Optimal Tracking Control for a Class of Nonlinear Discrete-Time ...

1862 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 22, NO. 12, DECEMBER 2011[43] A. Al-Tamimi, M. Abu-Khalaf, and F. L. Lewis, “Adaptive critic designsfor discrete-time zero-sum games with application to H ∞ control,”IEEE Trans. Syst., Man, Cybern., Part B: Cybern., vol. 37, no. 1, pp.240–247, Feb. 2007.[44] Q. L. Wei, H. G. Zhang, D. R. Liu, and Y. Zhao, “An optimal controlscheme for a class of discrete-time nonlinear systems with time delaysusing adaptive dynamic programming,” ACTA Autom. Sin., vol. 36,no. 1, pp. 121–129, Jan. 2010.[45] R. Beard, “Improving the closed-loop performance of nonlinearsystems,” Ph.D. dissertation, Dept. Electr. Eng., Rensselaer PolytechnicInst., Troy, NY, 1995.[46] D. H. Chyung, “Discrete optimal systems with time delay,” IEEE Trans.Autom. Control, vol. 13, no. 1, pp. 1–117, Feb. 1968.[47] D. H. Chyung, “Discrete systems with time delay,” presented at the 5thAnnual Allerton Conference on Circuit and System Theory, Urbana,IL, Oct. 1967.[48] D. H. Chyung and E. B. Lee, “Linear optimal systems with timedelays,” SIAM J. Control, vol. 4, no. 3, pp. 548–575, Nov. 1966.[49] P. Yang, G. Xie, and L. Wang. Controllability of Linear Discrete-TimeSystems with Time-Delay in State and Control [Online]. Available:http://dean.pku.edu.cn/bksky/1999tzlwj/4.pdf[50] V. N. Phat, “Controllability of discrete-time systems with multipledelays on controls and states,” Int. J. Control, vol. 49, no. 5, pp.1645–1654, 1989.[51] C. T. Chen, Linear System Theory and Design, 3rd ed. New York:Oxford Univ. Press, 1999.[52] D. Wang, D. R. Liu, and Q. L. Wei, “Finite-horizon neuro-optimal trackingcontrol for a class of discrete-time nonlinear systems using adaptivedynamic programming approach,” Neurocomputing, to be published.[53] X. Liao, L. Wang, and P. Yu, Stability of Dynamical Systems.Amsterdam, The Netherlands: Elsevier, 2007.[54] A. Al-Tamimi and F. L. Lewis, “Discrete-time nonlinear HJB solutionusing approximate dynamic programming: Convergence proof,” in Proc.IEEE Int. Symp. Approx. Dyn. Program. Reinforce. Learn., Honolulu,HI, Apr. 2007, pp. 38–43.Huaguang Zhang (SM’04) received the B.S. andM.S. degrees in control engineering from the NortheastDianli University of China, Jilin City, China, in1982 and 1985, respectively, and the Ph.D. degreein thermal power engineering and automation fromSoutheast University, Nanjing, China, in 1991.He joined the Department of Automatic Control,Northeastern University, Shenyang, China, in 1992,as a Post-Doctoral Fellow for two years. Since 1994,he has been a Professor and Head of the Institute ofElectric Automation, School of Information Scienceand Engineering, Northeastern University. He has authored and co-authoredover 200 journal and conference papers, four monographs, and co-inventedon 20 patents. His current research interests include fuzzy controls, stochasticsystem controls, neural networks-based controls, nonlinear controls, and theirapplications.Dr. Zhang is an Associate Editor of Automatica, the IEEE TRANSACTIONSON FUZZY SYSTEMS, the IEEE TRANSACTIONS ON SYSTEMS,MAN, ANDCYBERNETICS—PART B, and Neurocomputing. He was awarded the OutstandingYouth Science Foundation Award from the National Natural ScienceFoundation Committee of China in 2003. He was named the Cheung KongScholar by the Education Ministry of China in 2005.Ruizhuo Song (M’11) has been pursuing the Ph.D.degree with Northeastern University, Shenyang,China, since 2008.Her current research interests include neuralnetworks-basedcontrols, non-linear controls, fuzzycontrols, adaptive dynamic programming, and theirindustrial applications.Qinglai Wei (M’11) received the B.S. degree inautomation, the M.S. degree in control theory andcontrol engineering, and the Ph.D. degree in controltheory and control engineering from NortheasternUniversity, Shenyang, China, in 2002, 2005, and2008, respectively.He was a Post-Doctoral Fellow with the KeyLaboratory of Complex Systems and IntelligenceScience, Institute of Automation, Chinese Academyof Sciences, Beijing, China, from 2009 to 2011. Heis currently an Associate Researcher with the Instituteof Automation. His current research interests include neural-networksbasedcontrols, nonlinear controls, adaptive dynamic programming, and theirindustrial applications.Tieyan Zhang (M’08) was born in 1962. Hereceived the Ph.D. degree in control theory andcontrol engineering from Northeastern University,Shenyang, China, in 2007.He is currently a Professor and President of theShenyang Institute of Engineering, Shenyang. Hiscurrent research interests include fuzzy controls,fault diagnosis on electric power systems, and stabilityanalysis on smart grids.