Guaranteed Cost Output Feedback Control of Fuzzy Systems via LMI Approach

Shusaku Nishikawa; Jun Yoneyama

doi:10.20965/jaciii.2010.p0567

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JACIII Vol.14 No.6 pp. 567-573

doi: 10.20965/jaciii.2010.p0567

(2010)

Paper:

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Guaranteed Cost Output Feedback Control of Fuzzy Systems via LMI Approach

Shusaku Nishikawa and Jun Yoneyama

Department of Electrical Engineering and Electronics, Aoyama Gakuin University, 5-10-1 Fuchinobe, Chuo-ku, Sagamihara, Kanagawa 252-5258, Japan

Received:

December 2, 2009

Accepted:

April 14, 2010

Published:

September 20, 2010

Keywords:

Takagi-Sugeno fuzzy systems, premise variables, output feedback control, guaranteed cost

Abstract

This paper is concerned with the stability with guaranteed cost for a fuzzy system with immeasurable premise variables via output feedback. It is well known that Takagi-Sugeno fuzzy model describes a wide class of nonlinear systems especially when its premise variables include immeasurable functions. However, when it comes to output feedback control design of such a fuzzy system, a conventional Parallel Distributed Compensator (PDC) is not feasible because the PDC shares the same immeasurable premise variables as those of a fuzzy system. In this paper, we introduce an output feedback controller with the estimate of the premise variables of an original fuzzy system. We then formulate the stabilization problem with guaranteed cost for a fuzzy system with immeasurable premise variables. Our control design method is based on a set of strict LMI conditions. No tuning parameter is necessary a priori to solve them. The stability with guaranteed cost takes care of not only stabilization but also control performance. Our proposed method attempts to minimize the upper bound of the performance index, which results in the satisfactory trajectories of the system. Finally, numerical examples are given to illustrate our control design method.

Cite this article as:

S. Nishikawa and J. Yoneyama, “Guaranteed Cost Output Feedback Control of Fuzzy Systems via LMI Approach,” J. Adv. Comput. Intell. Intell. Inform., Vol.14 No.6, pp. 567-573, 2010.

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References

[1] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. on Systems, Man, Cybernetics, Vol.15, pp. 116-132, 1985.
[2] K. Tanaka and H. O. Wang, “Fuzzy Control Systems Design and Ana-lysis: A Linear Matrix Inequality Approach,” Wiley-Interscience, 2001.
[3] K. Tanaka and M. Sugeno, “Stability analysis and design of fuzzy control systems,” Fuzzy Sets and Systems, Vol.45, pp. 135-156, 1992.
[4] M. C. M. Teixiera, E. Assuncao, and R. G. Avellar, “On relaxed LMI-based designs for fuzzy regulators and fuzzy observers,” Vol.11, pp. 613-623, 2003.
[5] H. D. Tuan, P. Apkarian, T. Narikiyo, and Y. Yamamoto, “Parameterized linear matrix inequality techniques in fuzzy control system design,” IEEE Trans. on Fuzzy Systems, Vol.9, pp. 324-332, 2001.
[6] H. O.Wang, K. Tanaka, and M. Griffin, “An approach to fuzzy control of nonlinear system stability and design issues,” IEEE Trans. on Fuzzy Systems, Vol.4, No.1, pp. 14-21, 1996.
[7] J. Yoneyama, M. Nishikawa, H. Katayama, and A. Ichikawa, “Design of output feedback controllers for Takagi-Sugeno fuzzy systems,” Fuzzy Sets and Systems, Vol.121, No.1, pp. 127-148, 2001.
[8] J. Yoneyama, “H_∞ output feedback control for fuzzy systems with immeasurable premise variables: discrete-time case,” Applied Soft Computing, Vol.8, No.2, pp. 949-958, 2008.
[9] J. Yoneyama, “Robust output feedback guaranteed cost control of uncertain fuzzy systems with immeasurable premise variables,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.11, No.7, pp. 745-750, 2007.
[10] X. J. Ma, Z. Q. Sun, and Y. Y. He, “Analysis and design of fuzzy controller and fuzzy observer,” IEEE Trans. on Fuzzy Systems, Vol.6, No.1, pp. 41-50, 1998.
[11] K. Tanaka, T. Ikeda, and H. O. Wang, “Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs,” IEEE Trans. on Fuzzy Systems, Vol.6, No.2, pp. 1-16, 1998.
[12] K. Tanaka and M. Sano, “On the concepts of regulator and observer of fuzzy control systems,” 3rd IEEE Int. Conf. on Fuzzy Systems, pp. 767-772, 1994.
[13] W. Assawinchaichote, S. K. Nguang, and P. Shi, “Fuzzy Control and Filter Design for Uncertain Fuzzy Systems,” Springer, 2006.
[14] T. M. Guerra, A. Kruszewski, L. Vermeiren, and H. Tirmant, “Conditions of output stabilization for nonlinear models in the Takagi-Sugeno’s form,” Fuzzy Sets and Systems, Vol.157, pp. 1248-1259, 2006.
[15] H. N. Wu and K.-Y. Cai, “H₂ guaranteed cost fuzzy control for uncertain nonlinear systems via linear matrix inequalities,” Fuzzy Sets and Systems, Vol.148, pp.411-429, 2004.
[16] J. Yoneyama, “Robust guaranteed cost control of uncertain fuzzy systems under time-varying sampling,” Applied Soft Computing, Available online, November 24, 2009.
[17] C. Scherer, P. Gahinet, and M. Chilali, “Multiobjective outputfeedback control via LMI optimization,” IEEE Trans. on Automatic Control, Vol.42, pp. 896-911, 1997.

This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.

[1] [1] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. on Systems, Man, Cybernetics, Vol.15, pp. 116-132, 1985.

[2] [2] K. Tanaka and H. O. Wang, “Fuzzy Control Systems Design and Ana-lysis: A Linear Matrix Inequality Approach,” Wiley-Interscience, 2001.

[3] [3] K. Tanaka and M. Sugeno, “Stability analysis and design of fuzzy control systems,” Fuzzy Sets and Systems, Vol.45, pp. 135-156, 1992.

[4] [4] M. C. M. Teixiera, E. Assuncao, and R. G. Avellar, “On relaxed LMI-based designs for fuzzy regulators and fuzzy observers,” Vol.11, pp. 613-623, 2003.

[5] [5] H. D. Tuan, P. Apkarian, T. Narikiyo, and Y. Yamamoto, “Parameterized linear matrix inequality techniques in fuzzy control system design,” IEEE Trans. on Fuzzy Systems, Vol.9, pp. 324-332, 2001.

[6] [6] H. O.Wang, K. Tanaka, and M. Griffin, “An approach to fuzzy control of nonlinear system stability and design issues,” IEEE Trans. on Fuzzy Systems, Vol.4, No.1, pp. 14-21, 1996.

[7] [7] J. Yoneyama, M. Nishikawa, H. Katayama, and A. Ichikawa, “Design of output feedback controllers for Takagi-Sugeno fuzzy systems,” Fuzzy Sets and Systems, Vol.121, No.1, pp. 127-148, 2001.

[8] [8] J. Yoneyama, “H_∞ output feedback control for fuzzy systems with immeasurable premise variables: discrete-time case,” Applied Soft Computing, Vol.8, No.2, pp. 949-958, 2008.

[9] [9] J. Yoneyama, “Robust output feedback guaranteed cost control of uncertain fuzzy systems with immeasurable premise variables,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.11, No.7, pp. 745-750, 2007.

[10] [10] X. J. Ma, Z. Q. Sun, and Y. Y. He, “Analysis and design of fuzzy controller and fuzzy observer,” IEEE Trans. on Fuzzy Systems, Vol.6, No.1, pp. 41-50, 1998.

[11] [11] K. Tanaka, T. Ikeda, and H. O. Wang, “Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs,” IEEE Trans. on Fuzzy Systems, Vol.6, No.2, pp. 1-16, 1998.

[12] [12] K. Tanaka and M. Sano, “On the concepts of regulator and observer of fuzzy control systems,” 3rd IEEE Int. Conf. on Fuzzy Systems, pp. 767-772, 1994.

[13] [13] W. Assawinchaichote, S. K. Nguang, and P. Shi, “Fuzzy Control and Filter Design for Uncertain Fuzzy Systems,” Springer, 2006.

[14] [14] T. M. Guerra, A. Kruszewski, L. Vermeiren, and H. Tirmant, “Conditions of output stabilization for nonlinear models in the Takagi-Sugeno’s form,” Fuzzy Sets and Systems, Vol.157, pp. 1248-1259, 2006.

[15] [15] H. N. Wu and K.-Y. Cai, “H₂ guaranteed cost fuzzy control for uncertain nonlinear systems via linear matrix inequalities,” Fuzzy Sets and Systems, Vol.148, pp.411-429, 2004.

[16] [16] J. Yoneyama, “Robust guaranteed cost control of uncertain fuzzy systems under time-varying sampling,” Applied Soft Computing, Available online, November 24, 2009.

[17] [17] C. Scherer, P. Gahinet, and M. Chilali, “Multiobjective outputfeedback control via LMI optimization,” IEEE Trans. on Automatic Control, Vol.42, pp. 896-911, 1997.