Artículos

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

2 Technological Institute of Chihuahua, Mexico
pacosta@itchihuahua.edu.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.

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