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 \(A, E\).