LMIs in Control/Matrix and LMI Properties and Tools/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


Then since  


which is equivalent to


These expanded inequalities (1) and (2) are now in the form of the strict projection lemma, meaning they are equivalent to


where  and   By choosing


we can now rewrite the inequality (3) as


which is the new dilated inequality.


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


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