On the pole of non-square transfer function matrix Moore-Penrose pseudo-inverses

• Published in: International journal of systems science Vol. 46; no. 14; pp. 2560 - 2571
• Main Authors: Zhang, Wei, Ou, Linlin, He, Xing, Zhang, Weidong
• Format: Journal Article
• Language: English
• Published: Taylor & Francis 26.10.2015
• Subjects: Binet-Cauchy theorem; Moore-Penrose pseudo-inverse; non-square transfer function matrix; pole; stability
• ISSN: 0020-7721; 1464-5319
• DOI: 10.1080/00207721.2013.873835

An essential step in many controller design approaches is computing the inverse of the plant. For a square plant, its inverse is stable if the plant is minimum phase (MP). Nevertheless, this conclusion does not hold for a non-square plant. In this paper, the pole feature of the Moore-Penrose pseudo-inverse of a non-square transfer function matrix is analysed. Instead of complicated advanced mathematical tools, only basic results of polynomial theory and the Binet-Cauchy theorem are used in the analysing procedure. The condition for testing the stability of the Moore-Penrose pseudo-inverse of an MP non-square transfer function matrix is given. This condition implies that the Moore-Penrose pseudo-inverse of a non-square transfer function matrix cannot be directly used as the optimal controller. Numerical examples are provided to illustrate the correctness of the condition.