Estimación del dominio de atracción de sistemas no lineales mediante modelos borrosos polinomiales

J.L. Pitarch, A. Sala, C.V. Ariño, F. Bedate

Resumen

La mayor parte de referencias de la literatura en control borroso plantean condiciones LMI para un modelo Takagi-Sugeno y dan por terminado el problema una vez se obtienen resultados factibles. No obstante, dejan sin estudiar la región de atracción obtenida. Este tra-baajo propone probar que una zona, de forma prefijada, lo más grande posible, peertenece al dominio de atracción del origen de un sistema no lineal. Para ello, se usan modelos borrosos polinomiales cuyo análisis puede ser llevado a cabo mediante optimización convexa (su-mas de cuadrados). Asimismo, se utiliza información de la forma de las funciones de pertenencia para realizar iteraciones con la región de modelado borroso, maximizando la región de atracción probada, lo cual reduce el conservadurismo sobre otras propuestas.

Palabras clave

Función de Lyapunov; dominio de atracción; sistemas borrosos; Takagi-Sugeno; sistemas polinomiales; estabilidad local; sumas de cuadrados; conservadurismo

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Referencias

Amato, F., F. Calabrese, C. Cosentino, y A. Merola. 2011. «Stability analysis of nonlinear quadratic systems via polyhedral Lyapunov functions». Automatica 47: 614-617.

Boyd, Stephen. 1994. Linear matrix inequalities in system and control theory. Philadelphia: Society for Industrial and Applied Mathematics.

Chesi, Graziano. 2007. «On the Gap Between Positive Polynomials and SOS of Polynomials». IEEE Transactions on Automatic Control 52 (6) (Junio): 1066-1072. doi:10.1109/TAC.2007.899083.

Guerra, T. 2004. «LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi–Sugeno’s form». Automatica 40 (5) (Mayo): 823-829. doi:10.1016/j.automatica.2003.12.014.

Ichihara, Hiroyuki. 2008. State feedback synthesis for polynomial systems with bounded disturbances. En 2008 47th IEEE Conference on Decision and Control, 2520-2525. Cancun, Mexico. doi:10.1109/CDC.2008.4738610. http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=4738610.

Jarvis-Wloszek, Z., R. Feeley, W. Tan, K. Sun, y A. Packard. 2005. «Control applications of sum of squares programming». Positive Polynomials in Control. Lecture Notes in Control and Information Sciences. Springer Berlin / Heidelberg.

Khalil, Hassan. 2002. Nonlinear systems. 3o ed. Upper Saddle River N.J.: Prentice Hall.

Lofberg, J. 2009. «Pre- and Post-Processing Sum-of-Squares Programs in Practice». IEEE Transactions on Automatic Control 54 (Mayo): 1007- 1011. doi:10.1109/TAC.2009.2017144.

Luenberger, David. 2008. Linear and nonlinear programming. New York: Springer.

Neerhoff, F. L., y P. van der Kloet. 2001. The characteristic equation for timevarying models of nonlinear dynamic systems. En Proc. ECCTD, 28–31.

Papachristodoulou, A., y S. Prajna. 2002. On the construction of Lyapunov functions using the sum of squares decomposition. En Decision and Control, 2002, Proceedings of the 41st IEEE Conference on, 3:3482– 3487.

Pitarch, J. L., C. V. Ariño, F. Bedate, y A. Sala. 2010. Local fuzzy modeling: Maximising the basin of attraction. En International Conference on Fuzzy Systems, 1-7. Barcelona, Spain. doi:10.1109/FUZZY.2010.5584617. http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=5584617.

Pitarch, Jose Luis, Carlos Ariño, y Antonio Sala. 2011. Estimating domains of attraction of fuzzy polynomial systems. En, 680-685. Advances in Intelligent Systems Research. Aix-les-Bains: Atlantis Press. doi:10.2991/eusflat.2011.35. http://www.atlantis-press.com/php/paperdetails.php?id=2220.

Prajna, S., A. Papachristodoulou, P. Seiler, y P. A Parrilo. 2004a. SOSTOOLS: Control applications and new developments. En Computer Aided Control Systems Design, 2004 IEEE International Symposium on, 315–320.

Prajna, S., A. Papachristodoulou, P. Seiler, y P. A Parrilo. 2004b. Sum of Squares Optimization Toolbox for MATLAB User’s guide. Citeseer.

Prajna, S., A. Papachristodoulou, y P.A. Parrilo. 2002. Introducing SOSTOOLS: a general purpose sum of squares programming solver. En Proceedings of the 41st IEEE Conference on Decision and Control, 2002., 741-746. Las Vegas, NV, USA. doi:10.1109/CDC.2002.1184594. http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=1184594.

Prajna, S., A. Papachristodoulou, y F. Wu. 2004. Nonlinear control synthesis by sum of squares optimization: A Lyapunov-based approach. En Control Conference, 2004. 5th Asian, 1:157–165.

Reznick, B. 2000. «“Some concrete aspects of hilbert’s 17th problem». Real algebraic geometry and ordered structures: AMS Special Session on Real Algebraic Geometry and Ordered Algebraic Structures held at Louisiana State University, Baton Rouge, LA, April 17-21, 1996: Special Semester on Real Algebraic Geometry and Ordered Structures held at Louisiana State University and Southern University, Baton Rouge, LA, January-May 1996 253: 251.

Sala, A. 2007. Reducing the gap between fuzzyand nonlinear control (invited talk). En, 1-6. Valenciennes, France. http://personales.upv.es/asala/publics/papers/C81AFNC07PLEN.pdf.

Sala, A. 2008. Introducing shape-dependent relaxed conditions in fuzzy control of nonlinear systems in Takagi-Sugeno form. En Fuzzy Systems, 2008. FUZZ-IEEE 2008.(IEEE World Congress on Computational Intelligence). IEEE International Conference on, 512–517.

Sala, A. 2009. «On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems». Annual Reviews in Control 33 (1): 48–58.

Sala, A., y C. Ariño. 2009. «Polynomial Fuzzy Models for Nonlinear Control: A Taylor Series Approach». IEEE Transactions on Fuzzy Systems 17 (6) (Diciembre): 1284-1295. doi:10.1109/TFUZZ.2009.2029235.

Sala, A., y C. V. Ariño. 2006. Local stability of open-and closed-loop fuzzy systems. En Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control, 2006 IEEE, 2384–2389.

Sala, A., y T.M. Guerra. 2008. Stability analysis of fuzzy systems: membership-shape and polynomial approaches. En Proc. IFAC World Congress, 5605–5610. Seoul, Korea.

Takagi, T., y M. Sugeno. 1985. «Fuzzy identification of systems and its applications to modeling and control». IEEE transactions on systems, man, and cybernetics 15(1) (Febrero): 116–132.

Tanaka, K., H. Ohtake, y H.O. Wang. 2009. «Guaranteed Cost Control of Polynomial Fuzzy Systems via a Sum of Squares Approach». IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics) 39 (2) (Abril): 561-567. doi:10.1109/TSMCB.2008.2006639.

Tanaka, K., H. Yoshida, H. Ohtake, y H. O Wang. 2007a. A sum of squares approach to stability analysis of polynomial fuzzy systems. En American Control Conference, 2007. ACC’07, 4071–4076.

Tanaka, K., H. Yoshida, H. Ohtake, y H. O Wang. 2007b. Stabilization of polynomial fuzzy systems via a sum of squares approach. En Intelligent Control, 2007. ISIC 2007. IEEE 22nd International Symposium on, 160–165.

Tanaka, Kazuo, y Hua O. Wang. 2001. Fuzzy control systems design and analysis : a linear matrix inequality approach. New York: Wiley.

Toh, K. C, M. J Todd, y R. H Tutuncu. 1999. «SDPT3—a Matlab software package for semidefinite programming». Optimization Methods and Software 11 (12): 545–581.

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

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1. Inescapable-set Estimation for Nonlinear Systems with Non-vanishing Disturbances
J.L. Pitarch, A. Sala, F. Bedate, C.V. Ariño
IFAC Proceedings Volumes  vol: 46  num.: 20  primera página: 462  año: 2013  
doi: 10.3182/20130902-3-CN-3020.00067



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