# 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.

**Theorem**Edit

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

**Example**Edit

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

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.

- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.