# LMIs in Control/Matrix and LMI Properties and Tools/Continuous Time/Eigenvalue Problem

LMIs in Control/Matrix and LMI Properties and Tools/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 ${\displaystyle \gamma }$.

## The System

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

## The Data

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

## The Optimization Problem

${\displaystyle {\text{ Minimize }}\gamma {\text{ subject to the LMI below.}}}$

## The LMI:

{\displaystyle {\begin{aligned}{\text{Find}}\;&P>0:\\{\begin{bmatrix}-A^{T}P-PA-C^{T}C&PB\\B^{T}P&\gamma I\end{bmatrix}}>0\\\end{aligned}}}

## Conclusion:

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