Análisis de Estabilidad de Sistemas Borrosos

Basil M. Al-Hadithi, Fernando Matía, Agustín Jiménez

Resumen

En este trabajo se revisa el estado del arte sobre estabilidad de sistemas borrosos, poniéndose de manifiesto las dificultades para su análisis, debido a la característica falta de linealidad de los mismos. Se revisan los estudios basados en el criterio del círculo, las técnicas para calcular índices de estabilidad, así como técnicas basadas en aplicación del teorema de estabilidad de Lyapunov, que permite utilizar métodos numéricos de búsqueda de soluciones. Además, se revisan los trabajos de estabilidad mediante el uso del modelo borroso de Takagi-Sugeno (T-S), el enfoque de las Desigualdades Matriciales Lineales (LMI), que ha tenido un interés creciente en los últimos años, así como otra línea de investigación basada en estabilidad energética.

Palabras clave

Sistemas borrosos; Estabilidad global y asintótica; Modelo borroso afín de Takagi-Sugeno

Texto completo:

PDF

Referencias

Akar, M. and Ü. Özgüner (2000). Decentralized techniques for the analysis and control of takagi-sugeno fuzzy systems. IEEE Trans. Fuzzy Syst. 8(6), 691–704.

Al-Hadithi, B. M. (2002). Global Stability Analysis and Design of Fuzzy Control Systems. Universidad Politécnica de Madrid, Doctorate thesis (in spanish).

Al-Hadithi, B. M., F. Matía and A. Jiménez (2002). Global stability of fuzzy systems based variable structure control. In: NAFIPSFLINT 2002, the North American Fuzzy Information Processing Society, New Orleans, Louisiana.

Aracil, J., A. Garcia-Cerezo, A. Barreiro and A. Ollero (1993). Stability analysis of fuzzy control systems based on the conicity criterion. in: R. Lowen Ed., Fuzzy Logic: State of the Art. Series System Theory, Knowledge Engineering and Problem Solving (Norwell, MA: Kluwer Academic Publisher, Dordrecht), No.1, pages 487-496.

Aracil, J., A. Garcia-Cerezo and A. Ollero (1988). Stability analysis of fuzzy control systems-A geometric approach. in Artificial Intelligence, Expert Systems and languages in Modeling and Simulation, C. A. Kulikowski, R. M. Huber, and G. A. Ferraté, Eds. Amesterdam, The Netherlands: North-Holland, pages 323- 330.

Boyd, S., L. El Ghaoui, E. Feron and V. Balakrishnan (1994). Linear matrix inequalities in systems and control theory. Philadelphia, PA: Studies in Applied Mathematics. SIAM.

Braae, M. and D. A. Rutherford (1979). Theoretical and linguistic aspects of the fuzzy logic controller. Automatica 15, 553–577.

Cao, S. G., N. W. Rees and G. Feng (1996a). Analysis and design for a class of continuoustime fuzzy control systems. Int. J. Contr 64, 1069–1089.

Cao, S. G., N. W. Rees and G. Feng (1996b). Stability analysis of fuzzy control systems. IEEE Transactions on Systems, Man and Cybernetics 26-part B (1), 201–204.

Cao, S. G., N. W. Rees and G. Feng (1997). Analysis and design for a class of complex control systems-part ii: Fuzzy controller design. Automatica 33, 1029–1039.

Cao, S. G., N. W. Rees and G. Feng (1999). Analysis and design of fuzzy control systems using dynamic fuzzy-state space models. IEEE Transactions on Fuzzy Systems 7(2), 192– 200.

Castillo-Toledo, B. and J. A. Meda-Campana (2004). The fuzzy discrete-time robust regulation problem: An lmi approach. IEEE Trans. Fuzzy Syst 12(3), 360–367.

Chang, Y. C. and B. S. Chen (2005). Intelligent robust tracking controls for holonomic and nonholonomic mechanical systems using only position measurements. IEEE Trans. Fuzzy Syst 13(4), 491–507.

Chen, B. S., C. S. Tseng and H. J. Uang (2000). Mixed H2 / H∞ fuzzy output feedback control design for non-linear dynamic systems: An lmi approach. IEEE Trans. Fuzzy Syst 8(3), 249–265.

Chen, C. L., G. Feng and X. P. Guan (2005a). Delay-dependent stability analysis and controller synthesis for discrete time TS fuzzy systems with time delays. IEEE Trans. Fuzzy Syst 13(5), 630–643.

Chen, C. L., G. Feng, D. Sun and Y. Zhu (2005b). H-infinity output feedback control of discretetime fuzzy systems with application to chaos control. IEEE Trans. Fuzzy Syst 13(4), 531– 543.

Chen, J. Y. (2001). Rule regulation of fuzzy sliding mode controller design: Direct adaptive approach. Fuzzy Sets Syst 120, 159–168.

Chen, S. S., Y. C. Chang, S. F. Su, S. L. Chung and T. T. Lee (2005c). Robust static output-feedback stabilization for nonlinear discrete-time systems with time delay via fuzzy control approach. IEEE Trans. Fuzzy Syst 13(2), 263–272.

Chen, Y. Y. and T. C. Tsao (1989). A description of the dynamical behaviour of fuzzy systems. IEEE Transactions on Systems, Man and Cybernetics 19, 745–755.

Choi, D. J. and P. G. Park (2003). H-infinity state-feedback controller design for discretetime fuzzy systems using fuzzy weightingdependent lyapunov functions. IEEE Trans. Fuzzy Syst 11(2), 271–278.

Chung, S. K., J. Ryu and C. K. Park (1992). Stability analysis of fuzzy control system. In: Proc. Korea-Japan Joint Conf. Fuzzy Syst. Eng., Seoul, Korea. pp. 29–32.

Cuesta, F., F. Gordillo, J. Aracil and A. Ollero (1999). Stability analysis of nonlinear multivariable takagi-sugeno fuzzy control systems. IEEE Transactions on Fuzzy Systems 7(5), 508–520.

Feng, G. (2001). Approaches to quadratic stabilization of uncertain fuzzy dynamic systems. IEEE Trans. Circuits Syst. I 48(6), 760–769.

Feng, G. (2003). Controller synthesis of fuzzy dynamic systems based on piecewise lyapunov functions. IEEE Trans. Fuzzy Syst. 11(10), 605–612.

Feng, G. (2004a). H-infinity controller design of fuzzy dynamic systems based on piecewise lyapunov functions. IEEE Trans. Syst., Man, Cybern., B., Cybern. 34(1), 283–292.

Feng, G. (2004b). Stability analysis of discrete time fuzzy dynamic systems based on piecewise lyapunov functions. IEEE Trans. Fuzzy Syst. 12(1), 22–28.

Feng, G. (2005). Robust h-infinity ? ltering of fuzzy dynamic systems. IEEE Trans. Aerospace Electron. Syst. 41(2), 658–670.

Feng, G. (2006). A survey on analysis and design of model-based fuzzy control systems. IEEE Trans. Fuzzy Syst. 14(5), 676–697.

Feng, G. and D. Sun (2002). Generalized h2 controller synthesis of fuzzy dynamic systems based on piecewise lyapunov functions. IEEE Trans. Circuits Syst. I. 49(12), 1843–1850.

Feng, G. and J. Ma (2001). Quadratic stabilization of uncertain discrete-time fuzzy dynamic systems. IEEE Trans. Circuits Syst. I. 48(11), 1337–1343.

Feng, G., S. G. Cao, N. W. Rees and C. K. Chak (1997). Design of fuzzy control systems with guaranteed stability. Fuzzy Sets and Systems 85(1), 1–10.

Feng, M. and C. J. Harris (2001a). Feedback stabilization of fuzzy systems via linear matrix inequalities. Int. J. Syst. Sci. 32, 221–231.

Feng, M. and C. J. Harris (2001b). Piecewise lyapunov stability conditions of fuzzy systems. IEEE Trans. Syst., Man, Cybern., B, Cybern. 31, 259–262.

French, M. and E. Rogers (1998). Input/output stabilitytheory for direct neuro-fuzzy controllers. IEEE Trans. Fuzzy Syst. 6(3), 331– 345.

Gang, J. and C. Laijiu (1996). Linguistic stability analysis of fuzzy closed loop control systems. Fuzzy Sets and Systems 82, 27–34.

Glas, M. De (1984). invariance and stability of fuzzy systems. J. Mathematical Analysis and Applications 199, 299–319.

Guerra, T. M. and L. Vermeiren (2004). LMIbased relaxed nonquadratic stabilization conditions for nonlinear systems in the takagisugeno form. Automatica 40(5), 823–829.

Hojo, T., T. Terano and S. Masui (1992). Stability analysis of fuzzy control systems based on phase plane analysis. J. Japan Soc. Fuzzy Theory Syst. 4(6), 1133–1146.

Hong, S. K. and R. Langari (2000). An lmi-based h-infinity fuzzy control system design with ts framework. Inform. Sci. 123(3-4), 163–179.

Hsiao, F. H., C. W. Chen, Y. W. Liang, S. D. Xu and W. L. Chiang (2005). TS fuzzy controllers for nonlinear interconnected systems with multiple time delays. IEEE Trans. Circuits Syst. I. 52(9), 1883–1893.

Huang, S. J. and W. C. Lin (2003). Adaptive fuzzy controller with sliding surface for vehicle suspension control. IEEE Transactions on Fuzzy Systems 11(4), 550–559.

Hwang, C.-L. (2004). A novel takagi-sugenobased robust adaptive fuzzy sliding-mode controller. IEEE Transactions on Fuzzy Systems 12(5), 676–687.

Hwang, C.-L and H.-Y. Lin (2004). A fuzzy decentralized variable structure tracking control with optimal and improved robustness designs: theory and applications. IEEE Transactions on Fuzzy Systems 12(5), 615–630.

Ishikawa, T. (1988). A study on fuzzy control of an arm robot. B. E. thesis, Dept. Contr. Eng., Tokyo Inst. Technol.

Johansson, M., A. Rantzer and K.-E. ˚Arz´en (1999). Piecewise quadratic stability for fuzzy systems. IEEE Transactions on Fuzzy Systems 7(6), 713–722.

Kang, H.-J., C. Kwon, H. Lee and M. Park (1998). Robust stability analysis and design method for the feedback linearization regulator. IEEE Trans. Fuzzy Syst. 6(4), 464–472.

Kawamoto, S., K. Tada, A. Ishigame and T. Taniguchi (1993). An extension of tanakasugeno’s stability theorem for fuzzy systems. J. Japan Soc. Fuzzy Theory Syst. 5(5), 1190– 1199.

Khartinov, V. L. (1979). Asymptotic stability of an equilibrium position of a family of systems of linear differential equations. Differential Equations 14, 1483–1485.

Kickert, W. J. M. and E. H. Mamdani (1978). Analysis of a fuzzy logic controller. Int. J. Fuzzy Sets and Systems 1, 29–44.

Kim, E. and D. Kim (2001). Stability analysis and synthesis for an affine fuzzy system via lmi and ilmi: Discrete case. IEEE Trans. Syst., Man, Cybern., B, Cybern. 31(1), 132–140.

Kim, E. and S. Kim (2002). Stability analysis and synthesis for an affine fuzzy control system via lmi and ilmi: Continuous case. IEEE Trans. Fuzzy Syst. 10(3), 391–400.

Kim, E. and S. Lee (2005). Output feedback tracking control of mimo systems using a fuzzy disturbance observer and its application to the speed control of a pm synchronous motor. IEEE Trans. Fuzzy Syst. 13(6), 725– 741.

Kiszka, J. B., M. M. Gupta and P.N. Nikiforuk ˜ (1985). Energetistic stability of fuzzy dynamic systems. IEEE Transactions on Systems, Man and Cybernetics SMC-15(6), 783–792.

Kitamura, S. (1991). A stability condition for fuzzy ruled control systems-an extension of circle criterion. Trans. Soc. Instrum. Contr. Eng. 27, 532–537.

Kosko, B. (1997). Fuzzy Engineering. Prentice Hall.

Kung, C. C., T. H. Chen and C. H. Chen (May 2005). H-infinity state feedback controller design for TS fuzzy systems based on piecewise lyapunov function. In: in Proc. 14th IEEE Int. Conf. Fuzzy Systems, Reno, NV. pp. 708– 713.

Langari, G. and M. Tomizuka (1990). Stability of fuzzy linguistic control systems. In: Proc. 29th IEEE Conference on Decision and Control, Honolulu, HI. pp. 2185–2190.

Leung, F. H. F., P. K. S. Tam H. K. Lam and Y. S. Lee (2003). Stable fuzzy controller design for uncertain nonlinear systems: Genetic algorithm approach. In: in Proc. 12th IEEE Int. Conf. Fuzzy Systems, St Louis, MO. pp. 500– 505.

Li, T. H. S. and K. J. Lin (2004). Stabilization of singularly perturbed fuzzy systems. IEEE Trans. Fuzzy Syst 12(5), 579–595.

Lian, K. Y., C. S. Chiu, T. S. Chiang and P. Liu (2001). LMI-based fuzzy chaotic synchronization and communications. IEEE Trans. Fuzzy Syst 9(4), 539–553.

Liu, H. P., F. C. Sun and Z. Q. Sun (2005). Stability analysis and synthesis of fuzzy singularly perturbed systems. IEEE Trans. Fuzzy Syst 13(2), 273–284.

Lo, J.-C. and Y.-M. Chen (1999). Stability issues on takagi-sugeno fuzzy model-parametric approach. IEEE Trans. Fuzzy Syst. 7(5), 597– 607.

Mamdani, E. and S. Assilian (1975). An experiment in linguistic synthesis with a fuzzy logic controller. Int. J. Man-Machine Studies 7(1), 1–13.

Matía, F., B. M. Al-Hadithi and A. Jiménez (1999a). Necessary and Sufficient Conditions for Stability of Dynamic Fuzzy Systems. Computational Intelligence in Systems and Control Design and Applications, Spyros G. Tzafestas, Kluwer Academic Publications, Chapter 8, pp 81–91.

Matía, F., B. M. Al-Hadithi and A. Jiménez (1999b). On global stability of t-s general model. Mathware and Soft Computing VI (2- 3), 293–304.

Matía, F., B. M. Al-Hadithi and A. Jiménez (2000a). Stability analysis of discrete fuzzy systems. In: FUZZ-IEEE 2000, The 9th International Conference on Fuzzy System, San Antonio, Texas, USA. pp. 961–964.

Matía, F., B. M. Al-Hadithi and A. Jiménez (2002). Generalization of stability criterion for tanaka -sugeno continuous fuzzy model. Fuzzy Sets and System 129(3), 295–309.

Matía, F., B. M. Al-Hadithi and A. Jiménez (2000b). Global stability of affine takagisugeno fuzzy mode. In: X Congreso Español sobre Tecnologías y Lógica Fuzzy, ESTYLF 2000, Sevilla, España. pp. 31–36.

Meda-Campana, J. A. and B. Castillo-Toledo (2005). On the output regulation for takagisugeno fuzzy models using sliding modes. In: in Proc. Amer. Control Conf., Portland, OR. pp. 4062–4067.

Ohtake, H., K. Tanaka and H. O. Wang (Dec. 2003). Piecewise nonlinear control. In: in Proc. 42nd IEEE Conf. Decision and Control, Maui, HI. pp. 4735–4740.

Ollero, A., J. Aracil and A. Garcia-Cerezo (1995). Robust design of rule-based fuzzy controllers. Fuzzy Sets and Systems 70(1), 249–273.

Opitz, H.-P. (Sept. 1993). Fuzzy control and stability criteria. In: Proc. EUFIT’93, Aachen, Germany. pp. 130–136.

Palm, R. (1994). Robust control by fuzzy sliding mode. Automatica 30(9), 1429–1437.

Park, C.-W. and H.-J. Kang (2005). Graphical and numerical approach to robust stability analysis of fuzzy modeled systems with para metric uncertainty and disturbance. Fuzzy Sets Systems 151, 99–117.

Park, Y. M., M. J. Tahk and H. C. Bang (2004). Design and analysis of optimal controller for fuzzy systems with input constraint. IEEE Trans. Fuzzy Syst 12(6), 766–779.

Rantzer, A. and M. Johansson (1997). Piecewise linear quadratic optimal control. In: Proc. American Control Conf., Albuquerque, NM.

Ray, K. S., A. M. Ghosh and D. D. Majumder (1984). L2-stability and the related design concept for siso linear system associated with fuzzy logic controllers. IEEE Transactions on Systems, Man and Cybernetics 14(6), 932– 939.

Ray, K. S. and D. D. Majumder (1984). Application of circle criterion for stability analysis of linear siso and mimo systems associated with fuzzy logic controller. IEEE Transactions on Systems, Man and Cybernetics SMC-14(2), 345–349.

Su, J. P., T. M. Chen and C. C. Wang (2001). Adaptive fuzzy sliding mode control with ga-based reaching laws. Fuzzy Sets Syst. 120, 145–158.

Sugeno, M. (1985). An introductory survey of fuzzy control. Inform. Sci. 36(1), 59–83.

Sugeno, M. and K. Murakami (1984). Fuzzy parking control of model car. In: Proc. 23rd IEEE Conf. Decision Contr. (CDC), Las Vegas, NY. pp. 902–903.

Takagi, T. and M. Sugeno (1985). Fuzzy identification of systems and its applications to modeling and control. IEEE Transactions on Systems, Man and Cybernetics SMC15(1), 116–132.

Tanaka, K. and. H. O. Wang (2001). Fuzzy Control Systems Design and Analysis: A LMI Approach. New York: Wiley.

Tanaka, K. and M. Sano (1994). A robust stabilization problem of fuzzy control systems and its application to backing up control of a truck-trailer. IEEE Transactions on Fuzzy Systems 2(2), 119–134.

Tanaka, K. and M. Sugeno (1990). Stability analysis of fuzzy systems using lyapunov’s direct method. In: Proc. North Amer. Fuzzy Ingorm. Processing Soc. (NAFIPS), New York, NY. pp. 133–136.

Tanaka, K. and M. Sugeno (1992). Stability analysis and design of fuzzy control systems. Fuzzy Sets and Systems 12, 135–156.

Tanaka, K., T. Hori and H. O. Wang (2003). A multiple lyapunov function approach to stabilization of fuzzy control systems. IEEE Trans. Fuzzy Syst 11(4), 582–589.

Taniguchi, T. and M. Sugeno (Jul. 2004). Stabilization of nonlinear systems based on piecewise lyapunov functions. In: in Proc. 13th IEEE Int. Conf. on Fuzzy Systems, Budapest, Hungary. pp. 1607–1612.

Tao, C. W. and J. S. Taur (2005). Robust fuzzy control for a plant with fuzzy linear model. IEEE Trans. Fuzzy Syst. 13(1), 30–41.

Tao, C. W., J. S. Taur and M. L. Chan (2004). Adaptive fuzzy terminal sliding mode controller for linear systems with mismatched time-varying uncertainties. IEEE Trans. Syst., Man, Cybern., B, Cybern. 34(1), 255– 262.

Teixeira, M. C. M., E. Assuncao and R. G. Avellar (2003). On relaxed lmibased designs for fuzzy regulators and fuzzy observers. IEEE Trans. Fuzzy Syst. 11(5), 613–623.

Tian, E. and C. Peng (2006). Delay-dependent stability analysis and synthesis of uncertain t-s fuzzy systems with time-varying delay. Fuzzy Sets Systems 157, 544–559.

Tong, R. M., M. B. Beck and A. Latten (1980). Fuzzy control of the activated sludge wasteeater treatment process. Automatica 16(6), 695–701.

Tong, S. C. and H. X. Li (2003). Fuzzy adaptive sliding-mode control for mimo nonlinear systems. IEEE Trans. Fuzzy Syst. 11(3), 354– 360.

Tuan, H. D., P. Apkarian, T. Ñarikiyo and M. Kanota (2004). New fuzzy control model and dynamic output feed back parallel distributed compensation. IEEE Trans. Fuzzy Syst. 12(1), 13–21.

Wang, H. O., K. Tanaka and M. F. Griffin (1996). An approach to fuzzy control of nonlinear systems: Stability and design issues. IEEE Transactions on Fuzzy Systems 4(1), 14–23.

Wang, L. and G. Feng (2004). Piecewise h-infinity controller design of discrete time fuzzy systems. IEEE Trans. Syst., Man, Cybern., B, Cybern. 34(1), 682–686.

Wang, L., G. Feng and T. Hesketh (2003). Piecewise output feedback controller synthesis of discrete time fuzzy systems. In: in Proc. IEEE Conf. Decision and Control, Maui, HI. pp. 4741–4746.

Wang, L., G. Feng and T. Hesketh (2004a). Piecewise generalized H2 controller synthesis of discrete time fuzzy systems. In: in Proc. Inst. Elect. Eng. Control Theory Applications, no. 5, Part D. Vol. 151. pp. 554–560.

Wang, L.-X. (1997). A course in fuzzy systems and control. Englewood Cliffs, NJ: Prentice Hall.

Wang, W. J. and C. H. Sun (2005). Relaxed stability and stabilization conditions for a TS fuzzy discrete system. Fuzzy Sets Syst. 156(2), 208–225.

Wang, W. J. and L. Luoh (2004). Stability and stabilization of fuzzy large scale systems. IEEE Trans. Fuzzy Syst. 12(3), 309–315.

Wang, W. J. and W. W. Lin (2005). Decentralized pdc for large-scale TS fuzzy systems. IEEE Trans. Fuzzy Syst. 13(6), 779–786.

Wang, Y., Z. Q. Sun and F. C. Sun (2004 b). Stability analysis and control of discrete-time fuzzy systems: A fuzzy lyapunov function approach. In: in Proc. 5th Asian Control Conf., Melbourne, Australia. pp. 1855–1860.

Xiu, Zhi-Hong and Guang Ren (2005). Stability analysis and systematic design of takagi- sugeno fuzzy control systems. Fuzzy Sets Systems 151, 119–138.

Xu, S. Y. and J. Lam (2005). Robust hin? nity control for uncertain discrete-timedelay fuzzy systems via output feedback controllers. IEEE Trans. Fuzzy Syst. 13(1), 82– 93.

Yager, R. R. and D. P. Filev (1994). Essentials of Fuzzy Modeling and Control. New York: John Wiley.

Zak, S. H. (1999). Stabilizing fuzzy systems models using linear controllers. IEEE Transactions on Fuzzy Systems 7(2), 236–249.

Zhao, J., R. Gorez and V. Wertz (1997). Synthesis of fuzzy control systems based on linear Takagi-Sugeno fuzzy rnodels. Multiple Model Approaches to Nonlinear Modeling and Control, R. Murray-Smith and T. A. Johasen, Eds. London, U.K.: Taylor Francis.

Zhou, S. S., G. Feng, J. Lam and S. Y. Xu (2005). Robust h-infinity control for discrete fuzzy systems via basis-dependent lyapunov functions. Inform. Sci. 174(3-4), 197–217.

Abstract Views

392
Metrics Loading ...

Metrics powered by PLOS ALM




Creative Commons License

Esta revista se publica bajo una Licencia Creative Commons Attribution-NonCommercial-CompartirIgual 4.0 International (CC BY-NC-SA 4.0)

Universitat Politècnica de València     https://doi.org/10.4995/riai

e-ISSN: 1697-7920     ISSN: 1697-7912