# LMIs in Control/pages/H2 Deduced Condition

WIP, Description in progress

This gives LMI of the deduced condition for ${\displaystyle H_{2}}$-norm of a system.

## Deduced Condition

Given a positive ${\displaystyle \gamma }$ , the transfer function matrix

${\displaystyle G(s)=C(sI-A)^{-1}B}$

satisfies

${\displaystyle ||G(s)||_{2}<\gamma }$

if and only if there exists symmetric matrices ${\displaystyle Z,P}$  and a matrix ${\displaystyle V}$  such that,

${\displaystyle {\begin{cases}{\text{trace}}(Z)\leq \gamma ^{2}\\{\begin{bmatrix}-Z&C\\C^{T}&P\end{bmatrix}}<0\\{\begin{bmatrix}-(V+V^{T})&V^{T}A+P&V^{T}B&V^{T}\\A^{T}V+P&-P&0&0\\B^{T}V&0&-I&0\\V&0&0&-P\end{bmatrix}}<0\end{cases}}}$

${\displaystyle {\begin{cases}{\text{trace}}(Z)\leq \gamma ^{2}\\{\begin{bmatrix}-Z&B^{T}\\B&P\end{bmatrix}}<0\\{\begin{bmatrix}-(V+V^{T})&V^{T}A^{T}+P&V^{T}C^{T}&V^{T}\\AV+P&-P&0&0\\CV&0&-I&0\\V&0&0&-P\end{bmatrix}}<0\end{cases}}}$