Networked Stabilization of Multi-Input Systems with Channel Resource Allocation

Abstract: In this paper, we study the problem of stabilizing a linear time-invariant discretetime system with information constraints in the input channels. The information constraint in each input channel is modelled as a sector uncertainty. Equivalently, the transmission error of an input channel is modelled as an additive system uncertainty with a bound in the induced norm. We attempt to find the least information required, or equivalently the largest allowable uncertainty bound, in each input channel which renders the stabilization possible. The solution for the single-input case, which gives a typical H∞ optimal control problem, is available in the literature and is given analytically in terms of the Mahler measure or topological entropy of the plant. The main purpose of this paper is to address the multi-input case. In the multi-input case, if the information constraint in each input channel is given a priori, then our stabilization problem turns out to be a so-called μ synthesis problem, a notoriously hard problem. In this paper, we assume that the information constraints in the input channels are determined by the network resources assigned to the channels and they can be allocated subject to a total recourse constraint. With this assumption, the resource allocation becomes part of the design problem and a modified μ synthesis problem arises. Surprisingly, this modified μ-synthesis problem can be solved analytically and the solution is also given in terms of the Mahler measure or topological entropy as in the single-input case.