Metodología formal de análisis del comportamiento dinámico de sistemas no lineales mediante lógica borrosa

  1. Antonio Javier Barragán 1
  2. Basil Mohammed Al-Hadithi 2
  3. José Manuel Andújar 1
  4. Agustín Jiménez 2
  1. 1 Universidad de Huelva
    info

    Universidad de Huelva

    Huelva, España

    ROR https://ror.org/03a1kt624

  2. 2 Universidad Politécnica de Madrid
    info

    Universidad Politécnica de Madrid

    Madrid, España

    ROR https://ror.org/03n6nwv02

Journal:
Revista iberoamericana de automática e informática industrial ( RIAI )

ISSN: 1697-7920

Year of publication: 2015

Volume: 12

Issue: 4

Pages: 434-445

Type: Article

DOI: 10.1016/J.RIAI.2015.09.005 DIALNET GOOGLE SCHOLAR lock_openOpen access editor

More publications in: Revista iberoamericana de automática e informática industrial ( RIAI )

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.

Bibliographic References

  • Abraham, 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 Avello, A., Matía, F., Sep. 2012. 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.1014
  • Andújar, J. M., Aroba, J., Torre, M. L. d. l., Grande, J. A., Jan. 2006. 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/s00254-005-0103-2
  • Andújar, J. M., Barragán, A. J., Sep. 2005. A methodology to design stable nonlinear fuzzy control systems. Fuzzy Sets and Systems 154 (2), 157–181. DOI: 10.1016/j.fss.2005.03.006
  • 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, 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
  • Andújar, J. M., Barragán, A. J., Gegúndez, M. E., Oct. 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
  • Andújar, J. M., Bravo, J. M., Mar. 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.2004.03.023
  • Andújar, J. M., Bravo, J. M., Peregrín, A., Dec. 2004. Stability analysis and synthesis of multivariable fuzzy systems using interval arithmetic. Fuzzy Sets and Systems 148 (3), 337–353. DOI: 10.1016 issn = 0165-0114,/j.fss.2004.01.008
  • Angelov, P., Buswell, R., Oct. 2002. Identification of evolving fuzzy rule-based models. IEEE Transactions on Fuzzy Systems 10 (5), 667–677. DOI: 10.1109/TFUZZ.2002.803499
  • Angelov, P. P., Filev, D. P., Feb. 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.2003.817053
  • Aroba, J., Grande, J. A., Andújar, J. M., De La Torre, M. L., Riquelme, J. C., Sep. 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/s00254-006-0627-0
  • 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/FUZZY.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., May 2014. A general methodology for online TS fuzzy modeling by the extended kalman filter. Applied Soft Computing 18 (0), 277––289. DOI: 10.1016/j.asoc.2013.09.005
  • 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
  • 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, M. A., Palis, F., Zeghbib, A. H., Jun. 2007. Modeling and control of nonlinear systems using soft computing techniques. Applied Soft Computing 7 (3), 728–738. DOI: 10.1016/j.asoc.2005.12.005
  • Grande, J. A., Andújar, J. M., Aroba, J., De La Torre, M. L., Beltrán, R., Apr. 2005. Precipitation, pH and metal load in AMD river basins: An application of fuzzy clustering algorithms to the process characterization. Journal of Environmental Monitoring 7 (4), 325–334. DOI: 10.1039/b410795k
  • Horikawa, S.-I., Furuhashi, T., Uchikawa, Y., Sep. 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
  • Jang, J.-S. R., May 1993. ANFIS: adaptive-network-based fuzzy inference system. IEEE Transactions on Systems, Man, and Cybernetics 23 (3), 665–685. DOI: 10.1109/21.256541
  • Jiménez, A., Aroba, J., de la Torre, M. L. d. l., Andújar, 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 2 (11), 147–153. DOI: 10.2166/hydro.2009.015
  • Kosko, B., Nov. 1994. Fuzzy systems as universal approximators. IEEE Transactions on Computers 43 (11), 1329–1333. DOI: 10.1109/12.324566
  • Kreinovich, V., Hguyen, H. T., Yam, Y., Jun. 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;2-0
  • Levenberg, K., 1944. A method for the solution of certain problems in least squares. En: Quart. Appl. Math. Vol. 2. pp. 164–168.
  • López-Baldán, M. J., García-Cerezo, A., Cejudo, J. M., Romero, A., Apr. 2002. Fuzzy modeling of a thermal solar plant. International Journal of Intelligent Systems 17 (4), 369–379. DOI: 10.1002/int.10026
  • Marquez, H. J., 2003. Nonlinear control systems. Analysis and design. John Wiley & Sons, Inc.
  • Mencattini, A., Salmeri, M., Salsano, A., Aug. 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.2004.841742
  • Moré, 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 15 (1), 116–132.
  • 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, 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