Generalized KYP lemma: unified frequency domain inequalities with design applications

The celebrated Kalman-Yakubovic/spl caron/-Popov (KYP) lemma establishes the equivalence between a frequency domain inequality (FDI) and a linear matrix inequality, and has played one of the most fundamental roles in systems and control theory. This paper first develops a necessary and sufficient co...

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Bibliographic Details
Published in:IEEE transactions on automatic control 2005-01, Vol.50 (1), p.41-59
Main Authors: Iwasaki, T., Hara, S.
Format: Article
Language:English
Subjects:
Applied sciences
Computer science
control theory
systems
Control design
Control theory
Control theory. Systems
Detection, estimation, filtering, equalization, prediction
digital filter
Digital filters
Educational technology
Exact sciences and technology
Fault detection
Frequency domain analysis
frequency domain inequality
Information science
Information, signal and communications theory
Kalman-YakuboviČ-Popov (KYP) lemma
Linear matrix inequalities
linear matrix inequality (LMI)
Polynomials
Signal and communications theory
Signal, noise
Sufficient conditions
Telecommunications and information theory
Transfer functions
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Summary:The celebrated Kalman-Yakubovic/spl caron/-Popov (KYP) lemma establishes the equivalence between a frequency domain inequality (FDI) and a linear matrix inequality, and has played one of the most fundamental roles in systems and control theory. This paper first develops a necessary and sufficient condition for an S-procedure to be lossless, and uses the result to generalize the KYP lemma in two aspects-the frequency range and the class of systems-and to unify various existing versions by a single theorem. In particular, our result covers FDIs in finite frequency intervals for both continuous/discrete-time settings as opposed to the standard infinite frequency range. The class of systems for which FDIs are considered is no longer constrained to be proper, and nonproper transfer functions including polynomials can also be treated. We study implications of this generalization, and develop a proper interface between the basic result and various engineering applications. Specifically, it is shown that our result allows us to solve a certain class of system design problems with multiple specifications on the gain/phase properties in several frequency ranges. The method is illustrated by numerical design examples of digital filters and proportional-integral-derivative controllers.
ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2004.840475