Lack of Separation Principle for Quantized Linear Quadratic Gaussian Control

This technical note studies the quantized linear quadratic Gaussian (LQG) control problem which is generalized from the classical LQG control but with the constraint that the feedback signal is quantized with a fixed bit rate. We show that state feedback control, state estimation and quantization can not be fully separated in general. Only a weak separation principle holds which converts the quantized LQG control problem into a quantized state estimation problem. Further separation of estimation and quantization is not possible in general. A concrete example is provided to demonstrate this fact. It is also shown that the so-called "whitening" approach to quantized state estimation is not optimal.