Parametrization of the regular equivalences of the canonical controller

We study control problems for linear systems in the behavioral framework. Our focus is a class of regular controllers that are equivalent to the canonical controller. The canonical controller is a particular controller that is guaranteed to solve the control problem whenever a solution exists. However, it has been shown that, in most cases, the canonical controller is not regular. The main result of the note is a parametrization of all regular controllers that are equivalent to the canonical controller. The parametrization is then used to solve two control problems. The first problem is related to designing a regular controller that uses as few control variables as possible. The second problem is to design a regular controller that satisfies a predefined input-output partitioning constraint. In both problems, based on the parametrization, we present algorithms for designing the controllers.