Relaxed LMI conditions for local stability and local stabilization of continuous-time takagi-sugeno fuzzy systems

IEEE Transactions on Cybernetics

Volumn 44, Issue 3, 2014, Pages 394-405

  Lee, Dong Hwan ^a   Kim, Do Wan ^b  

^a Yonsei University   (South Korea)

^b Hanbat National University   (South Korea)

Author keywords

Convex optimization; domain of attraction (DA); local stability; nonlinear systems; Takagi Sugeno (T S) fuzzy systems

Indexed keywords

DOMAIN OF ATTRACTION; LINEAR MATRIX INEQUALITY OPTIMIZATIONS; LOCAL STABILITY; LOCAL STABILITY ANALYSIS; MINIMIZATION PROBLEMS; REDUCTION OF CONSERVATISMS; STABILITY AND STABILIZATIONS; TAKAGI SUGENO FUZZY SYSTEMS;

CONTINUOUS TIME SYSTEMS; CONVEX OPTIMIZATION; EIGENVALUES AND EIGENFUNCTIONS; FEEDBACK CONTROL; LINEAR MATRIX INEQUALITIES; NONLINEAR SYSTEMS;

STABILIZATION;

EID: 84896899741     PISSN: 21682267     EISSN: None     Source Type: Journal    
DOI: 10.1109/TCYB.2013.2256781     Document Type: Article

References (57)

1. T. Takagi and M. Sugeno, "Fuzzy identification of systems and its applications to modelling and control," IEEE Trans. Syst., Man, Cybern., vol. SMC-15, no. 1, pp. 116-132, Jan.-Feb. 1985.
2. L. X. Wang and J. M. Mendel, "Fuzzy basis functions, universal approximation, and orthogonal least-squares learning," IEEE Trans. Neural Netw., vol. 3, no. 5, pp. 807-814, Sep. 1992.
3. Z. Huaguang and Q. Yongbing, "Modelling, identification, and control of a class of nonlinear systems," IEEE Trans. Fuzzy Syst., vol. 9, no. 2, pp. 349-354, Apr. 2001. (Pubitemid 32505591)
4. K. Tanaka, T. Ikeda, and H. O. Wang, "Fuzzy regulators and fuzzy observers: Relaxed stability conditions and LMI-based designs," IEEE Trans. Fuzzy Syst., vol. 6, no. 2, pp. 250-265, May 1998. (Pubitemid 128750059)
5. K. Tanaka and H. O. Wang, Fuzzy Control Systems Design and Analysis. New York: Wiley, 2001.
6. H. D. Tuan, P. Apkarian, T. Narikiyo, and Y. Yamamoto, "Parameterized linear matrix inequality techniques in fuzzy control system design," IEEE Trans. Fuzzy Syst., vol. 9, no. 2, pp. 324-332, Apr. 2001. (Pubitemid 32505588)
7. E. Kim and H. Lee, "New approaches to relaxed quadratic stability condition of fuzzy control systems," IEEE Trans. Fuzzy Syst., vol. 8, no. 5, pp. 523-534, Oct. 2000.
8. C. H. Fang, Y. S. Liu, S. W. Kau, L. Hong, and C. H. Lee, "A new LMIbased approach to relaxed quadratic stabilization of T-S fuzzy control systems," IEEE Trans. Fuzzy Syst., vol. 14, no. 3, pp. 386-397, Jun. 2006. (Pubitemid 43898881)
9. 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. Syst., Man, Cybern. B, Cybern., vol. 37, no. 5, pp. 1396-1406, Oct. 2007. (Pubitemid 47534603)
10. K. Y. Lian, Hui. W. Tu, and J. J. Liou, "Stability conditions for LMIbased fuzzy control from viewpoint of membership functions," IEEE Trans. Fuzzy Syst., vol. 14, no. 6, pp. 874-884, Dec. 2006.
11. A. Sala and C. Ariño, "Relaxed stability and performance conditions for Takagi-Sugeno fuzzy systems with knowledge on membership function overlap," IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 37, no. 3, pp. 727-732, Jun. 2007. (Pubitemid 46784606)
12. M. Narimani and H. K. Lam, "Relaxed LMI-based stability conditions for Takagi-Sugeno fuzzy control systems using regional-membershipfunction-shape- dependent analysis approach," IEEE Trans. Fuzzy Syst., vol. 17, no. 5, pp. 1221-1228, Oct. 2009.
13. A. Kruszewski, A. Sala, T. M. Guerra, and C. M. Ariño, "A triangulation approach to asymptotically exact conditions for fuzzy summations," IEEE Trans. Fuzzy Syst., vol. 17, no. 5, pp. 985-994, Oct. 2009.
14. J. Johansson, A. Rantzer, and M. Arzen, "Piecewise quadratic stability of fuzzy systems," IEEE Trans. Fuzzy Syst., vol. 7, no. 6, pp. 713-722, Dec. 1999. (Pubitemid 32212162)
15. G. Feng, "Controller synthesis of fuzzy dynamic systems based on piecewise Lyapunov functions," IEEE Trans. Fuzzy Syst., vol. 11, no. 5, pp. 605-612, Oct. 2003.
16. M. Bernal, T. M. Guerra, and A. Kruszewski, "A membershipfunction- dependent approach for stability analysis and controller synthesis of Takagi-Sugeno models," Fuzzy Sets Syst., vol. 160, no. 19, pp. 2776-2795, OCt. 2009.
17. Y. J. Chen, H. Ohtake, K. Tanaka, W. J. Wang, and H. O. Wang, "Relaxed stabilization criterion for T-S fuzzy systems by minimum-type piecewise Lyapunov function based switching fuzzy controller," IEEE Trans. Fuzzy Syst., vol. 20, no. 6, pp. 1166-1173, Dec. 2012.
18. K. Tanaka, T. Hori, and H. O. Wang, "A fuzzy Lyapunov approach to fuzzy control system design," in Proc. Amer. Control Conf., 2001, pp. 4790-4795. (Pubitemid 32837085)
19. K. Tanaka, T. Hori, and H. O. Wang, "A multiple Lyapunov function approach to stabilization of fuzzy control systems," IEEE Trans. Fuzzy Syst., vol. 11, no. 4, pp. 582-589, Aug. 2003.
20. 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, vol. 45, no. 6, pp. 1580-1583, 2009.
21. L. A. Mozelli, R. M. Palhares, and G. S. C. Avellar, "A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems," Inf. Sci., vol. 179, no. 8, pp. 1149-1162, 2009.
22. T. M. Guerra and L. Vermeiren, "LMI-based relaxed nonquadratic stabilization condttions for nonlinear systems in the Takagi-Sugeno's form," Automatica, vol. 40, no. 5, pp. 823-829, 2004.
23. B. C. Ding, H. X. Sun, and P. Yang, "Further studies on LMIbased relaxed stabilization conditions for nonlinear systems in Takagi-Sugeno's form," Automatica, vol. 42, no. 3, pp. 503-508, 2006. (Pubitemid 43131462)
24. A. Kruszewski, R. Wang, and T. M. Guerra, "Nonquadratic stabilization conditions for a class of uncertain nonlinear discrete time T-S fuzzy models: A new approach," IEEE Trans. Autom. Control, vol. 53, no. 2, pp. 606-611, Mar. 2008. (Pubitemid 351493901)
25. T. M. Guerra, A. Kruszewski, and M. Bernal, "Control law proposition for the stabilization of discrete Takagi-Sugeno models," IEEE Trans. Fuzzy Syst., vol. 17, no. 3, pp. 724-731, Jun. 2009.
26. D. H. Lee, J. B. Park, and Y. H. Joo, "Approaches to extended non-quadratic stability and stabilization conditions for discretetime Takagi-Sugeno fuzzy systems," Automatica, vol. 47, no. 3, pp. 534-538, 2011.
27. D. H. Lee, J. B. Park, and Y. H. Joo, "Further theoretical justification of the k-samples variation approach for discrete-time Takagi-Sugeno fuzzy systems," IEEE Trans. Fuzzy Syst., vol. 19, no. 3, pp. 594-597, Jun. 2011.
28. D. W. Kim, H. J. Lee, and M. Tomizuka, "Fuzzy stabilization of nonlinear systems under sampled-data feedback: An exact discrete-time model approach," IEEE Trans. Fuzzy Syst., vol. 18, no. 2, pp. 251-260, Apr. 2010.
29. D.W. Kim and H. J. Lee, "Sampled-data observer-based output-feedback fuzzy stabilization of nonlinear systems: Exact discrete-time design approach," Fuzzy Sets Syst., vol. 201, pp. 20-39, Aug. 2012.
30. B.-J. Rhee and S. Won, "A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design," Fuzzy Sets Syst., vol. 157, no. 9, pp. 1211-1228, 2006.
31. J. Li, S. Zhou, and S. Xu, "Fuzzy control system design via fuzzy Lyapunov functions," IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 38, no. 6, pp. 1657-1661, Dec. 2008.
32. M. Bernal, A. Sala, A. Jaadari, and T. M. Guerra, "Stability analysis of polynomial fuzzy models via polynomial fuzzy Lyapunov functions," Fuzzy Sets Syst., vol. 185, no. 1, pp. 5-14, 2011.
33. H. K. Lam, "Polynomial fuzzy-model-based control systems: Stability analysis via piecewise-linear membership functions," IEEE Trans. Fuzzy Syst., vol. 19, no. 3, pp. 588-593, Jun. 2011.
34. A. Sala and C. Ariño, "Polynomial fuzzy models for nonlinear control: A Taylor series approach," IEEE Trans. Fuzzy Syst., vol. 17, no. 6, pp. 1284-1295, Dec. 2009.
35. A. Sala, "On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems," Annu. Rev. Control, vol. 33, no. 1, pp. 48-58, 2009.
36. K. Tanaka, H. Yoshida, H. Ohtake, and H. O. Wang, "A sum-of-squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems," IEEE Trans. Fuzzy Syst., vol. 17, no. 4, pp. 911-922, Aug. 2009.
37. H. K. Lam, "Stabilization of nonlinear systems using sampled-data output-feedback fuzzy controller based on polynomial-fuzzy-modelbased control approach," IEEE Trans. Syst., Man Cybern., vol. 42, no. 1, pp. 258-267, Feb. 2012.
38. A. Sala and C. Ariño, "Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Pólya's theorem," Fuzzy Sets Syst., vol. 158, no. 24, pp. 2671-2686, 2007. (Pubitemid 47513206)
39. B. Ding, "Homogeneous polynomially nonquadratic stabilization of discrete-time Takagi-Sugeno systems via nonparallel distributed compensation law," IEEE Trans. Fuzzy Syst., vol. 18, no. 5, pp. 994-1000, Oct. 2010.
40. D. H. Lee, J. B. Park, and Y. H. Joo, "Improvement on nonquadratic stabilization of discrete-time Takagi-Sugeno fuzzy systems: Multiple- parameterization approach," IEEE Trans. Fuzzy Syst. vol. 18, no. 2, pp. 425-429, Apr. 2010.
41. H. Zhang and X. Xie, "Relaxed stability conditions for continuoustime T-S fuzzy-control systems via augmented multi-indexed matrix approach," IEEE Trans. Fuzzy Syst., vol. 19, no. 3, pp. 478-492, Jun. 2011.
42. X. Xie, H. Ma, Y. Zhao, D. W. Ding, and Y. Wang, "Control synthesis of discrete-time T-S fuzzy systems based on a novel non-PDC control scheme," IEEE Trans. Fuzzy Syst., vol. 21, no. 1, pp. 147-157, Feb. 2013.
43. 2 and H∞ stabilization of Takagi-Sugeno fuzzy systems," IEEE Trans. Fuzzy Syst., vol. 19, no. 5, pp. 890-900, Oct. 2011.
44. T. M. Guerra and M. Bernal, "A way to escape from the quadratic framework," in Proc. IEEE Int. Conf. Fuzzy Syst., Aug. 2009, pp. 784-789.
45. M. Bernal and T. M. Guerra, "Generalized nonquadratic stability of continuous-time Takagi-Sugeno models," IEEE Trans. Fuzzy Syst., vol. 18, no. 4, pp. 815-822, Aug. 2010.
46. T. M. Guerra, A. Jaadari, J. Pan, and A. Sala, "Some refinements for non quadratic stabilization of continuous TS models," in Proc. IEEE Int. Conf. Fuzzy Syst., Jun. 2011, pp. 329-333.
47. M. Bernal, A. Soto-Cota, and J. Cortez, "Local non-quadratic H-infinity control for continuous-time Takagi-Sugeno models," in Proc. IEEE Int. Conf. Fuzzy Syst., Jun. 2011, pp. 1615-1620.
48. J. T. Pan, T. M. Guerra, S. M. Fei, and A. Jaadari, "Nonquadratic stabilization of continuous T-S fuzzy models: LMI solution for a local approach," IEEE Trans. Fuzzy Syst., vol. 20, no. 3, pp. 594-602, Jun. 2012.
49. A. Jaadari, T. M. Guerra, and M. Bernal, "New controllers and new designs for continuous-time Takagi-Sugeno models," in Proc. IEEE World Congr. Comput. Intell., Jun. 2012, pp. 1-7.
50. D. H. Lee, J. B. Park, and Y. H. Joo, "A fuzzy Lyapunov function approach to estimating the domain of attraction for continuous-time Takagi-Sugeno fuzzy systems," Inf. Sci., vol. 185, no. 1, pp. 230-248, 2012.
51. T. Coleman, M. Branch, and A. Grace, Optimization Toolbox for Use with Matlab, Natick, MA, USA: MathWorks, 1999.
52. S. Boyd, L. EI Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory. Philadelphia, PA, USA: SIAM, 1994.
53. H. K. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ, USA: Prentice-Hall, 2001.
54. P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox, Natick, MA, USA: MathWorks, 1995.
55. J. F. Sturm, "Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cons," Optim. Methods Softw., vol. 11, no. 12, pp. 625-653, 1999.
56. J. Löfberg. (2004, Sep.). "YALMIP: A toolbox for modeling and optimization in MATLAB," in Proc. IEEE Int. Symp. Comput. Aided Control Syst. Des., pp. 284-289 [Online]. Available: http://control.ee.ethz.ch/joloef/ yalmip.php (Pubitemid 40788529)
57. S. Prajna, A. Papachristodoulou, P. Seiler, and P. A. Parrilo, SOSTOOLS: Sum of Squares Optimization Toolbox for MATLAB, Calif. Inst. Technol., Pasadena, 2004.