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LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Dualization Lemma
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Dualization Lemma
Consider
and the subspaces , where is invertible and . The following are equivalent.
for all \ and for all .
for all \ and for all .
Example
Consider the matrices where which define the quadratic matrix inequality
Define
where . Notice that is equivalent to for all \.Additionally, for all is euaivalent to
which is satisfied based on the definition of . By the dualization lemma, is satisfied with
if and only if
where
, and
.
External Links
LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Norm-Preserving Dilations - Norm-preserving dilations and their applications to optimal error bounds (Davis, Kahan, Weinberger).