Modal truncation

Description

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

$E\dot{x}(t)=Ax(t)+Bu(t),y(t)=Cx(t)+Du(t)(1)$

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 ${\lambda }_i\in \mathrm{\Lambda }(A,E)$. The eigentriples $({\lambda }_i,x_i,y_i)$ satisfy.

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

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.

One advantage of such eigenvalue based MOR methods is that they can be easily adapted to more general time invariant systems, for instance, systems in second order form

$M\ddot{x}(t)+D\dot{x}(t)+Kx(t)=Bu(t),y(t)=C_{p}x(t)+C_v\dot{x}(t)+Du(t)(2)$

which occur frequently in vibration analysis for mechanical systems. There, $M,D,K$ being referred to as Mass, Damping, and Stiffness matrix are represent a finite element discretization of the mechanical system. In fact, a large variety of modal truncation type approaches originated in this application, e.g., Condensation (Guyan reduction) and Component Mode Synthesis (Craig-Bampton).