On Linear Coding Schemes for Stabilizing LTI Control with Multiple Sensors

We examine the problem of designing the encoding and control policies of a linear stochastic control system, where the communication channel between the plant state observer (sensor) and the controller is a lossy wireless channel that is constrained in terms of transmit power and bandwidth. For a first-order ARMA modeled plant with Gaussian statistics, when there are two sensors observing the plant, nonlinear encoding is shown to result in smaller cost at time instant T = 1 compared to the linear schemes, if transmissions are carried out over parallel Gaussian independent channels. In this paper, optimal linear coding schemes for the case of multiple sensors are examined. They are shown to minimize the control cost at the infinite time horizon, when the wireless channel is accessed using time division multiplexing. Our analysis is carried out for when separation between the state estimation and control is possible, and the optimal steady state control law is certainty equivalent. The distortion lower bound for estimating the plant state is derived, along with the necessary conditions on the transmit power that minimize the steady state control cost. We also propose a linear scheme that reaches the distortion bound asymptotically under relaxed conditions.