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Research Papers

Stability Analysis of Nonlinear Dynamic Systems by Nonlinear Takagi–Sugeno–Kang Fuzzy Systems

[+] Author and Article Information
Zahra Namadchian

Islamic Azad University,
Gonabad Branch, Iran
e-mail: Zahra.namadchian@gmail.com

Assef Zare

Islamic Azad University,
Gonabad Branch, Iran
e-mail: Assefzare@gmail.com

Ali Namadchian

Islamic Azad University,
Mashhad Branch, Iran
e-mail: Ali_namadchian@yahoo.com

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received May 19, 2013; final manuscript received October 14, 2013; published online December 16, 2013. Assoc. Editor: Hashem Ashrafiuon.

J. Dyn. Sys., Meas., Control 136(2), 021019 (Dec 16, 2013) (6 pages) Paper No: DS-13-1205; doi: 10.1115/1.4025803 History: Received May 19, 2013; Revised October 14, 2013

This paper proposes a systematic procedure to address the limit cycle prediction of a Nonlinear Takagi–Sugeno–Kang (NTSK) fuzzy control system with adjustable parameters. NTSK fuzzy can be linearized by describing function method. The stability of the equivalent linearized system is then analyzed using the stability equations and the parameter plane method. After that the gain–phase margin (PM) tester has been added, then gain margin (GM) and phase margin for limit cycle are analyzed. Using NTSK fuzzy control system can help to have fewer rules. In order to analyze the stability with the same technique of stability analysis, the results of NTSK fuzzy control system will be compared with Dynamic fuzzy control system [1]. Computer simulations show differences between both systems.

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References

Perng, J. W., Wu, B. F., Chin, H., and Lee, T. T., 2004, “Gain-Phase Margin Analysis of Dynamic Fuzzy Control Systems,” IEEE Trans. Syst., Man, Cybern. Part B: Cybern., 34(5), pp. 2133–2139. [CrossRef]
Ban, X., Gao, X. Z., Huang, X., and Vasilakosc, A. V., 2007, “Stability Analysis of the Simplest Takagi–Sugeno Fuzzy Control System Using Circle Criterion,” Inf. Sci., 177(20), pp. 4387–4409. [CrossRef]
Chuang, C. C., Su, C. F., and Chen, S. S., 2001, “Robust TSK Fuzzy Modeling for Function Approximation With Outliers,” IEEE Trans. Fuzzy Syst., 9(6), pp. 810–821. [CrossRef]
Rajesh, R., and Kaimal, M., 2006, “Variable Gain Takagi–Sugeno Fuzzy Logic Controllers,” Informatica, 17(3), pp. 427–444. Available at http://dl.acm.org/citation.cfm?id=1413860
Ding, Y., Ying, H., and Shao, S., 2003, “Typical Takagi–Sugeno PI and PD Fuzzy Controllers: Analytical Structures and Stability Analysis,” Inf. Sci., 151, pp. 245–262. [CrossRef]
Precup, R. E., and Preitl, S., 2004, “Describing Function Method for PI-Fuzzy Controled Systems Stability Analysis,” The Annals of Dunarea de Jos University of Galati Fascicle III.
Kim, E., Lee, H., and Park, M., 2000, “Limit-Cycle Prediction of a Fuzzy Control System Based on Describing Function Method,”IEEE Trans. Fuzzy Syst., 8(1), pp. 11–22. [CrossRef]
Perng, J. W., Wu, B. F., Liao, T. Y., and Lee, T. T., 2007, Robust Stability Analysis of a Fuzzy Vehicle Lateral Control System Using Describing Function Method, ASC Vol. 42, Springer–Verlag, Berlin, pp. 769–779.
Perng, J. W., Ma, L. S., and Wu, B. F., 2010, “Limit Cycle Prediction of a Neurocontrol Vehicle System Based on Gain-Phase Margin Analysis,” Neural Comput. Appl., 19(4), pp. 565–571. [CrossRef]

Figures

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Fig. 1

NTSK fuzzy control system

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Fig. 2

Premise membership functions of NTSK fuzzy controller

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Fig. 3

A general linearized system including gain–phase margin tester

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Fig. 4

Premise membership functions of NTSK fuzzy controller (x(t) = x1(t))

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Fig. 5

Premise membership functions of NTSK fuzzy controller (x(t) = x2(t))

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Fig. 6

Limit cycle loci in parameter plane (NTSK fuzzy control system)

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Fig. 7

Limit cycle loci in parameter plane (dynamic fuzzy control system)

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Fig. 11

Limit cycle loci in parameter plane (NTSK fuzzy control system)

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Fig. 12

Limit cycle loci in parameter plane (Dynamic fuzzy control system)

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Fig. 13

Input signal x(t)

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Fig. 14

Sets of GM and PM

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