# Abstract:

This paper proposes new stability conditions for interval type-2 fuzzy-model-based (FMB) control systems. The type-1 T-S fuzzy model has been widely studied because it can represent a wide class of nonlinear systems. Many favorable results for type-1 T-S fuzzy model have been reported. However, most of conclusions for type-1 T-S fuzzy model can not be applied to nonlinear systems subject to parameter uncertainties. In fact, Most of the practical applications are subject to parameters uncertainties. To address above problem, an interval type-2 T-S fuzzy model has been proposed to approximate nonlinear systems subject to parameter uncertainties, and stability conditions for interval type-2 FMB control systems have also been presented in the form of linear matrix inequalities (LMIs). The aim of this paper is to relax the existing stability conditions. The new stability conditions in terms of LMIs are derived to guarantee the stability of interval type-2 FMB control systems. The theoretical poof is given to show the proposed conditions reduce the conservativeness in stability analysis. Several numerical examples are also provided to illustrate the effectiveness of the proposed conditions.

# Keywords:

stability analysis, interval type-2 fuzzy set, interval type-2 T-S fuzzy system, linear matrix inequalities

93E12, 62A10

# References:

1. M. Bernal, T. M. Guerra and A. Kruszewski: A membership-function-dependent approach for stability analysis and controller synthesis of Takagi-Sugeno models. Fuzzy Sets and Systems 160 (2009), 19, 2776-2795.   CrossRef
2. B. C. Ding, H. X. Sun and P. Yang: Further studies on LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in Takagi-Sugeno form. Automatica 42 (2006), 3, 503-508.   CrossRef
3. C. H. Fang, Y. S. Liu, S. W. Kau, L. Hong and C. H. Lee: A new LMI-based approach to relaxed quadratic stabilization of Takagi-Sugeno fuzzy control systems. IEEE Trans. Fuzzy Systems 14 (2006), 3, 386-397.   CrossRef
4. G. Feng: Controller synthesis of fuzzy dynamical systems based on piecewise Lyapunov functions. IEEE Trans. Fuzzy Systems 11 (2003), 5, 605-612.   CrossRef
5. M. Johansson, A. Rantzer and K. Arzen: Piecewise quadratic stability of fuzzy systems. IEEE Trans. Fuzzy Systems 7 (1999), 6, 713-722.   CrossRef
6. E. Kim and H. Lee: New approaches to relaxed quadratic stability condition of fuzzy control systems. IEEE Trans. Fuzzy Systems 8 (2000), 5, 523-534.   CrossRef
7. H. K. Lam and F. H. F. Leung: Stability analysis of fuzzy control systems subject to uncertain grades of membership. IEEE Trans. Systems, Man and Cybernetics, Part B: Cybernetics 35 (2005), 6, 1322-1325.   CrossRef
8. H. K. Lam and F. H. F. Leung: LMI-based stability and performance conditions for continuous-time nonlinear systems in Takagi-Sugeno's form. IEEE Trans. Systems, Man and Cybernetics, Part B: Cybernetics 137 (2007), 5, 1396-1406.   CrossRef
9. H. K. Lam and F. H. F. Leung: Stability Analysis of Fuzzy-Model-Based Control Systems. Studies on Fuzziness and Soft Computing. Springer, 2010.   CrossRef
10. H. K. Lam, M. Narimani and L. D. Seneviratne: LMI-based stability conditions for interval type-2 fuzzy-model-based control systems. In: 2011 IEEE International Conference on Fuzzy Systems, Taipei, pp. 298-303.   CrossRef
11. H. K. Lam and L. D. Seneviratne: Stability analysis of interval type-2 fuzzy-model-based control systems. IEEE Trans. Systems, Man and Cybernetics, Part B: Cybernetics 38 (2008), 3, 617-628.   CrossRef
12. D. H. Lee, J. B. Park and Y. H. Joo: A new fuzzy Lyapunov function for relaxed stability condition of continuous-time Takagi-Sugeno fuzzy systems. IEEE Trans. Fuzzy Systems 19 (2011), 4, 785-791.   CrossRef
13. X. D. Liu and Q. L. Zhang: New approaches to $H_{\infty}$ controller designs based on fuzzy observers for T-S fuzzy systems via LMI. Automatica 38 (2003), 9, 1571-1582.   CrossRef
14. V. F. Montagner, R. C. L. F. Oliveira and P. L. D. Peres: Convergent LMI relaxtions for quadratic stabilizability and $H_{\infty}$ control of Takagi-Sugeno fuzzy systems. IEEE Trans. Fuzzy Systems 17 (2009), 4, 863-873.   CrossRef
15. L. A. Mozelli, R. M. Palhares, F. O. Souza and E. M. A. M. Mendes: Reducing conservativeness in recent stability conditions of T-S fuzzy systems. Automatica 45 (2009),6, 1580-1583.   CrossRef
16. A. Sala and C. Arino: Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Poly's theorem. Fuzzy Sets and Systems 158 (2007), 24, 2671-2686.   CrossRef
17. K. Tanaka, T. Hori and H. O. Wang: A multiple Lyapunov function approach to stabilization of fuzzy control systems. IEEE Trans. Fuzzy Systems 11 (2003), 4, 582-589.   CrossRef
18. K. Tanaka, T. Hori and H. O. Wang: A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions. IEEE Trans. Fuzzy Systems 15 (2007), 3, 333-341.   CrossRef
19. K. Tanaka and M. Sugen: Stability analysis and design of fuzzy control systems. Fuzzy Sets and Systems 42 (1992), 2, 135-156.   CrossRef
20. M. C. M. Teixeira, E. Assuncao and R. G. Vellar: On relaxed LMI-based designs for fuzzy regulators and fuzzy observers. IEEE Trans. Fuzzy Systems 11 (2003), 5, 613-623.   CrossRef
21. H. D. Tuan and P. Apkarian: Parameterized linear matrix inequalities in fuzzy control system design. IEEE Trans. Fuzzy Systems 9 (2001), 2, 324-332.   CrossRef
22. W. J. Wang, Y. J. Chen and C. H. Sun: Relaxed stabilization criteria for discrete-time T-S fuzzy control systems based on a switching fuzzy model and piecewise Lyapunov function. IEEE Trans. Systems, Man and Cybernetics, Part B: Cybernetics 37 (2007), 3, 551-559.   CrossRef
23. Z. C. Wei and Z. Wang: Chaotic behavior and modified function projective synchronization of a simple system with one stable equilibrium. Kybernetika 49 (2013), 2, 359-374.   CrossRef
24. Z. Wei and Q. Yang: Anti-control of Hopf bifurcation in the new chaotic system with two stable node-foci. Appl. Math. Comput. 27 (2010), 1, 422-429.   CrossRef
25. H. G. Zhang and X. P. Xie: Relaxed stability conditions for continuous-time T-S fuzzy-control systems via augmented multi-indexed matrix approach. IEEE Trans. Fuzzy Systems 19 (2011), 3, 478-492.   CrossRef