Modal truncation

Description

Model truncation is one of the oldest MOR methods for linear time invariant systems

$E\frac{dx(t)}{dt}=Ax(t)+Bu(t),y(t)=Cx(t)+Du(t)(1)$

The main idea is to construct the projection matrices as $V=[x_1,\dots ,x_r],W=[y_1,\dots ,y_r]$ where the $x_i,y_i$ are right and left eigenvectors corresponding to certain eigenvalues ${\lambda }_i$ of $(A,E)$. The eigentriples $({\lambda }_i,x_i,y_i)$ satisfy.

$A{x}_i={\lambda }_{i}E{x}_i,A^H{y}_i=\overline{{\lambda }_i}{E}^H{y}_i,i=1,\dots ,r.$