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

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

The LMI:


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

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