How good is quantized model predictive control with horizon one?

Model Predictive Control is increasingly being used in areas where decision variables are constrained to finite or countably infinite sets. Well known fields include Power Electronics, Signal Processing, and Telecommunications. Typically, the applications utilize high speed sampling and, thus, there is an incentive to reduce computational burden. One way of achieving this is to use small optimization horizons. This raises the question as to the optimality and performance of control laws with short horizons. In this paper, we give necessary and sufficient conditions for horizon one quantized model predictive control to be equivalent to the use of larger horizons. We also explore situations where horizon one is near optimal.