Volume 04 Issue 12-2024
42
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
12
Pages:
42-46
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
ABSTRACT
This article presents a model for optimizing the number of control function parameters for objects with delay, which
are expressed by a system of differential equations. It also discusses the model of the stability of the robot's motion
trajectory after the practical process of a specific object, as well as algorithms for improving positional accuracy.
KEYWORDS
Control function parameters, optimal solution, kinematic chain, system of differential-difference equations.
INTRODUCTION
This article presents a generalized algorithm for optimizing the number of parameters of the control function in the
optimal control of systems of linear differential-difference equations. The algorithm is built upon the "K-
controllability" algorithm for autonomous control systems described in [1]. Starting from the second condition of the
theorem provided in [2], the algorithm fully utilizes the procedure outlined in [1] (Figure 1). This includes finding the
eigenvalues of the system matrix for "K-controllability" and transforming the system into Jordan normal form, as well
as detailing the algorithms for achieving full controllability of the system.
The general form of a system of linear differential-difference equations is as follows:
( )
(
)
( )
( )
( )
x t
Ax t
h
Bu t
y t
Hx t
=
− +
=
(1)
Research Article
ON THE GENERALIZED ALGORITHM FOR OPTIMAL CONTROL OF
SYSTEMS OF LINEAR DIFFERENTIAL-DIFFERENCE EQUATIONS
Submission Date:
December 08, 2024,
Accepted Date:
December 13, 2024,
Published Date:
December 18, 2024
Crossref doi:
https://doi.org/10.37547/ajast/Volume04Issue12-08
Dilshod Davronovich Aroev
PhD, Associate Professor, Kokand State Pedagogical Institute, Uzbekistan
Orchid: - https://orcid.org/0009-0004-0392-7597
Journal
Website:
https://theusajournals.
com/index.php/ajast
Copyright:
Original
content from this work
may be used under the
terms of the creative
commons
attributes
4.0 licence.
Volume 04 Issue 12-2024
43
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
12
Pages:
42-46
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
where:
( )
x
x t
n
=
− −
a vector representing the state of the system at time
t
;
A
−
constant matrices of specific dimensions;
(
)
n n
−
ўлчовли ўзгармас матрица;
(
)
B
n m
−
−
A constant matrix with non-zero elements;;
h
−
an infinitesimally small quantity;;
( )
(1
)
u t
m
−
−
a control function vector;
0
0,
h
t
t
T
;
( )
y t
n
− −
a vector of specific dimensions depending on the state vector
(1
)
n
−
;
(
)
H
r n
− −
A constant matrix of specified dimensions.
The application of the second theorem on optimization
provided in [2] to the process of industrial robot
operation indicates that the selection of
k
C
and
k
B
matrices can be represented through an
(
)
n n
−
dimensional
matrix
expressed
via
an
((
1) (
1))
n
n
+ +
−
dimensional matrix, based on
the kinematic equations of the industrial robot's
motion. The appearance of the
k
C
matrix is as
follows:
0
0
1
k
C
C
=
,
The appearance of the
k
B
matrix is as follows:
0
0
1
k
B
B
=
To satisfy the first condition of the theorem, the
P
matrix is selected based on the technical and structural characteristics
of the industrial robot. Then, it is verified that the equation
k
k
PB
C P
=
holds true.
0
0
1
k
B
B
=
,
0
0
1
k
C
C
=
The problem in [2], i.e., restoring damaged points using the recovery equation, similarly reduces to a system of linear
equations, as in the first problem. Due to the repetitive nature of the algorithms constructed for this, we do not
present the algorithms here.
1 - the beginning:
Volume 04 Issue 12-2024
44
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
12
Pages:
42-46
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
2 - initial data: he dimension of the space under consideration
)
(
N
, the initial and final time of the process:
)
,
(
0
T
t
, the delay time
)
(
h
, system matrices
);
)(
,
1
,
,
1
,
{
};
,
1
,
,
{
(
,
N
M
M
j
N
i
b
B
N
j
i
a
A
j
i
ij
=
=
=
=
=
))
,
1
,
,
{
,
N
j
i
p
P
j
i
=
=
;
)
}(
,
1
,
,
1
,
{
,
N
K
K
l
N
i
c
C
l
i
=
=
=
control function values
)
,
1
(
N
i
u
i
=
, Additional
matrices
}
,
1
,
{
};
,
1
,
{
N
ij
c
C
N
ij
b
ij
k
ij
k
B
=
=
=
=
, initial values of the system at
0
t
t
=
.
3 - Determining the direction of the damaged point in the system.
4 - transforming the matrix
A
, which represents the system's state, into its Jordan normal form
)
(
A
J
.
5 - Generating the matrices
BC
P
C
PB
k
k
,
,
.
6 - the condition
P
C
PB
k
k
=
is checked.
7
–
If "yes", move to block 8; if "no", move to block 2.
8
–
the second condition of the theorem is checked based on the algorithm provided in [2].
9
–
end.
Volume 04 Issue 12-2024
45
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
12
Pages:
42-46
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
Figure 1. Block diagram of the optimization algorithm for the number of control function parameters in the control of a system
of linear differential equations.
No
Beginning:
, ,
,
Determining the direction of
the damage at the point.
Transforming the matrix A
into its Jordan normal form
J(A) .
Generating the matrices-
. Finding the
eigenvalues of the matrix.
End
DB
Continuation of the algorithm in [2].
Volume 04 Issue 12-2024
46
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
12
Pages:
42-46
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
REFERENCES
1.
Онорбоев Б.О. Разработка и применение метода
“К
-
управляемости”
для
оптимального
управления
сложными
технологическими
системами. Дисс. докт. техн. наук.
-
Ташкент,
2005.
–
С. 230.
2.
D.Aroev. On the method of controlling objects
whose motion is described by a system of linear
differential-difference
equations.
American
Journal of Applied Science and Technology.
https://doi.org/10.37547/ajast/Volume 04 Issue11-15.
2024-11-30.
3.
Ароев Д. Д. Об оптимизации параметров
функции управления объектами описываемым
системой
дифференциально
-
разностных
уравнений //Научные исследования молодых
ученых. –
2020.
–
С. 10
-12.
4.
Aroev, Dilshod Davronovich. "On optimization of
parameters of the object control function
describeed by a system of differential-difference
equations." Scientific research of young scientists
(2020).
