Moment-matching method

Description

The moment-matching methods are also called the Krylov subspace methods, as well as Padé approximation methods. They belong to the Projection based MOR methods. These methods are applicable to non-parametric linear time invariant systems, often descriptor systems, e.g.

$E \dot{x}(t)=A x(t)+B u(t),$

$y(t)=Cx(t).$

They are very efficient in many engineering applications, such as circuit simulation, Microelectromechanical systems (MEMS) simulation, etc..

The basic steps are as follows. First, the transfer function

$H(s)=Y(s)/U(s)=C(sE-A)^{-1}B$

is expanded into a power series at an expansion point $s_0\in\mathbb{C}\cup \infty$.

Let $s=s_0+\sigma$, then, within the convergence radius of the series, we have

$H(s_0 + \sigma)= C[(s_{0}+\sigma){E}-A]^{-1}B$

$=C[\sigma { E}+(s_{0}{ E}-{ A})]^{-1}B$

$=C[{ I}+\sigma(s_0{ E}-{ A})^{-1}E]^{-1}(s_0{ E}-{ A})^{-1}B$

$=C[{ I}-\sigma(s_0{ E}- A )^{-1}E+\sigma^2[(s_0{ E}-{ A})^{-1}E]^{2}+\ldots] (s_0{E}-{ A})^{-1}B$

$=\sum \limits^\infty_{i=0}\underbrace{C[-(s_0{ E}-{A})^{-1}E]^i(s_0{ E}-{ A})^{-1}B}_{:= m_i(s_0)} \, \sigma^i,$

where $m_i(s_0)$ are called the moments of the transfer function about $s_0$ for $i=0,1,2,\ldots$. If the expansion point is chosen as zero, then the moments simplify to $m_i(0)=C(A^{-1}E)^i(-A^{-1}B)$.

The goal in moment-matching model reduction is the construction of a reduced order system where some moments $\hat m_i$ of the associated transfer function $\hat H$ match some moments of the original transfer function $H$.

The matrices $V$ and $W$ for model order reduction can be computed from the vectors which are associated with the moments, for example, using a single expansion point $s_0=0$, by

$\textrm{range}(V)=\textrm{span}\{\tilde B,({ A}^{-1}E)^2 \tilde B, \ldots,({ A}^{-1}E)^{r}{\tilde B}\}, \quad \quad \quad \quad \quad \quad \quad \quad\quad \quad \quad \ (1)$

$\textrm{range}(W)=\textrm{span}\{C^T, E^T{ A}^{-T}C^T,(E^T{A}^{-T})^2C^T, \ldots ,(E^T{A}^{-T})^{r-1}C^T\}, \quad \quad (2)$

where $\tilde B=-A^{-1}B$.

The transfer function $\hat H$ of the reduced model has good approximation properties around $s_0$, which matches the first $2r$ moments of $H(s)$ at $s_0$.

Using a set of $k$ distinct expansion points $\{s_1,\cdots,s_k\}$, the reduced model obtained by, e.g.,

$\textrm{range}(V)=\textrm{span}\{(A-s_1 {E})^{-1}E\tilde B,\ldots,(A-s_k {E})^{-1}E\tilde B \}, \quad \quad \quad \quad \quad \quad \quad \quad (3)$

$\textrm{range}(W)=\textrm{span}\{E^T(A-s_1 {E})^{-T}C^T,\ldots,E^T(A-s_k {E})^{-T}C^T \},\quad \quad \quad \quad \quad (4)$

matches the first two moments at each $s_j$, $j=1,\ldots,k$, see . The reduced model is in the form as below

$W^TE \frac{d{V z}}{t}=W^TA Vz +W^T B u(t), \quad \hat{y}(t)=CVz.$

For the case of one expansion point in (1)(2), it can be seen that the columns of $V$, $W$ span Krylov subspaces which can easily be computed by Arnoldi or Lanczos methods. The matrices $V$ and $W$ in (3)(4) can be computed with the rational Krylov algorithm in or with the modified Gram-Schmidt process. In these algorithms only a few number of linear systems need to be solved, where matrix-vector multiplications are only used if using iterative solvers, which are simple to implement and the complexity of the resulting methods is roughly $O(n r^2)$ for sparse matrices $A, E$.