Optimal tracking over an additive white noise feedback channel

This paper studies the best achievable tracking performance of multiple-input multiple-output (MIMO), finite dimensional, linear and time-invariant (FDLTI) system over a additive white noise (AWN) channel in the feedback path. The plant under consideration is assumed to be minimum phase but unstable. The reference input is a vector valued wide-sense stationary (WSS) random process. The power of the tracking error is adopted as the measure of the performance and is to be minimized subject to the channel power constraint. We investigate a joint design of the controller and the scaling factor, which can be interpreted as a simple compensation for the channel. It is shown explicitly that the tracking performance, which in a noise-free setting can be made as zero, depends on plant unstable poles and the power constraint of the AWN channel. It is also shown in the sequel that when the constant scaling is not available, the channel is not exploited to the maximum extent allowable under the power constraint, and the tracking performance deteriorates.