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Duality theory of the optimal two‐block H ∞ problem
Author(s) -
Djouadi S. M.,
Birdwell J. D.
Publication year - 2008
Publication title -
international journal of robust and nonlinear control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.361
H-Index - 106
eISSN - 1099-1239
pISSN - 1049-8923
DOI - 10.1002/rnc.1287
Subject(s) - mathematics , duality (order theory) , optimal control , toeplitz matrix , pure mathematics , hilbert space , duality gap , operator (biology) , norm (philosophy) , block (permutation group theory) , optimization problem , mathematical optimization , combinatorics , biochemistry , chemistry , repressor , political science , transcription factor , law , gene
This paper provides the Banach duality theory structure of the optimal two‐block H ∞ problem. Alignment conditions are obtained and show that the optimal solution is flat or allpass in general, and is unique in the single‐input single‐output case. The optimal solution is shown to satisfy an extremal identity, which gives a necessary and sufficient condition for a controller to be optimal. Moreover, nearly optimal control laws are shown to satisfy an approximate allpass condition. It is also proved that under specific conditions a Hankel–Toeplitz operator achieves its norm on the discrete spectrum, therefore generalizing a similar result obtained formerly for finite‐dimensional (rational) systems. The norm of this Hankel–Toeplitz operator corresponds to the optimal two‐block H ∞ performance. The dual description leads naturally to a numerical solution based on convex programming for linear time‐invariant (possibly infinite‐dimensional) systems. Copyright © 2007 John Wiley & Sons, Ltd.