A Fully Probabilistic Design for Stochastic Systems with Input Delay

This paper is concerned with the regulation problem of discrete-time stochastic systems involving input delays which is relevant to networked control systems. The problem is formulated in a fully probabilistic framework, and the control solution is obtained by minimising the Kullback–Leibler Divergence (KLD) between the actual and desired joint probability density functions of the system dynamics. A closed-form solution for the randomised controller is obtained for stochastic systems that can be described by arbitrary probability density functions. Furthermore, the analytic solution for a class of linear Gaussian stochastic systems is obtained. For this class of systems, the optimal randomised controller is shown to be a state feedback controller which is modified by an extra linear term that is related to the lagged and future control inputs. The developed method is demonstrated on a simulation example, and the results are compared with the standard fully probabilistic design control method.