LMIs in Control/Matrix and LMI Properties and Tools/Dilation

Dilation Matrix inequalities can be dilated in order to obtain a larger matrix inequality. This can be a useful technique to separate design variables in a BMI (bi-linear matrix inequality), as the dilation often introduces additional design variables. A common technique of LMI dilation involves using the projection lemma in reverse, or the "reciprocal projection lemma." For instance, consider the matrix inequality where , , with This can be rewritten as (1) Then since which is equivalent to (2) These expanded inequalities (1) and (2) are now in the form of the strict projection lemma, meaning they are equivalent to (3) where and By choosing we can now rewrite the inequality (3) as which is the new dilated inequality. Examples Some useful examples of dilated matrix inequalities are presented here. Example 1 Consider matrices where and The following matrix inequalities are equivalent: Example 2 Consider matrices and where The matrix inequality implies the inequality Example 3 Consider matrices and where The matrix inequality implies the inequality Related Pages Projection Lemma - The projection lemma. Reciprocal Projection Lemma - The reciprocal projection lemma. 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).