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203
BASICS OF THE THEORY OF ADAPTIVE IDENTIFICATION FOR AUTOMATION
OF MULTI- CONNECTED OBJECTS
Kholmatov Umid Sadirdinovich
Associate Professor of the Department of Transport Logistics,
Andijan State Technical Institute,Uzbekistan, Andijan
E-mail address:
https://orcid.org/0000-0003-2295-502X
Annotation:
The article proposes solutions to the problem of using the theory of adaptive
identification for automation of multiply connected objects and shows the possibilities of
applying the theory of adaptive identification of multiply connected objects using the example of
wastewater treatment plants.
Keywords:
Discrete systems, drainage and treatment facilities, control of multiply connected
objects, adaptive identification.
Introduction
It is known that numerous tasks of managing production processes and complex installations,
which include chemical and biological wastewater treatment, are multi-connected objects that
require a transition from automation of individual processes to automation of production
complexes.
Automation of industrial complexes. leads to the need to take into account the
interconnectedness of the input and output coordinates of individual processes, and,
consequently, the structural links between them. The lack of sufficiently complete a priori
information about the object, the laws of distribution of random parameters and random
influences makes it necessary to apply the theory of adaptive identification. In the future,
adaptive identification of multiply connected objects will be understood as the determination of
the parameters and structure of objects under conditions of initial uncertainty, based on the
results of monitoring the change in input and output values during normal operation. From this
point of view, of particular interest are the electric power systems of drainage and treatment
facilities, in which the frequency and voltage, active and reactive power flows, the performance
of turbocompressors of pumping stations are simultaneously regulated, and according to the
technological mode they are treated as multi-connected objects with separate control channels,
operating modes [1-2].
Methods
The task of adaptive identification arises due to the fact that, in the general case, the internal and
external influence that acts on the object is of a random nature. For water treatment facilities as
objects [3], this randomness is due to the random nature of the disturbing moments and other
factors caused by the uneven distribution of pump motor power, the instability of pressure in
turbocompressors from cycle to cycle, the concentration of activated sludge, the dose of active
chlorine, etc [4, 6-8]. For treatment facilities, such impacts are: filling of sedimentation tanks and
aerotanks, failure of one of the symmetrically located engines and pumps, etc.
It is easy to determine the distribution laws for each of these factors separately [5], but it is
almost impossible to determine the resulting distribution law for the entire set of factors, and,
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accordingly, the identifiable object parameters that depend on them. In this regard, the problem
of identifying multiply connected objects is reduced to the problem of adaptive identification.
Currently, there is no complete theory of adaptive identification of multiply connected objects. In
this article, some questions of the theory of adaptive identification of multiply connected objects
containing forward and reverse cross-links are presented.
Results and Discussion
Generalization of the equation of dynamics of multiply connected objects.
Let us describe processes in multiply connected objects of a system of linear inhomogeneous
l
-th
order differential equations with
r
unknown variables x1, x2, . . . , xr of the argument t with
constant coefficients
j
=1
r
a
ij
D x
j
=
j
=1
r
b
ij
(
D)v
j
(1)
where the set of coordinates
x= x
1
,
x
2
, . . .,
x
r
;
v= v
1
,
v
2
,. . .,
v
r
- vectors - columns of object state
and control, respectively;
i
- number of a separate channel;
D=d/dt
– differentiation operator;
a
ij
D , b
ij
(
D)
- are polynomials in
D,
that have the form
a
ij
D =a
ij
l
D
l
+
a
ij
l
−1
D
l
−1
+ . . .+
a
ij
1
D+a
ij
0
;
(2)
b
ij
D =b
ij
l
1
D
j
1
+
b
ij
l
1
−1
D
l
1
−1
+ . . .+
b
ij
1
D+b
ij
0
;
Here
i, j=1,2, . . .,r; l, l
1
- the order of the polynomial of the coefficients a and b, respectively;
r
is the number of separate channels of the controlled object. It is assumed that the number of
direct cross-links is equal to the number of reverse ones; the order of differential equations of
reverse cross-links is equal to the order of differential equations of direct cross-links. These
assumptions do not reduce the generality of the problem, since in the presence of any other
options and combinations of cross-couplings, as well as the order of differential equations, it is
reduced to special cases. Let us introduce numerous matrices of operator coefficients [3-4]:
A(D)=
a
ij
D ;
B(D)=
b
ij
D ;
(3)
or expanded
A D =
a
11
D a
12
D . . . a
1
r
D
a
21
D a
22
D . . . a
2
r
D
…………………………….
a
r
1
D a
r
2
D . . . a
rr
D
;
(4)
B D =
b
11
D b
12
D . . . b
1
r
D
b
21
D b
22
D . . . b
2
r
D
…………………………….
b
r
1
D b
r
2
D . . . b
rr
D
;
Sloping
A
k
=
a
ij
k
(
i, j=1, 2, . . . , r;
k
=0, 1, 2,. . . , l;
(5)
B
q
=
b
ij
q
q=0, 1, 2, . . . l
1
),
one can represent multiple matrices A (D) in B (D) as polynomials with matrix coefficients
A(D)=A
l
D
l
+A
l-1
D
l-1
+. . . +A
1
D+A
0
;
B(D)=B
l1
D
l1
+B
l-1
D
l1-1
+. . . +B
1
D+B
0
;
(6)
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Then, in matrix form, the system of differential equations (1) takes the form
k
=0
l
A
k
D
k
x=
q
=1
l
1
B
q
D
q
u
(7)
In expanded form, for any separate channel, one can write
k
=0
l
j
=1
r
a
ij
(
k
)
D
k
x
j
=
q
=0
l
1
j
=1
r
b
ij
(
q
)
D
q
u
j
(8)
Let us rewrite equation (8) in a difference form (in a recurrent form):
x
i
n =
k
=0
l
j
=1
r
c
ij
(
k
)
x
i
n−k +
q
=1
l
1
j
=1
r
d
ij
(
q
)
u[n−q]
(9)
The matrix coefficients of the equations are interconnected by relations [1-2, 5].
c
ij
(
l
−
k
)
=−
v
=0
k
a
ij
l
−
v
(−1)
k
−
v
c
l
−
v
k
−
v
;
a
ij
(
l
1
−
q
)
=−
v
=0
q
a
ij
l
1
−
v
(−1)
q
−
v
c
l
1
−
v
q
−
v
;
where
с
l
−
v
k
−
v
=
(1−
v)!
k−v ! l−k ! ;
с
l
1
−
v
k
−
v
=
(1
1
−
v)!
q−v ! l
1
−
k ! ;
For a controlled object in the presence of only direct cross-links, equation (9) has the form
x
i
n =
m
=1
l
c
ii
(
m
)
x
i
n−m +
j
=1
r
m
=1
S
d
ij
(
m
)
v
j
n−m ,
(9.а)
and in the presence of only inverses -
x
i
n =
j
=1
r
m
=1
S
c
ij
(
m
)
x
i
n−m +
m
=1
l
d
ij
(
m
)
v
i
n−m ,
(9.б)
In some cases, some of the coefficients
c
ii
m
and
d
ij
m
may be equal, which corresponds to the
absence of any links.
Conclusion
The above algorithms allow solving problems from the transition of automation of individual
processes to automation of industrial complexes, and determine the possibilities of applying the
theory of adaptive identification of multiply connected objects, as well as consider complex
issues of compiling identification algorithms by using an iterative probabilistic method.
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