LMIs in Control/Matrix and LMI Properties and Tools/Congruence Transformation

LMIs in Control/Matrix and LMI Properties and Tools/Congruence Transformation


This methods uses change of variable and some matrix properties to transform Bilinear Matrix Inequalities to Linear Matrix Inequalities. This method preserves the definiteness of the matrices that undergo the transformation.

TheoremEdit

Consider  , where  . The matrix inequality   is satisfied if and only if   or equivalently,  .

ExampleEdit

Consider   and  , where   and  . The matrix inequality given by


 


is linear in variable   and bilinear in the variable pair  . Choose the matrix   to obtain the equivalent BMI given by


 


Using a change of variable   and  , the above equation becomes

 


which is an LMI of variables   and  . The original variable   is recovered by doing a reverse change of variable  .

ConclusionEdit

A congruence transformation preserves the definiteness of a matrix by ensuring that   and   are equivalent. A congruence transformation is related, but not equivalent to a similarity transformation  , which preserves not only the definiteness, but also the eigenvalues of a matrix. A congruence transformation is equivalent to a similarity transformation in the special case when  .

External LinksEdit

A list of references documenting and validating the LMI.


Return to Main Page:Edit