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 $\gamma$.

The System

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

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:

$\begin{array}{cc}\text{Find}& P>0:\\ \left[\begin{array}{cc}-A^{T}P-PA-C^{T}C& PB\\ B^{T}P& \gamma I\end{array}\right]>0\\ \end{array}$

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

External Links

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

• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.