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,

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

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

WIP, additional references to be added