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Discrete Dynamics in Nature and Society
Volume 2016 (2016), Article ID 7217364, 11 pages
http://dx.doi.org/10.1155/2016/7217364
Review Article

A Review of Fuzzy Logic and Neural Network Based Intelligent Control Design for Discrete-Time Systems

1Key Lab of Autonomous Systems and Networked Control (MOE), School of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, China
2Zienkiewicz Centre for Computational Engineering, Swansea University, Swansea SA1 8EN, UK
3State Key Lab of Intelligent Control and Decision of Complex Systems, Beijing Institute of Technology, Beijing 100081, China

Received 5 November 2015; Accepted 29 December 2015

Academic Editor: Juan R. Torregrosa

Copyright © 2016 Yiming Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. G. Goodwin and K. Sin, Adaptive Filtering, Prediction and Control, Prentice-Hall, Englewood Cliffs, NJ, USA, 1984.
  2. K. Aström and B. Wittenmark, Adaptive Control, Addison-Wesley, 1989.
  3. M. Krstić, I. Kanellakopoulos, and P. V. Kokotović, Nonlinear and Adaptive Control Design, John Wiley & Sons, New York, NY, USA, 1995.
  4. G. Tao and P. Kokotovic, Aaptive Control of Systems with Actuator and Sensor Non-Linearities, John Wiley & Sons, Hoboken, NJ, USA, 1996.
  5. S. S. Ge, C. C. Hang, T. H. Lee, and T. Zhang, Stable Adaptive Neural Network Control, Kluwer Academic Publishers, Norwell, Mass, USA, 2001.
  6. C. Yang, Adaptive control and neural network control of nonlinear discrete-time systems [Ph.D. thesis], National University of Singapore, 2009.
  7. C. Yang, H. Ma, and M. Fu, “Adaptive predictive control of periodic non-linear auto-regressive moving average systems using nearest-neighbour compensation,” IET Control Theory & Applications, vol. 7, no. 7, pp. 936–951, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. C. Yang, L. Zhai, S. S. Ge, T. Chai, and T. H. Lee, “Adaptive model reference control of a class of MIMO discrete-time systems with compensation of nonparametric uncertainty,” in Proceedings of the American Control Conference, pp. 4111–4116, IEEE, Seattle, Wash, USA, June 2008. View at Publisher · View at Google Scholar
  9. S.-L. Dai, C. Yang, S. S. Ge, and T. H. Lee, “Robust adaptive output feedback control of a class of discrete-time nonlinear systems with nonlinear uncertainties and unknown control directions,” International Journal of Robust and Nonlinear Control, vol. 23, no. 13, pp. 1472–1495, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. K. J. Åström and B. Wittenmark, “On self tuning regulators,” Automatica, vol. 9, no. 2, pp. 185–199, 1973. View at Publisher · View at Google Scholar · View at Scopus
  11. L. Ljung, “Analysis of recursive stochastic algorithms,” IEEE Transactions on Automatic Control, vol. 22, no. 4, pp. 551–575, 1977. View at Google Scholar · View at MathSciNet
  12. G. C. Goodwin, P. J. Ramadge, and P. E. Caines, “Discrete time multivariable adaptive control,” IEEE Transactions on Automatic Control, vol. 25, no. 3, pp. 449–456, 1980. View at Publisher · View at Google Scholar · View at MathSciNet
  13. L. Guo and H. F. Chen, “The Åström-Wittenmark self-tuning regulator revisited and ELS-based adaptive trackers,” IEEE Transactions on Automatic Control, vol. 36, no. 7, pp. 802–812, 1991. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. L. Guo, Time-Varing Stochastic Systems, Jilin Science and Technology Press, Changchun, China, 1993 (Chinese).
  15. H. F. Chen and L. Guo, Identification and Stochastic Adaptive Control, Birkhäuser, Boston, Mass, USA, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  16. F. P. Skantze, A. Kojic, A.-P. Loh, and A. M. Annaswamy, “Adaptive estimation of discrete-time systems with nonlinear parameterization,” Automatica, vol. 36, no. 12, pp. 1879–1887, 2000. View at Google Scholar · View at MathSciNet
  17. L. Chen and K. S. Narendra, “Nonlinear adaptive control using neural networks and multiple models,” Automatica, vol. 37, no. 8, pp. 1245–1255, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. L. Guo and C. Wei, “LS-based discrete-time adaptive nonlinear control: feasibility and limitations,” Science in China Series E: Technological Sciences, vol. 39, no. 3, pp. 255–269, 1996. View at Google Scholar · View at MathSciNet
  19. L. L. Xie and L. Guo, “Adaptive control of discrete-time nonlinear systems with structural uncertainties,” in Lectures on Systems, Control, and Information, vol. 17 of AMS/IP Studies in Advanced Mathematics, American Mathematical Society, International Press, Providence, RI, USA, 2000. View at Google Scholar
  20. J. D. Boskovic, “Stable adaptive control of a class of first-order nonlinearly parameterized plants,” IEEE Transactions on Automatic Control, vol. 40, no. 2, pp. 347–350, 1995. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. A. L. Fradkov, I. V. Miroshnik, and V. O. Nikiforov, Nonlinear and Adaptive Control of Complex Systems: Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2004.
  22. D. Angeli and E. Mosca, “Adaptive switching supervisory control of nonlinear systems with no prior knowledge of noise bounds,” Automatica, vol. 40, no. 3, pp. 449–457, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. H. B. Ma, “Finite-model adaptive control using an LS-like algorithm,” International Journal of Adaptive Control and Signal Processing, vol. 21, no. 5, pp. 391–414, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. H. B. Ma, “Finite-model adaptive control using WLS-like algorithm,” Automatica, vol. 43, no. 4, pp. 677–684, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. H. B. Ma, “Several algorithms for finite-model adaptive control: partial answers to finite-model adaptive control problem,” Mathematics of Control, Signals, and Systems, vol. 20, no. 3, pp. 271–303, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. S. S. Ge, C. C. Hang, and T. Zhang, “A direct adaptive controller for dynamic systems with a class of nonlinear parameterizations,” Automatica, vol. 35, no. 4, pp. 741–747, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. C. Y. Li and L. Guo, “On feedback capability in a class of nonlinearly parameterized uncertain systems,” IEEE Transactions on Automatic Control, vol. 56, no. 12, pp. 2946–2951, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. H. Ma, K.-Y. Lum, and S. S. Ge, “Adaptive control for a discrete-time first-order nonlinear system with both parametric and non-parametric uncertainties,” in Proceedings of the 46th IEEE Conference on Decision and Control (CDC '07), pp. 4839–4844, IEEE, New Orleans, La, USA, December 2007. View at Publisher · View at Google Scholar · View at Scopus
  29. L. Guo, “Exploring the capability and limits of the feedback mechanism,” in Proceedings of the International Congress of Mathematicians (ICM '02), Beijing, China, August 2002.
  30. H.-B. Ma, “An ‘impossibility’ theorem on a class of high-order discrete-time nonlinear control systems,” Systems and Control Letters, vol. 57, no. 6, pp. 497–504, 2008. View at Publisher · View at Google Scholar · View at Scopus
  31. I. Kanellakopoulos, P. V. Kokotovic, and A. S. Morse, “Systematic design of adaptive controllers for feedback linearizable systems,” IEEE Transactions on Automatic Control, vol. 36, no. 11, pp. 1241–1253, 1991. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  32. L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965. View at Publisher · View at Google Scholar
  33. L.-X. Wang and J. M. Mendel, “Fuzzy basis functions, universal approximation, and orthogonal least-squares learning,” IEEE Transactions on Neural Networks, vol. 3, no. 5, pp. 807–814, 1992. View at Publisher · View at Google Scholar · View at Scopus
  34. L.-X. Wang, “Stable adaptive fuzzy control of nonlinear systems,” IEEE Transactions on Fuzzy Systems, vol. 1, no. 2, pp. 146–155, 1993. View at Publisher · View at Google Scholar · View at Scopus
  35. W. S. McCulloch and W. Pitts, “A logical calculus of the ideas immanent in nervous activity,” The Bulletin of Mathematical Biophysics, vol. 5, pp. 115–133, 1943. View at Google Scholar · View at MathSciNet
  36. K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Networks, vol. 2, no. 5, pp. 359–366, 1989. View at Publisher · View at Google Scholar · View at Scopus
  37. T. Khanna, Foundations of Neural Networks, Addison-Wesley, Reading, Mass, USA, 1990.
  38. R. M. Sanner and J.-J. E. Slotine, “Gaussian networks for direct adaptive control,” IEEE Transactions on Neural Networks, vol. 3, no. 6, pp. 837–863, 1992. View at Publisher · View at Google Scholar · View at Scopus
  39. Y. J. Liu, Y. J. Fang, and M. A. Bao-Ping, “Sliding-data-window-driven Bayesian-Gaussian neural network and its application to modeling of nonlinear system,” Control Theory & Applications, vol. 26, no. 12, pp. 1435–1438, 2009. View at Google Scholar
  40. D. Wang and J. Huang, “Adaptive neural network control for a class of uncertain nonlinear systems in pure-feedback form,” Automatica, vol. 38, no. 8, pp. 1365–1372, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  41. Y. Song and J. W. Grizzle, “Adaptive output-feedback control of a class of discrete-time nonlinear systems,” in Proceedings of the American Control Conference, pp. 1359–1363, June 1993. View at Scopus
  42. B.-S. Chen, C.-S. Tseng, and H.-J. Uang, “Robustness design of nonlinear dynamic systems via fuzzy linear control,” IEEE Transactions on Fuzzy Systems, vol. 7, no. 5, pp. 571–585, 1999. View at Publisher · View at Google Scholar · View at Scopus
  43. T. Chai and S. Tong, “Fuzzy direct adaptive control for a class of nonlinear systems,” Fuzzy Sets and Systems, vol. 103, no. 3, pp. 379–387, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  44. K. Tanaka, T. Ikeda, and H. O. Wang, “Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: quadratic stabilizability, H  control theory, and linear matrix inequalities,” IEEE Transactions on Fuzzy Systems, vol. 4, no. 1, pp. 1–13, 1996. View at Publisher · View at Google Scholar · View at Scopus
  45. S. Jagannathan, M. W. Vandegrift, and F. L. Lewis, “Adaptive fuzzy logic control of discrete-time dynamical systems,” Automatica, vol. 36, no. 2, pp. 229–241, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  46. Y. Jiang, Z. Liu, C. Chen, and Y. Zhang, “Adaptive robust fuzzy control for dual arm robot with unknown input deadzone nonlinearity,” Nonlinear Dynamics, vol. 81, no. 3, pp. 1301–1314, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  47. M. W. Vandegrift, F. L. Lewis, S. Jagannathan, and K. Liu, “Adaptive fuzzy logic control of discrete-time dynamical systems,” in Proceedings of the IEEE International Symposium on Intelligent Control, pp. 395–401, IEEE, Monterey, Calif, USA, August 1995. View at Publisher · View at Google Scholar
  48. S. Jagannathan, “Adaptive fuzzy logic control of feedback linearizable discrete-time dynamical systems under persistence of excitation,” Automatica, vol. 34, no. 11, pp. 1295–1310, 1998. View at Publisher · View at Google Scholar · View at Scopus
  49. R. Qi and M. A. Brdys, “Stable indirect adaptive control based on discrete-time T-S fuzzy model,” Fuzzy Sets and Systems, vol. 159, no. 8, pp. 900–925, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  50. T.-C. Lin, S.-W. Chang, and C.-H. Hsu, “Robust adaptive fuzzy sliding mode control for a class of uncertain discrete-time nonlinear systems,” International Journal of Innovative Computing, Information and Control, vol. 8, no. 1, pp. 347–359, 2012. View at Google Scholar · View at Scopus
  51. G. Feng and G. Chen, “Adaptive control of discrete-time chaotic systems: a fuzzy control approach,” Chaos, Solitons and Fractals, vol. 23, no. 2, pp. 459–467, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  52. H. J. Lee, J. B. Park, and G. Chen, “Robust fuzzy control of nonlinear systems with parametric uncertainties,” IEEE Transactions on Fuzzy Systems, vol. 9, no. 2, pp. 369–379, 2001. View at Publisher · View at Google Scholar · View at Scopus
  53. Y.-Y. Cao and P. M. Frank, “Robust H disturbance attenuation for a class of uncertain discrete-time fuzzy systems,” IEEE Transactions on Fuzzy Systems, vol. 8, no. 4, pp. 406–415, 2000. View at Publisher · View at Google Scholar · View at Scopus
  54. S. Zhou, G. Feng, J. Lam, and S. Xu, “Robust H control for discrete-time fuzzy systems via basis-dependent Lyapunov functions,” Information Sciences, vol. 174, no. 3-4, pp. 197–217, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  55. S. Xu and J. Lam, “Robust H control for uncertain discrete-time-delay fuzzy systems via output feedback controllers,” IEEE Transactions on Fuzzy Systems, vol. 13, no. 1, pp. 82–93, 2005. View at Publisher · View at Google Scholar · View at Scopus
  56. C.-S. Tseng and B.-S. Chen, “Robust fuzzy observer-based fuzzy control design for nonlinear discrete-time systems with persistent bounded disturbances,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 3, pp. 711–723, 2009. View at Publisher · View at Google Scholar · View at Scopus
  57. S. Xu, B. Song, J. Lu, and J. Lam, “Robust stability of uncertain discrete-time singular fuzzy systems,” Fuzzy Sets and Systems, vol. 158, no. 20, pp. 2306–2316, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  58. Z.-G. Wu, P. Shi, H. Su, and J. Chu, “Reliable H control for discrete-time fuzzy systems with infinite-distributed delay,” IEEE Transactions on Fuzzy Systems, vol. 20, no. 1, pp. 22–31, 2012. View at Publisher · View at Google Scholar · View at Scopus
  59. G. Feng and J. Ma, “Quadratic stabilization of uncertain discrete-time fuzzy dynamic systems,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 48, no. 11, pp. 1337–1344, 2001. View at Publisher · View at Google Scholar · View at Scopus
  60. A. Kruszewski, R. Wang, and T. M. Guerra, “Nonquadratic stabilization conditions for a class of uncertain nonlinear discrete time TS fuzzy models: a new approach,” IEEE Transactions on Automatic Control, vol. 53, no. 2, pp. 606–611, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  61. W.-J. Wang, Y.-J. Chen, and C.-H. Sun, “Relaxed stabilization criteria for discrete-time T-S fuzzy control systems based on a switching fuzzy model and piecewise Lyapunov function,” IEEE Transactions on Systems, Man, & Cybernetics Part B: Cybernetics, vol. 37, no. 3, pp. 551–559, 2007. View at Publisher · View at Google Scholar · View at Scopus
  62. G. Feng, “Stability analysis of discrete-time fuzzy dynamic systems based on piecewise Lyapunov functions,” IEEE Transactions on Fuzzy Systems, vol. 12, no. 1, pp. 22–28, 2004. View at Publisher · View at Google Scholar · View at Scopus
  63. H. Gao, X. Liu, and J. Lam, “Stability analysis and stabilization for discrete-time fuzzy systems with time-varying delay,” IEEE Transactions on Systems, Man, and Cybernetics Part B: Cybernetics, vol. 39, no. 2, pp. 306–317, 2009. View at Publisher · View at Google Scholar · View at Scopus
  64. L. Wu, X. Su, P. Shi, and J. Qiu, “A new approach to stability analysis and stabilization of discrete-time T-S fuzzy time-varying delay systems,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 41, no. 1, pp. 273–286, 2011. View at Publisher · View at Google Scholar · View at Scopus
  65. X. Su, P. Shi, L. Wu, and Y.-D. Song, “A novel control design on discrete-time takagi-sugeno fuzzy systems with time-varying delays,” IEEE Transactions on Fuzzy Systems, vol. 21, no. 4, pp. 655–671, 2013. View at Publisher · View at Google Scholar · View at Scopus
  66. X. Su, P. Shi, L. Wu, and Y.-D. Song, “A novel approach to filter design for T-S fuzzy discrete-time systems with time-varying delay,” IEEE Transactions on Fuzzy Systems, vol. 20, no. 6, pp. 1114–1129, 2012. View at Publisher · View at Google Scholar · View at Scopus
  67. C.-S. Tseng, “Model reference output feedback fuzzy tracking control design for nonlinear discrete-time systems with time-delay,” IEEE Transactions on Fuzzy Systems, vol. 14, no. 1, pp. 58–70, 2006. View at Publisher · View at Google Scholar · View at Scopus
  68. X. Su, P. Shi, L. Wu, and S. K. Nguang, “Induced l2 filtering of fuzzy stochastic systems with time-varying delays,” IEEE Transactions on Cybernetics, vol. 43, no. 4, pp. 1257–1264, 2013. View at Publisher · View at Google Scholar · View at Scopus
  69. L. Wu, X. Su, P. Shi, and J. Qiu, “Model approximation for discrete-time state-delay systems in the TS fuzzy framework,” IEEE Transactions on Fuzzy Systems, vol. 19, no. 2, pp. 366–378, 2011. View at Publisher · View at Google Scholar · View at Scopus
  70. F. L. Lewis, S. Jagannathan, and A. Yesildirek, Neural Network Control of Robot Manipulators and Nonlinear Systems, Taylor & Francis, London, UK, 1999.
  71. A. M. Shaw and F. J. Doyle III, “Multivariable nonlinear control applications for a high purity distillation column using a recurrent dynamic neuron model,” Journal of Process Control, vol. 7, no. 4, pp. 255–268, 1997. View at Publisher · View at Google Scholar · View at Scopus
  72. K. Najim, Process Modeling and Control in Chemical Engineering, Marcel Dekker, New York, NY, USA, 1989.
  73. B. Xu, D. Wang, F. Sun, and Z. Shi, “Direct neural discrete control of hypersonic flight vehicle,” Nonlinear Dynamics, vol. 70, no. 1, pp. 269–278, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  74. B. Xu and Y. Zhang, “Neural discrete back-stepping control of hypersonic flight vehicle with equivalent prediction model,” Neurocomputing, vol. 154, pp. 337–346, 2015. View at Publisher · View at Google Scholar · View at Scopus
  75. B. Xu, F. Sun, H. Liu, and J. Ren, “Adaptive Kriging controller design for hypersonic flight vehicle via back-stepping,” IET Control Theory & Applications, vol. 6, no. 4, pp. 487–497, 2012. View at Publisher · View at Google Scholar · View at Scopus
  76. B. Xu, “Robust adaptive neural control of flexible hypersonic flight vehicle with dead-zone input nonlinearity,” Nonlinear Dynamics, vol. 80, no. 3, pp. 1509–1520, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  77. B. Xu, X. Huang, D. Wang, and F. Sun, “Dynamic surface control of constrained hypersonic flight models with parameter estimation and actuator compensation,” Asian Journal of Control, vol. 16, no. 1, pp. 162–174, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  78. B. Xu and Z. Shi, “An overview on flight dynamics and control approaches for hypersonic vehicles,” Science China Information Sciences, vol. 58, no. 7, pp. 1–19, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  79. D. E. Rumelhart, G. E. Hinton, and R. J. Williams, “Learning internal representations by error propagation,” in Parallel Distributed Processing, vol. 1, pp. 318–362, MIT Press, 1986. View at Google Scholar
  80. S. Jagannathan and F. L. Lewis, “Discrete-time neural net controller for a class of nonlinear dynamical systems,” IEEE Transactions on Automatic Control, vol. 41, no. 11, pp. 1693–1699, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  81. S. Jagannathan and F. L. Lewis, “Multilayer discrete-time neural-net controller with guaranteed performance,” IEEE Transactions on Neural Network, vol. 7, no. 1, pp. 107–130, 1996. View at Publisher · View at Google Scholar
  82. P. He and S. Jagannathan, “Neuro-controller for reducing cyclic variation in lean combustion spark ignition engines,” Automatica, vol. 41, no. 7, pp. 1133–1142, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  83. S. S. Ge, G. Y. Li, and T. H. Lee, “Adaptive NN control for a class of strict-feedback discrete-time nonlinear systems,” Automatica, vol. 39, no. 5, pp. 807–819, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  84. S. S. Ge, T. H. Lee, G. Y. Li, and J. Zhang, “Adaptive NN control for a class of discrete-time non-linear systems,” International Journal of Control, vol. 76, no. 4, pp. 334–354, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  85. C. J. Goh, “Model reference control of non-linear systems via implicit function emulation,” International Journal of Control, vol. 60, no. 1, pp. 91–115, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  86. C. J. Goh and T. H. Lee, “Direct adaptive control of nonlinear systems via implicit function emulation,” Control Theory and Advanced Technology, vol. 10, no. 3, pp. 539–552, 1994. View at Google Scholar · View at MathSciNet
  87. A. U. Levin and K. S. Narendra, “Control of nonlinear dynamical systems using neural networks—part II: observability, identification, and control,” IEEE Transactions on Neural Networks, vol. 7, no. 1, pp. 30–42, 1996. View at Publisher · View at Google Scholar · View at Scopus
  88. S. S. Ge, J. Zhang, and T. H. Lee, “Adaptive MNN control for a class of non-affine NARMAX systems with disturbances,” Systems & Control Letters, vol. 53, no. 1, pp. 1–12, 2004. View at Publisher · View at Google Scholar · View at Scopus
  89. S. S. Ge, J. Zhang, and T. H. Lee, “Adaptive neural network control for a class of MIMO nonlinear systems with disturbances in discrete-time,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 34, no. 4, pp. 1630–1645, 2004. View at Publisher · View at Google Scholar · View at Scopus
  90. J. Zhang, S. S. Ge, and T. H. Lee, “Output feedback control of a class of discrete MIMO nonlinear systems with triangular form inputs,” IEEE Transactions on Neural Networks, vol. 16, no. 6, pp. 1491–1503, 2005. View at Publisher · View at Google Scholar · View at Scopus
  91. F. C. Sun, Z. Sun, and P.-Y. Woo, “Stable neural-network-based adaptive control for sampled-data nonlinear systems,” IEEE Transactions on Neural Networks, vol. 9, no. 5, pp. 956–968, 1998. View at Publisher · View at Google Scholar · View at Scopus
  92. C. Yang, S. S. Ge, C. Xiang, T. Chai, and T. H. Lee, “Output feedback NN control for two classes of discrete-time systems with unknown control directions in a unified approach,” IEEE Transactions on Neural Networks, vol. 19, no. 11, pp. 1873–1886, 2008. View at Publisher · View at Google Scholar · View at Scopus
  93. S. S. Ge, C. Yang, and T. H. Lee, “Adaptive predictive control using neural network for a class of pure-feedback systems in discrete time,” IEEE Transactions on Neural Networks, vol. 19, no. 9, pp. 1599–1614, 2008. View at Publisher · View at Google Scholar · View at Scopus
  94. Y. Li, C. Yang, S. S. Ge, and T. H. Lee, “Adaptive output feedback NN control of a class of discrete-time MIMO nonlinear systems with unknown control directions,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 41, no. 2, pp. 507–517, 2011. View at Publisher · View at Google Scholar · View at Scopus
  95. Y.-J. Liu, C. L. P. Chen, G.-X. Wen, and S. Tong, “Adaptive neural output feedback tracking control for a class of uncertain discrete-time nonlinear systems,” IEEE Transactions on Neural Networks, vol. 22, no. 7, pp. 1162–1167, 2011. View at Publisher · View at Google Scholar · View at Scopus
  96. C. Yang, S. S. Ge, and T. H. Lee, “Output feedback adaptive control of a class of nonlinear discrete-time systems with unknown control directions,” Automatica, vol. 45, no. 1, pp. 270–276, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  97. A. Al-Tamimi, F. L. Lewis, and M. Abu-Khalaf, “Discrete-time nonlinear HJB solution using approximate dynamic programming: convergence proof,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 38, no. 4, pp. 943–949, 2008. View at Publisher · View at Google Scholar · View at Scopus
  98. P. Werbos, “Approximate dynamic programming for real-time control and neural modeling,” in Handbook of Intelligent Control Neural Fuzzy & Adaptive Approaches, Van Nostrand Reinhold, 1992. View at Google Scholar
  99. P. He and S. Jagannathan, “Reinforcement learning neural-network-based controller for nonlinear discrete-time systems with input constraints,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 37, no. 2, pp. 425–436, 2007. View at Publisher · View at Google Scholar · View at Scopus
  100. B. Xu, C. Yang, and Z. Shi, “Reinforcement learning output feedback NN control using deterministic learning technique,” IEEE Transactions on Neural Networks and Learning Systems, vol. 25, no. 3, pp. 635–641, 2014. View at Publisher · View at Google Scholar · View at Scopus
  101. D. Liu, D. Wang, D. Zhao, Q. Wei, and N. Jin, “Neural-network-based optimal control for a class of unknown discrete-time nonlinear systems using globalized dual heuristic programming,” IEEE Transactions on Automation Science and Engineering, vol. 9, no. 3, pp. 628–634, 2012. View at Publisher · View at Google Scholar · View at Scopus
  102. D. Liu and Q. Wei, “Finite-approximation-error-based optimal control approach for discrete-time nonlinear systems,” IEEE Transactions on Cybernetics, vol. 43, no. 2, pp. 779–789, 2013. View at Publisher · View at Google Scholar · View at Scopus
  103. X. Zhong, H. He, H. Zhang, and Z. Wang, “Optimal control for unknown discrete-time nonlinear markov jump systems using adaptive dynamic programming,” IEEE Transactions on Neural Networks and Learning Systems, vol. 25, no. 12, pp. 2141–2155, 2014. View at Publisher · View at Google Scholar · View at Scopus
  104. D. Liu, D. Wang, and X. Yang, “An iterative adaptive dynamic programming algorithm for optimal control of unknown discrete-time nonlinear systems with constrained inputs,” Information Sciences, vol. 220, pp. 331–342, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  105. F.-Y. Wang, N. Jin, D. Liu, and Q. Wei, “Adaptive dynamic programming for finite-horizon optimal control of discrete-time nonlinear systems with ε-error bound,” IEEE Transactions on Neural Networks, vol. 22, no. 1, pp. 24–36, 2011. View at Publisher · View at Google Scholar · View at Scopus
  106. H. Zhang, Y. Luo, and D. Liu, “Neural-network-based near-optimal control for a class of discrete-time affine nonlinear systems with control constraints,” IEEE Transactions on Neural Networks, vol. 20, no. 9, pp. 1490–1503, 2009. View at Publisher · View at Google Scholar · View at Scopus