Conditions for optimality of naïve quantized finite horizon control

This paper presents properties of a control law which quantizes the unconstrained solution to a unitary horizon quadratic programme. This naïve quantized control law underlies many popular algorithms, such as ΣΔ-converters and decision feedback equalities, and is easily shown to be globally optimal for horizon one. However, the question arises as to whether it is also globally optimal for horizons greater than one, i.e. whether it solves a finite horizon quadratic programme, where decision variables are restricted to belonging to a quantized set. By using dynamic programming, we develop sufficient conditions for this to hold. The present analysis is restricted to first order plants. However, this case already raises a number of highly non-trivial issues. The results can be applied to arbitrary horizons and quantized sets, which may contain a finite or an infinite (though countable) number of elements.