A Fully Probabilistic Design for Tracking Control for Stochastic Systems With Input Delay

This article studies model reference adaptive control (MRAC) for a class of stochastic discrete time control systems with time delays in the control input. In particular, a unified fully probabilistic control framework is established to develop the solution to the MRAC, where the controller is the minimizer of the Kullback-Leibler divergence between the actual and desired joint probability density functions of the tracking error and the controller. The developed framework is quite general, where all the components within this framework, including the controller and system tracking error, are modeled using probabilistic models. The general solution for arbitrary probabilistic models of the framework components is first obtained and then demonstrated on a class of linear Gaussian systems with time delay in the main control input, thus obtaining the desired results. The contribution of this article is twofold. First, we develop a fully probabilistic design framework for MRAC, referred to as model reference fully probabilistic design, for stochastic dynamical systems. Second, we establish a systematic pedagogic procedure that is based on deriving explicit forms for the required predictive distributions for obtaining the causal form of the randomized controller when input delays are present.