LMIs in Control/Tools/Eigenvalue Problem

LMIs in Control/Tools/Eigenvalue Problem

The System

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

The Data

The matrices $A,B,C$.

The Optimization Problem

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

The LMI: The Lyapunov Inequality

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

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.