# LMIs in Control/Matrix and LMI Properties and Tools/Change of Subject

LMIs in Control/Matrix and LMI Properties and Tools/Change of Subject

A Bilinear Matrix Inequality (BMI) can sometimes be converted into a Linear Matrix Inequality (LMI) using a change of variables. This is a basic mathematical technique of changing the position of variables with respect to equal signs and the inequality operators. The change of subject will be demonstrated by the example below.

## Example

Consider ${\displaystyle A\in \mathbb {R} ^{n\times n},B\in \mathbb {R} ^{n\times m},K\in \mathbb {R} ^{m\times n}}$ , and ${\displaystyle Q\in \mathbb {S} ^{n}}$ , where ${\displaystyle Q>0}$ .

The matrix inequality given by:

${\displaystyle QA^{T}+AQ-QK^{T}B{T}-BKQ<0}$  is bilinear in the variables ${\displaystyle Q}$  and ${\displaystyle K}$ .

Defining a change of variable as ${\displaystyle F=KQ}$  to obtain

${\displaystyle QA^{T}+AQ+-F^{T}B^{T}-BF<0}$ ,

which is an LMI in the variables ${\displaystyle Q}$  and ${\displaystyle F}$ .

Once this LMI is solved, the original variable can be recovered by ${\displaystyle K=FQ^{-1}}$ .

## Conclusion

It is important that a change of variables is chosen to be a one-to-one mapping in order for the new matrix inequality to be equivalent to the original matrix inequality. The change of variable ${\displaystyle F=KQ}$  from the above example is a one-to-one mapping since ${\displaystyle Q^{-1}}$  is invertible, which gives a unique solution for the reverse change of variable ${\displaystyle K=FQ^{-1}}$ .