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

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{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.

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.