Solutions to matrix equation and its application in control

It is well-known that the matrix equation plays an important role in stability analysis and controller design in linear systems. In this article, we develop a Smith normal form reduction method for computing the solution to pair (X, Y) of the nonhomogeneous matrix equation XB — AX = CY + R. The closed-form solutions to the nonhomogeneous matrix equation are presented. Finally, a numerical applied example is provided to support the theoretical findings and to testify the effectiveness and usefulness of the developed algorithm.