| Line 29: | Line 29: | ||
<math>=\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,</math> |
<math>=\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,</math> |
||
| − | where |
+ | where <math>m_i(s_0)</math> are called the moments of the transfer function about <math>s_0</math> for <math>i=0,1,2,\ldots</math>. |
| − | If the expansion point is chosen as zero then the moments simplify to |
+ | If the expansion point is chosen as zero then the moments simplify to <math>m_i(0)=L^\mathrm{T}(-A^{-1})^{i+1}B</math>. |
| − | For |
+ | For <math>s_0=\infty</math> the moments are also called Markov parameters which can be computed by <math>L^\mathrm{T} \bA^{i-1}B</math>. |
The goal in moment-matching model reduction is the construction of a reduced order |
The goal in moment-matching model reduction is the construction of a reduced order |
||
| − | system where some moments |
+ | system where some moments <math>\hat m_i</math> of the associated transfer function <math>\hat H</math> match some moments |
| − | of the original transfer function |
+ | of the original transfer function <math>H</math>. |
| − | A few important classes of approximations are listed in Table~\ref{tab:moments}. |
||
| − | % |
||
| − | % |
||
| − | \begin{center} |
||
| − | \begin{table*}[ht] |
||
| − | \hfill{} |
||
| − | \begin{tabular}{l|ll} |
||
| − | %\hline |
||
| − | Name of reduced order system& Matched moments &\\\hline |
||
| − | %\cline{3-4} |
||
| − | %\hline |
||
| − | Pad\'e approximation~\cite{Bak75} & $m_i(s_0) = \hat m_i(s_0)$, & $i=0,1,\cdots,2r-1$ \\ |
||
| − | Partial realization~\cite{GraL83} & $m_i(\infty) = \hat m_i(\infty)$, & $i=0,1,\cdots,2r-1$ \\ |
||
| − | Multipoint Pad\'e approximation or & $m_i(s_j) = \hat m_i(s_j)$, & $i=0,1,\cdots,2r_j-1$, for $j=1,\cdots,k$, and $r_1+\dots+r_k = r$ \\ |
||
| − | rational interpolation~\cite{AndA90,Bak75} & & |
||
| − | %\hline |
||
| − | \end{tabular} |
||
| − | \hfill{} |
||
| − | \caption{Some examples for model reduction by moment-matching.} |
||
| − | \label{tab:moments} |
||
| − | \end{table*} |
||
| − | \end{center} |
||
Revision as of 09:51, 13 March 2013
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 \(L^\mathrm{T} \bA^{i-1}B\).
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 \(H\).