Optimal and Robust Sliding Mode Regulator for Linear Systems with Delayed Control
Regulador Óptimo y Robusto con Modos Deslizantes para Sistemas Lineales con Control Retardado
Michael Basin1, Jesus Rodríguez Gonzalez1, Pedro Acosta2 and Leonid Fridman3
1 Autonomous University of Nuevo Leon, México
mbasin@fcfm.uanl.mx, jgrg17@yahoo.com.mx
3 National Autonomous University of Mexico, México
lfridman@verona.fi–p.unam.mx
Article received on March 10, 2003; accepted July 08, 2005
Abstract
This paper presents the optimal regulator for a linear system with time delay in control input and a quadratic criterion. The optimal regulator equations are obtained using the duality principle, which is applied to the optimal filter for linear systems with time delay in observations. Performance of the obtained optimal regulator is verified in the illustrative example against the best linear regulator available for linear systems without delays. Simulation graphs and comparison tables demonstrating better performance of the obtained optimal regulator are included. The paper then presents a robustification algorithm for the obtained optimal regulator based on integral sliding mode compensation of disturbances. The general principles of the integral sliding mode compensator design are modified to yield the basic control algorithm oriented to time–delay systems, which is then applied to robustify the optimal regulator. As a result, the sliding mode compensating control leading to suppression of the disturbances from the initial time moment is designed. The obtained robust control algorithm is verified by simulations in the illustrative example.
Keywords: Linear time–delay system, Optimal control, Filtering, Sliding mode regulator.
]]>Resumen
Este artículo presenta el regulador óptimo para un sistema lineal con retardo en la entrada de control y un criterio cuadrático. Las ecuaciones del regulador óptimo se obtienen usando el principio de dualidad, el cual es aplicado al filtro óptimo para sistemas lineales con tiempo de retardo en las observaciones. El desempeño de el regulador óptimo obtenido es verificado en el ejemplo ilustrativo contra el mejor regulador disponible para sistemas lineales sin retardo. Se incluyen las gráficas de simulación mostrando mejor desempeño del regulador óptimo obtenido. El artículo también presenta un algoritmo de robustificación para el regulador óptimo obtenido basado en compensación de perturbaciones con modos deslizantes integrales. Los principios generales del diseño del compensador con modos deslizantes integrales se modifican para dar el algoritmo básico de control orientado a sistemas con retardo en el tiempo. Como resultado, se diseña el control compensador con modos deslizantes llevando a la supresión de las perturbaciones desde el momento del tiempo inicial. El algoritmo de control robusto obtenido es verificado con simulaciones en el ejemplo ilustrativo.
Palabras Clave: Sistemas lineales, Retardo, Control óptimo, Filtrado, Regulador, Modos deslizantes.
DESCARGA ARTICULO EN FORMATO PDF
References
1. Abdallah; K. Gu, Proc. 3rd IFAC Workshop on Time Delay Systems. Edited by C. T., (Madison: OMNIPRESS, 2001). [ Links ]
2. G. Bartolini, W. et al., "Vibration damping in elastic robotic structure via sliding modes," Int. J. Robotic Systems, 14: 675–696 (1997). [ Links ]
3. Basin, M. V.; and Alcorta–Garcia, M. A., "Optimal control for third degree polynomial systems and its automotive application," in Proc. 41st Conf. on Decision and Control, (Las Vegas, NV, December 10–13, 2002), 1745–1750. [ Links ]
4. Basin, M. V. et al., "Optimal and robust integral sliding mode filters over observations with delay," in Proc. 7th International Workshop on Variable Structure Systems, (Sarajevo, July 17–19, 2002), 163–174. [ Links ]
5. Basin, M. V. ; and Martinez–Zuniga, R., Optimal linear filtering over observations with multiple delays, accepted for Proc. American Control Conference 2003, (Denver, CO, June 4–6, 2003), Ref. no. WA05–2. [ Links ]
6. Bellman, R., \emph, Dynamic Programming, (Princeton: University Press, 1957). [ Links ]
7. Boukas, E. K.; and Liu, Z. K., Deterministic and Stochastic Time–Delay Systems, (Birkhauser, 2002). [ Links ]
8. Cao, W. J.; and Xu, J. X., "Nonlinear integral type sliding surfaces for both matched and unmatched uncertain systems," IEEE Trans. Automat. Control, submitted. [ Links ]
9. Delfour, M. C, "The linear quadratic control problem with delays in space and control variables: a state space approach," SIAM. J. Contr. Optim, 24: 835–883 (1986). [ Links ]
10. Linear Time Delay Systems. Edited by Dion \emph, J. M. et al., (London: Pergamon,1999). [ Links ]
11. Drakunov, S. V.; and Utkin, V. I., "Sliding mode control in dynamic systems," Int. J. Control, 55: 1029–1037 (1993). [ Links ]
12. Stability and Control of Time–Delay Systems. Edited by J. L. Dugard ; and E.I. Verriest, (Springer, 1998). [ Links ]
13. Eller, D. H.; Aggarwal, J. K.; and Banks, H. T., "Optimal control of linear time–delay systems," IEEE Trans. Aut. Contr, AC–14: 678–687 (1969). [ Links ]
14. Filippov, A. F., \emph, Differential Equations with Discontinuous Right–Hand Sides, (Kluwer, 1989). [ Links ]
15. Fleming, W. H.; and Rishel, R. W., Deterministic and Stochastic Optimal Control, (New York: Springer, 1975). [ Links ]
16. Fridman, L.; Fridman, E.; and Shustin, E., "Steady modes and sliding modes in relay control systems with delay," in \emph Sliding Mode in Engineering. Edited by J. P. Barbot ; and W. Perruquetti, (New York: Marcel Dekker,2002) 263–294. [ Links ]
17. Fridman, E.; Fridman, L.; and Shustin, E., "Steady modes in the relay control systems with delay and periodic disturbances," ASME Journal of Dynamical Systems Measurement and Control, 122 (4): 732–737 (2000). [ Links ]
18. Fridman, L.; Acosta, P.; and Polyakov, A., "Robust eigenvalue assignment for uncertain delay control systems," in Proc. 3rd IFAC Workshop on Time Delay Systems, (Santa Fe, NM, December 10–12, 2001), 239–244. [ Links ]
19. Kharatashvili, G. L., "A maximum principle in external problems with delays," in Mathematical Theory of Control. Edited by A.V. Balakrishnan and L. W. Neustadt (New York: Academic Press, 1967). [ Links ]
20. Kolmanovskii, V. B.; and Shaikhet, L. E., Control of Systems with Aftereffect, (Providence: American Mathematical Society, 1996). [ Links ]
21. Kolmanovskii, V. B.; and Myshkis, A. D., Introduction to the Theory and Applications of Functional Differential Equations, (New York: Kluwer, 1999). [ Links ]
22. Kwakernaak, H.; and Sivan, R., Linear Optimal Control Systems, (New York: Wiley–Interscience, 1972). [ Links ]
23. Li, X.; and Yurkovitch, S., "Sliding Mode Control of Systems with Delayed States and Controls," in Variable Structure Systems, Sliding Mode and Nonlinear Control. Edited by K.D. Young ; and U. Ozguner, \emph (Lecture Notes in Control and Information Sciences), (Berlin: Springer Verlag, 1999) 247: 93–108. [ Links ]
24. Mahmoud, M. S., Robust Control and Filtering for Time–Delay Systems, (Marcel Dekker, 2000). [ Links ]
25. Malek–Zavarei, M.; and Jamshidi, M., Time–Delay Systems: Analysis, Optimization and Applications, (North–Holland, Amsterdam, 1987). [ Links ]
26. Oguztoreli, M. N., Time–Lag Control Systems, (New York: Academic Press, 1966). [ Links ]
27. Oguztoreli, M. N., "A time optimal control problem for systems described by differential difference equations," SIAM J. Control, 1: 290–310 (1963). [ Links ]
28. Orlov, Y. et al., "On identifiability of linear time–delay systems," IEEE Trans. Automat. Control, 47: 1319–1324 (2002). [ Links ]
29. Pontryagin, L. S. et al., The Mathematical Theory of Optimal Processes, (New York: Interscience, 1962). [ Links ]
30. Poznyak, A. S., "Stochastic output noise effects in sliding mode state observation," Int. J. Control (2003), in print. [ Links ]
31. Richard, J. P.; Gouaisbaut, F.; and Perruquetti, W., "Sliding mode control in the presence of delay," Kybernetica, 37: 277–294 (2001). [ Links ]
32. Roh, Y. H. ; and Oh, J. H., "Robust stabilization of uncertain input delay systems by sliding mode control with delay compensation," Automatica, 37: 1861–1865 (1999). [ Links ]
33. Nguang, S. K., "Comments on "Robust stabilization ofuncertain input delay systems by sliding mode control with delay compensation," Automatica, 37: 1677 (2001). [ Links ]
34. Shtessel, Yu B.; Zinober, A. S. I.; and Shkolnikov, I., "Sliding mode control using method of stable system centre for nonlinear systems with output delay," in Proc.41st Conf. On Decision and Control (Las Vegas, NV, December 10–13, 2002), 993–998. [ Links ]
35. Uchida, K. et al., "The linear–quadratic optimal control approach to feedback control design for systems with delay," Automatica, 24: 773–780 (1988). [ Links ]
36. Utkin, V. I.; Guldner, J.; and Shi, J., Sliding Mode Control in Electromechanical Systems, (London: Taylor and Francis, 1999). [ Links ] ]]>