LMIs in Control/Matrix and LMI Properties and Tools/Concatenation Of Matrices

Introduction

Matrix concatenation is the process of joining one or more matrices to make a new matrix. This process is similar for concatenating the LMIs as well.

Concatenation of LMIs

A useful property of LMIs is that multiple LMIs can be concatenated together to form a single LMI. For example, satisfying the LMIs

$A<0$ and,
$B<0$ 

is equivalent to satisfying the concatenated LMI:

${\begin{bmatrix}A&&0\\0&&B\end{bmatrix}}$ ${\begin{aligned}<0\end{aligned}}$,

More generally, satisfying the LMIs $Ai<0$, where $i=1,...,n$ is equivalent to satisfying the concatenated LMI:

${\begin{bmatrix}A&&0&&0&&.&&.&&0\\0&&B&&0&&.&&.&&0\\0&&0&&C&&.&&.&&0\\.&&.&&.&&.&&.&&0\\.&&.&&.&&.&&.&&0\\0&&0&&0&&0&&0&&A_{n}\\\end{bmatrix}}$ ${\begin{aligned}<0\end{aligned}}$,

External Links

A list of references documenting and validating the LMI.