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The aim of system identification is to determine system models which fit
well to the measured data.
This toolbox seeks the transfer function of linear dynamic systems,
in the form

for continuous domain (*s*-domain), or in the form

for discrete domain (*z*-domain),
where *nn* is the order of the numerator, *dn* is that of the denominator,
*T*_{d} is the delay, and *f*_{s} is the sampling frequency.
Since the problem is to be formulated in the frequency domain, each measured
signal
is transformed from time to frequency domain.
For the Fourier transform to be error-free, periodic excitation signals are
used, the measurement is accomplished in steady-state of the system, and the
record length is equal to an integer multiple of the period length.
For transformation `fft` is used, and the excited frequency lines are
selected from the Fourier transforms by simple indexing.
In this toolbox it is generally assumed that the frequency domain samples of
the noises are independent from each other at different frequencies, and
each of them has
circularly symmetric Gaussian distribution (these assumptions are usually
reasonable).
Using these assumptions, the maximum likelihood estimation method for the
unknown transfer function coefficients leads to a nonlinear least squares
(LS) problem
[10].
The function `elis` solves this problem
using the Levenberg-Marquardt scheme
[8].
Since it makes sense to keep the toolbox stand-alone, the LM scheme is
directly programmed, so the Optimization Toolbox is not required.
The last step in identification is the validation of the
results.
We always have to check whether the result really satisfies our
requirements, is in no contradiction with the preliminary assumptions,
and corresponds to the data.
A program can only offer tools for this purpose: the validation itself is
the task of the person who performs the identification.

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*István Kollár*

*1999-07-06*