Convergence characterisation of an iterative algorithm for periodic Lyapunov matrix equations

ABSTRACT

This paper is concerned with convergence characterisation of an iterative algorithm for a class of reverse discrete periodic Lyapunov matrix equation associated with discrete-time linear periodic systems. Firstly, a simple necessary condition is given for this algorithm to be convergent. Then, a necessary and sufficient condition is presented for the convergence of the algorithm in terms of the roots of polynomial equations. In addition, with the aid of the necessary condition explicit expressions of the optimal parameter such that the algorithm has the fastest convergence rate are provided for two special cases. The advantage of the proposed approaches is illustrated by numerical examples.

Keywords:

• Periodic Lyapunov matrix equations
• discrete-time linear periodic systems
• iterative algorithms

Acknowledgments

We are very grateful to anonymous reviewers for their valuable comments and suggestions, which have significantly improve the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Shenzhen Municipal Basic Research Project for Discipline Layout with Project No. JCYJ20170811160715620, and by National Natural Science Foundation of China under Grant Nos. 61822305, 61690210 and 61690212.

Notes on contributors

Ying Zhang

Ying Zhang was born in Jilin Province, P. R. China. She received her M. Eng. degree in Control Theory and Control Engineering from Harbin University of Science and Technology in 2003, and Ph.D. degree in Control Science and Engineering in 2007 from Harbin Institute of Technology. From 2007 to 2010, she was a postdoctoral researcher in Harbin Institute of Technology Shenzhen Graduate School, where she became an assistant professor in 2010, and an associate professor in 2011. Her main research interests include robust control and filter theory, iteration based control methods.

Ai-Guo Wu

Ai-Guo Wu was born in Gong'an County, Hubei Province, P. R. China on September 20, 1980. He received his B. Eng. degree in Automation in July 2002, M. Eng. degree in Navigation, Guidance and Control in July 2004, and Ph.D. degree in Control Science and Engineering in November 2008 all from Harbin Institute of Technology. In October 2008, he joined Harbin Institute of Technology Shenzhen Graduate School, where he is now a professor. Prof. Wu visited City University of Hong Kong from March 2009 to March 2011 as a Research Fellow. His research interests include descriptor systems, conjugate product of polynomials, switched systems. Prof. Wu is a Reviewer for American Mathematical Review. He was an Outstanding Reviewer for IEEE Transactions on Automatic Control. He received the National Natural Science Award (Second Prize) in 2015 from P. R. China, and the National Excellent Doctoral Dissertation Award in 2011 from the Academic Degrees Committee of the State Council and the Ministry of Education of P. R. China. He was supported by the Program for New Century Excellent Talents in University in 2011.

Yu Wang

Yu Wang was born in Anhui Province, P. R. China. She received her M. Eng. degree in Control Science and Engineering from Harbin Institute of Technology Shenzhen Graduate School in 2016. She is currently a Ph.D. student at Harbin Institute of Technology, Shenzhen. Her main research interests include control system with time delay and iteration based control methods.