Stabilization of Networked Multi-Input Systems With Channel Resource Allocation

Abstract

In this paper, we study the problem of state feedback stabilization of a linear time-invariant (LTI) discrete-time multi-input system with imperfect input channels. Each input channel is modeled in three different ways. First it is modeled as an ideal transmission system together with an additive norm bounded uncertainty, introducing a multiplicative uncertainty to the plant. Then it is modeled as an ideal transmission system together with a feedback norm bounded uncertainty, introducing a relative uncertainty to the plant. Finally it is modeled as an additive white Gaussian noise channel. For each of these models, we properly define the capacity of each channel whose sum yields the total capacity of all input channels. We aim at finding the least total channel capacity for stabilization. Different from the single-input case that is available in the literature and boils down to a typical <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> or <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> optimal control problem, the multi-input case involves allocation of the total capacity among the input channels in addition to the design of the feedback controller. The overall process of channel resource allocation and the controller design can be considered as a case of channel-controller co-design which gives rise to modified nonconvex optimization problems. Surprisingly, the modified nonconvex optimization problems, though appear more complicated, can be solved analytically. The main results of this paper can be summarized into a universal theorem: The state feedback stabilization can be accomplished by the channel-controller co-design, if and only if the total input channel capacity is greater than the topological entropy of the open-loop system.