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Research Papers

# Robust Stability and Stabilization of Fractional Order Systems Based on Uncertain Takagi-Sugeno Fuzzy Model With the Fractional Order $1≤v < 2$

[+] Author and Article Information
Li Junmin

e-mail: jmli@mail.xidian.edu.cn

Li Yuting

e-mail: yutinglee@126.com
Department of Mathematics,
Xidian University,
Xi'an 710071, PRC

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 20, 2012; final manuscript received January 25, 2013; published online March 26, 2013. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 8(4), 041005 (Mar 26, 2013) (7 pages) Paper No: CND-12-1147; doi: 10.1115/1.4023739 History: Received September 20, 2012; Revised January 25, 2013

## Abstract

This paper addresses the problems of the robust stability and stabilization for fractional order systems based on the uncertain Takagi-Sugeno fuzzy model. A sufficient and necessary condition of asymptotical stability for fractional order uncertain T-S fuzzy model is given, and a parallel distributed compensate fuzzy controller is designed to asymptotically stabilize the model. The results are obtained in terms of linear matrix inequalities. Finally, a numerical example and fractional order Van der Pol system are given to show the effectiveness of our results.

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## References

Bagley, R. L., and Calico, R. A., 1991, “Fractional Order State Equations For The Control of Viscoelastic Structures,” J. Guid. Control Dyn., 14(2), pp. 304–311.
Skaar, S. B., Michel, A. N., and Miller, R. K., 1988, “Stability of Viscoelastic Control Systems,” IEEE Trans. Automat. Control, 33(4), pp. 348–357.
Hartley, T. T., and Lorenzo, C. F., 2002, “Dynamics and Control of Initialized Fractional-Order Systems,” Nonlinear Dyn., 29(1), pp. 201–233.
Sun, H. H., Abdelwahab, A. A., and Onaral, B., 1984, “Linear Approximation of Transfer Function With a Pole of Fractional Order,” IEEE Trans. Automat. Control, 29(5), pp. 441–444.
Goldberger, A. L., Bhargava, V., West, B. J., and Mandell, A. J., 1985, “On a Mechanism of Cardiac Electrical Stability: The Fractal Hypothesis,” Biophys. J., 48(3), pp. 525–528. [PubMed]
Hartley, T. T., Lorenzo, C. F., and Qammar, H. K., 1995, “Chaos in a Fractional Order Chua System”. IEEE Trans. Circuits Syst., 42(8), pp. 485–490.
Matignon, D., 1996, “Stability Results for Fractional Differential Equations With Applications to Control Processing,” in Computational Eng. in Sys. Appl. Proceedings, Vol. 2, pp. 963–968.
Matignon, D., 1998, “Stability Properties for Generalized Fractional Differential Systems,” in ESAIM Proceedings, Vol. 5, pp. 145–158.
Petras, I., Chen, Y. Q., Vinagre, B. M., and Podlubny, I., 2005, “Stability of Linear Time Invariant Systems With Interval Fractional Orders and Interval Coefficients,” in Proceedings of the International Conference on Comput. Cybern. (ICCC’04), Vienna, Austria, pp. 1–4.
Chen, Y. Q., Ahn, H. S., and Podlubny, I., 2006, “Robust Stability Check of Fractional Order Linear Time Invariant Systems With Interval Uncertainties,” Signal Process., 86(10), pp. 2611–2618.
Ahn, H. S., Chen, Y. Q., and Podlubny, I., 2007, “Robust Stability Test of A Class of Linear Time-Invariant Interval Fractional Order System Using Lyapunov Inequality,” Appl. Math. Comput., 187(1), pp. 27–34.
Ahn, H. S., and Chen, Y. Q., 2008, “Necessary and Sufficient Stability Condition of Fractional-Order Interval Linear Systems,” Automatica, 44(11), pp. 2985–2988.
Lu, J. G., and Chen, G. R., 2009, “Robust Stability and Stabilization of Fractional-Order Interval Systems: An LMI Approach,” IEEE Trans. Automat. Control, 54(6), pp. 1294–1299.
Lu, J. G., and Chen, Y. Q., 2010, “Robust Stability and Stabilization of Fractional-Order Interval Systems With The Fractional Orderα: The0<α<1 Case,” IEEE Trans. Automat. Control, 55(1), pp. 152–158.
Lazarevic, M. P., 2006, “Finite Time Stability Analysis ofPDα Fractional Control of Robotic Time-Delay Systems,” Mech. Res. Commun., 33(2), pp. 269–279.
Zhang, X. Y., 2008, “Some Results of Linear Fractional Order Time-Delay System,” Appl. Math. Comput., 197(1), pp. 407–411.
Lazarevic, M. P., and Spasic, A. M., 2009, “Finite Time Stability Analysis of Fractional Order Time Delay Systems: Gronwall's Approach,” Math. Comput. Model., 49(3-4), pp. 475–481.
Boyd, S., and Vandenberghe, L., 2004, Convex Optimization, Cambridge University Press, Cambridge.
Moze, M., Sabatier, J., and Oustaloup, A., 2005, “LMI Tools for Stability Analysis of Fractional Systems,” in IDETC/CIE2005 Proceedings, Long Beach, CA, pp. 1611–1619.
Sabatier, J., Moze, M., and Farges, C., 2010, “LMI Stability Conditions for Fractional Order Systems,” Comput. Math. Appl., 59(5), pp. 1594–1609.
Farges, C., Moze, M., and Sabatier, J., 2010, “Pseudo-State Feedback Stabilization of Commensurate Fractional Order Systems,” Automatica, 46(10), pp. 1730–1734.
Lan, Y. H., 2012, “Observer-Based Robust Control ofα (0≤α<1) Fractional-Order Uncertain Systems: A Linear Matrix Inequality Approach,” IET Control Theory A, 6, pp. 229–234.
Lan, Y. H., and Zhou, Y., 2011, “LMI-Based Robust Control of Fractional-Order Uncertain Linear Systems,” Comput. Math. Appl., 62(3), pp. 1460–1471.
Wang, Z., Huang, X., and Shen, H., 2012, “Control of an Uncertain Fractional Order Economic System via Adaptive Sliding Mode,” Neurocomputing, 83, pp. 83–88.
Takagi, T., and Sugeno, M., 1985, “Fuzzy Identification of Systems and Its Application to Modeling and Control,” IEEE Trans. Systems Man and Cybernetics, 15(1), pp. 116–132.
Tanaka, K., and Sugeno, M., 1992, “Stability Analysis and Design of Fuzzy Control Systems,” Fuzzy Set. Syst., 45(2), pp. 135–156.
Lee, H. J., Park, J. B., and Chen, G., 2001, “Robust Fuzzy Control of Nonlinear Systems With Parametric Uncertainties,” IEEE Trans. Fuzzy Syst., 9(2), pp. 369–379.
Tanaka, K., and Sano, M., 1996, “A Robust Stabilization Problem of Fuzzy Controller Systems and its Applications to Backing up Control of a Truck-Trailer,” IEEE Trans. Fuzzy Syst., 2(2), pp. 119–134.
Tian, E., and Pang, C., 2006, “Delay-Dependent Stability Analysis and Synthesis of Uncertain T–S Fuzzy Systems With Time-Varying Delay,” Fuzzy Set. Syst., 157(4), pp. 544–559.
Yoneyama, J., 2008, “New Delay-Dependent Approach to Robust Stability and Stabilization for Takagi–Sugeno Fuzzy Time-Delay Systems,” Fuzzy Set. Syst., 158(20), pp. 2225–2237.
Han, Q. L., and Gu, K. Q., 2001, “On Robust Stability of Time-Delay Systems With Norm Bounded Uncertainty,” IEEE Trans. Automat. Control, 46(9), pp. 1426–1431.
Huang, H., and Ho, D. W. C., 2007,“Delay-Dependent Robust Control of Uncertain Stochastic Fuzzy Systems With Time-Varying Delay,” IET Control Theory A, 1(4), pp. 1075–1085.
Liu, X., and Zhang, H., 2008, “Delay-Dependent Robust Stability of Uncertain Fuzzy Large-Scale Systems With Time-Varying Delays,” Automatica, 44, pp. 193–198.
Zhang, H., Wang, Y., and Liu, D., 2008, “Delay-Dependent Guaranteed Cost Control for Uncertain Stochastic Fuzzy Systems With Multiple Time Delays,” IEEE Trans. Syst. Man Cybern. Part B Cybern., 38(1), pp. 126–140.
Feng, G., 2006, “A Survey on Analysis and Design of Model-Based Fuzzy Control Systems,” IEEE Trans. Fuzzy Syst., 14(5), pp. 676–697.
Zheng, Y. G., Nian, Y. B., and Wang, D. J., 2010, “Controlling Fractional Order Chaotic Systems Based on Takagi-Sugeno Fuzzy Model and Adaptive Adjustment Mechanism,” Phys. Lett. A, 375(21), pp. 125–129.
Podlubny, I., 1999, Fractional Differential Equations, Academic Press, San Diego.
Xie, L. H., 1996, “Output FeedbackH∞ Control of Systems With Parameter Uncertainty,” Int. J. Control, 63(4), pp. 741–750.
Diethelm, K., Ford, N. J., and Freed, A. D., 2002, “A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations,” Nonlinear Dyn., 29(1-4), pp. 3–22.

## Figures

Fig. 2

Time response of the closed-loop system states in example 1

Fig. 3

Control curve of the system in example 1

Fig. 4

States of Van der Pol system in example 2

Fig. 1

Time response of the autonomous system in example 1

Fig. 5

Control curve of Van der Pol system in example 2

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