Modal truncation

Description

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

$E \frac{dx(t)}{dt}=A x(t)+B u(t), \quad y(t)=Cx(t)+Du(t) \quad \quad (1)$

The main idea is to construct the projection matrices as $V=[x_1,\ldots,x_r], W=[y_1,\ldots,y_r]$ 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.

$Ax_i=\lambda_iEx_i,\quad A^Hy_i=\overline{\lambda_i}E^Hy_i,\quad i=1,\ldots,r.$