Transfer Functions of Infinite-Dimensional Systems: Positive Realness and Stabilization

Guiver, C.; Logemann, H.; Opmeer, M. R.

Abstract

We consider a general class of operator-valued irrational positive-real functions with an emphasis on their frequency-domain properties and the relation with stabilization by output feedback. Such functions arise naturally as the transfer functions of numerous infinite-dimensional control systems, including examples specified by PDEs. Our results include characterizations of positive realness in terms of imaginary axis conditions, as well as characterizations in terms of stabilizing output feedback, where both static and dynamic output feedback are considered. In particular, it is shown that stabilizability by all static output feedback operators belonging to a sector can be characterized in terms of a natural positive-real condition and, furthermore, we derive a characterization of positive realness in terms of a mixture of imaginary axis and stabilization conditions. Finally, we introduce concepts of strict and strong positive realness, prove results which relate these notions and analyse the relationship between the strong positive realness property and stabilization by feedback. The theory is illustrated by examples, some arising from controlled and observed partial differential equations.

• Journal Article Type: Article
• Acceptance Date: Sep 17, 2017
• Online Publication Date: Dec 16, 2017
• Publication Date: Dec 16, 2017
• Deposit Date: Jul 23, 2020
• Publicly Available Date: Aug 5, 2020
• Journal: Mathematics of Control, Signals, and Systems
• Print ISSN: 0932-4194
• Electronic ISSN: 1435-568X
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 29
• Article Number: 20 (2017)
• DOI: https://doi.org/10.1007/s00498-017-0203-z
• Public URL: http://researchrepository.napier.ac.uk/Output/2677297