State feedback [formula omitted] controllers can be dynamic

Abstract

This paper considers ℓ 1- optimal control problems given by discrete-time systems with full state feedback, scalar control and scalar disturbance. Motivation stems from the central role that this problem structure played in the development of the H 2 and H ∞ theories. First, systems with a scalar regulated output are studied (singular problems). Sufficient conditions are given, based on the non-minimum phase zeros of the transfer function from the control to the regulated output, for the existence of a static ℓ 1- optimal controller. A simple way to compute the static gain is provided, using pole placement ideas. It is shown, however, that having full state information does not prevent the ℓ 1- optimal controller from being dynamic in general, and that examples with arbitrarily high order optimal controllers can be easily constructed. Second, problems with two regulated outputs, one of them being the scalar control, are considered (non-singular problems). It is shown, by means of a class of fairly general examples for which exact ℓ 1- optimal solutions are constructed, that such problems may not have static controllers that are ℓ 1- optimal , thus concluding that a ‘separation structure’ does not occur in these problems in general.