Rate-cost tradeoffs in scalar LQG control and tracking with side information

Consider a control problem in which a remote controller chooses its control action based on two kinds of information about the system state: the information it receives from the system via a rate-constrained feedback link, and side information - a noisy measurement of the system state it observes directly. The goal of the controller is to minimize a quadratic cost function in the state variables and control signal, known as the linear quadratic regulator (LQR). We study the fundamental tradeoff between the communication rate, the expected cost b and the quality of side information. Due to a separation principle between estimation and control, we focus on the tracking problem, where the goal is to track the system state rather than to control it. We introduce the causal rate-distortion function with side information at the decoder. It is expressed in terms of directed mutual information, and it extends the classical (noncausal) Wyner-Ziv rate-distortion function to real-time tracking problems with causality constraints and memory of the past at both encoder and decoder. We compute that function in the scalar linear Gaussian setting; we draw a link with the Kalman filter; we show that making side information available also at the encoder does not help to improve the optimal tradeoffs