Parametric solutions to the generalized discrete Sylvester matrix equation MXN - X = TY and their applications

In this paper, an explicit, analytical and complete solution to the generalized discrete Sylvester matrix equation M X N − X = T Y which is closely related with several types of matrix equations in control theory is obtained. The proposed solution has a neat and elegant form in terms of the Krylov matrix, a block Hankel matrix and an observability matrix. Based on the proposed solution, an explicit solution to the general discrete Lyapunov matrix equation is also derived. As an application, the parametric pole assignment for descriptor linear systems by proportional-plus-derivative state feedback is considered. The results presented here are parallel to our earlier results on the generalized Sylvester matrix equation AX − X F = BY .