User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Continuous Time/Eigenvalue Problem

The maximum eigenvalue of a matrix is going to have the most impact on system performance. This LMI allows for minimization of the maximum eigenvalue by minimizing $$ \gamma $$.

The System


\begin{align} \dot x(t)&=Ax(t)+Bu(t), \\ y(t) &= Cx(t) \end{align}$$

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

The Optimization Problem
$$ \text{ Minimize } \gamma \text{ subject to the LMI below.} $$

The LMI:


\begin{align} \text{Find} \; &P>0:\\ \begin{bmatrix} -A^TP-PA-C^TC & PB \\ B^TP & \gamma I\end{bmatrix} > 0\\ \end{align}$$

Conclusion:
The eigenvalue problem can be utilized to minimize the maximum eigenvalue of a matrix that depends affinely on a variable.

Implementation
https://github.com/mcavorsi/LMI

Related LMIs
Structured Singular Value