The Fenchel duality, S-procedure, and the Yakubovich-Kalman lemma
- 1 February 2006
- journal article
- Published by Pleiades Publishing Ltd in Automation and Remote Control
- Vol. 67 (2) , 293-310
- https://doi.org/10.1134/s0005117906020081
Abstract
The role of the Fenchel duality in the theorem on the losslessness of the S-procedure and in Yakubovich-Kalman lemma is studied. The Fenchel duality theorem implying the well-known results on the losslessness of the S-procedure is formulated. A relation between the Yakubovich-Kalman lemma and a special extremal problem defined on a set of positive-demidefinite solutions of the generalized Lyapunov inclusion is derived. Every assertion of the lemma is shown to be necessary and sufficient for the value of the extremal problem to be bounded.Keywords
This publication has 11 references indexed in Scilit:
- Generalized KYP lemma: unified frequency domain inequalities with design applicationsIEEE Transactions on Automatic Control, 2005
- Convex Analysis and Variational ProblemsPublished by Society for Industrial & Applied Mathematics (SIAM) ,1999
- Convexity of Quadratic Transformations and Its Use in Control and OptimizationJournal of Optimization Theory and Applications, 1998
- On interconnections, control, and feedbackIEEE Transactions on Automatic Control, 1997
- On the Kalman—Yakubovich—Popov lemmaSystems & Control Letters, 1996
- Linear Matrix Inequalities in System and Control TheoryPublished by Society for Industrial & Applied Mathematics (SIAM) ,1994
- A frequency theorem in control theorySiberian Mathematical Journal, 1973
- Duality theorems for certain nonconvex extremal problemsSiberian Mathematical Journal, 1973
- Convex AnalysisPublished by Walter de Gruyter GmbH ,1970
- LYAPUNOV FUNCTIONS FOR THE PROBLEM OF LUR'E IN AUTOMATIC CONTROLProceedings of the National Academy of Sciences, 1963