Infinite horizon LQG control with fixed-rate quantization for scalar systems

We study the infinite-horizon LQG control systems with the constraint that the measurement signal is quantized by a fixed-rate quantizer before going into the controller. It has been shown recently that only weak separation principle holds for the LQG control system with communication channels. In this paper, we study the problem of quantized LQG for a scalar system. An adaptive fixed-rate quantizer is designed to achieve the mean-square stability and the good long term average performance. The long term average cost is divided into two parts. The first part depends on the classical LQG cost, and the second part depends on the distortion of the quantizer. For a quantizer with a fixed bit rate of R (per sample), we show that the quantization distortion order is R2^-2R for a large R.