Optimal quantized feedback control for linear quandratic Gaussian systems with input delay

This paper is concerned with the optimal quantized feedback linear quadratic Gaussian (LQG) control problem for a discrete-time stochastic system with input delay as well as the measurements to be quantized before transmitted to the controller. In this scenario, the system is presented with several choices of quantizers, along with the cost of using each quantizer. The objective is to jointly select the quantizers and synthesize the controller to maintain an optimal balance between control performance and quantization cost. It is shown that this problem can be decoupled into two optimization problems when the innovation signal is quantized instead of state: one for optimal controller synthesis and the other for optimal quantizer selection. More specifically, a necessary and sufficient condition is derived for the optimal control problem based on Pontryagin's maximum principle. On the other hand, the optimal quantizer selection policy is established by dealing with a certain Markov decision process (MDP).