Modal truncation

Description

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

$E\dot{x}(t)=A x(t)+B u(t), \quad y(t)=Cx(t)+Du(t),\quad E,A\in\mathbb{R}^{n\times n},~B\in\mathbb{R}^{n\times m},~C\in\mathbb{R}^{p\times n},~D\in\mathbb{R}^{p\times m}.\qquad (1)$ (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.

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

$M\ddot{x}(t)+D\dot{x}(t)+K x(t)=B u(t), \quad y(t)=C_px(t)+C_v\dot x(t)+Du(t)$

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). Modal truncation methods can also be generalized to handle linear, time-invariant system with higher time derivatives and even with time-delays, e.g.

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

where $\tau\geq0$ is the time-delay.

Other advantages are that modal truncation can in principle applied to DAEs of arbitrary index and to unstable systems. They do, however, preserve stability since they do not change the eigenvalues of the original system. They rely on eigenvalue algorithms to compute the required eigentriplets. There are several algorithms available for this purpose for large and sparse matrices.

Disadvantages are the lack of a computationally feasible error bound and the often observed lower approximation accuracy compared to other MOR methods.

Dominant pole based modal truncation

Figure 1: 3D Bode plot of transfer function.

Figure 2: 2D Bode plot of transfer function.

This modal truncation variant aims at the identification of eigentriplets $(\lambda_i,x_i,y_i)$ which have a strong contribution to the input-output behavior of the dynamical. For this purpose, let all eigenvalues be semisimple and consider the residue expansion of the transfer function matrix (exemplary of (1))

$H(s)=C(sE-A)^{-1}B+D=\sum\limits_{j=1}^{n_f}\frac{(Cx_j)(y_j^HB)}{s-\lambda_j}+D+R_{\infty},\qquad(2)$ (2)

where $n_f$ is the number of finite eigenvalue of $(A,E)$ and $R_{\infty}$ is the residue corresponding to the infinite eigenvalues. For simplicity we assume $R_{\infty}=0$. The quantities $R_j:=(Cx_j)(y_j^HB)$ in the numerator of the above series are the residues w.r.t. $\lambda_i$. Each finite eigenvalue $\lambda_i$ is pole of $H(s)$ and is called dominant pole if its scaled residue norm $\frac{\|R_j\|}{|\real{\lambda_j}|}$ is larger than the ones of the other eigentriples which corresponds in some sense to the largest summand in the residue expansion of $H(s)$. Dominant poles can be observed in the Bode, or sigma plot. Fig. 1 and Fig. 2 illustrates this phenomenon for the New England test system.

The upper figure shows a three dimensional surface plot of $H(s)$ in a region in the left half plane. The poles of $H(s)$ (eigenvalues of $(A,E)$) are marked as black dots in the $\real(s)-\Im(s)$-plane. Observe that the function values grow in the limit towards infinity as $s$ reaches an eigenvalue $\lambda\in\Lambda(A,E)$. However, the poles marked as thick blue dots elevate the function values in a stronger way and are the one with the largest scaled residues, i.e., the dominant poles. The cutsection of this plot along the $\Im(s)$ axis gives the Bode plot and is shown in the bottom figure where peaks occur near the imaginary parts of the dominant poles which are marked by the blue dots.

Dominant pole based model truncation then amounts to compute the, say $r\ll n_f$, most dominant poles (i.e., the dominant eigentriplets) and take the associated right and left eigenvectors as columns of the truncation matrices $V, W$. Equivalently, the reduced order model is obtained by truncating the residue expansion (2):

$H(s)\approx\tilde{H}(s)=\sum\limits_{j=1}^{r}\frac{(Cx_j)(y_j^HB)}{s-\lambda_j}+D$.

Note that $\tilde{H}(s)$ constructed in that way is often called modal equivalent of $H(s)$. A specially tailored eigenvalue algorithm to compute the sought dominant eigentriplets for this task is the (Subspace Accelerated) Dominant Pole Algorithm .

A MATLAB implementation of this algorithms and certain variants thereof can be found at https://sites.google.com/site/rommes/software.