7.5.3 Least–square identiﬁcation using the step response

The least–square identiﬁcation of the system parameters of the model (1.8) can be formulated as the problem of
ﬁnding a quadruple $\left(a,{R}_{a},{R}_{0},{R}_{1}\right)$,
which minimizes the performance index

where $y\left(mh\right)$
stands for the theoretical value of the step response
$y$ at
time $mh$,
$m=1,2,\dots ,M$, evaluated by means of
formulae (7.26) $\xf7$ (7.28),
and ${y}_{e}\left(mh\right)$, is a practical value of
the step response at time $mh$,
$m=1,2,\dots ,M$, obtained from
measurements; $h$ denotes the
time discretization step, and $M$
is the ﬁnal horizon of observation expressed by the total number of trials. In experiments we have
assumed $h=0.1$,
$M=3000$. The unknown
coefficient ${y}_{0}$
is determined from the requirement that the steady–state values of
$y$ and
${y}_{e}$
coincide. To be more precise, we have

In numerical calculations we used truncation of series (7.26) to
$25$ terms, which
seems to be a reasonable compromise between the time of calculations and the accuracy. The optimal
values of $a$,
${R}_{a}$,
${R}_{0}$ and
${R}_{1}$ were determined
by means of the procedure fmins from the Matlab/Optimization Toolboxpackage. Since the
performance index $\mathcal{\mathcal{J}}$
is not unimodal the procedure fmins of the optimization without constraints was used
several times and with various starting points. Final results are presented in Table 7.1.

Table 7.1:

The optimal values of the heat exchange coefficients

$a$

${R}_{a}$

${R}_{0}$

${R}_{1}$

$\mathcal{\mathcal{J}}$

0.000945

0.02709999

0

0

0.0877040968

The experimental and theoretical step responses are depicted in Figure 7.1.

Figure 7.1:

Comparison of the experimental and the optimal theoretical step responses