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Inicio Revista Iberoamericana de Automática e Informática Industrial RIAI Metodología formal de análisis del comportamiento dinámico de sistemas no lin...
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Vol. 12. Núm. 4.
Páginas 434-445 (Octubre - Diciembre 2015)
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Vol. 12. Núm. 4.
Páginas 434-445 (Octubre - Diciembre 2015)
DOI: 10.1016/j.riai.2015.09.005
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Metodología formal de análisis del comportamiento dinámico de sistemas no lineales mediante lógica borrosa
Formal methodology for analyzing the dynamic behavior of nonlinear systems using fuzzy logic
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Antonio Javier Barragána,
Autor para correspondencia
antonio.barragan@diesia.uhu.es

Autor para correspondencia.
, Basil Mohammed Al-Hadithib, José Manuel Andújara, Agustín Jiménezb
a Dep. de Ing. Electrónica, de Sistemas Electrónicos y Automática, Universidad de Huelva.
b Groupo de Control Inteligente, Universidad Politécnica de Madrid, Centro de Automatización y Robótica UPM - CSIC
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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.

Palabras clave:
Análisis dinámico
estabilidad
estado de equilibrio
linealización
metodología de Poincaré
modelado borroso
sistemas dinámicos
Takagi-Sugeno (TS) model
Abstract

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.

Keywords:
Dynamic analysis dynamic systems equilibrium state fuzzy control linearization fuzzy modeling Poincaré’s methodology stability Takagi-Sugeno (TS) model
Referencias
[Abraham and Shaw, 1997]
Abraham, R.H., Shaw, C.D., 1997. Dynamics: The Geometry of Behavior. Aerial Press, Incorporated.
[Al-Hadithi et al., 2014]
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 et al., 2012]
B.M. Al-Hadithi, A. Jiménez Avello, F. Matía.
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 (2012), pp. 552-575
[Andújar et al., 2006]
J.M. Andújar, J. Aroba, M.L.dl. Torre, J.A. Grande.
Contrast of evolution models for agricultural contaminants in ground waters by means of fuzzy logic and data mining.
Environmental Geology, 49 (2006), pp. 458-466
[Andújar and Barragán, 2005]
J.M. Andújar, A.J. Barragán.
A methodology to design stable nonlinear fuzzy control systems.
Fuzzy Sets and Systems, 154 (2005), pp. 157-181
[Andújar and Barragán, 2014]
Andújar, 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 et al., 2014a]
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 et al., 2014b]
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.
[Andújar et al., 2009]
J.M. Andújar, A.J. Barragán, M.E. Gegúndez.
A general and formal methodology for designing stable nonlinear fuzzy control systems.
IEEE Transactions on Fuzzy Systems, 17 (2009), pp. 1081-1091
[Andújar and Bravo, 2005]
J.M. Andújar, J.M. Bravo.
Multivariable fuzzy control applied to the physical-chemical treatment facility of a cellulose factory.
Fuzzy Sets and Systems, 150 (2005), pp. 475-492
[Andújar et al., 2004]
J.M. Andújar, J.M. Bravo, A. Peregrín.
Stability analysis and synthesis of multivariable fuzzy systems using interval arithmetic.
Fuzzy Sets and Systems, 148 (2004), pp. 337-353
[Angelov, 2002]
P. Angelov, R. Buswell.
Identification of evolving fuzzy rule-based models.
IEEE Transactions on Fuzzy Systems, 10 (2002), pp. 667-677
[Angelov and Filev, 2004]
P.P. Angelov, D.P. Filev.
An approach to online identification of Takagi-Sugeno fuzzy models.
IEEE Transactions on Systems, Man, and Cybernetics—Part B: Cybernetics, 34 (2004), pp. 484-498
[Aroba et al., 2007]
J. Aroba, J.A. Grande, J.M. Andújar, M.L. De La Torre, J.C. Riquelme.
Application of fuzzy logic and data mining techniques as tools for qualitative interpretation of acid mine drainage processes.
Environmental Geology, 53 (2007), pp. 135-145
[Babuška, 1995a]
Babuška, 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, 1995b]
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 et al., 2014]
A.J. Barragán, B.M. Al-Hadithi, A. Jiménez, J.M. Andújar.
A general methodology for online TS fuzzy modeling by the extended kalman filter.
Applied Soft Computing, 18 (2014), pp. 277-289
[Bezdek et al., 1984]
J.C. Bezdek, R. Ehrlich, W.E. Full.
FCM: The fuzzy c-means clustering algorithm.
Computers and Geosciences, 10 (1984), pp. 191-203
[Chua et al., 1987]
Chua, 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 et al., 2007]
M.A. Denaï, F. Palis, A.H. Zeghbib.
Modeling and control of nonlinear systems using soft computing techniques.
Applied Soft Computing, 7 (2007), pp. 728-738
[Grande et al., 2005]
J.A. Grande, J.M. Andújar, J. Aroba, M.L. De La Torre, R. Beltrán.
Precipitation, pH and metal load in AMD river basins: An application of fuzzy clustering algorithms to the process characterization.
Journal of Environmental Monitoring, 7 (2005), pp. 325-334
[Horikawa et al., 1992]
S.-I. Horikawa, T. Furuhashi, Y. Uchikawa.
On fuzzy modeling using fuzzy neural networks with the back-propagation algorithm.
IEEE Transactions on Neural Networks, 3 (1992), pp. 801-806
[Jang, 1993]
J.-S.R. Jang.
ANFIS: adaptive-network-based fuzzy inference system. IEEE Transactions on Systems.
Man, and Cybernetics, 23 (1993), pp. 665-685
[Jiménez et al., 2009]
A. Jiménez, J. Aroba, M.L.l. de la Torre, J.M. Andújar, J.A. Grande.
Model of behaviour of conductivity versus pH in acid mine drainage water, based on fuzzy logic and data mining techniques.
Journal of Hydroinformatics, 2 (2009), pp. 147-153
[Kosko, 1994]
B. Kosko.
Fuzzy systems as universal approximators.
IEEE Transactions on Computers, 43 (1994), pp. 1329-1333
[Kreinovich et al., 2000]
V. Kreinovich, H.T. Hguyen, Y. Yam.
Fuzzy systems are universal approximators for a smooth function and its derivatives.
International journal of Intelligent Systems, 15 (2000), pp. 565-574
[Levenberg, 1944]
K. Levenberg.
A method for the solution of certain problems in least squares.
En: Quart. Appl. Math., 2 (1944), pp. 164-168
[López-Baldán et al., 2002]
M.J. López-Baldán, A. García-Cerezo, J.M. Cejudo, A. Romero.
Fuzzy modeling of a thermal solar plant.
International Journal of Intelligent Systems, 17 (2002), pp. 369-379
[Marquez, 2003]
Marquez, H.J., 2003. Nonlinear control systems. Analysis and design. John Wiley & Sons, Inc.
[Mencattini, 2005]
A. Mencattini, M. Salmeri, A. Salsano.
Sufficient conditions to impose derivative constraints on MISO Takagi–Sugeno fuzzy logic systems.
IEEE Transactions on Fuzzy Systems, 13 (2005), pp. 454-467
[Moré, 1977]
Moré, J.J., 1977. The Levenberg-Marquardt algorithm: Implementation and theory. En: Watson, G. (Ed.), Numerical Analysis. Springer Verlag, Berlin, pp. 105-116.
[Nguyen et al., 1995]
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 and Schaft, 1990]
Nijmeijer, H., Schaft, A. v. d., 1990. Nonlinear dynamical control systems. Springer Verlag, Berlin.
[Sastry, 1999]
Sastry, S., 1999. Nonlinear system: analysis, stability, and control. Springer, New York.
[Slotine and Li, 1991]
Slotine, J.-J. E., Li, W., 1991. Applied nonlinear control. Prentice-Hall, NJ.
[Takagi and Sugeno, 1985]
T. Takagi, M. Sugeno.
Fuzzy identification of systems and its applications to modeling and control.
IEEE Transactions on Systems, Man, and Cybernetics, 15 (1985), pp. 116-132
[Wang, 1992]
Wang, 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, 1994]
Wang, L.X., 1994. Adaptive fuzzy systems and control. Prentice Hall, New Jersey.
[Wang, 1997]
Wang, L.-X., 1997. A course in fuzzy systems and control. Prentice Hall, New Yersey, USA.
[Wiggins, 2003]
Wiggins, S., Oct. 2003. Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd Edición. Texts in Applied Mathematics. Springer.
[Wong, 1997]
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.
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