Consider the problem of parametric optimization of the system consisting of a
proportional controller (operational ampliﬁer) and a distributed parameter plant (the
$RC$–transmission
line) discussed in Chapter 2. It relies on ﬁnding minimum of the function

in the interval $\left(-cosh\pi ,1\right)$
being the stability region in the space of the parameter
$K$ and the
function $f$ is
given by its Laplace transform

In Section 2.7, to solve the problem we employed spectral properties
of the operator describing the closed–loop system in the state space
${\text{L}}^{2}\left(0,1\right)$. It
was shown that the formula expressing the performance index as a function of
$K$ has
the form (2.53). Recall that the formula (2.53) admits a simple, highly stable computer
implementation which is due to the fast convergence of the series in (2.53).

The efforts to implement a reasonable approximation of
${\u2225{P}_{n}f\u2225}_{{\text{L}}^{2}\left(0,\infty \right)}^{2}$
with the aid of both (5.4) and (5.10) failed down on Matlab 6 out of
ill–conditioning. Recall that the standard Matlabcalculations are carried on
$32$
positions. Therefore to control the accuracy of ﬂoating–point calculations the formula (5.10) was
implemented on Maple 6 rather than on Matlab 6. This was achieved by Digits:=90;
command. The comparison of the exact values of the performance index (2.53) with their
approximations are depicted in Figure 5.3.

Figure 5.3:

Plots of the performance index for the system examined in
Example 2. The upper curve is exact, the lower ones are the approximations for
$n=20$,
$n=30$,
$n=40$
and $n=50$