LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Concatenation of LMIs

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 $A_{i}<0$, where $i=1,...,n$ is equivalent to satisfying the concatenated LMI:

${\begin{bmatrix}A_{1}&&0&&0&&.&&.&&0\\0&&A_{2}&&0&&.&&.&&0\\0&&0&&A_{3}&&.&&.&&0\\.&&.&&.&&.&&.&&0\\.&&.&&.&&.&&.&&0\\0&&0&&0&&0&&0&&A_{n}\\\end{bmatrix}}$ ${\begin{aligned}<0\end{aligned}}$,

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.