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values of the parameters to intervals. These have the known boundaries. Actually the
interval model of a system mirrors a real situation with the information on values of
its parameters, when a priori only the boundaries of interval are known. Therefore,
using the rules and nomenclature of interval mathematics we can represent it as
mathematical model.
This paper we have used two conceptions of mathematical description of complex
systems. The first conception — “state space” — describes a system by means of
using differential or differences (deterministic, stochastic or interval) equations. The
second one — “input-output” mapping- describes a system by means of using the
deterministic (stochastic) Volterra series or discrete analogues of the Volterra series
known as Gabor-Kolmogorov polynomials known as power series.
For mathematical modeling the dynamics of two interacting
Th1(t) -Th2(t)
systems
the Genetic Programming approach have used with Gabor- Kolmogorov polynomials
which are defined by the power series:
=
+
∑
∏
,
=
=
where
a
m
are the term coefficients,
m
ranges up to a pre-selected maximum number of
terms
M: m
≤
≤
M
;
are
the independent variable values of the input vector
x
,
j
d
numbers; and
r
jm
= 0, 1, are the powers with which the
j-th
element
participates in
the
m-th
term. It is assumed that
r
jm
is bound by a maximum polynomial order (degree)
s:
∑
≤
for every
m.
The results of solving the task of parameter identification
for dynamics of changing
Th1 (t)
and
Th2 (t)
on the base of Genetic Programming
have been
produced.
The results of solving the task of parameter identification for dynamics of two in-
teraction
Th1(t)
and
Th2(t)
systems have been used for receiving the dynamics
mathematical model. This mathematical model is given in the state space as interval
functional-differential delay type inclusions having the following vector-matrix view:
=
∈
X
'
(t)
A
X(t) +
A
X(t -
b
u(t),
X(t
0
) = X
0
, t
∈
∞
[t
0
,
),
∈
T
( ),
[t
0
-
0
],
X(t)
,
(1)
where
X(t) = (Th1(t), Th2(t))
- states vector,
Th1, Th2
are continuous functions at
[t
0
,
∞
∈
∞
),
i = 1, 2;
A, A
-
interval matrices the corresponding
dimension;
b
- an interval vector,
b =
(
b
i
)
,
b
i
=
[b
i-
, b
i+
], i = 1,…, m, b
i-
< b
i+
- the low
and upper boundaries
of interval;
u(t)
)
, i.e.
Thi(t)
C[t
0
,
∈
∞
C [t
0
,
)
- nonlinear function that satisfies
the relation
u(t) = ( ),
= r
T
X(t),
( )
- a continuous differentiable function,
:
→
≥
R
R
, such, that true the following inequality is
F(X,u)
0,
where
F(X,u)
- a real
∈
quadratic form of the variables
X
and
u
; the value
R
is determined according to
∈
the expression
= r
T
X(t), r
R
m
, -
delay
.
The absolute stability conditions of
(1)
is founded on using the functional defined
on segments of integral curves of motion tubes of the nonlinear system
:
