MIMO control using MIMO communication: A Majorization condition for networked stabilizability

In this paper, we initiate the study of networked stabilization via a MIMO communication scheme between the controller and the plant. Specifically, the communication system is modeled as a MIMO transceiver, which consists of three parts: an encoder, a MIMO channel, and a decoder. In the spirit of MIMO communication, the number of SISO subchannels in the transceiver is often greater than the number of data streams to be transmitted. Moreover, the subchannel capacities are assumed to be fixed a priori. In this case, the encoder/decoder pair gives an additional design freedom on top of the controller, leading to a stabilization problem via coding/control co-design. The controller designer needs to design the encoder/decoder pair and the controller jointly so as to stabilize the system. We arrive at a necessary and sufficient condition on the subchannel capacities under which the coding/control co-design problem is solvable. Quite surprisingly, the condition is given in terms of a majorization type relation. As we go along, a systematic procedure is also put forward to perform the coding/control co-design. A numerical example is presented to illustrate our results.