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
Consider $$ Q \in \mathbb{S}^{n}, W \in \mathbb{R}^{n\times n}$$, where $$\operatorname{rank}(W) = n$$. The matrix inequality $$Q < 0$$ is satisfied if and only if $$WQW^{T} < 0$$ or equivalently, $$W^{T}QW < 0$$.

Example
Consider $$ A \in \mathbb{R}^{n\times n}, B \in \mathbb{R}^{n\times m}, K \in \mathbb{R}^{m\times p}, C^{T} \in \mathbb{R}^{n\times p}, P \in \mathbb{S}^{n}$$ and $$ V \in \mathbb{S}^{p} $$, where $$P > 0$$ and $$V > 0 $$. The matrix inequality given by

$$

\begin{align} \ Q &= \begin{bmatrix} \ A^{T}P + PA & -PBK + C^{T}V \\ \ * & -2V \\ \end{bmatrix} \ < 0, \end{align} $$

is linear in variable $$V$$ and bilinear in the variable pair $$(P.K)$$. Choose the matrix $$\operatorname{diag}(P^{-1},V^{-1})$$ to obtain the equivalent BMI given by

$$

\begin{align} \ WQW^{T} &= \begin{bmatrix} \ P^{-1}A^{T} + AP^{-1} & -BKV^{-1} + P^{-1}C^{T} \\ \ * & -2V^{-1} \\ \end{bmatrix} \ < 0, \end{align} $$

Using a change of variable $$X = P^{-1}, U = V^{-1}$$ and $$F = KV^{-1}$$, the above equation becomes

$$

\begin{align} \ WQW^{T} &= \begin{bmatrix} \ XA^{T} + AX & -BF + XC^{T} \\ \ * & -2U \\ \end{bmatrix} \ < 0, \end{align} $$

which is an LMI of variables $$X, U$$ and $$F$$. The original variable $$K$$ is recovered by doing a reverse change of variable $$K = FU^{-1}$$.

Conclusion
A congruence transformation preserves the definiteness of a matrix by ensuring that $$Q < 0$$ and $$W^{T}QW < 0$$ are equivalent. A congruence transformation is related, but not equivalent to a similarity transformation $$TQT^{-1} $$, 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 $$W^{T} = W^{-1} $$.