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Discovering the dynamic behavior of unknown systems using fuzzy logic. (English) Zbl 1422.93107

Summary: To know the dynamic behavior of a system it is convenient to have a good dynamic model of it. However, in many cases it is not possible either because of its complexity or because of the lack of knowledge of the laws involved in its operation. In these cases, obtaining models from input-output data is shown as a highly effective technique. Specifically, intelligent modeling techniques have become important in recent years in this field. Among these techniques, fuzzy logic is especially interesting because it allows to incorporate to the model the knowledge that is possessed of the system, besides offering a more interpretable model than other techniques. A fuzzy model is, formally speaking, a mathematical model. Therefore, this model can be used to analyze the original system using known systems analysis techniques. In this paper a methodology for extract information from unknown systems using fuzzy logic is presented. More precisely, it is presented the exact linearization of a Takagi-Sugeno fuzzy model with no restrictions in use or distribution of its membership functions, as well as obtaining its equilibrium states, the study of its local behavior and the search for periodic orbits by the application of Poincaré.

MSC:

93C42 Fuzzy control/observation systems
93B18 Linearizations
93B15 Realizations from input-output data

Software:

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References:

[1] Abraham, R. H., & Shaw, C. D. (1997). Dynamics: The geometry of behavior. Santa Cruz: Aerial Press, Incorporated.
[2] Al-Hadithi, BM; Jiménez, A.; Matía, F.; Andújar, JM; Barragán, AJ; Matía, F. (ed.); Marichal, GN (ed.); Jiménez, E. (ed.), New concepts for the estimation of Takagi-Sugeno model based on extended Kalman filter, No. 1, 3-24 (2014), Paris
[3] Andújar, J. M., & Barragán, A. J. (2005). A methodology to design stable nonlinear fuzzy control systems. Fuzzy Sets and Systems, 154(2), 157-181. · Zbl 1068.93030 · doi:10.1016/j.fss.2005.03.006
[4] Andújar, JM; Barragán, AJ; Al-Hadithi, BM; Matía, F.; Jiménez, A.; Matía, F. (ed.); Marichal, GN (ed.); Jiménez, E. (ed.), Suboptimal recursive methodology for Takagi-Sugeno fuzzy models identification, No. 2, 25-47 (2014), Paris
[5] Andújar, J. M., & Bravo, J. M. (2005). Multivariable fuzzy control applied to the physical-chemical treatment facility of a cellulose factory. Fuzzy Sets and Systems, 150(3), 475-492. · Zbl 1060.93541 · doi:10.1016/j.fss.2004.03.023
[6] Andújar, J. M., Bravo, J. M., & Peregrín, A. (2004). Stability analysis and synthesis of multivariable fuzzy systems using interval arithmetic. Fuzzy Sets and Systems, 148(3), 337-353. · Zbl 1057.93025 · doi:10.1016/j.fss.2004.01.008
[7] Barragán, A. J., Al-Hadithi, B. M., Jiménez, A., & Andújar, J. M. (2014). A general methodology for online TS fuzzy modeling by the extended Kalman filter. Applied Soft Computing, 18, 277-289. · doi:10.1016/j.asoc.2013.09.005
[8] Barragán, A. J., & Andújar, J. M. (2012). Fuzzy logic tools reference manual v1.0. University of Huelva. http://uhu.es/antonio.barragan/flt
[9] Bezdek, J. C., Ehrlich, R., & Full, W. E. (1984). FCM: The fuzzy c-means clustering algorithm. Computers and Geosciences, 10(2-3), 191-203. · doi:10.1016/0098-3004(84)90020-7
[10] Cazarez-Castro, N. R., Aguilar, L. T., Cardenas-Maciel, S. L., Goribar-Jimenez, C. A., & Odreman-Vera, M. (2017). Design of a fuzzy controller via fuzzy Lyapunov synthesis for the stabilization of an inertial wheel pendulum. Revista Iberoamericana de Automática e Informática Industrial (RIAI), 14(2), 13-140.
[11] Dorigo, M., Di Caro, G., & Gambardella, L. M. (1999). Ant algorithms for discrete optimization. Artificial Life, 5(2), 137-172. · doi:10.1162/106454699568728
[12] González Fontanet, J. G., Lussón Cervantes, A., & Bausa Ortiz, I. (2016). Alternatives of control for a Furuta’s pendulum. Revista Iberoamericana de Automática e Informática Industrial (RIAI), 13(4), 410-420. · doi:10.1016/j.riai.2016.05.008
[13] Grande, J. A., Andújar, J. M., Aroba, J., Beltrán, R., de la Torre, M. L., Cerón, J. C., et al. (2010). Fuzzy modeling of the spatial evolution of the chemistry in the Tinto River (SW Spain). Water Resources Management, 24(12), 3219-3235. · doi:10.1007/s11269-010-9603-2
[14] Horikawa, S. I., Furuhashi, T., & Uchikawa, Y. (1992). On fuzzy modeling using fuzzy neural networks with the back-propagation algorithm. IEEE Transactions on Neural Networks, 3(5), 801-806. · doi:10.1109/72.159069
[15] Jang, J. S. R. (1993). ANFIS: Adaptive-network-based fuzzy inference system. IEEE Transactions on Systems, Man, and Cybernetics, 23(3), 665-685. · doi:10.1109/21.256541
[16] Khalil, H. K. (2000). Nonlinear systems. Upper Saddle River, NJ: Prentice-Hall. · Zbl 0973.93010
[17] Kóczy, L. T., & Hirota, K. (1997). Size reduction by interpolation in fuzzy rule bases. IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics, 27(1), 14-25. · doi:10.1109/3477.552182
[18] Kreinovich, V., Hguyen, H. T., & Yam, Y. (2000). Fuzzy systems are universal approximators for a smooth function and its derivatives. International Journal of Intelligent Systems, 15(6), 565-574. · Zbl 0957.93050 · doi:10.1002/(SICI)1098-111X(200006)15:6<565::AID-INT6>3.0.CO;2-0
[19] Levenberg, K. (1944). A method for the solution of certain problems in least squares. Quarterly of Applied Mathematics, 2, 164-168. · Zbl 0063.03501 · doi:10.1090/qam/10666
[20] Li, S. Y., & Yang, Y. P. (2003). On-line constrained predictive control algorithm using multi-objective fuzzy-optimization and a case study. Fuzzy Optimization and Decision Making, 2(2), 123-142. · Zbl 1510.90304 · doi:10.1023/A:1023495100920
[21] López-Baldán, M. J., García-Cerezo, A., Cejudo, J. M., & Romero, A. (2002). Fuzzy modeling of a thermal solar plant. International Journal of Intelligent Systems, 17(4), 369-379. · Zbl 1002.68574 · doi:10.1002/int.10026
[22] Márquez, J. M. A., Piña, A. J. B., & Arias, M. E. G. (2009). A general and formal methodology for designing stable nonlinear fuzzy control systems. IEEE Transactions on Fuzzy Systems, 17(5), 1081-1091. · doi:10.1109/TFUZZ.2009.2021984
[23] McCulloch, W. S., & Pitts, W. (1943). A logical calculus of the ideas immanent in nervous activity. The Bulletin of Mathematical Biophysics, 5(4), 115-133. · Zbl 0063.03860 · doi:10.1007/BF02478259
[24] Mora, L. A., & Amaya, J. E. (2017). A new identification method based on open loop step response of overdamped system. Revista Iberoamericana de Automática e Informática Industrial (RIAI), 14(1), 31-43. · doi:10.1016/j.riai.2016.09.006
[25] Penedo, F., Haber, R. E., Gajate, A., & del Toro, R. M. (2012). Hybrid incremental modeling based on least squares and fuzzy \[KK\]-NN for monitoring tool wear in turning processes. IEEE Transactions on Industrial Informatics, 8(4), 811-818. · doi:10.1109/TII.2012.2205699
[26] Shin, H., Choi, C., Kim, E., & Park, M. (2004). Fuzzy partial state feedback control of discrete nonlinear systems with unknown time-delay. Fuzzy Optimization and Decision Making, 3(1), 83-92. · Zbl 1084.93015 · doi:10.1023/B:FODM.0000013073.24535.7d
[27] Shouyu, C., & Yu, G. (2006). Variable fuzzy sets and its application in comprehensive risk evaluation for flood-control engineering system. Fuzzy Optimization and Decision Making, 5(2), 153-162. · Zbl 1169.86303 · doi:10.1007/s10700-006-7333-y
[28] Slotine, J. J. E., & Li, W. (1991). Applied nonlinear control. Upper Saddle River, NJ: Prentice-Hall. · Zbl 0753.93036
[29] Takagi, T., & Sugeno, M. (1985). Fuzzy identification of systems and its applications to modeling and control. IEEE Transactions on Systems, Man, and Cybernetics, 15(1), 116-132. · Zbl 0576.93021 · doi:10.1109/TSMC.1985.6313399
[30] Vélez, M. A., Sánchez, O., Romero, S., & Manuel, A. J. (2010). A new methodology to improve interpretability in neuro-fuzzy TSK models. Applied Soft Computing, 10(2), 578-591. · doi:10.1016/j.asoc.2009.08.027
[31] Wang, L. X. (1992). Fuzzy systems are universal approximators. In IEEE international conference on fuzzy systems, San Diego, CA, USA (pp. 1163-1170).
[32] Wang, L. X. (1994). Adaptive fuzzy systems and control. Upper Saddle River, NJ: Prentice Hall.
[33] Wong, L., Leung, F., & Tam, P. (1997). Stability design of TS model based fuzzy systems. In IEEE international conference on fuzzy systems, Barcelona, Spain (Vol. 1, pp. 83-86).
[34] Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338-353. · Zbl 0139.24606 · doi:10.1016/S0019-9958(65)90241-X
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