Appendix: Basic Ideas of the Toolbox

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

$$H(s)=e^{-sT_d}\frac{b_0s^0+b_1s^1+\ldots+b_{nn}s^{nn}} {a_0s^0+a_1s^1+\ldots+a_{dn}s^{dn}}$$

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

$$H(z)=z^{-f_sT_d}\frac{b_0z^0+b_1z^{-1}+\ldots+b_{nn}z^{-nn}} {a_0z^0+a_1z^{-1}+\ldots+a_{dn}z^{-dn}}$$

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.