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,

$\{\begin{array}{c}\text{trace}(Z)\le {\gamma }^2\\ \left[\begin{array}{cc}-Z& C\\ C^T& P\end{array}\right]<0\\ \left[\begin{array}{cccc}-(V+V^T)& V^{T}A+P& V^{T}B& V^T\\ A^{T}V+P& -P& 0& 0\\ B^{T}V& 0& -I& 0\\ V& 0& 0& -P\end{array}\right]<0\end{array}$

$\{\begin{array}{c}\text{trace}(Z)\le {\gamma }^2\\ \left[\begin{array}{cc}-Z& B^T\\ B& P\end{array}\right]<0\\ \left[\begin{array}{cccc}-(V+V^T)& V^T{A}^T+P& V^T{C}^T& V^T\\ AV+P& -P& 0& 0\\ CV& 0& -I& 0\\ V& 0& 0& -P\end{array}\right]<0\end{array}$

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