LMIs in Control/pages/H2 Deduced Condition

WIP, Description in progress

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

Deduced Condition

Given a positive $\gamma$, the transfer function matrix

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

satisfies

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

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

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

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

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