On the recursive reduction of the order of stabilizing controllers for LTI systems

We first prove the following result: if there exists a proper stabilizing controller of order "r", there is a strictly proper stabilizing controller of order "r+1"; moreover, the set of strictly proper stabilizing controllers contains an infinite line segment in the controller parameter space. Using this result, we (1) provide a sufficient condition, based on pole-zero cancellation, for reducing the order of a stabilizing controller and (2) show that the minimal order of a proper stabilizing controller is "r" if and only if the following two conditions hold: (a) the set of rational, strictly proper stabilizing controllers of order "r" is bounded (can even be empty) in the controller parameter space and (b) the set of proper stabilizing controllers of order "r" is not empty. This result holds even for complex stabilization and hence, for the minimal order of stabilizing controllers that guarantee a performance describable through a complex stabilization problem.