Optimal Control and Quantization for Discrete‐Time Stochastic Systems With Input Delay

ABSTRACT This paper mainly considers the optimal quantized feedback linear quadratic Gaussian (LQG) control problem for a discrete‐time stochastic system with input delay. Note that the quantization part of the system consists of several selectable quantizers, the cost of using each quantizer and the quantized information transmitted to the controller. The goal is to jointly select the quantizer and design the controller to maintain the best balance between the control performance and the quantization cost. In this paper, the problem can be divided into two problems: one is the optimal controller synthesis problem, and the other is the optimal quantizer selection problem. More specifically, a sufficient and necessary condition for the optimal control problem is obtained by applying Pontryagin's maximum principle. Besides, the dynamic programming principle is used to find the optimal quantizer selection policy. A numerical example is provided to illustrate our conclusion.