Communication-limited stabilization of linear systems

This paper investigates the problem of stabilizing a linear, discrete-time plant using a digital link with a finite data rate. The plant model is infinite-dimensional and time-varying, with a real-valued output which is zero at negative times and distributed according to a probability density p at time zero. Finite and infinite horizon costs in terms of the m-th output moment are defined and the equations of the optimal, finite horizon coder-controller derived. Asymptotic quantization theory is then used to obtain the solution as the horizon tends to infinity, without needing to explicitly solve the finite horizon problem. It is shown that this limiting coder-controller is optimal with respect to the infinite horizon cost, provided that p satisfies certain technical conditions. This immediately leads to a necessary and sufficient condition for the existence of a coder-controller that takes the m-th output moment to zero asymptotically with time. If the open-loop plant is finite-dimensional and time-invariant, this condition simplifies to an inequality involving the data rate and the open-loop pole with greatest magnitude. Analogous results automatically hold for the related problem of state estimation with a finite data rate.