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:

$\left[\begin{array}{ccc}A& & 0\\ 0& & B\end{array}\right]$ $\begin{array}{c}<0\end{array}$,

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

$\left[\begin{array}{ccccccccccc}A& & 0& & 0& & .& & .& & 0\\ 0& & B& & 0& & .& & .& & 0\\ 0& & 0& & C& & .& & .& & 0\\ .& & .& & .& & .& & .& & 0\\ .& & .& & .& & .& & .& & 0\\ 0& & 0& & 0& & 0& & 0& & A_n\end{array}\right]$ $\begin{array}{c}<0\end{array}$,

External Links

A list of references documenting and validating the LMI.