LMIs in Control/Matrix and LMI Properties and Tools/Dualization Lemma
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Dualization Lemma
[edit | edit source]Consider and the subspaces , where is invertible and . The following are equivalent.
for all \ and for all .
for all \ and for all .
Example
[edit | edit source]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
[edit | edit source]- 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).