User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Continuous Time/Structured Singular Value

User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Continuous Time/Structured Singular Value

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

The System

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

The Data

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

The Optimization Problem

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

The LMI:

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

Conclusion:

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

Implementation

https://github.com/mcavorsi/LMI

Related LMIs

Eigenvalue Problem

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.