# LMIs in Control/Matrix and LMI Properties and Tools/Continuous Time/Structured Singular Value

LMIs in Control/Matrix and LMI Properties and Tools/Continuous Time/Structured Singular Value

The LMI can be used to find a ${\displaystyle \Theta }$ that belongs to the set of scalings ${\displaystyle P\Theta }$. Minimizing ${\displaystyle \gamma }$ allows to minimize the squared norm of ${\displaystyle \Theta M\Theta ^{-}1}$.

## The System

{\displaystyle {\begin{aligned}M{\text{ with transfer function }}{\hat {M}}(s)=C(sI-A)^{-1}B+D,&&{\hat {M}}\in H_{\infty }\end{aligned}}}

## The Data

The matrices ${\displaystyle A\in R^{n\times n},B\in R^{n\times m},C\in R^{o\times n},D\in R^{o\times m}}$.

## The Optimization Problem

{\displaystyle {\begin{aligned}{\text{There exists }}\Theta \in \Theta {\text{ such that }}||\Theta M\Theta ^{-1}||^{2}<\gamma .\end{aligned}}}

## The LMI:

{\displaystyle {\begin{aligned}{\text{Find}}\;&X>0:\\{\begin{bmatrix}A^{T}X+XA&XB\\B^{T}X&-\Theta \end{bmatrix}}+\gamma ^{-2}{\begin{bmatrix}C^{T}\\D^{T}\end{bmatrix}}\Theta {\begin{bmatrix}C&D\end{bmatrix}}<0\\\end{aligned}}}

## Conclusion:

${\displaystyle {\text{The optimization problem and the LMI are equivalent. }}\gamma {\text{ must be optimized using bisection.}}}$