Modal truncation

Description

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

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

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 $\lambda_i\in\Lambda(A,E)$. The eigentriples $(\lambda_i,x_i,y_i)$ satisfy.

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

They are different ways to select this typically small subset of eigenvalues. An often used criterion is to take the eigenvalue closest to the imaginary axis, i.e. the ones with the smallest real part, and their associated eigenvectors into account. Dominant pole based modal truncation selects $(\lambda_i,x_i,y_i)$ with respect to their contribution in the transfer function and is described below.