The moment-matching methods are also called the Krylov subspace methods, as well as \(Pade\) approximation methods. These methods are applicable for non-parametric linear time invariant systems, often the descriptor systems, e.g.
\( E \frac{dx(t)}{dt}=A x(t)+B u(t), \quad y(t)=Cx(t), \quad \quad (1) \)
They are very efficient in many engineering applications, 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
\(H(s_0 + \sigma)= L^T[(s_{0}+\sigma){I}-A]^{-1}B\)
\(=L^T[\sigma { I}+(s_{0}{ I}-{ A})]^{-1}B\)
\(=L^T[{ I}-\sigma(s_0{ I}-{ A})^{-1}]^{-1}[-(s_0{ I}-{ A})]^{-1}B\)
\(=L^T[{ I}+\sigma(s_0{ I}- A )^{-1}+\sigma^2[(s_0{ I}-{ A})^{-1}]^{2}+\ldots]\times \quad({ A}-s_0{I})^{-1}B\)
\(=\sum \limits^\infty_{i=0}\underbrace{L^T[(s_0{ I}-{A})^{-1}]^i({ A}-s_0{ I})^{-1}B}_{:= m_i(s_0)} \, \sigma^i,\)
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 \(m_i(0)=L^\mathrm{T}(-A^{-1})^{i+1}B\). For \(s_0=\infty\) the moments are also called Markov parameters which can be computed by \(L^\mathrm{T} A^{i-1}B\).
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
\(\textrm{range}(V)=\textrm{span}\{{A}^{-1}B,({ A}^{-1})^2B, \ldots,({ A}^{-1})^{r}{B}\},\)
\(\textrm{range}(W)=\textrm{span}\{L, { A}^{-T}L,({ A}^{-T})^2L, \ldots \hfill\ldots,({A}^{-T})^{r-1}L\}.\)
The reduced model is
The derived reduced order system matches the first \(2r\) moments; the corresponding transfer function \(\hat H\) has good approximation properties around $0$. % % Using a set of $k$ distinct expansion points $\{s_1,\cdots,s_k\}$, the reduced order system obtained by, e.g., % \begin{eqnarray*} \textrm{range}(V)&=&\textrm{span}\{(\bA-s_1 {I})^{-1}B,\ldots,(\bA-s_k {I})^{-1}B \},\\ \textrm{range}(W)&=&\textrm{span}\{(\bA-s_1 {I})^{-T}L,\ldots,(\bA-s_k {I})^{-T}L \}, \end{eqnarray*} % has order $r=k$ and matches the first two moments at each $s_j$, $j=1,\ldots,k$, see~\cite{Gri97}. % It can be seen that the columns of $V$, $W$ span Krylov subspaces which can easily be computed by Arnoldi or Lanczos methods. In these algorithms only matrix-vector multiplications are used which are simple to implement and the complexity of the resulting methods is only $O(n r^2)$. % for general systems, $O(nq)$ for a sparse matrix $\bA$. A reduced order system~(\ref{e2.5}) is obtained following (\ref{e2.2}) and (\ref{e2.3}).