struct. Dyn. vol. 30, pp. 1639–1653, 2001.
C. Hu, K. Yu, and L. Wu, “Robust H ∞ switching control and switching signal design for uncertain discrete switched systems with interval time-varying delay,” J. Franklin Inst., vol. 351, no. 1, pp. 565–578, January 2014.
Y. Li and J. Qi, “Robust H ∞ control of uncertain stochastic time-delay linear repetitive processes,” J. Control Theory Appl., vol. 8, no. 4, pp. 491–495, November 2010.
M. Abbaszadeh and H. J. Marquez, “Nonlinear robust H ∞ filtering for a class of uncertain systems via convex optimization,” J. Control Theory Appl., vol. 10, no. 2, pp. 152–158, May 2012.
L. K. Wang and X. D. Liu, “Robust H ∞ fuzzy control for discrete-time nonlinear systems,” Int. J. Control Autom. Syst., vol. 8, no. 1, pp. 118–126, February 2010.
D. Wang, W. Wang, and P. Shi, “Design on H ∞-filtering for discrete-time switched delay systems,” Int. J. Syst. Sci., vol. 42, no. 12, pp. 1965–1973, December 2010.
S. Chae, D. Huang, and S. K. Nguang, “Robust partially mode delay dependent H ∞ control of discrete-time networked control systems,” Int. J. Syst. Sci., vol. 43, pp. 1764–1773, February 2011.
Q. Song and Z. Wang, “A delay-dependent LMI approach to dynamics analysis of discrete-time recurrent neural networks with time-varying delays,” Phys. Lett. A, vol. 368, pp. 134–145, August 2007.
B. Zhang, S. Xu, and Y. Zou, “Improved delay-dependent exponential stability criteria for discrete-time recurrent neural networks with timevarying delays,” Neurocomputing, vol. 72, no. 1–3, pp. 321–330, December 2008.
J. Yu, K. Zhang, and S. Fei, “Exponential stability criteria for discrete-time recurrent neural networks with timevarying delay,” Nonlinear Anal. Real World Appl., vol. 11, no. 1, pp. 207–216, February 2010.
S. Ramasamy, G. Nagamani, and Radhika, “Further results on dissipativity criterion for Markovian jump discrete-time neural networks with two delay components via discrete wirtinger inequality approach,” Neural Process. Lett., vol. 45, no. 3, pp. 939–965, June 2017.
R. Saravanakumar, G. Rajchakit, M. Syed Ali, Z. Xiang and Y. Hoon Joo, “Robust extended dissipativity criteria for discrete-time uncertain neural networks with time-varying delays,” Neural Comput & Applic., May 2017.
C. K. Zhang, Y. He, L. Jiang, Q. Wang, and M. Wu, “Stability analysis of discrete-time neural networks with time-varying delay via an extended reciprocally convex matrix inequality,” IEEE Trans. Cybern., vol. 47, no. 10, pp. 3040–3049, February 2017.
Author information
Authors and Affiliations
Corresponding author
Additional information
Recommended by Associate Editor Yongping Pan under the direction of Editor Myo Taeg Lim. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2016R1D1A109917886) and by the Brain Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2017M3C7A1044815. This work was also supported by CSIR. 25(0274)/17/EMR-II dated 27/04/2017.
M. Syed Ali graduated in 2002 and post-graduated in 2005 from Bharathiar University, India. He was conferred with Doctor of Philosophy in 2010 in Gandhigram Rural University, Gandhigram, India. Since March 2011 he is working as an Assistant Professor in Department of Mathematics, Thiruvalluvar University, Vellore, Tamil Nadu, India. He was awarded Young Scientist Award 2016 by The Academy of Sciences, Chennai. He has published more than 85 research papers in various SCI journals holding impact factors. He has also published research articles in national journals and international conference proceedings. He also serves as a reviewer for several SCI journals. His research interests include stochastic differential equations, dynamical systems, fuzzy neural networks, complex networks and cryptography.
K. Meenakshi received the B.Sc. degree in 2007. She received the M.Sc. in 2011. She was awarded Master of Philosophy in the year 2012 from the Department of Mathematics, Voorhees College, Vellore, which is affiliated to the Thiruvalluvar University in 2011.Currently she is pursuing Ph.D. degree under the supervision of an Assistant Professor Dr. M. Syed Ali, in the Department of Mathematics, Thiruvalluvar University, Tamil Nadu, India.
R. Vadivel received the B.Sc., M.Sc., and M.Phil. degrees in Mathematics from Sri Ramakrishna Mission Vidyalaya College of Arts and Science affiliated to Bharathiar University, Coimbatore, Tamil Nadu, India, in 2007, 2010, and 2012, respectively. He is currently pursuing the Ph.D. degree in Department of Mathematics, Thiruvalluvar University, Vellore, Tamil Nadu, India.
O. M. Kwon received the B.S. degree in electronic engineering from Kyungbuk National University, Daegu, Republic of Korea, in 1997, and the Ph.D. degree in electrical and electronic engineering from POSTECH, Pohang, South Korea, in 2004. He was a Senior Researcher with the Mechatronics Center of Samsung Heavy Industries, Daejeon, Republic of Korea, from 2004 to 2006. He is currently a Professor with the School of Electrical Engineering, Chungbuk National University, Cheongju, South Korea. His current research interests include time delay systems, cellular neural networks, robust control and filtering, large-scale systems, secure communication through synchronization between two chaotic systems, complex dynamical networks, multiagent systems, and sampled data control. He has presented over 160 international papers in the above areas. Dr. Kwon was a recipient of the One of the Highly Cited Researchers in the field of mathematics, in 2015, 2016, and 2017. He currently serves as an Associate Editor for Neural Networks, the International Journal of Control, Automation and Systems, Journal of Institute of Control, Robotics and Systems, and Journal of Applied Mathematics and Informatics.
Rights and permissions
About this article
Cite this article
Syed Ali, M., Meenakshi, K., Vadivel, R. et al. Robust H∞ Performance of Discrete-time Neural Networks with Uncertainty and Time-varying Delay. Int. J. Control Autom. Syst. 16, 1637–1647 (2018). https://doi.org/10.1007/s12555-017-0416-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12555-017-0416-4