Optimal Tracking Control and Stabilization for Stochastic Systems with Multi-Step Input Delay

Abstract

This paper mainly investigates the linear quadratic tracking (LQT) control problems for the discrete-time systems with multiplicative noises and input delays. Both finite-horizon and infinite-horizon cases are considered. As for the finite-horizon case, an equivalent augmented problem is constructed with an augmented state which consists of the system state and the reference trajectory. A necessary and sufficient condition for the existence of the optimal LQT controller is given explicitly from a set of generalized difference Riccati equations (GDRE) by a stochastic maximum principle method. As for the infinite-horizon case, a necessary and sufficient condition for the existence of the stabilization solution is established based on a set of generalized algebraic Riccati equations (GARE). The optimal controller and optimal cost function in infinite-horizon case are expressed explicitly. It is worthy to point out that both the GDRE and GARE have the same dimensions as that of the original system's state.