# 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**Edit

Consider , and , where .

The matrix inequality given by:

is bilinear in the variables and .

Defining a change of variable as to obtain

,

which is an LMI in the variables and .

Once this LMI is solved, the original variable can be recovered by .

**Conclusion**Edit

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 from the above example is a one-to-one mapping since is invertible, which gives a unique solution for the reverse change of variable .

## 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.