LMIs in Control/Tools/Elimination of Variables/Variable Elimination in a Partition Matrix

WIP, Description in progress

Two basic lemmas treat the elimination of variables in a partitioned matrix.

Lemma 1

Consider the matricies $Z={\begin{bmatrix}Z_{11}&Z_{12}\\Z_{12}^{T}&Z_{22}\end{bmatrix}}$, $Z_{11}\in \mathbb {R} ^{n\times n}$, be symmetrix. Then, there exists a symmetric matrix $X$ such that

${\begin{bmatrix}Z_{11}-X&Z_{12}&X\\Z_{12}^{T}&Z_{22}&0\\X&0&-X\end{bmatrix}}<0$

if and only if

$Z={\begin{bmatrix}Z_{11}&Z_{12}\\Z_{12}^{T}&Z_{22}\end{bmatrix}}<0$.

Lemma 2

Let $Z_{ij}$, $i=1,2,3,j=i,\cdots ,3$, be given matrices of appropriate dimensions. Then there exists a matrix $X$ such that

${\begin{bmatrix}Z_{11}&Z_{12}&Z_{13}\\Z_{12}^{T}&Z_{22}&Z_{23}+X^{T}\\Z_{13}^{T}&Z_{23}^{T}+X&Z_{33}\end{bmatrix}}<0$

if and only if

$Z={\begin{bmatrix}Z_{11}&Z_{12}\\Z_{12}^{T}&Z_{22}\end{bmatrix}}<0$, $Z={\begin{bmatrix}Z_{11}&Z_{13}\\Z_{13}^{T}&Z_{33}\end{bmatrix}}<0$

In this case, $X$ is given by

$X=Z_{13}^{T}Z_{11}^{-1}Z_{12}-Z_{23}^{T}$.

WIP, additional references to be added

External Links

A list of references documenting and validating the LMI.

• [1] - LMI in Control Systems Analysis, Design and Applications