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IMA Journal of Mathematical Control and Information Advance Access originally published online on January 19, 2009
IMA Journal of Mathematical Control and Information 2009 26(1):59-78; doi:10.1093/imamci/dnn013
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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

Bin Zhou{dagger} and Guang-Ren Duan

Center for Control Theory and Guidance Technology, Harbin Institute of Technology, PO Box 416, Harbin 150001, Heilongjiang, People's Republic of China

{dagger} Email: binzhou{at}hit.edu.cn

Received on September 9, 2006; Revision received October 25, 2006. Accepted on November 8, 2006

In this paper, an explicit, analytical and complete solution to the generalized discrete Sylvester matrix equation MXN – X = TY 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 – XF = BY.

Keywords: generalized discrete Sylvester matrix equations; parametric solutions; controllability and observability; parametric pole assignment; descriptor linear systems.


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