Invariant distributions of linear systems under finite communication bandwidth feedback

In the paper, we study the asymptotic probabilistic behavior of a system stabilized by finite communication bandwidth feedback control in the form of an essentially symmetric 1-bit control law. It is shown that the state orbits eventually converge to an invariant interval under the proposed coded control law. If the resulting closed-loop system is a Markov transformation, the invariant density is piecewise constant and can be associated with the left eigenvector of a non-negative matrix induced by the transformation. The optimal control law that minimizes an asymptotic expected cost function is also derived when the transformation is a covering.