5.4.1 Example 1: A simple time–delay control system

Consider the control system containing a time delay in the feedback loop, depicted in Figure 5.1. The disturbance
$z$ has a form of the
Dirac $\delta $–pseudofunction.

Figure 5.1:

The automatic control system with a time lag

We wish to solve the parametric optimization problem of ﬁnding minimum of the
function

where $\kappa =\frac{1}{T}\sqrt{\left|{m}^{2}-1\right|}$.

Exercise 5.4.1. Conﬁrm (5.11) using the method presented in Section 3.2.

For $T=1$
and $r=0.5$
the exact values of the performance index calculated from (5.11) are compared in
Figures 5.2 with the results derived by applying the approximate method presented in
Sections 5.2 and 5.3. The lower curve of Figure 5.2 has been obtained by plotting
${\u2225{P}_{n}f\u2225}_{{\text{L}}^{2}\left(0,\infty \right)}^{2}$ versus
$m$.
This was achieved by implementing formula (5.4), with
$A$,
$\widehat{f}\left(-{A}^{\ast}\right)$ and
$H$
given by (5.6) and (5.7), on Matlab 6. The ill–conditioning was observed for large
$n$, however
$n=32$–dimensional
approximation ensures satisfactory results. The optimal value obtained by the approximation method is
$0.372138$ while by the exact value
is $0.372194$. The approximate
value of optimal $m$ is
$m=1.220738$ while the exact one is
$1.218772$. The relative error of
ﬁnding the optimal value of $m$
is less than $0.17\%$.

Figure 5.2:

Plots of the performance index for the system examined in Example 1