International Journal of Control, Automation, and Systems

Institute of Control, Robotics and Systems (제어로봇시스템학회)
권호 표지

A Novel Stabilizing Control for Neural Nonlinear Systems with Time Delays by State and Dynamic Output Feedback

  • Liu, Mei-Qin (College of Electrical Engineering, Zhejiang University) ;
  • Wang, Hui-Fang (College of Electrical Engineering, Zhejiang University)
  • Published : 2008.02.28
Download PDF
⟨ Previous Next ⟩

Abstract

A novel neural network model, termed the standard neural network model (SNNM), similar to the nominal model in linear robust control theory, is suggested to facilitate the synthesis of controllers for delayed (or non-delayed) nonlinear systems composed of neural networks. The model is composed of a linear dynamic system and a bounded static delayed (or non-delayed) nonlinear operator. Based on the global asymptotic stability analysis of SNNMs, Static state-feedback controller and dynamic output feedback controller are designed for the SNNMs to stabilize the closed-loop systems, respectively. The control design equations are shown to be a set of linear matrix inequalities (LMIs) which can be easily solved by various convex optimization algorithms to determine the control signals. Most neural-network-based nonlinear systems with time delays or without time delays can be transformed into the SNNMs for controller synthesis in a unified way. Two application examples are given where the SNNMs are employed to synthesize the feedback stabilizing controllers for an SISO nonlinear system modeled by the neural network, and for a chaotic neural network, respectively. Through these examples, it is demonstrated that the SNNM not only makes controller synthesis of neural-network-based systems much easier, but also provides a new approach to the synthesis of the controllers for the other type of nonlinear systems.

Keywords

References

  1. K. S. Narendra and K. Parthasarathy, "Identification and control of dynamical systems using neural networks," IEEE Trans. on Neural Networks, vol. 1, no. 1, pp. 4-27, March 1990 https://doi.org/10.1109/72.80202
  2. C. Sun, K. Zhang, S. Fei, and C.-B. Feng, "On exponential stability of delayed neural networks with a general class of activation functions," Physics Letters A, vol. 298, no. 2-3, pp. 122-132, June 2002 https://doi.org/10.1016/S0375-9601(02)00471-1
  3. J. A. K. Suykens, J. P. L. Vandewalle, and B. L. R. De Moor, Artificial Neural Networks for Modeling and Control of Non-linear Systems, Kluwer Academic Publishers, Norwell, MA, 1996
  4. K. Tanaka, "An approach to stability criteria of neural-network control system," IEEE Trans. on Neural Networks, vol. 7, no. 3, pp. 629-642, May 1996 https://doi.org/10.1109/72.501721
  5. S. Limanond and J. Si, "Neural-network-based control design: An LMI approach," IEEE Trans. on Neural Networks, vol. 9, no. 6, pp. 1422-1429, June 1998 https://doi.org/10.1109/72.728392
  6. C. L. Lin and T. Y. Lin, "An $H_{\infty}$ design approach for neural net-based control schemes," IEEE Trans. on Automatic Control, vol. 46, no. 10, pp. 1599-1605, October 2001 https://doi.org/10.1109/9.956056
  7. X. F. Liao, G. R. Chen, and E. N. Sanchez, "LMI-based approach for asymptotically stability analysis of delayed neural networks," IEEE Trans. CAS-I, vol. 49, no. 7, pp. 1033-1039, July 2002 https://doi.org/10.1109/TCSI.2002.800842
  8. X. F. Liao, G. R. Chen, and E. N. Sanchez, "Delay-dependent exponential stability analysis of delayed neural networks: An LMI approach," Neural Networks, vol. 15, no. 7, pp. 855-866, September 2002 https://doi.org/10.1016/S0893-6080(02)00041-2
  9. P. C. Chandrasekharan, Robust Control of Linear Dynamical Systems, Academic Press, London, 1996
  10. J. B. Moore and B. D. O. Anderson. "A generalization of the Popov criterion," Journal of the Franklin Institute, vol. 285, no. 6, pp. 488-492, 1968 https://doi.org/10.1016/0016-0032(68)90053-7
  11. E. Rios-Patron, A General Framework for the Control of Nonlinear Systems, Ph.D. Thesis, University of Illinois, 2000
  12. M. Liu, "Discrete-time delayed standard neural network model and its application," Science in China Ser.E-Information Sciences, vol. 35, no. 10, pp. 1031-1048, October 2005
  13. M. Liu, "Delayed standard neural network models for control systems," IEEE Trans. on Neural Networks, vol. 18, no. 5, pp. 1376-1391, September 2007 https://doi.org/10.1109/TNN.2007.894084
  14. S. Zhang and M. Liu, "LMI-based approach for global asymptotic stability analysis of continuous BAM neural networks," Journal of Zhejiang University Science, vol. 6A, no. 1, pp. 32-37, January 2005 https://doi.org/10.1631/jzus.2005.A32
  15. M. Liu, "Delayed standard neural network model and its application," Acta Automatica Sinica, vol. 31, no. 5, pp. 750-758, September 2005
  16. M. Liu, "Dynamic output feedback stabilization for nonlinear systems based on standard neural network models," International Journal of Neural Systems, vol. 16, no. 4, pp. 305-317, August 2006 https://doi.org/10.1142/S0129065706000706
  17. S. P. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA, 1994
  18. O. Toker and H. Ozbay, "On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback," Proc. of Amer. Control Conference, Seattle, WA, vol. 4, pp. 2525-2526, June 1995
  19. A. Hassibi, J. How, and S. Boyd, "A path following method for solving BMI problems in control," Proc. Amer. Control Conference, San Diego, CA, vol. 2, pp. 1385-1389, June 1999
  20. S. Ibaraki, Nonconvex Optimization Problems in $H_{\infty}$ Optimization and Their Applications, Ph.D. Thesis, UC Berkeley, CA, 2000
  21. H. D. Tuan and P. Apkarian, "Low nonconvexity-rank bilinear matrix inequalities: Algorithms and applications in robust controller and structure designs," IEEE Trans. on Automatic Control, vol. 45, no. 11, pp. 2111-2117, November 2000 https://doi.org/10.1109/9.887636
  22. M. Fukuda and M. Kojima, "Branch-and-cut algorithms for the bilinear matrix inequality eigenvalue problem," Computational Optimization and Application, vol. 19, no. 1, pp. 79-105, January 2001 https://doi.org/10.1023/A:1011224403708
  23. A. Delgado, C. Kambhampati, and K. Warwick, "Dynamic recurrent neural network for system identification and control," IEE Proceedings of Control Theory and Applications, vol. 142, no. 4, pp. 307-314, July 1995 https://doi.org/10.1049/ip-cta:19951873
  24. A. Delgado, C. Kambhampati, and K. Warwick, "Identification of nonlinear systems with a dynamic recurrent neural network," Proc. of Fourth International Conference on Artificial Neural Networks, pp. 318-322, June 1995
  25. P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox- for Use with Matlab, The MATH Works, Inc., Natick, MA, 1995