A solution of polynomial matrix equations using extended division algorithm and another description of all-stabilizing controllers

In this paper, an extended division algorithm for polynomial matrices is proposed. This algorithm can be executed using a state-space representation of polynomial matrix fractional description. Using the proposed algorithm, the solution methods of some polynomial matrix equations will be shown. Then, the Diophantine equation which can be used in the design of pole placement systems will be considered. The degrees of freedom in this method are then shown to be equivalent to the freedom of Youla parametrization. The presented method is then another description of all-stabilizing controllers. This description is especially useful for optimization after assigning the pole location of closed-loop systems. Exact model matching for a system with measurement noise is considered as an application of the proposed all-stabilizing controllers.