MIMO Control Over Additive White Noise Channels: Stabilization and Tracking by LTI Controllers

In this paper, we study the limitations in stabilization and tracking of multi-input, multi-output (MIMO) networked feedback systems. We adopt a parallel additive white noise (AWN) model for MIMO communication channels, and consider as our performance measure the mean square error for a system's output to track in the mean square sense a random reference signal with finite power. We derive necessary and sufficient conditions for the system to be mean square stabilizable and obtain analytical expressions of the optimal performance achievable by linear time-invariant (LTI) controllers subject to channel input power constraint. We show that the AWN channel power constraint imposes fundamental limits on the system's stabilizability and tracking performance, which depend on the unstable poles and nonminimum phase zeros of the system. In particular, for MIMO systems, these limits are seen to be dependent on the directions of the unstable poles and nonminimum phase zeros, and especially in how these directions are aligned with noise power distribution; in order to achieve the optimal tracking performance, the channel input power must be allocated to individual channels in ways accounting for pole/zero directions, a scheme that departs from the Shannon's classical "water-filling" strategy. Channel scalings are investigated as a means of realigning pole/zero directions and redistributing the channel power, which are found to be capable of improving fundamentally a system's stabilizability and tracking performance.