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

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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.

Theorem[edit | edit source]

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

Example[edit | edit source]

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 .

Conclusion[edit | edit source]

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 Links[edit | edit source]

A list of references documenting and validating the LMI.


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