Resumen de tesis

 

Localización de conjuntos compactos invariantes de sistemas físicos y electromecánicos y sus aplicaciones

 

Localization of Compact Invariant Sets of Physical and Electromechanical Systems and their Applications¹

 

Luis Néstor Coria de los Rios y Konstantin E. Starkov

 

Centro de Investigación y Desarrollo de Tecnología Digital, Instituto Politécnico Nacional Ave. del Parque 1310, Mesa de Otay, Tijuana,B.C, México luis.coria@gmail.com, konst@citedi.mx

 

Artículo recibido el 26/01/2011;
aceptado el 10/08/2011.

 

Resumen

Con la localización de conjuntos compactos invariantes se pretende entender la dinámica a largo plazo de un sistema caótico. En esta investigación se aplica el método de localización por medio de condiciones de extrema y el teorema iterativo para estudiar la dinámica de un modelo de un motor síncrono de imán permanente (PMSM). De esta forma, se determina una localización dada por un elipsoide que es cortada por un paraboloide elíptico y un cilindro. Para los resultados obtenidos se realizan simulaciones numéricas en las que se observan las superficies de localización respecto al atractor caótico del sistema en estudio. Los resultados son contribuciones útiles en el análisis de la dinámica compleja de los sistemas estudiados. La aplicación de resultados de localización corresponde al diseño de dos observadores de Thau para el PMSM.

Palabras clave: Caos, conjuntos compactos invariantes, sistema electromecánico, motor síncrono de imán permanente.

 

Abstract

Localization of compact invariant sets allows understanding the long-time behavior of a chaotic system. In this paper we apply the solution of the conditional extremum problem to the study of a model of a permanent magnet synchronous motor. The localization set is given by a one-parameter set of ellipsoids, crossed by an elliptical paraboloid and a cylinder. This improves the initial ellipsoidal localization. Numerical simulations are made to show the effectiveness of the method. The results are useful for analyzing the complex behavior of the systems under study. The application of the localization results corresponds to the design of two Thau observers for the PMSM.

Keywords: Chaos, compact invariant set, electromechanical system, permanent magnet synchronous motor.

 

DESCARGAR ARTÍCULO EN FORMATO PDF

 

Agradecimientos

El trabajo de tesis doctoral fue realizado en el marco del proyecto SEP-CONACYT 78890 y el proyecto DGEST TIJ-IET-2009-217.

 

Referencias

1. Coria, L.N. & Starkov, K.E. (2009). Bounding a domain containing all compact invariant sets of the permanent-magnet motor system. Communications in Nonlinear Science and Numerical Simulation, 14 (11), 3879-3888.         [ Links ]

2. Fitzgerald, A.E., Kingsley, C, & Umans, S.D. (1990). Electric machinen/. New York: McGraw-Hill.         [ Links ]

3. Gao, Y. & Chau, K.T. (2003). Design of permanent magnets to avoid chaos in PM synchronous. IEEE Transactions on Magnetics, 39 (5), 2995-997.         [ Links ]

4. Hemati, N. (1994). Strange attractors in brushless DC motors. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 41(1), 40-45.         [ Links ]

5. Hemati, N. & Kwatny, H. (1993). Bifurcation of equilibria and chaos in permanent-magnet machines. 32nd IEEE Conference on Decisión and Control, San Antonio, TX, USA, 1, 475-479.         [ Links ]

6. Jing, Z., Yu, C, & Chen, G. (2004). Complex dynamics in a permanent-magnet synchronous motor model. Chaos, Solitons and Fractals, 22(4), 831-848.         [ Links ]

7. Khalil, H.K. (1996). Nonlinear systems (2^nd ed). UpperSaddle River, NJ: Prentice Hall.         [ Links ]

8. Krishchenko, A.P. (1997). Estimation of domains with cycles. Computers & Mathematics with Applications, 34(2-4), 325-332.         [ Links ]

9. Krishchenko, A.P. & Starkov, K.E. (2006). Localization of compact invariant sets of nonlinear systems with applications to the Lanford system. International Journal of Bifurcations and Chaos, 16(11), 3249-3256.         [ Links ]

10. Krishchenko, A.P. & Starkov, K.E. (2006). Localization of compact invariant sets of the Lorenz system. Physics Letters A, 353(5), 383-388.         [ Links ]

11. Krishchenko, A. & Starkov, K. (2007). Estimation of the domain containing all compact invariant sets of a system modelling the amplitude of a plasma instability. Physics Letters A, 367(1-2), 65-72.         [ Links ]

12. Krishchenko, A.P. & Starkov, K.E. (2008). Localization of compact invariant sets of nonlinear time-varying systems. International Journal of Bifurcation and Chaos, 18(5), 1599-1604.         [ Links ]

13. Kuroe, Y. & Hayashi, S. (1989). Analysis of bifurcation in power electronic induction motor drive system. 20^th Annual IEEE Power Electronics Specialists Conference (PESC'89), Milwaukee, Wl, USA, 923-930.         [ Links ]

14. Li, Z., Park, J.B., Joo, Y.H., Zhang, B., & Chen, G. (2002). Bifurcations and chaos in a permanent-magnet synchronous motor. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 49(3), 383-387        [ Links ]

15. Mahboobi, S.H., Shahrokhi, M., & Pishkenari, H.N. (2006). Observer- based control design for three well-known chaotic systems. Chaos, Solitons and Fractals, 29(2), 381-392.         [ Links ]

16. Márquez, H.J. (2003). Nonlinear control systems: analysis and design. Hoboken, N.J.: John Wiley.         [ Links ]

17. Pikovskii, A.S., Rabinovich, M.I., & Trakhtengerts, V.Y. (1978). Onset of stochastic in decay confinement of parametric instability. SovietPhysics-JETP, 47(4), 715-719.         [ Links ]

18. Starkov, K.E. (2007). Estimation of the domain containing all compact invariant sets ofthe optically injected láser system. International Journal of Bifurcations and Chaos, 17(11), 4213-4217.         [ Links ]

19. Starkov, K.E. (2009). Bounding a domain that contains all compact invariant sets of the Bloch system. International Journal of Bifurcations and Chaos. 19(3), 1037-1042.         [ Links ]

20. Starkov, K.E. (2009). Bounds for a domain containing all compact invariant sets of the system describing the laser- plasma interaction. Chaos, Solitons and Fractals, 39(4), 1671-1676.         [ Links ]

21. Starkov, K. & Coria, L.N. (2005). Examples of localization of periodic orbits of polynomial systems. 2005 International Conference Physics and Control, Saint Petersburg, Russia, 606-609.         [ Links ]

22. Starkov, K.E. & Coria, L. (2005). Localization of periodic orbits of polimomial sprott systems with one or two quadratic monomials. Intenational Journal of Nonlinear Sciences and Numerical Simulation, 6(3), 271-277.         [ Links ]

23. Starkov, K.E. & Krishchenko, A.P. (2005). Localization of periodic orbits of polynomial systems by ellipsoidal estimates. Chaos, Solitons and Fractals, 23(3), 981-988.         [ Links ]

24. Starkov, K.E. & Starkov Jr., K., K. (2007). Localization of periodic orbits ofthe Róssier system under variation of its parameters. Chaos, Solutions and Fractals, 33(5), 1445-49.         [ Links ]

25.Strogatz, S.H. (1994). Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Reading, Mass.: Addison Wesley.         [ Links ]

26. Yu, P. & Liao, X. (2006). Globally attractive and positive invariant set of the Lorenz system. International Journal of Bifurcations and Chaos, 16(3), 754-764.         [ Links ]

 

Nota

Resumen extendido de tesis doctoral. Graduado: Luis N. Coria. Director: Konstantin E. Starkov. Fecha de graduación: 18/03/2010

Extended abstract of PhD thesis. Graduated: Luis N. Coria. Advisor: Konstantin E. Starkov. Graduation date: 18/03/2010.