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

The matrix inequality given by:

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

Defining a change of variable as $$ F = KQ $$ to obtain

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

which is an LMI in the variables $$Q$$ and $$F$$.

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

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 $$F = KQ$$ from the above example is a one-to-one mapping since $$Q^{-1}$$ is invertible, which gives a unique solution for the reverse change of variable $$ K = FQ^{-1}$$.