An implicit iterative algorithm with a tuning parameter for Itô Lyapunov matrix equations

ABSTRACT

In this paper, an implicit iterative algorithm is proposed for solving a class of Lyapunov matrix equations arising in Itô stochastic linear systems. A tuning parameter is introduced in this algorithm, and thus the convergence rate of the algorithm can be changed. Some conditions are presented such that the developed algorithm is convergent. In addition, an explicit expression is also derived for the optimal tuning parameter, which guarantees that the obtained algorithm achieves its fastest convergence rate. Finally, numerical examples are employed to illustrate the effectiveness of the given algorithm.

KEYWORDS:

• Iterative algorithms
• discrete Lyapunov matrix equations
• Smith iteration
• convergence rate

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 61603111]; Guangdong Natural Science Foundation [grant number 2017A030313340]; Shenzhen Municipal Basic Research Project for Discipline Layout [project number JCYJ20170413112722597]; Shenzhen Municipal Project for Basic Research [project number JCYJ20170307150227897].

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 post-doctoral 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 Institue 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 Ounstanding 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.

Hui-Jie Sun

Hui-Jie Sun was born in Inner Mo. He recieved the B. Eng. degree in Automation in 2011, M. Eng. degree in Control Science and Engineering from Harbin Institute of Technology Shenzhen Graduate School, Shenzhen, China, in 2014. He is currently pursuing the Ph. D. degree in Harbin Institute of Technology Shenzhen Graduate School. His research interest covers system identification, stochastic systems, and iterative algorithms.