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Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 2452746, 9 pages
Research Article

Recursive Reduced-Order Algorithm for Singularly Perturbed Cross Grammian Algebraic Sylvester Equation

1Electrical and Computer Engineering Department, Rutgers University, New Brunswick, NJ, USA
2Electrical Engineering Department, College of Engineering, University of Basrah, Basrah, Iraq

Received 10 July 2016; Accepted 24 October 2016

Academic Editor: Guangming Xie

Copyright © 2016 Intessar Al-Iedani and Zoran Gajic. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A new recursive algorithm is developed for solving the algebraic Sylvester equation that defines the cross Grammian of singularly perturbed linear systems. The cross Grammian matrix provides aggregate information about controllability and observability of a linear system. The solution is obtained in terms of reduced-order algebraic Sylvester equations that correspond to slow and fast subsystems of a singularly perturbed system. The rate of convergence of the proposed algorithm is , where is a small singular perturbation parameter that indicates separation of slow and fast state variables. Several real physical system examples are solved to demonstrate efficiency of the proposed algorithm.