SYSTEM CONTROL IN CONDITIONS OF DISCRETE STOCHASTIC INPUT PROCESS

Abstract

In this article, a mathematical model of control for a discrete stochastic system, which can also be power supply system, is presented. Analytical approaches that can be used to describe the mutual impact of output and stocks (additional capacities) on hierarchically distributed occurrence/usage/variation or demand already exist. We add dynamics to the system. For this, we use discrete time functions, i.e. dynamic processes, which are of a random (stochastic) form, since our goal is to describe actual, concrete system as accurately as possible. We build a dynamic discrete model of control for this system with a system of difference equations, which can be solved with a one-part z-transform. Due to the stochastic inputs of the system, we can use a Wiener filter for discrete random processes to meet the requirement of optimality. With a z-transform, we convert the system of difference equations in a real time zone into system of algebraic equations in a complex area. First, we derive the Wiener-Hopf equation in a complex area; then we use spectral factorisation to obtain its solution, which has to be transformed back to real time zone with inverse z-transform. This is how we find the optimal solution of a given control problem. At the end of the article, we also demonstrate a numerical example with concrete form of input discrete random process.