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\begin{document}

\title{The LMS force reconstruction algorithm}


\author{Maarten van der Seijs}
\affil{Affiliation not available}



\date{\today}

\maketitle

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\begin{abstract}
A short note on the least-mean-square force reconstruction algorithm in the time domain.
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\end{abstract}%



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\section{LMS Force reconstruction}
Given a system with $n_i$ output DoFs and $n_j$ input DoFs. Impulse responses can be measured / computed such that:
\[
\vu(t) = \mH(t) \ast \vf(t) = \int_0^t{\mH(t-\tau) \vf(\tau) d\tau}
\]
For a time-discritised system\footnote{All time-discrete vectors have first index $i=1$.}, the output $\vu\est$ at time step $n$ caused by forcing vector $\vf\est$ writes:
\[
\vu_n\est = \dt \sum_{i=n-I+1}^n\mH_{n-i+1} \, \vf\est_i 
= \sum_{i=1}^I\mH_{i} \, \vf\est_{n-i+1}
\qquad \text{with} \qquad I \leq n \leq N
\]
$N$ is the length of the force input $\vf$; $I$ the length (or order) of the impulse response functions in $\bsm{H}$. The response $\vu_n$ is only known and causal in the interval from $n = I$, \ie after a full length of the impulse responses, to $n = N$, \ie the length of the force history. Let us define the following notation:
\[
\vf[m \dots n] = \left[ f_{m}, f_{m+1}, \dots, f_{n} \right] 
\]

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