3.3.1 Example 1: Nuclear reactor temperature control II
The performance index (3.4) has a form (3.16) with
Treating and
as fixed constants,
we seek for a pair
belonging to the domain of stability which minimizes the performance
index. The stability domain of the characteristic quasipolynomial
, in the plane
is an open bounded
set with boundary
where the curve
is given parametrically
Here denotes the smallest
positive root of the equation
and is an interval of
axis with ends coinciding
with the ends of .
Taking into account the nature of initial conditions we conclude that
But, since to
find from (3.33)
we ought to put
in its right hand side. To determine the system (3.33) we first calculate the polynomial
(3.29),
It has eight roots but we write down only four of them because others may differ of
signs,
From (3.31) we determine the corresponding eigenvectors of (3.28)
The third equation is redundant. Adding both sides of the fifth equation multiplied by
and the sixth
equation multiplied by
one obtains ,
whence .
Next adding both sides of the seventh equation multiplied by
and the eighth
equation multiplied by
one obtains , whence
, provided that
. Since the common
part of the straight line
and the stability domain is of planar measure zero we can neglect the possible degeneration
getting a preliminary reduction of the linear system to be solved,
The final reduced form arises by elimination of
and
,
This yields
(3.34)
with
The results of minimization of the performance index (3.34) for
,
are
presented at Figure 3.4 and they agree with calculations obtained in [36, Formula
(44), p. 1049] with the use of the frequency–domain method. The minimal value of
is
and it is
achieved at ,
.
Figure 3.4:
The level curves of the performance index
as a function of
and