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Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 4362641, 11 pages
Research Article

Efficient Techniques for Solving the Periodic Projected Lyapunov Equations and Model Reduction of Periodic Systems

Department of Mathematics and Physics, North South University, Dhaka, Bangladesh

Correspondence should be addressed to M. Monir Uddin

Received 8 September 2016; Revised 17 January 2017; Accepted 22 January 2017; Published 14 February 2017

Academic Editor: Masoud Hajarian

Copyright © 2017 Mohammad-Sahadet Hossain and M. Monir Uddin. 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.


We have presented the efficient techniques for the solutions of large-scale sparse projected periodic discrete-time Lyapunov equations in lifted form. These types of problems arise in model reduction and state feedback problems of periodic descriptor systems. Two most popular techniques to solve such Lyapunov equations iteratively are the low-rank alternating direction implicit (LR-ADI) method and the low-rank Smith method. The main contribution of this paper is to update the LR-ADI method by exploiting the ideas of the adaptive shift parameters computation and the efficient handling of complex shift parameters. These approaches efficiently reduce the computational cost with respect to time and memory. We also apply these iterative Lyapunov solvers in balanced truncation model reduction of periodic discrete-time descriptor systems. We illustrate numerical results to show the performance and accuracy of the proposed methods.