Abstract

In this paper, sliding mode control for discrete time systems with stochastic noise in their input channel has been discussed. The idea of process control using control charts has influenced this new approach towards dealing with systems with stochastic noise. The new approach approximates the stochastic noise as a bounded uncertainty, similar to having bounds in the control charts for stochastic process control data. For discrete time systems, this results in a bounded stability in probability of the quasi sliding mode, which is referred to as the N-sigma bounded stability. The probability associated with the stability notions is not fixed and the control engineer may desire lower or higher degrees of stability in terms of this probability. Thus one has design flexibility while implementing the theory in practice, where one might have to adjust the desired degree of stability due to hardware limitations.

Introduction

Stochastic systems have been finding quite a lot of interest in the control community over the years. Researchers have attempted to develop the theory and control for stabilization of such systems in both continuous time and discrete time [1,2,4–۸]. Several approaches have been taken by researchers, which can be broadly separated into their dealing of the system dynamics using ordinary difference [5–۸] in case of discrete time systems and stochastic differential [1,2,4] in case of continuous time systems. All of them have been able to achieve either a notion of stability with certain probability [7,8] or have been able to assign the mean and covariance to them [6]. Some have proposed mean square stability or stochastic stability of the system [1,2,5]. Such and other stability ideas had been discussed in [13] in details.