
weighed intermediate crossover:
11 2 2
12
() ()
() ()
offspring
ii
i
itness op op fitness op op
op
fitness op fitness op
⋅+ ⋅
=
+
;

discrete crossover:
12
(1 )
offspring
iii
op z op z op=−⋅ +⋅ ;
 randomly weighted crossover. Let
(0,1)Rv U
be the uniformly distributed random value:
12
(1 )
offspring
iii
op op Rv op Rv=⋅+⋅−
.
The mutation of every offspring’s gene is
executed with the chosen probability
m
p . If we have
the random value
{0,1}, ( 1)
m
zPzp===, which is
generated for every objective gene and its strategic
parameter then
(0, )
offspring offspring offspring
ii i
op op z N sp=+⋅ ;
(0,1)
offspring offspring
ii
sp sp z N=+⋅,
where
2
(, )Nmσ is normally distributed random
value with the mean m and the variance
2
σ
.
We suggest a new operation that could increase
the efficiency of the given algorithm. For every
individual, the real value is rounded to integer. That
provides searching for solutions with near the same
structure. This modification is made to decrease the
destructive effect of the mutation on the forming the
structure.
Also for
1
N randomly chosen individuals and
for
2
N randomly chosen objective gene we make
3
N iterations of the local optimization with the step
l
h to determine the better solution. It is the random
coordinatewise optimization. Local optimization is
executed until fitness function increases.
4 TESTING THE ALGORITHMS
WITH DIFFERENT SETTINGS
To make an investigation 50 systems were
generated. It means that for every order of the
differential equation from the first to the ninth we
have 5 different systems. Parameters of the systems
were randomly generated:
(5,5),
i
k
aU=−
(5,5),
k
bU=−
______
2,10,i =
___
1,ki
, where (5,5)U
is
the uniform distribution. The solution of every
system was found with the RungeKutta integration
method with the step
0.05
i
h
. The time of the
process was set to 5. The control function was the
step excitation and we know what was the control
for every system, so
() 1ut
. Let
{}
,, 1,/
ii i
ti Th=
be the numerical solution for the system. We take
/, 100
i
sThs
= points randomly. For every
system 10 runs of the algorithm were executed with
every combination of its parameters. Now, to
estimate the efficiency of different approaches we
consider the identification without any noise.
Having different types of the selection and the
crossover, we would also vary the
151
,,,1
11 11 5
m
p
⎫
∈
⎬
⎩⎭
to find out the most effective
combination of the algorithm settings. As a preset
we use population size in 50, number of populations
in 50,
1
50N
,
2
50N
and
3
1N = with
0.05
l
h
.
Now we can compare the efficiency of following
algorithms: 1 – the evolutionary strategies (ES)
algorithm; 2 – ES with the local optimization, hybrid
evolutionary strategies (HES); 3 – HES with
modified mutation; 4 – HES with turning real
numbers into integer numbers; 5  HES with
modified mutation and turning real numbers to
integer ones.
After testing the algorithms on different
samples of the systems, the efficient presets were
found: modified HES algorithm with turning the real
numbers to integer ones, 50 individuals for 50
populations,
1
50N
,
2
50N = and
3
1N = with
0.05
l
h
, the tournament selection with the
tournament size 25%, the discrete crossover and the
mutation with the probability
5
11
m
p =
.
Table 1: Mean criterion values for different algorithms and
system orders.
Algorithm
Order 1 2 3 4 5
1 0,63 0,72 0,93 0,92 0,93
2 0,69 0,73 0,74 0,79 0,85
3 0,74 0,76 0,90 0,88 0,91
4 0,69 0,79 0,99 0,98 0,99
5 0,89 0,96 0,99 0,99 0,99
6 0,76 0,80 0,82 0,83 0,86
7 0,89 0,96 0,96 0,98 0,99
8 0,85 0,89 0,93 0,91 0,93
9 0,99 0,99 0,99 0,99 0,99
10 0,99 0,99 0,99 0,99 0,99
ICINCO20129thInternationalConferenceonInformaticsinControl,AutomationandRobotics
620