Control with time-varying feedback communication constraints - the case of dimension greater than one

Recently, there has been a great deal of interest in the so called data-rate inequality which establishes of tight bound on the closed-loop data rate required to stabilize an open-loop unstable linear feedback system. An important feature of the theory behind this bound is that there is complete freedom to trade off coarseness in temporal quantization against coarseness in assigning control levels (say by using A/D conversion). In recent work by the authors, it has been shown that for feedback systems that must operate using data-rate-constrained feedback loops in which the severity of the constraints varies with time (as in congestion prone data-networks) there are significant differences in how well control designs with different numbers of quantization levels will perform. In particular, we have shown in the case of scalar systems that binary control (i.e. two-level control, as would result from 1bit A/D conversion) presents the most robust control quantization under data rate constraints imposed by time-varying congestion on the feedback communication channel. The aim of the present paper is to extend this result to systems of higher dimension. Systems with distinct unstable modes are treated, with a continued emphasis on applications where there are uncertain delays between the generation of control commands by the controller and their actual application by the plant.