Transfer functions of infinite-dimensional systems: positive realness and stabilization

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, i...

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Bibliographic Details
Published in:Mathematics of control, signals, and systems signals, and systems, 2017-12, Vol.29 (4), p.1-61, Article 20
Main Authors: Guiver, C., Logemann, H., Opmeer, M. R.
Format: Article
Language:English
Subjects:
Communications Engineering
Control
Feedback control systems
Mathematics
Mathematics and Statistics
Mechatronics
Networks
Operators (mathematics)
Original Article
Output feedback
Partial differential equations
Robotics
Stabilization
Systems Theory
Transfer functions
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Summary: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.
ISSN:0932-4194
1435-568X
DOI:10.1007/s00498-017-0203-z