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 \(H\).