# A new fuzzy Lyapunov approach to non-quadratic stabilization of Takagi-Sugeno fuzzy models

Ibtissem Abdelmalek; Noureddine Goléa; Mohamed Hadjili

International Journal of Applied Mathematics and Computer Science (2007)

- Volume: 17, Issue: 1, page 39-51
- ISSN: 1641-876X

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topAbdelmalek, Ibtissem, Goléa, Noureddine, and Hadjili, Mohamed. "A new fuzzy Lyapunov approach to non-quadratic stabilization of Takagi-Sugeno fuzzy models." International Journal of Applied Mathematics and Computer Science 17.1 (2007): 39-51. <http://eudml.org/doc/207820>.

@article{Abdelmalek2007,

abstract = {In this paper, new non-quadratic stability conditions are derived based on the parallel distributed compensation scheme to stabilize Takagi-Sugeno (T-S) fuzzy systems. We use a non-quadratic Lyapunov function as a fuzzy mixture of multiple quadratic Lyapunov functions. The quadratic Lyapunov functions share the same membership functions with the T-S fuzzy model. The stability conditions we propose are less conservative and stabilize also fuzzy systems which do not admit a quadratic stabilization. The proposed approach is based on two assumptions. The first one relates to a proportional relation between multiple Lyapunov functions and the second one considers an upper bound to the time derivative of the premise membership functions. To illustrate the advantages of our proposal, four examples are given.},

author = {Abdelmalek, Ibtissem, Goléa, Noureddine, Hadjili, Mohamed},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {linear matrix inequalities; non-quadratic stability conditions; T-S fuzzy systems; stabilization; parallel distributed compensation},

language = {eng},

number = {1},

pages = {39-51},

title = {A new fuzzy Lyapunov approach to non-quadratic stabilization of Takagi-Sugeno fuzzy models},

url = {http://eudml.org/doc/207820},

volume = {17},

year = {2007},

}

TY - JOUR

AU - Abdelmalek, Ibtissem

AU - Goléa, Noureddine

AU - Hadjili, Mohamed

TI - A new fuzzy Lyapunov approach to non-quadratic stabilization of Takagi-Sugeno fuzzy models

JO - International Journal of Applied Mathematics and Computer Science

PY - 2007

VL - 17

IS - 1

SP - 39

EP - 51

AB - In this paper, new non-quadratic stability conditions are derived based on the parallel distributed compensation scheme to stabilize Takagi-Sugeno (T-S) fuzzy systems. We use a non-quadratic Lyapunov function as a fuzzy mixture of multiple quadratic Lyapunov functions. The quadratic Lyapunov functions share the same membership functions with the T-S fuzzy model. The stability conditions we propose are less conservative and stabilize also fuzzy systems which do not admit a quadratic stabilization. The proposed approach is based on two assumptions. The first one relates to a proportional relation between multiple Lyapunov functions and the second one considers an upper bound to the time derivative of the premise membership functions. To illustrate the advantages of our proposal, four examples are given.

LA - eng

KW - linear matrix inequalities; non-quadratic stability conditions; T-S fuzzy systems; stabilization; parallel distributed compensation

UR - http://eudml.org/doc/207820

ER -

## References

top- Bernal M. and Huvsek P. (2005): Non-quadratic performance design for Takagi-Sugeno fuzzy systems. - Int. J. Appl. Math. Comput. Sci., Vol.15, No.3, pp.383-391. Zbl1169.93355
- Blanco Y., Perruquetti W. and Borne P. (2001): Non-quadratic stability of nonlinear systems in the Takagi-Sugeno form. - Proc. European Control Conference, ECC, Porto, Portugal, pp.3917-3922.
- Boyd S., El Ghaoui L., Feron E. and Balakrishnan B. (1994): Linear Matrix Inequalities in Systems and Control Theory. - Philadelphia: SIAM. Zbl0816.93004
- Cao Y-Y and Lin Z. (2003): Robust stability analysis and fuzzy schduling control for nonlinear systems subject to actuator saturation. - IEEE Trans. Fuzzy Syst., Vol.11, No.1, pp.57-67.
- Cao S.G., Rees N.W. and Feng G. (1997): Analysis and design of a class of complex control systems. - Automatica, Vol.33, No.6, pp.1029-1039. Zbl0887.93036
- Chadli M., Maquin D., Ragot J. (2000): Relaxed stability conditions for Takagi-Sugeno fuzzy systems. - Proc. IEEE Int. Conf. Systems, Man and Cybernetics, SMC, Nashville, TN, pp.3514-3519.
- Chadli M., Maquin D., Ragot J. (2001): On the stability analysis of multiple model systems. - Proc. European Control Conference, ECC, Porto, Portugal, pp.1894-1899.
- Chadli M., Maquin D., Ragot J. (2005): Stability analysis and design for continuous-time Takagi-Sugeno control systems. -Int. J. Fuzzy Syst., Vol.7, No.3, pp.101-109.
- Feng G. (2002): An approach to adaptive control of fuzzy dynamic systems. - IEEE Trans. Fuzzy Syst., Vol.10, No.2, pp.268-275.
- Hadjili M.L. (2002): Fuzzy logic in nonlinear modeling and control. - Ph.D. thesis, CESAME, UCL, Belgium.
- Jadbabaie A. (1999): A reduction in conservatism in stability and L2 gain analysis of Takagi-Sugeno fuzzy systems via linear matrix inequalities. - Proc. 14th IFAC Triennial World Congress, Beijing, China, pp.285-289.
- Johansson M. (1999): Piecewise linear control systems . - Ph.D. thesis, Lund Institute of Technology, Lund, Sweden.
- Johansson M., Rantzer A. and Arzen K-E. (1999): Piecewise quadratic stability of fuzzy systems. - IEEE Trans. Fuzzy Syst., Vol.4, No.6, pp.713-722.
- Khalil H.K. (1996): Nonlinear Systems, 2nd Ed.. - Upper Saddle River, NJ: Prentice-Hall.
- Korba P., Babuvska R., Verbruggen H.B. and Frank P.M. (2003): Fuzzy gain scheduling: controller and observer design based on Lyapunov method and convex optimization. - IEEE Trans. Fuzzy Syst., Vol.11, No.3, pp.285-298.
- Lee H.J., Park J.B. and Chen G. (2001): Robust fuzzy control of nonlinear systems with parametric uncertainties. - IEEE Trans. Fuzzy Syst.,Vol.9, No.2, pp.369-379.
- Morère Y. (2001): Mise en oeuvre de lois decommande pour les modèles flous de type Takagi-Sugeno. - Ph.D. thesis, LAMIH, Universite de Valenciennes et du Haut Cambresis, France.
- Ohtake H., Tanaka K. and Wang H.O. (2003): Piecewise nonlinear control . - Proc. 42nd IEEEConf. Decision and Control, Maui, HI, (CD-ROM).
- Takagi T. and Sugeno M. (1985): Fuzzy identification of systems and its applications to modeling andcontrol. - IEEE Trans. Syst. Man Cybern., Vol.SMC-15, No.1, pp.116-132. Zbl0576.93021
- Tanaka K. and Sugeno M. (1992): Stability analysis and design of fuzzy control systems. - Fuzzy Sets Syst., Vol.45, No.2, pp.135-156. Zbl0758.93042
- Tanaka K. and Wang H.O. (2001): Fuzzy Control Systems Design and Analysis. A Linear Matrix Inequalities Approach. - New York: Wiley.
- Tanaka K., Ikeda T. and Wang H.O. (1996): Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: Quadratic stabilizability, H^∞ control theory, and linear matrix inequalities. - IEEE Trans. Fuzzy Syst., Vol.4, No.1, pp.1-13.
- Tanaka K., Hori T. and Wang H.O. (2001a): Adual design problem via multiple Lyapunov functions. - Proc. 10th IEEE Int. Conf. Fuzzy Systems, Melbourne, Australia.
- Tanaka K., Hori T. and Wang H.O. (2001b): A fuzzy Lyapunov approach to fuzzy control system design. -Proc. American Control Conf., Arlington VA, Washington CD, pp.4790-4795.
- Tanaka K., Hori T. and Wang H.O. (2001c): New parallel distributed compensation using time derivative membership functions: A fuzzy Lyapunov approach. - Proc. 40th IEEE Conf. Decision and Control, Orlando, FL, pp.3942-3947.
- Tanaka K., Hori T. and Wang H.O. (2003): A multiple Lyapunov function approach to stabilization of fuzzy control systems. - IEEE Trans. Fuzzy Syst., Vol.11, No.4, pp.582-589.
- Tsen C-S, Chen B-S and Uang H-J (2001): Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model. - IEEE Trans. Fuzzy Syst., Vol.9, No.3, pp.381-392.
- Wang H.O., Tanaka K. and Griffin M. (1995): Parallel distributed compensation of nonlinear systems by Takagi and Sugeno's fuzzy model. - Proc. 4th IEEE, Int. Conf. Fuzzy Systems, Yokohama, Japan, pp.531-538.
- Wang H.O., Tanaka K. and Griffin M.F. (1996): An approach to fuzzy control of nonlinear systems: Stability and design issues. - IEEE Trans. Fuzzy Syst., Vol.4, No.1, pp.14-23.
- Wong L.K., Leung F.H.F. and Tam P.K.S. (1998): Lyapunov-function-based design of fuzzy logic controllers and its application on combining controllers. - IEEE Trans. Industr. Electron.,Vol.45, No.3, pp.502-509.
- Zhao J. (1995): System modeling, identificationand control using fuzzy logic. - Ph.D. thesis, CESAME, UCL, Belgium

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