A majorization condition for MIMO stabilizability via MIMO transceivers with pure fading subchannels

This paper aims at characterizing a fundamental limitation on the information constraints required for multi-input networked stabilization. A MIMO communication system is deployed for information exchange between the controller and the plant. The communication system is modeled as a MIMO transceiver, which consists of three parts: an encoder, a MIMO channel consisting of parallel SISO subchannels, and a decoder. We focus on the pure fading subchannels in this paper while the case of pure AWGN subchannels has been discussed in our previous work. Inheriting the spirit of MIMO communication, the number of SISO subchannels in the transceiver is often greater than the number of control inputs to be transmitted. The subchannel capacities are assumed to be fixed a priori. With the encoder/decoder pair at hand, the controller designer gains an additional design freedom on top of the controller, leading to a stabilization problem via coding/control co-design. A necessary and sufficient condition is obtained for the solvability of this coding/control co-design problem given in terms of a majorization type relation. A numerical example is presented to illustrate our results.