Common eigenvector approach to exact order reduction for multidimensional Fornasini–Marchesini state-space models

ABSTRACT

This paper proposes an exact order reduction approach for the multidimensional (n-D) Fornasini–Marchesini (F–M) model by making use of the common eigenvector. Specifically, by introducing the concept of common eigenvectors, sufficient conditions of exact order reductions are developed for an n-D F–M model, which are able to simultaneously deal with n eigenvalues of the system matrices $A_1,\dots ,A_n$ of the n-D F–M model. The obtained results reveal, for the first time, the internal connection between the multiple eigenvalues of the system matrices and the reducibility of the considered n-D F–M model. Then, a corresponding algorithm is proposed to exactly reduce the order of an n-D F–M model as much as possible. Examples are given to illustrate the details as well as the effectiveness of the proposed approach.

Keywords:

• Exact order reduction
• Fornasini–Marchesini model
• common eigenvectors
• eigenvalues

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Dongdong Zhao http://orcid.org/0000-0002-5305-1616

Additional information

Funding

This work was partly supported by the Japan Society for the Promotion of Science (JSPS.KAKENHI15K06072), the Natural Science Foundation of Gansu Province, China (18JR3RA286) and the Fundamental Research Funds for the Central Universities (lzujbky-2018-128).

Notes on contributors

Dongdong Zhao

Dongdong Zhao received the B.Eng. degree from Zhengzhou University, Zhengzhou, China, in 2012, and the M. Eng. degree from Lanzhou University, Lanzhou, China, 2016. He is currently pursuing Dr Eng. degree at Akita Prefectural University. His research interests include multidimensional systems theory and signal processing.

Shi Yan

Shi Yan received the B.S. and M.Eng. degrees from Lanzhou University, Lanzhou, China, in 2001 and 2004, and the Dr Eng degree from Akita Prefectural University, Akita, Japan, in 2010, respectively. From July 2004 to March 2007, he was an Assistant Professor at the School of Information Science and Engineering, Lanzhou University. He was a Visiting Researcher from January 2007 to March 2007 and a Visiting Research Fellow from April 2010 to September 2010 at the Department of Electronics and Information Systems, Akita Prefectural University. He was a Lecturer from April 2007 to April 2012 and is currently an Associate Professor since May 2012 at the School of Information Science and Engineering, Lanzhou University. His research interests include multidimensional system control and signal processing. Dr Yan is a member of the Young Scientists Working Committee, Chinese Association of Automation.

Shinya Matsushita

Shin-ya Matsushita received the B.Sci. and M.Sci. degrees from Shimane University, Shimane, Japan, in 2000 and 2002, respectively, and Dr.Sci. degree from Tokyo Institute of Technology, Tokyo, Japan, in 2005. From April 2005 to March 2006, he was a research fellow at the Department of Mathematical and Computing Sciences, Tokyo Institute of Technology. From April 2006 to March 2009, he was an Assistant Professor at the Department of Information Engineering, Matsue National College of Technology, Shimane, Japan. Since April 2009, he has been with the Faculty of Systems Science and Technology, Akita Prefectural University, Akita, Japan, where he is currently an Associate Professor at the Department of Intelligent Mechatronics. His research interests include fixed-point theory and system control theory.

Li Xu

Li Xu received the B.Eng. degree from Huazhong University of Science and Technology, Wuhan, China, in 1982, and the M.Eng. and Dr.Eng. degrees from Toyohashi University of Technology, Toyohashi, Japan, in 1990 and 1993, respectively. From April 1993 to March 1998, he was an Assistant Professor at the Department of Knowledge-Based Information Engineering, Toyohashi University of Technology. From April 1998 to March 2000, he was a Lecturer at the Department of Information Management, Asahi University, Gifu, Japan. He is currently a Professor at the Department of Intelligent Mechatronics, the Faculty of Systems Science and Technology, Akita Prefectural University, Akita, Japan. His research interests include multidimensional system theory, signal processing and the applications of computer algebra to system theory. Dr. Xu was an Associate Editor from April 2000 to November 2014, and has been an Editorial Board Member since December 2014 for the international journal of Multidimensional Systems and Signal Processing (MSSP).