On the dynamics of receding horizon linear quadratic finite alphabet control loops

This paper addresses the issue of dynamics of receding horizon linear quadratic control with finite input constraint sets. Under special circumstances it is possible to exploit standard tools from model predictive control to establish asymptotic stability. However, in general, the notion of asymptotic stability is too strong when used in the context of finite alphabet control. We are thus led to weaker notions, including that of ultimate boundedness of closed loop trajectories, in which interalia is related to the existence of positively invariant sets. We utilize a recent characterization of finite alphabet receding horizon control to establish a link to parallel literature on predictive analog-to-digital converters. By borrowing ideas from this latter area, we are able to develop several methods for determining positively invariant sets, hence ensuring ultimate boundedness. The efficacy of these methods is illustrated by examples.