Formal methodology for analyzing the dynamic behavior of nonlinear systems using fuzzy logic

[ES] Tener la capacidad para analizar un sistema desde un punto de vista dinámico puede ser muy útil en muchas circunstancias (sistemas industriales, biológicos, económicos,. ..). El análisis dinámico de un sistema permite conocer su comportamiento y la respuesta que presentará a distintos estímulos de entrada, su estabilidad en lazo abierto, tanto local como global, o si está afectado por fenómenos no lineales, como ciclos límites o bifurcaciones, entre otros. Si el sistema es desconocido o su dinámica es lo suficientemente compleja como para no poder obtener un modelo matemático del mismo, en principio no sería posible realizar un análisis dinámico formal del sistema. En estos casos la lógica borrosa, y más concretamente los modelos borrosos de tipo Takagi-Sugeno (TS), se presentan como una herramienta muy poderosa de análisis y diseño. Los modelos borrosos TS son aproximadores universales tanto de una función como de su derivada, por lo que permiten modelar sistemas no lineales en base a datos de entrada/salida. Puesto que un modelo borroso es un modelo matemático formalmente hablando, a partir del mismo es posible estudiar aspectos de la dinámica del sistema real que modela tal como se hace en la teoría de control no lineal. En este artículo se presenta una metodología para la obtención de los estados de equilibrio de un sistema no lineal, la linealización exacta de su modelo borroso de estado completamente general, el estudio de la estabilidad local de los equilibrios a partir de dicha linealización, y la utilización de la metodología de Poincare para el estudio de órbitas periódicas en modelos borrosos. A partir de esa información, es posible estudiar la estabilidad local de los estados de equilibrio, así como la dinámica del sistema en su entorno y la presencia de oscilaciones, obteniéndose una valiosa información del comportamiento dinámico del sistema.[EN] Having the ability to analyze a system from a dynamic point of view can be very useful in many circumstances (industrial systems, biological, economical, . . .). The dynamic analysis of a system allows to understand its behavior and response to different inputs, open loop stability, both locally and globally, or if it is affected by nonlinear phenomena, such as limit cycles, or bifurcations, among others. If the system is unknown or its dynamic is complex enough to obtain its mathematical model, in principle it would not be possible to make a formal dynamic analysis of the system. In these cases, fuzzy logic, and more specifically fuzzy TS models is presented as a powerful tool for analysis and design. The TS fuzzy models are universal approximators both of a function and its derivative, so it allows modeling highly nonlinear systems based on input/output data. Since a fuzzy model is a mathematical model formally speaking, it is possible to study the dynamic aspects of the real system that it models such as in the theory of nonlinear control.This article describes a methodology for obtaining the equilibrium states of a generic nonlinear system, the exact linearization of a completely general fuzzy model, and the use of the Poincaré’s methodology for the study of periodic orbits in fuzzy models. From this information it is possible to study the local stability of the equilibrium states, the dynamics of the system in its environment, and the presence of oscillations, yielding valuable information on the dynamic behavior of the system.Este artículo es una contribución del proyecto DPI2013-43870-R financiado por el Ministerio de Economía y Competitividad, y del proyecto TEP-6124 financiado por la Junta de Andalucía. Ambos proyectos están cofinanciados con fondos FEDERBarragán, AJ.; Al-Hadithi, BM.; Andújar, JM.; Jiménez, A. (2015). Metodología formal de análisis del comportamiento dinámico de sistemas no lineales mediante lógica borrosa. Revista Iberoamericana de Automática e Informática industrial. 12(4):434-445. https://doi.org/10.1016/j.riai.2015.09.005OJS434445124Abraham, R.H., Shaw, C.D., 1997. Dynamics: The Geometry of Behavior. Aerial Press, Incorporated.Al-Hadithi, B.M., Jiménez, A., Matía, F., Andújar, J.M., Barragán, A.J., Aug. 2014. New concepts for the estimation of Takagi-Sugeno model based on extended Kalman filter. En: Matía, F., Marichal, G.N., Jiménez, E., (Eds.), Fuzzy Modeling and Control: Theory and Applications. Vol. 9 of Atlantis Computational Intelligence Systems. Atlantis Press, pp. 3-24. DOI: 10.2991/978-94-6239-082-9_1.Al-Hadithi, B. M., Jiménez, A., & Matía, F. (2011). New methods for the estimation of Takagi-Sugeno model based extended Kalman filter and its applications to optimal control for nonlinear systems. Optimal Control Applications and Methods, 33(5), 552-575. doi:10.1002/oca.1014Andujar, J. M., Aroba, J., de Torre, M. L. la, & Grande, J. A. (2005). Contrast of evolution models for agricultural contaminants in ground waters by means of fuzzy logic and data mining. Environmental Geology, 49(3), 458-466. doi:10.1007/s00254-005-0103-2Andújar, J. M., & Barragán, A. J. (2005). A methodology to design stable nonlinear fuzzy control systems. Fuzzy Sets and Systems, 154(2), 157-181. doi:10.1016/j.fss.2005.03.006Andújar, J.M., Barragán, A.J., Apr. 2014. Hybridization of fuzzy systems for modeling and control. Revista Iberoamericana de Automática e Informática Industrial {RIAI} 11 (2), 127-141. DOI:http://dx.doi.org/10.1016/j.riai.2014.03.004.Andújar, J.M., Barragán, A.J., Al-Hadithi, B.M., Matía, F., Jiménez, A., Aug. 2014a. Stable fuzzy control system by design. En: Matía, F., Marichal, G.N., Jiménez, E., (Eds.), Fuzzy Modeling and Control: Theory and Applications. Vol. 9 of Atlantis Computational Intelligence Systems. Atlantis Press, pp. 69-94. DOI: 10.2991/978-94-6239-082-9_4.Andújar, J.M., Barragán, A.J., Al-Hadithi, B.M., Matía, F., Jiménez, A., Aug. 2014b. Suboptimal recursive methodology for Takagi-Sugeno fuzzy models identification. En: Matía, F., Marichal, G.N., Jiménez, E., (Eds.), Fuzzy Modeling and Control: Theory and Applications. Vol. 9 of Atlantis Computational Intelligence Systems. Atlantis Press, pp. 25-47. DOI: http://dx.doi.org/10.2991/978-94-6239-082-9_2.Marquez, J. M. A., Pina, A. J. B., & Arias, M. E. G. (2009). A General and Formal Methodology to Design Stable Nonlinear Fuzzy Control Systems. IEEE Transactions on Fuzzy Systems, 17(5), 1081-1091. doi:10.1109/tfuzz.2009.2021984Andújar, 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. doi:10.1016/j.fss.2004.03.023Andújar, J. M., Bravo, J. M., & Peregrı́n, A. (2004). Stability analysis and synthesis of multivariable fuzzy systems using interval arithmetic. Fuzzy Sets and Systems, 148(3), 337-353. doi:10.1016/j.fss.2004.01.008Angelov, P., & Buswell, R. (2002). Identification of evolving fuzzy rule-based models. IEEE Transactions on Fuzzy Systems, 10(5), 667-677. doi:10.1109/tfuzz.2002.803499Angelov, P. P., & Filev, D. P. (2004). An Approach to Online Identification of Takagi-Sugeno Fuzzy Models. IEEE Transactions on Systems, Man and Cybernetics, Part B (Cybernetics), 34(1), 484-498. doi:10.1109/tsmcb.2003.817053Aroba, J., Grande, J. A., Andújar, J. M., de la Torre, M. L., & Riquelme, J. C. (2007). Application of fuzzy logic and data mining techniques as tools for qualitative interpretation of acid mine drainage processes. Environmental Geology, 53(1), 135-145. doi:10.1007/s00254-006-0627-0Babuška, R., Mar. 1995. Fuzzy modeling - a control engineering perspective. En: Proceedings of, FUZZ-IEEE/IFES’95., Vol. 4., Yokohama, Japan, pp. 1897-1902. DOI: 10.1109/FUZZ.Y. 1995.409939.Babuška, R., Verbruggen, H.B., Mar. 1995. A new identification method for linguistic fuzzy models. En: Proceedings of FUZZ-IEEE/IFES’95. Vol. 4. Yokohama, Japan, pp. 905-912. DOI: 10.1109/FUZZY. 1995.409939.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.2013.09.005Bezdek, J. C., Ehrlich, R., & Full, W. (1984). FCM: The fuzzy c-means clustering algorithm. Computers & Geosciences, 10(2-3), 191-203. doi:10.1016/0098-3004(84)90020-7Chua, L.O., Desoer, C.A., Kuh, E.S., 1987. Linear and nonlinear circuits. McGraw-Hill series in electrical and computer engineering: Circuits and systems. McGraw-Hill Book Company, New York.Denaï, M. A., Palis, F., & Zeghbib, A. (2007). Modeling and control of non-linear systems using soft computing techniques. Applied Soft Computing, 7(3), 728-738. doi:10.1016/j.asoc.2005.12.005Grande, J. A., Andújar, J. M., Aroba, J., de la Torre, M. L., & Beltrán, R. (2005). Precipitation, pH and metal load in AMD river basins: an application of fuzzy clustering algorithms to the process characterization. J. Environ. Monit., 7(4), 325-334. doi:10.1039/b410795kHorikawa, 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:10.1109/72.159069Jang, J.-S. R. (1993). ANFIS: adaptive-network-based fuzzy inference system. IEEE Transactions on Systems, Man, and Cybernetics, 23(3), 665-685. doi:10.1109/21.256541Jiménez, A., Aroba, J., de la Torre, M. L., Andujar, J. M., & Grande, J. A. (2009). Model of behaviour of conductivity versus pH in acid mine drainage water, based on fuzzy logic and data mining techniques. Journal of Hydroinformatics, 11(2), 147-153. doi:10.2166/hydro.2009.015Kosko, B. (1994). Fuzzy systems as universal approximators. IEEE Transactions on Computers, 43(11), 1329-1333. doi:10.1109/12.324566Kreinovich, V., Nguyen, 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. doi:10.1002/(sici)1098-111x(200006)15:63.0.co;2-0López-Baldán, M. J., García-Cerezo, A., López, J. M. C., & Gallego, A. R. (2002). Fuzzy modeling of a thermal solar plant. International Journal of Intelligent Systems, 17(4), 369-379. doi:10.1002/int.10026Marquez, H.J., 2003. Nonlinear control systems. Analysis and design. John Wiley & Sons, Inc.Mencattini, A., Salmeri, M., & Salsano, A. (2005). Sufficient conditions to impose derivative constraints on MISO Takagi-Sugeno Fuzzy logic systems. IEEE Transactions on Fuzzy Systems, 13(4), 454-467. doi:10.1109/tfuzz.2004.841742Moré, J.J., 1977. The Levenberg-Marquardt algorithm: Implementation and theory. En: Watson, G. (Ed.), Numerical Analysis. Springer Verlag, Berlin, pp. 105-116.Nguyen, H.T., Sugeno, M., Tong, R.M., Yager, R.R., 1995. Theoretical aspects of fuzzy control. John Wiley Sons, New York, NY, USA.Nijmeijer, H., Schaft, A. v. d., 1990. Nonlinear dynamical control systems. Springer Verlag, Berlin.Sastry, S., 1999. Nonlinear system: analysis, stability, and control. Springer, New York.Slotine, J.-J. E., Li, W., 1991. Applied nonlinear control. Prentice-Hall, NJ.Takagi, T., & Sugeno, M. (1985). Fuzzy identification of systems and its applications to modeling and control. IEEE Transactions on Systems, Man, and Cybernetics, SMC-15(1), 116-132. doi:10.1109/tsmc.1985.6313399Wang, L.-X., 1992. Fuzzy systems are universal approximators. En: IEEE International Conference on Fuzzy Systems. San Diego, CA, USA, pp. 1163-1170. DOI: 10.1109/FUZZY. 1992.258721.Wang, L.X., 1994. Adaptive fuzzy systems and control. Prentice Hall, New Jersey.Wang, L.-X., 1997. A course in fuzzy systems and control. Prentice Hall, New Yersey, USA.Wiggins, S., Oct. 2003. Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd Edición. Texts in Applied Mathematics. Springer.Wong, L., Leung, F., Tam, P., Jul. 1997. Stability design of TS model based fuzzy systems. En: IEEE International Conference on Fuzzy Systems. Vol. 1. Barcelona, Spain, pp. 83-86. DOI: 10.1109/FUZZY. 1997.616349