State Boundedness in Discrete-Time Sliding Mode Control

Abstract

Discrete-time sliding mode control strategies are an excellent way of ensuring robustness of the plant with respect to disturbance and model uncertainties. Such strategies mitigate the effect of perturbations on the system by confining its representative point to a specific, narrow quasi-sliding mode band defined in the state space. Sliding mode controller design is typically focused on ensuring a specific evolution of the sliding variable, which is a constructed output of the plant. On the other hand, such strategies give no explicit information about the evolution of individual state variables. Motivated by this problem, in this chapter, we have calculated ultimate bounds of each state variable during system sliding motion. In particular, we have considered conventional sliding variables with relative degree one as well as more sophisticated ones with arbitrary relative degrees. The obtained bounds of all state variables are applicable to any discrete-time sliding mode control strategy as long as its quasi-sliding mode band width is known. Moreover, we have analyzed three specific cases in which state error can be further reduced: the case of matched perturbations, dead-beat sliding hyperplane, and switching-type quasi-sliding motion.