Stabilization of Unknown Nonlinear Discrete-Time Delay Systems Based on Neural Network

Affiliation(s)

Department of Electronics & Communications Engineering, G. L. A. University, Mathura, India.

Department of Electrical Engineering, Motilal Nehru National Institute of Technology (MNNIT), Allahabad, India.

Department of Electronics & Communications Engineering, G. L. A. University, Mathura, India.

Department of Electrical Engineering, Motilal Nehru National Institute of Technology (MNNIT), Allahabad, India.

ABSTRACT

This paper discusses about the stabilization of unknown nonlinear discrete-time fixed state delay systems. The unknown system nonlinearity is approximated by Chebyshev neural network (CNN), and weight update law is presented for approximating the system nonlinearity. Using appropriate Lyapunov-Krasovskii functional the stability of the nonlinear system is ensured by the solution of linear matrix inequalities. Finally, a relevant example is given to illustrate the effectiveness of the proposed control scheme.

This paper discusses about the stabilization of unknown nonlinear discrete-time fixed state delay systems. The unknown system nonlinearity is approximated by Chebyshev neural network (CNN), and weight update law is presented for approximating the system nonlinearity. Using appropriate Lyapunov-Krasovskii functional the stability of the nonlinear system is ensured by the solution of linear matrix inequalities. Finally, a relevant example is given to illustrate the effectiveness of the proposed control scheme.

Cite this paper

V. Deolia, S. Purwar and T. Sharma, "Stabilization of Unknown Nonlinear Discrete-Time Delay Systems Based on Neural Network,"*Intelligent Control and Automation*, Vol. 3 No. 4, 2012, pp. 337-345. doi: 10.4236/ica.2012.34039.

V. Deolia, S. Purwar and T. Sharma, "Stabilization of Unknown Nonlinear Discrete-Time Delay Systems Based on Neural Network,"

References

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[2] J. Chen, D. M. Xu and B. Shafai, “On Sufficient Conditions for Stability Independent of Delay,” IEEE Transactions on Automatic Control, Vol. 40, No. 9, 1995, pp. 1675-1680. doi:10.1109/9.412644

[3] E. Fridman and U. Shaked, “Stability and Guaranteed Cost Control of Uncertain Discrete-Delay Systems,” International Journal of Control, Vol. 78, No. 4, 2005, pp. 235-246. doi:10.1080/00207170500041472

[4] C. Lin, Q.-G. Wang and T. H. Lee, “A Less Conservative Robust Stability Test for Linear Uncertain Time-Delay Systems,” IEEE Transactions on Automatic Control, Vol. 51, No. 1, 2006, pp. 87-91. doi:10.1109/TAC.2005.861720

[5] Z. Wang, B. Huang and H. Unbehauen, “Robust Hx Observer Design of Linear State Delayed Systems with Parameteric Uncertainty: The Discrete-Time Case,” Automatica, Vol. 35, 1999, pp. 1161-1167 doi:10.1016/S0005-1098(99)00008-4

[6] Y. He, Q. G. Wang, C. Lin and M. Wu, “Delay-RangeDependent Stability for Systems with Time Varying Delay,” Automatica, Vol. 43, No. 2, 2007, pp. 371-376. doi:10.1016/j.automatica.2006.08.015

[7] X. Li and C. de Sauza, “Criteria for Robust Stability and Stabilization of Uncertain Linear Systems with State Delay,” Automatica, Vol. 33, 1997, pp. 1697-1662. doi:10.1016/S0005-1098(97)00082-4

[8] V. Kolmanvoskii and J.-P. Richard, “Stability of Some Linear Systems with Delays,” IEEE Transactions on Automatic Control, Vol. 44, 1999, pp. 984-989. doi:10.1109/9.763213

[9] E. Fridman, “New Lyapunov-Krasovskii Functional for Stability of Linear Retarted and Neutral Type Systems,” Systems and Control Letters, Vol. 43, 2001, pp. 309-319. doi:10.1016/S0167-6911(01)00114-1

[10] S.-I. Niculescu, “Delay Effects on Stability: A Robust Control Approach,” Lecture Notes in Control and Information Sciences, Vol. 269, Springer-Verlag, London, 2001.

[11] E. Fridman and U. Shaked, “An Improved Stabilization Method for Linear Systems with Time-Delay,” IEEE Transactions on Automation Control, Vol. 47, 2002, pp. 1931-1937. doi:10.1109/TAC.2002.804462

[12] E. Fridman, “Stability of Linear Functional Differential Equations: A New Lyapunov Technique,” Proceedings of Mathematical Theory of Networks and Systems, Leuven, 5-9 July 2004.

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[14] E. Fridman and U. Shaked, “An LMI Approach to Stability of Discrete Delay Systems,” Proceedings of 7th IEE European Control Conference, Cambridge, 2003.

[15] Y. S. Lee and W. H. Kwon, “Delay Dependent Robust Stabilization of Uncertain Discrete-Time State-Delayed Systems,” 15th Triennial World Congress, Barcelona, 2002.

[16] S. S. Ge, F. Hong and T. H. Lee, “Adaptive Neural Network Control of Nonlinear Systems with Unknown Time Delays,” IEEE Transactions on Automatic Control, Vol. 48, No. 11, 2003, pp. 2004-2010. doi:10.1109/TAC.2003.819287

[17] S. S. Ge, F. Hong and T. H. Lee, “Adaptive Neural Control of Nonlinear Time-Delay Systems with Unknown Virtual Control Coefficients,” IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics, Vol. 34, No. 1, 2004, pp. 499-516. doi:10.1109/TSMCB.2003.817055

[18] D. W. C. Ho, J. Li and Y. Niu, “Adaptive Neural Control for a Class of Nonlinearly Parametric Time-Delay Systems,” IEEE Transactions on Neural Networks, Vol. 16, No. 3, 2005, pp. 625-635. doi:10.1109/TNN.2005.844907

[19] A. Kulkarni and S. Purwar, “Backstepping Control for a Class of Delayed Nonlinear Systems with Input Constraints,” IEEE Conference, Vol. 1, 2008, pp. 274-279.

[20] B. Chen, X. P. Liu, K. F. Liu and C. Lin, “Fuzzy-Approximation-Based Adaptive Control of Strict-Feedback Nonlinear Systems with Time Delays,” IEEE Transactions on Fuzzy Systems, Vol. 18, No. 5, 2010, pp. 883-891. doi:10.1109/TFUZZ.2010.2050892

[21] W. Aggoune, R. Kharel and K. Busawon, “On Feedback Stabilization of Nonlinear Discrete-Time State-Delayed Systems,” ECC Conference, 23-26 August 2009, Budapest.

[22] H. J. Gao and T. W. Chen, “New Results on Stability of Discrete-Time Systems with Time-Varying State Delay,” IEEE Transactions on Automatic Control, Vol. 52, No. 2, 2007, pp. 328-334. doi:10.1109/TAC.2006.890320

[23] S. Ibrir, W. F. Xie and C.-Y. Su, “Observer Design for Discrete-Time Systems Subject to Time Delay Nonlinearities,” International Journal of Systems Science, Vol. 37, No. 9, 2006, pp. 629-641. doi:10.1080/00207720600774289

[24] Q.-L. Wei, H.-G. Zhang, D.-R. Liu and Y. Zhao, “An Optimal Control Scheme for a Class of Discrete-Time Nonlinear Systems with t Delays Using Adaptive Dynamic Programming,” ACTA Automatica Sinica, Vol. 36, No. 1, 2010, pp. 121-129. doi:10.3724/SP.J.1004.2010.00121

[25] K. E. Bouazza, M. Boutayeb and M. Darouach, “State Feedback Stabilization of Discrete-Time Delay Nonlinear Systems,” Proceedings of Mathematical Theory of Networks and Systems, Leuven, 5-9 July 2004.

[26] E. J. Hartman, J. D. Keelar and J. M. Kowalski, “Layered Neural Networks with Gaussian Hidden Units as Universal Approximation,” Neural Computation, Vol. 2, 1990, pp. 210-215. doi:10.1162/neco.1990.2.2.210

[27] S. Chen, S. A. Billings and P. M. Grant, “Recursive Hybrid Algorithm for Nonlinear System Identification Using Radial Basis Function Networks,” International Journal of Control, Vol. 55, No. 5, 1992, pp. 1051-1070. doi:10.1080/00207179208934272

[28] S. V. T. Elanayar and Y. C. Shin, “Radial Basis Function Neural Network for Approximation and Estimation of Nonlinear Stochastic Dynamic Systems,” IEEE Transactions on Neural Networks, Vol. 5, 1994, pp. 594-603. doi:10.1109/72.298229

[29] J. C. Patra and A. C. Kot, “Nonlinear Dynamic System Identification Using Chebyshev Functional Link Artificial Neural Networks,” IEEE Transactions on Systems, Man, and Cybernetics, Vol. 32, No. 4, 2002, pp. 505-511. doi:10.1109/TSMCB.2002.1018769

[30] A. Namatame and N. Ueda, “Pattern Classification with Chebyshev Neural Network,” International Journal Neural Network, Vol. 3, 1992, pp. 23-31.

[31] T. T. Lee and J. T. Jeng, “The Chebyshev Polynomial Based Unified Model Neural Networks for Functions Approximations,” IEEE Transactions on Systems, Man & Cybernetics, Part B, Vol. 28, No. 6, 1998, pp. 925-935.

[32] S. Purwar, I. N. Kar and A. N. Jha, “Adaptive Output Feedback Tracking Control of Robot Manipulators Using Position Measurements Only,” Expert Systems with Applications, Vol. 34, 2008, pp. 2789-2798. doi:10.1016/j.eswa.2007.05.030

[33] S. Purwar, I. N. Kar and A. N. Jha, “On-Line System Identification of Complex Systems Using Chebyshev Neural Networks,” Applied Soft Computing, Vol. 7, No. 1, 2005, pp. 364-372. doi:10.1016/j.asoc.2005.08.001

[34] C. Kwan and F. L. Lewis, “Robust Backstepping Control of Nonlinear Systems Using Neural Networks,” IEEE Transactions on Systems, Man, and Cybernetics, Vol. 30, No. 6, 2000, pp. 753-766. doi:10.1109/3468.895898

[1] M. Basin, J. Perez and R. Martinenz-Zuniga, “Optimal Filtering for Nonlinear Polynomial Systems over Linear Observations with Delay,” International Journal of Innovative Computing Information and Control, Vol. 2, 2006, pp. 863-874.

[2] J. Chen, D. M. Xu and B. Shafai, “On Sufficient Conditions for Stability Independent of Delay,” IEEE Transactions on Automatic Control, Vol. 40, No. 9, 1995, pp. 1675-1680. doi:10.1109/9.412644

[3] E. Fridman and U. Shaked, “Stability and Guaranteed Cost Control of Uncertain Discrete-Delay Systems,” International Journal of Control, Vol. 78, No. 4, 2005, pp. 235-246. doi:10.1080/00207170500041472

[4] C. Lin, Q.-G. Wang and T. H. Lee, “A Less Conservative Robust Stability Test for Linear Uncertain Time-Delay Systems,” IEEE Transactions on Automatic Control, Vol. 51, No. 1, 2006, pp. 87-91. doi:10.1109/TAC.2005.861720

[5] Z. Wang, B. Huang and H. Unbehauen, “Robust Hx Observer Design of Linear State Delayed Systems with Parameteric Uncertainty: The Discrete-Time Case,” Automatica, Vol. 35, 1999, pp. 1161-1167 doi:10.1016/S0005-1098(99)00008-4

[6] Y. He, Q. G. Wang, C. Lin and M. Wu, “Delay-RangeDependent Stability for Systems with Time Varying Delay,” Automatica, Vol. 43, No. 2, 2007, pp. 371-376. doi:10.1016/j.automatica.2006.08.015

[7] X. Li and C. de Sauza, “Criteria for Robust Stability and Stabilization of Uncertain Linear Systems with State Delay,” Automatica, Vol. 33, 1997, pp. 1697-1662. doi:10.1016/S0005-1098(97)00082-4

[8] V. Kolmanvoskii and J.-P. Richard, “Stability of Some Linear Systems with Delays,” IEEE Transactions on Automatic Control, Vol. 44, 1999, pp. 984-989. doi:10.1109/9.763213

[9] E. Fridman, “New Lyapunov-Krasovskii Functional for Stability of Linear Retarted and Neutral Type Systems,” Systems and Control Letters, Vol. 43, 2001, pp. 309-319. doi:10.1016/S0167-6911(01)00114-1

[10] S.-I. Niculescu, “Delay Effects on Stability: A Robust Control Approach,” Lecture Notes in Control and Information Sciences, Vol. 269, Springer-Verlag, London, 2001.

[11] E. Fridman and U. Shaked, “An Improved Stabilization Method for Linear Systems with Time-Delay,” IEEE Transactions on Automation Control, Vol. 47, 2002, pp. 1931-1937. doi:10.1109/TAC.2002.804462

[12] E. Fridman, “Stability of Linear Functional Differential Equations: A New Lyapunov Technique,” Proceedings of Mathematical Theory of Networks and Systems, Leuven, 5-9 July 2004.

[13] K. J. Astrom and B. Wittenmark, “Computer Controlled Systems: Theory and Design,” Prentice-Hall Inc., Englewood Cliffs, 1984.

[14] E. Fridman and U. Shaked, “An LMI Approach to Stability of Discrete Delay Systems,” Proceedings of 7th IEE European Control Conference, Cambridge, 2003.

[15] Y. S. Lee and W. H. Kwon, “Delay Dependent Robust Stabilization of Uncertain Discrete-Time State-Delayed Systems,” 15th Triennial World Congress, Barcelona, 2002.

[16] S. S. Ge, F. Hong and T. H. Lee, “Adaptive Neural Network Control of Nonlinear Systems with Unknown Time Delays,” IEEE Transactions on Automatic Control, Vol. 48, No. 11, 2003, pp. 2004-2010. doi:10.1109/TAC.2003.819287

[17] S. S. Ge, F. Hong and T. H. Lee, “Adaptive Neural Control of Nonlinear Time-Delay Systems with Unknown Virtual Control Coefficients,” IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics, Vol. 34, No. 1, 2004, pp. 499-516. doi:10.1109/TSMCB.2003.817055

[18] D. W. C. Ho, J. Li and Y. Niu, “Adaptive Neural Control for a Class of Nonlinearly Parametric Time-Delay Systems,” IEEE Transactions on Neural Networks, Vol. 16, No. 3, 2005, pp. 625-635. doi:10.1109/TNN.2005.844907

[19] A. Kulkarni and S. Purwar, “Backstepping Control for a Class of Delayed Nonlinear Systems with Input Constraints,” IEEE Conference, Vol. 1, 2008, pp. 274-279.

[20] B. Chen, X. P. Liu, K. F. Liu and C. Lin, “Fuzzy-Approximation-Based Adaptive Control of Strict-Feedback Nonlinear Systems with Time Delays,” IEEE Transactions on Fuzzy Systems, Vol. 18, No. 5, 2010, pp. 883-891. doi:10.1109/TFUZZ.2010.2050892

[21] W. Aggoune, R. Kharel and K. Busawon, “On Feedback Stabilization of Nonlinear Discrete-Time State-Delayed Systems,” ECC Conference, 23-26 August 2009, Budapest.

[22] H. J. Gao and T. W. Chen, “New Results on Stability of Discrete-Time Systems with Time-Varying State Delay,” IEEE Transactions on Automatic Control, Vol. 52, No. 2, 2007, pp. 328-334. doi:10.1109/TAC.2006.890320

[23] S. Ibrir, W. F. Xie and C.-Y. Su, “Observer Design for Discrete-Time Systems Subject to Time Delay Nonlinearities,” International Journal of Systems Science, Vol. 37, No. 9, 2006, pp. 629-641. doi:10.1080/00207720600774289

[24] Q.-L. Wei, H.-G. Zhang, D.-R. Liu and Y. Zhao, “An Optimal Control Scheme for a Class of Discrete-Time Nonlinear Systems with t Delays Using Adaptive Dynamic Programming,” ACTA Automatica Sinica, Vol. 36, No. 1, 2010, pp. 121-129. doi:10.3724/SP.J.1004.2010.00121

[25] K. E. Bouazza, M. Boutayeb and M. Darouach, “State Feedback Stabilization of Discrete-Time Delay Nonlinear Systems,” Proceedings of Mathematical Theory of Networks and Systems, Leuven, 5-9 July 2004.

[26] E. J. Hartman, J. D. Keelar and J. M. Kowalski, “Layered Neural Networks with Gaussian Hidden Units as Universal Approximation,” Neural Computation, Vol. 2, 1990, pp. 210-215. doi:10.1162/neco.1990.2.2.210

[27] S. Chen, S. A. Billings and P. M. Grant, “Recursive Hybrid Algorithm for Nonlinear System Identification Using Radial Basis Function Networks,” International Journal of Control, Vol. 55, No. 5, 1992, pp. 1051-1070. doi:10.1080/00207179208934272

[28] S. V. T. Elanayar and Y. C. Shin, “Radial Basis Function Neural Network for Approximation and Estimation of Nonlinear Stochastic Dynamic Systems,” IEEE Transactions on Neural Networks, Vol. 5, 1994, pp. 594-603. doi:10.1109/72.298229

[29] J. C. Patra and A. C. Kot, “Nonlinear Dynamic System Identification Using Chebyshev Functional Link Artificial Neural Networks,” IEEE Transactions on Systems, Man, and Cybernetics, Vol. 32, No. 4, 2002, pp. 505-511. doi:10.1109/TSMCB.2002.1018769

[30] A. Namatame and N. Ueda, “Pattern Classification with Chebyshev Neural Network,” International Journal Neural Network, Vol. 3, 1992, pp. 23-31.

[31] T. T. Lee and J. T. Jeng, “The Chebyshev Polynomial Based Unified Model Neural Networks for Functions Approximations,” IEEE Transactions on Systems, Man & Cybernetics, Part B, Vol. 28, No. 6, 1998, pp. 925-935.

[32] S. Purwar, I. N. Kar and A. N. Jha, “Adaptive Output Feedback Tracking Control of Robot Manipulators Using Position Measurements Only,” Expert Systems with Applications, Vol. 34, 2008, pp. 2789-2798. doi:10.1016/j.eswa.2007.05.030

[33] S. Purwar, I. N. Kar and A. N. Jha, “On-Line System Identification of Complex Systems Using Chebyshev Neural Networks,” Applied Soft Computing, Vol. 7, No. 1, 2005, pp. 364-372. doi:10.1016/j.asoc.2005.08.001

[34] C. Kwan and F. L. Lewis, “Robust Backstepping Control of Nonlinear Systems Using Neural Networks,” IEEE Transactions on Systems, Man, and Cybernetics, Vol. 30, No. 6, 2000, pp. 753-766. doi:10.1109/3468.895898