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Mathematical Problems in Engineering
Volume 2011, Article ID 361640, 21 pages
http://dx.doi.org/10.1155/2011/361640
Research Article

Less Conservative Fuzzy Control for Discrete-Time Takagi-Sugeno Systems

1Federal University of São João del-Rei, Campus Alto Paraopeba, Rodovia MG 443 Km 7, 36420 Ouro Branco, MG, Brazil
2Department of Electronics Engineering, Federal University of Minas Gerais, Avenida Antônio Carlos 6627, 31270-010 Belo Horizonte, MG, Brazil

Received 23 August 2010; Accepted 24 January 2011

Academic Editor: Geraldo Silva

Copyright © 2011 Leonardo Amaral Mozelli and Reinaldo Martinez Palhares. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. G. Feng, “A survey on analysis and design of model-based fuzzy control systems,” IEEE Transactions on Fuzzy Systems, vol. 14, no. 5, pp. 676–697, 2006. View at Publisher · View at Google Scholar · View at Scopus
  2. T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Transactions on Systems, Man and Cybernetics, vol. 15, no. 1, pp. 116–132, 1985. View at Google Scholar · View at Scopus
  3. K. Tanaka and H. O. Wang, Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach, John Wiley & Sons, New York, NY, USA, 2001.
  4. K. Zeng, N. Y. Zhang, and W. L. Xu, “A comparative study on sufficient conditions for Takagi-Sugeno fuzzy systems as universal approximators,” IEEE Transactions on Fuzzy Systems, vol. 8, no. 6, pp. 773–780, 2000. View at Google Scholar · View at Scopus
  5. T. A. Johansen, R. Shorten, and R. Murray-Smith, “On the interpretation and identification of dynamic Takagi-Sugeno fuzzy models,” IEEE Transactions on Fuzzy Systems, vol. 8, no. 3, pp. 297–313, 2000. View at Publisher · View at Google Scholar · View at Scopus
  6. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15 of SIAM Studies in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1994. View at Zentralblatt MATH
  7. S. K. Nguang and P. Shi, “Robust output feedback control design for fuzzy dynamic systems with quadratic 𝒟 stability constraints: an LMI approach,” Information Sciences, vol. 176, no. 15, pp. 2161–2191, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Y. Y. Cao and P. M. Frank, “Robust disturbance attenuation for a class of uncertain discrete-time fuzzy systems,” IEEE Transactions on Fuzzy Systems, vol. 8, no. 4, pp. 406–415, 2000. View at Publisher · View at Google Scholar · View at Scopus
  9. D. J. Choi and P. Park, “ state-feedback controller design for discrete-time fuzzy systems using fuzzy weighting-dependent lyapunov functions,” IEEE Transactions on Fuzzy Systems, vol. 11, no. 2, pp. 271–278, 2003. View at Publisher · View at Google Scholar · View at Scopus
  10. H. Gao, Z. Wang, and C. Wang, “Improved control of discrete-time fuzzy systems: a cone complementarity linearization approach,” Information Sciences, vol. 175, no. 1-2, pp. 57–77, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. L. A. Mozelli, C. D. Campos, R. M. Palhares, L. A. B. Tôrres, and E. M. A. M. Mendes, “Chaotic synchronization and information transmission experiments: a fuzzy relaxed control approach,” Circuits, Systems, and Signal Processing, vol. 26, no. 4, pp. 427–449, 2007. View at Publisher · View at Google Scholar · View at Scopus
  12. S. Zhou, G. Feng, J. Lam, and S. Xu, “Robust control for discrete-time fuzzy systems via basis-dependent Lyapunov functions,” Information Sciences, vol. 174, no. 3-4, pp. 197–217, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. S. Zhou, J. Lam, and W. X. Zheng, “Control design for fuzzy systems based on relaxed nonquadratic stability and performance conditions,” IEEE Transactions on Fuzzy Systems, vol. 15, no. 2, pp. 188–199, 2007. View at Publisher · View at Google Scholar · View at Scopus
  14. E. Kim and H. Lee, “New approaches to relaxed quadratic stability condition of fuzzy control systems,” IEEE Transactions on Fuzzy Systems, vol. 8, no. 5, pp. 523–534, 2000. View at Google Scholar · View at Scopus
  15. M. C. M. Teixeira, E. Assunção, and R. G. Avellar, “On relaxed LMI-based designs for fuzzy regulators and fuzzy observers,” IEEE Transactions on Fuzzy Systems, vol. 11, no. 5, pp. 613–623, 2003. View at Publisher · View at Google Scholar · View at Scopus
  16. 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 T-S fuzzy control systems,” IEEE Transactions on Fuzzy Systems, vol. 14, no. 3, pp. 386–397, 2006. View at Publisher · View at Google Scholar · View at Scopus
  17. V. F. Montagner, R. C. L. F. Oliveira, and P. L. D. Peres, “Necessary and sufficient LMI conditions to compute quadratically stabilizing state feedback controllers for Takagi-Sugeno systems,” in Proceedings of the American Control Conference (ACC '07), pp. 4059–4064, New York, NY, USA, July 2007. View at Publisher · View at Google Scholar · View at Scopus
  18. V. F. Montagner, R. C. L. F. Oliveira, and P. L. D. Peres, “Convergent LMI relaxations for quadratic stabilizability and control of Takagi-Sugeno fuzzy systems,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 4, pp. 863–873, 2009. View at Publisher · View at Google Scholar · View at Scopus
  19. A. Sala and C. Arino, “Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: applications of Polya's theorem,” Fuzzy Sets and Systems, vol. 158, no. 24, pp. 2671–2686, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. M. Johansson, A. Rantzer, and K. E. Årzén, “Piecewise quadratic stability of fuzzy systems,” IEEE Transactions on Fuzzy Systems, vol. 7, no. 6, pp. 713–722, 1999. View at Google Scholar · View at Scopus
  21. S. G. Cao, N. W. Rees, and G. Feng, “Analysis and design for a class of complex control systems. II. Fuzzy controller design,” Automatica, vol. 33, no. 6, pp. 1029–1039, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. P. Borne and J.-Y. Dieulot, “Fuzzy systems and controllers: Lyapunov tools for a regionwise approach,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 5–7, pp. 653–665, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. K. Tanaka, T. Hori, and H. O. Wang, “A multiple Lyapunov function approach to stabilization of fuzzy control systems,” IEEE Transactions on Fuzzy Systems, vol. 11, no. 4, pp. 582–589, 2003. View at Publisher · View at Google Scholar · View at Scopus
  24. B.-J. Rhee and S. Won, “A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design,” Fuzzy Sets and Systems, vol. 157, no. 9, pp. 1211–1228, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. K. Tanaka, H. Ohtake, and H. O. Wang, “A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions,” IEEE Transactions on Fuzzy Systems, vol. 15, no. 3, pp. 333–341, 2007. View at Publisher · View at Google Scholar · View at Scopus
  26. 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,” Information Sciences, vol. 179, no. 8, pp. 1149–1162, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. L. A. Mozelli, R. M. Palhares, F. O. Souza, and E. M. A. M. Mendes, “Reducing conservativeness in recent stability conditions of TS fuzzy systems,” Automatica, vol. 45, no. 6, pp. 1580–1583, 2009. View at Publisher · View at Google Scholar · View at Scopus
  28. L. A. Mozelli, G. S. C. De Avellar, R. M. Palhares, and R. F. Dos Santos, “Condições LMIs alternativas para sistemas takagi-sugeno via função de lyapunov fuzzy,” Controle & Automacao, vol. 21, no. 1, pp. 96–107, 2010. View at Google Scholar · View at Scopus
  29. F. O. Souza, L. A. Mozelli, and R. M. Palhares, “On stability and stabilization of T-S fuzzy time-delayed systems,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 6, Article ID 5238534, pp. 1450–1455, 2009. View at Publisher · View at Google Scholar · View at Scopus
  30. L. A. Mozelli, R. M. Palhares, and E. M. A. M. Mendes, “Equivalent techniques, extra comparisons and less conservative control design for Takagi-Sugeno (TS) fuzzy systems,” IET Control Theory and Applications, vol. 4, no. 12, pp. 2813–2822, 2010. View at Publisher · View at Google Scholar
  31. L. A. Mozelli, F. O. Souza, and R. M. Palhares, “A new discretized Lyapunov-Krasovskii functional for stability analysis and control design of time-delayed TS fuzzy systems,” International Journal of Robust and Nonlinear Control, vol. 21, no. 1, pp. 93–105, 2011. View at Publisher · View at Google Scholar
  32. B. Ding, H. Sun, and P. Yang, “Further studies on LMI-based relaxed stabilization conditions for nonlinear systems in Takagi-Sugeno's form,” Automatica, vol. 42, no. 3, pp. 503–508, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. H. N. Wu and H. Y. Zhang, “Reliable fuzzy control for a class of discrete-time nonlinear systems using multiple fuzzy lyapunov functions,” IEEE Transactions on Circuits and Systems II, vol. 54, no. 4, pp. 357–361, 2007. View at Publisher · View at Google Scholar · View at Scopus
  34. F. O. Souza, R. M. Palhares, E. M. A. M. Mendes, and L. A. B. Torres, “Further results on master-slave synchronization of general Lur'e systems with time-varying delay,” International Journal of Bifurcation and Chaos, vol. 18, no. 1, pp. 187–202, 2008. View at Publisher · View at Google Scholar
  35. F. O. Souza, R. M. Palhares, E. M. A. M. Mendes, and L. A. B. Torres, “Robust control for master-slave synchronization of Lur'e systems with time-delay feedback control,” International Journal of Bifurcation and Chaos, vol. 18, no. 4, pp. 1161–1173, 2008. View at Publisher · View at Google Scholar
  36. C. D. Campos, R. M. Palhares, E. M. A. M. Mendes, L. A. B. Torres, and L. A. Mozelli, “Experimental results on Chua's circuit robust synchronization via LMIs,” International Journal of Bifurcation and Chaos, vol. 17, no. 9, pp. 3199–3209, 2007. View at Publisher · View at Google Scholar · View at Scopus
  37. F. G. Guimaraes, R. M. Palhares, F. Campelo, and H. Igarashi, “Design of mixed 2/ control systems using algorithms inspired by the immune system,” Information Sciences, vol. 177, no. 20, pp. 4368–4386, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. J. Daafouz and J. Bernussou, “Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties,” Systems & Control Letters, vol. 43, no. 5, pp. 355–359, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. M. C. de Oliveira, J. Bernussou, and J. C. Geromel, “A new discrete-time robust stability condition,” Systems & Control Letters, vol. 37, no. 4, pp. 261–265, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. M. C. de Oliveira, J. C. Geromel, and J. Bernussou, “Extended 2 and norm characterizations and controller parametrizations for discrete-time systems,” International Journal of Control, vol. 75, no. 9, pp. 666–679, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. T. Matsumoto, “A chaotic attractor from Chua's circuit,” IEEE Transactions on Circuits and Systems, vol. 31, no. 12, pp. 1055–1058, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  42. R. M. Palhares and P. L. D. Peres, “Robust filtering with guaranteed energy-to-peak performance—an LMI approach,” Automatica, vol. 36, no. 6, pp. 851–858, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  43. J. F. Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,” Optimization Methods and Software, vol. 11-12, no. 1–4, pp. 625–653, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  44. J. Lofberg, “YALMIP: a toolbox for modeling and optimization in MATLAB,” in Proceedings of the IEEE International Symposium on Computer Aided Control Systems Design (CACSD '04), pp. 284–289, Taipei, Taiwan, 2004. View at Publisher · View at Google Scholar
  45. L. O. Chua, M. Komuro, and T. Matsumoto, “The double scroll family. I. Rigorous proof of chaos,” IEEE Transactions on Circuits and Systems, vol. 33, no. 11, pp. 1072–1097, 1986. View at Publisher · View at Google Scholar
  46. L. A. B. Tôrres and L. A. Aguirre, “Pcchua—a laboratory setup for real-time control and synchronization of chaotic oscillations,” International Journal of Bifurcation and Chaos, vol. 15, no. 8, pp. 2349–2360, 2005. View at Publisher · View at Google Scholar · View at Scopus
  47. E. M. A. M. Mendes and S. A. Billings, “A note on discretization of nonlinear differential equations,” Chaos, vol. 12, no. 1, pp. 66–71, 2002. View at Publisher · View at Google Scholar · View at PubMed · View at Scopus