Розширення математичного апарату дискретно-неперервних мереж для автоматизації процедур їх синтезу
The paper deals with a model of an intelligent system related to the automatic synthesis of Petri nets and presents a certain stage of developing this model. The peculiarity of the extended mathematical apparatus is that it contains a combination of Petri net incidence matrices to represent various...
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| author | Gurskiy, Alexander Denisenko, Andrey Goncharenko, Alexander |
| author_facet | Gurskiy, Alexander Denisenko, Andrey Goncharenko, Alexander |
| author_institution_txt_mv | [
{
"author": "Alexander Gurskiy",
"institution": "Odesa National University of Technology, Odesa"
},
{
"author": "Andrey Denisenko",
"institution": "Odesa Polytechnic National University, Odesa"
},
{
"author": "Alexander Goncharenko",
"institution": "Odesa National University of Technology, Odesa"
}
] |
| author_sort | Gurskiy, Alexander |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
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| datestamp_date | 2024-08-11T01:12:49Z |
| description | The paper deals with a model of an intelligent system related to the automatic synthesis of Petri nets and presents a certain stage of developing this model. The peculiarity of the extended mathematical apparatus is that it contains a combination of Petri net incidence matrices to represent various algorithms. This combination of matrices is part of the equations describing the logic control device of a complex system. Accordingly, the work also presents a well-known mathematical description of discrete-continuous systems with a controlled structure, which includes certain logical control devices. This mathematical description, based on means of discrete-continuous networks, is associated with the incidence matrix of the Petri net, which is formed as a result of a particular synthesis algorithm. At the same time, the formed Petri net represents the corresponding logical control algorithm that should ensure the effective functioning of the corresponding system. The final part of the work presents various structural schemes of logic-dynamic models of systems related to the automatic synthesis of Petri nets. Here, we determine the features of the advanced mathematical apparatus based on discrete-continuous networks to develop an intelligent system that forms logical control algorithms. It is also noted that such systems can be used to create certain control algorithms that ensure increased efficiency of the functioning of some objects in difficult and unpredictable conditions. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2024.2.07 |
| first_indexed | 2025-07-17T10:28:34Z |
| format | Article |
| fulltext |
A.A. Gurskiy, A.V. Denisenko, A.E. Goncharenko, 2024
Системні дослідження та інформаційні технології, 2024, № 2 93
UDC 681.513
DOI: 10.20535/SRIT.2308-8893.2024.2.07
EXPANSION OF THE MATHEMATICAL APPARATUS
OF DISCRETE-CONTINUOUS NETWORKS
FOR THE AUTOMATION OF THEIR SYNTHESIS PROCEDURES
A.A. GURSKIY, A.V. DENISENKO, A.E. GONCHARENKO
Abstract. The paper deals with a model of an intelligent system related to the auto-
matic synthesis of Petri nets and presents a certain stage of developing this model.
The peculiarity of the extended mathematical apparatus is that it contains a combina-
tion of Petri net incidence matrices to represent various algorithms. This combina-
tion of matrices is part of the equations describing the logic control device of a com-
plex system. Accordingly, the work also presents a well-known mathematical
description of discrete-continuous systems with a controlled structure, which in-
cludes certain logical control devices. This mathematical description, based on
means of discrete-continuous networks, is associated with the incidence matrix of
the Petri net, which is formed as a result of a particular synthesis algorithm. At the
same time, the formed Petri net represents the corresponding logical control algo-
rithm that should ensure the effective functioning of the corresponding system. The
final part of the work presents various structural schemes of logic-dynamic models
of systems related to the automatic synthesis of Petri nets. Here, we determine the
features of the advanced mathematical apparatus based on discrete-continuous net-
works to develop an intelligent system that forms logical control algorithms. It is
also noted that such systems can be used to create certain control algorithms that en-
sure increased efficiency of the functioning of some objects in difficult and unpre-
dictable conditions.
Keywords: Petri nets, system with controlled structure, discrete-continuous
network, automatic synthesis of Petri nets.
INTRODUCTION
Petri nets, as an applied mathematical apparatus, are quite well-known in the field
of modeling and analysis of discrete dynamic or logic-dynamic systems. Petri nets
were first proposed by Carl Adam Petri in 1962 as part of his dissertation work –
“Communication with automata”. Petri's work became a significant contribution
to the development of parallel and distributed computing. Such a concept as the
automatic synthesis of Petri nets can be found in the work of James Peterson [1]
as a direction related to the development of certain algorithms. At the same time,
the development of methods for the automatic synthesis of Petri nets entails the
need to expand the mathematical apparatus for describing complex systems,
functioning algorithms that can be represented by Petri nets.
REVIEW OF DISCRETE-CONTINUOUS NETWORKS
Nowadays, Petri nets are greatly expanded. So, for example, there are many
varieties of Petri nets, such as: time Petri nets, inhibitor Petri nets, colored Petri
nets, hybrid Petri nets, and others [2; 3].
A.A. Gurskiy, A.V. Denisenko, A.E. Goncharenko
ISSN 1681–6048 System Research & Information Technologies, 2024, № 2 94
The history of the corresponding scientific direction and the corresponding
scientific thought, starting with Carl Adam Petri, is quite long, so there is no need
to consider separate stages of development or less important elements. But, in this
case, it is necessary to note the invention of a discrete-continuous network [4].
Discrete continuous network (DC-net) proposed in 1990–1993, is essentially a
synthesis of structural schemes of automatic control systems and Petri nets, which
are not the usual extension such as, for example, hybrid Petri nets. DC-net is
primarily a tool for describing, modeling and analyzing logic-dynamic systems
and systems with a controlled structure.
The description and modeling of systems by means of discrete-continuous
networks allows us to imagine a certain class of systems called discrete-
continuous with a controlled structure (DCCS). In English-language publications,
such systems are called hybrid systems, and both traditional Petri nets and
their variants, in individual cases, hybrid Petri nets are used to study such
systems [5; 6].
DC-nets, like Petri nets, is an applied mathematical apparatus and we extent
it in order to develop the technique of automatic synthesis of Petri nets.
The development of models based on such an advanced mathematical
apparatus will allow solving complex problems related to the development of
certain algorithms. As an example, it is worth noting the so-called “smart ant”
problem, which is presented in works [7; 8]. An ant builds an automaton of its
behavior with the help of trial and error and mutations. Thus, assume that a model
built with the use of DC-nets, can synthesize an automaton of its behavior or an
algorithm of some logical control; then we advise to use such a model at the stage
of automated development of control algorithms, control systems, or as a certain
intelligent system [9].
So, it can be noted that in this case it is necessary to expand the
mathematical apparatus while developing methods of automatic synthesis of Petri
nets. The paper is relevant due to the development of certain systems that provide
the synthesis of Petri nets with the use of modern intelligent technologies such as
fuzzy logic, artificial neural networks, genetic algorithms, etc. [10; 11].
Purpose of work: The purpose of this work is to minimize time and auto-
mate the process of synthesis of some control algorithms of complex systems.
To achieve the goal, we expand the mathematical apparatus of discrete-
continuous networks, taking into account the procedure of automatic synthesis of
Petri nets.
MAIN PART
Description of the system with a controlled structure
Modeling tools of DC-nets allow to present a model of complex technical systems
consisting of two parts: continuous-event part (CEP) and discrete-event part
(DEP) in a structural unity. Such a system was called a system with a controlled
structure (SCS). The continuous-event part of the model represents the control
object with a controlled structure (COCS) and the DEP of DC-net represents the
logical control device (LCD).
The COCS is represented by the state and output equations:
Expansion of the mathematical apparatus of discrete-continuous networks…
Системні дослідження та інформаційні технології, 2024, № 2 95
))(),(),(()( tutxtftx k ; (1)
))(),(()( txtfty k , (2)
where )(tu is continuous control vector; )(tx is state vector; )(ty is output
vector; )( kt is vector function for controlling the structure of COCS
(functioning modes). Accordingly, in such a system it is possible to identify a
generalized input effect:
))(),(( kttuU .
The LCD is represented by a finite state machine (3),(4) characterized by the
equations:
),( 1 kkk aa , (3)
),( kkk a , (4)
where },...,...,,{ 21 ik aaaaA is a finite set of internal states, },...,,...,,{ 21 jk
is an iput alphabet, },...,,...,,{ 21 vk is an output alphabet, is a transition
function (from state to another state), is an output function.
The presented formal form of equations (1)–(4) is appropriate without taking
into account the procedure for automatic synthesis of LCD control algorithm and,
accordingly, the automatic generation of a Petri net during the operation of SCS.
Mathematical description of discrete-continuous systems with a controlled
structure with the use of the DC-net
The mathematical description of a discrete-continuous system with a controlled
structure, taking into account the means of DC-net, can be obtained from a set of
equations.
Dynamics of COCS in continuous space )( , kttX can be represented in
matrix-differential form by the equation of state:
)())(()())(()( ,2,1, kko
d
okko
d
ok ttutuBttXtuAttX , (5)
output equation
)())(()( ,3, kko
d
ok ttXtuCttY , (6)
equation of state of the LCD
)()()()()( 1 kL
d
kL
d
kL
d
LkL
d
kL
d twtutvAtXtX , (7)
and the LCD output equation
)()( kL
d
kL
d tXtY , (8)
where )( , kttX is vector of continuous event state of COCS; )( , kttY is
continuous event output vector; )( , kttu is continuous exposure vector;
00
2
0
10 ... NAAAA ; 00
2
0
10 ... NBBBB ; 00
2
0
10 ... NCCCC ;
00
2
0
1 ..., NAAA , 00
2
0
1 ..., NBBB and 00
2
0
1 ..., NССС — matrices of states, controls and
A.A. Gurskiy, A.V. Denisenko, A.E. Goncharenko
ISSN 1681–6048 System Research & Information Technologies, 2024, № 2 96
outputs of different structural operating modes; ))(( ko
d tu — vector function for
managing structural changes (functioning modes of SCS); ))((1 ko
d tu
T11
2
1
1 ... N and
T22
2
2
12 ...))(( Nko
d tu is vector-functions of
structure control depending on the discrete state )(0 k
d tX continuous event part.
They implement the selection of a specific structure from a variety of structures
N
ii 1}{ using matrix multiplication 000 ,, CBA . Vector function control
))(( ko
d tu is a matrix of dimension 1n , that contains only one non-zero
element. The dimension of the control function vector is consistent with the
dimension of the matrices 000 ,, CBA ; )( ko
d tu is a discrete component of the
control influence on CEP, transferring the system from one structure
to another; )( kL
d tw is external control action; )( kL
d tX , )( 1kL
d tX —
preliminary and subsequent discrete state (labeling) of the Petri net; LA is
incidence matrix reflecting the relationship of elements in DC-net; )( kL
d tv is
control vector in DC-net. A simplified block diagram of the SCS, according to the
given mathematical description, is presented in Fig. 1.
Expansion of the mathematical apparatus for describing complex systems
based on procedures for automatic synthesis of Petri nets
These equations (5)–(8) are the basis for expanding the mathematical apparatus
taking into account the procedures for automatic synthesis of Petri nets.
During the automatic synthesis of a Petri net, the dimension of vectors
)( kL
d tX , )( kL
d tu and matrices LA may not change, but the elements of the
matrix must change during different runs of the system model. In this case, the
incidence matrix LA may be 1LA , 2LA . Thus LnLLL AAAA ,...,, 21 , in
this case, the value of n is unknown in advance and depends on the task at hand.
Fig. 1. Simplified block diagram of the logic-dynamic model
Expansion of the mathematical apparatus of discrete-continuous networks…
Системні дослідження та інформаційні технології, 2024, № 2 97
Matrix selection LiA , where ni ,,1 , depends on initial conditions S ,
where T
21 ]...[ nsssS — a vector of initial conditions generated by some
expert system. Taking into account the automatic synthesis of the Petri net,
equation (7) will take the following form:
)()()()))((()()( 1 kL
d
kL
d
kL
d
kS
dd
LkL
d
kL
d twtutvtuSWtXtX , (9)
where ]...[ 21 LnLLL AAAW ; according LikS
dd
L AtuSW ))(( , where
ni ,,1 , T
nkS
dd ssstuS ]...[))(( 21 , 1
1
n
i
is , 0
1
n
i
is .
Vector function ))(( kS
d tuS contains elements
,)()(at0
;)()(at1
givenkS
d
kS
d
givenkS
d
kS
d
i
tutu
tutu
s
where ]...[)( 21 Sm
d
S
d
S
d
kS
d uuutu ; )),(( wtJfu DCSi
d ; givenkS
d tu )(
— tasks vector;
4
3
2
1
),,(),,()(
t
t
t
t
dttuyfdttuyftJ is an/the increment of the
system perfomance criterion; w — external influence from the expert; DCf —
function of continuous-discrete transformation of variables.
The equation (9) implies that the simplified block diagram presented in
Fig. 1 is transformed as shown in Fig. 2.
To visualize the Petri net synthesis process, state variables ,)( kL
d tX
)( 1kL
d tX and increase in the value of the system performance criterion J can
be output using a parametric file to the visualization platform, as shown in [12].
Visualization of the process of Petri net synthesis based on the corresponding
Flash or Unity platform is an important component of the interaction of an expert
with an intelligent system.
Fig. 2. Simplified block diagram of the logical-dynamic model of the system,
implementing the automatic synthesis of Petri nets
A.A. Gurskiy, A.V. Denisenko, A.E. Goncharenko
ISSN 1681–6048 System Research & Information Technologies, 2024, № 2 98
Research results
The consideration of the extended mathematical description of complex systems
provides the basis for constructing a model of an intelligent system intended for
the formation of some logical control algorithms.
These algorithms are formed according to certain methods through the
automatic synthesis of Petri nets. Combination of different incidence matrices
LiA , where ni ,,1 , that are included in equation (9) represents a set of
various algorithms that can be adjusted during the functioning of the intelligent
system. The block diagram presented in Fig. 2 can represent this intelligent
system, taking into account the additional procedure for correcting the incidence
matrix; LiA implemented, as shown in [11], taking into account the functioning
of an artificial neural network and its training. Such a system can be used to form
certain control algorithms that provide increased operating efficiency in some objects.
CONCLUSIONS
In this work we expand the mathematical description of complex technological
systems with the use of DC-nets, taking into account the automatic synthesis of
Petri nets.
Expanding the mathematical description of complex technological systems
with the use of DC-nets makes it possible to approach the solution of a practical
problem associated with the development of an intelligent system that
automatically synthesizes some logical control algorithms.
In turn, the use of such an intelligent system makes it possible to achieve a
certain goal of work, which is to minimize the time and material costs for the
development of logical control algorithms.
Further scientific research should be related to the development of methods
for the synthesis of Petri nets and some algorithms used in control systems.
REFERENCES
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strukturi [Discrete-continuous system with controlled structure]. Kyiv: Naukova
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gramming for automatic construction of an automaton in the problem of “intelligent
ants”]. M.: Fizmatlit, 2007, pp. 590–597.
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baze sredstv diskretno-nepreryvnyh setej pri formirovanii algoritma avtomaticheskoj
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tuning of the coordinating control system],” Electrotechnic and Computer Systems,
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tion of Petri nets for manufacturing system control and scheduling,” International
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functioning of artificial neural network,” Radio Electronics, Computer Science, Con-
trol, issue 2/2021, pp. 84–92, 2021. doi: 10.15588/1607-3274-2021-2-9
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10.20535/SRIT.2308-8893.2021.2.06
Received 10.01.2024
INFORMATION ON THE ARTICLE
Alexander A. Gurskiy, ORCID: 0000-0001-5158-2125, Odessa National University of
Technology, Ukraine, e-mail: gurskiya2017@gmail.com
Andrey V. Denisenko, ORCID: 0000-0002-8610-0082, National University “Odessa
Polytechnics”, Ukraine, e-mail: denisenko.a.v@op.edu.ua
Alexander E. Goncharenko, ORCID: 0000-0003-4959-6469, Odessa National
University of Technology, Ukraine, e-mail: kholod.automatic@gmail.com
РОЗШИРЕННЯ МАТЕМАТИЧНОГО АПАРАТУ ДИСКРЕТНО-НЕПЕРЕРВНИХ
МЕРЕЖ ДЛЯ АВТОМАТИЗАЦІЇ ПРОЦЕДУР ЇХ СИНТЕЗУ/ O.O. Гурський,
А.В. Денисенко, О.Є. Гончаренко
Анотація. Подано певний етап розроблення моделі інтелектуальної системи,
пов’язаної з автоматичним синтезом мереж Петрі. Розглянуто розширений ма-
тематичний опис складних систем на основі засобів дискретно-неперервних
мереж, який покладено в основу розроблення такої інтелектуальної системи,
спрямованої передусім на формування алгоритмів логічного керування. Особ-
ливість розширеного математичного апарату полягає у тому, що у його складі
є комбінація матриць інцидентності мереж Петрі для подання різноманітних
алгоритмів. Ця комбінація матриць входить до складу рівнянь, що описує при-
стрій логічного керування складної системи. Відповідно подано відомий мате-
матичний опис дискретно-неперервних систем із керованою структурою, що
включають певні пристрої логічного керування. Цей математичний опис на
основі засобів дискретно-неперервних мереж, пов’язаний з матрицею інциден-
тності мережі Петрі, що формується в результаті певного алгоритму синтезу.
Сформовано мережу Петрі — відповідний алгоритм логічного керування для
забезпечення процесу ефективного функціонування відповідної системи. По-
дано різні структурні схеми логіко-динамічних моделей систем з автоматич-
ним синтезом мереж Петрі. Визначено особливість розширеного математично-
го апарату на основі дискретно-неперервних мереж для розроблення
інтелектуальної системи, що формує алгоритми логічного керування. Такі сис-
теми можна використовувати для формування певних алгоритмів керування,
які забезпечують підвищену ефективність функціонування деяких об’єктів.
Ключові слова: мережі Петрі, система з керованою структурою, дискретно-
неперервна мережа, автоматичний синтез мереж Петрі.
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| id | journaliasakpiua-article-309718 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:28:34Z |
| publishDate | 2024 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/87/28ff21cf418bc19ba1d2935fd13c2987.pdf |
| spelling | journaliasakpiua-article-3097182024-08-11T01:12:49Z Expansion of the mathematical apparatus of discrete-continuous networks for the automation of their synthesis procedures Розширення математичного апарату дискретно-неперервних мереж для автоматизації процедур їх синтезу Gurskiy, Alexander Denisenko, Andrey Goncharenko, Alexander мережі Петрі система з керованою структурою дискретно-неперервна мережа автоматичний синтез мереж Петрі Petri nets system with controlled structure discrete-continuous network automatic synthesis of Petri nets The paper deals with a model of an intelligent system related to the automatic synthesis of Petri nets and presents a certain stage of developing this model. The peculiarity of the extended mathematical apparatus is that it contains a combination of Petri net incidence matrices to represent various algorithms. This combination of matrices is part of the equations describing the logic control device of a complex system. Accordingly, the work also presents a well-known mathematical description of discrete-continuous systems with a controlled structure, which includes certain logical control devices. This mathematical description, based on means of discrete-continuous networks, is associated with the incidence matrix of the Petri net, which is formed as a result of a particular synthesis algorithm. At the same time, the formed Petri net represents the corresponding logical control algorithm that should ensure the effective functioning of the corresponding system. The final part of the work presents various structural schemes of logic-dynamic models of systems related to the automatic synthesis of Petri nets. Here, we determine the features of the advanced mathematical apparatus based on discrete-continuous networks to develop an intelligent system that forms logical control algorithms. It is also noted that such systems can be used to create certain control algorithms that ensure increased efficiency of the functioning of some objects in difficult and unpredictable conditions. Подано певний етап розроблення моделі інтелектуальної системи, пов’язаної з автоматичним синтезом мереж Петрі. Розглянуто розширений математичний опис складних систем на основі засобів дискретно-неперервних мереж, який покладено в основу розроблення такої інтелектуальної системи, спрямованої передусім на формування алгоритмів логічного керування. Особливість розширеного математичного апарату полягає у тому, що у його складі є комбінація матриць інцидентності мереж Петрі для подання різноманітних алгоритмів. Ця комбінація матриць входить до складу рівнянь, що описує пристрій логічного керування складної системи. Відповідно подано відомий математичний опис дискретно-неперервних систем із керованою структурою, що включають певні пристрої логічного керування. Цей математичний опис на основі засобів дискретно-неперервних мереж, пов’язаний з матрицею інцидентності мережі Петрі, що формується в результаті певного алгоритму синтезу. Сформовано мережу Петрі — відповідний алгоритм логічного керування для забезпечення процесу ефективного функціонування відповідної системи. Подано різні структурні схеми логіко-динамічних моделей систем з автоматичним синтезом мереж Петрі. Визначено особливість розширеного математичного апарату на основі дискретно-неперервних мереж для розроблення інтелектуальної системи, що формує алгоритми логічного керування. Такі системи можна використовувати для формування певних алгоритмів керування, які забезпечують підвищену ефективність функціонування деяких об’єктів. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2024-06-28 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/309718 10.20535/SRIT.2308-8893.2024.2.07 System research and information technologies; No. 2 (2024); 93-99 Системные исследования и информационные технологии; № 2 (2024); 93-99 Системні дослідження та інформаційні технології; № 2 (2024); 93-99 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/309718/301145 |
| spellingShingle | мережі Петрі система з керованою структурою дискретно-неперервна мережа автоматичний синтез мереж Петрі Gurskiy, Alexander Denisenko, Andrey Goncharenko, Alexander Розширення математичного апарату дискретно-неперервних мереж для автоматизації процедур їх синтезу |
| title | Розширення математичного апарату дискретно-неперервних мереж для автоматизації процедур їх синтезу |
| title_alt | Expansion of the mathematical apparatus of discrete-continuous networks for the automation of their synthesis procedures |
| title_full | Розширення математичного апарату дискретно-неперервних мереж для автоматизації процедур їх синтезу |
| title_fullStr | Розширення математичного апарату дискретно-неперервних мереж для автоматизації процедур їх синтезу |
| title_full_unstemmed | Розширення математичного апарату дискретно-неперервних мереж для автоматизації процедур їх синтезу |
| title_short | Розширення математичного апарату дискретно-неперервних мереж для автоматизації процедур їх синтезу |
| title_sort | розширення математичного апарату дискретно-неперервних мереж для автоматизації процедур їх синтезу |
| topic | мережі Петрі система з керованою структурою дискретно-неперервна мережа автоматичний синтез мереж Петрі |
| topic_facet | мережі Петрі система з керованою структурою дискретно-неперервна мережа автоматичний синтез мереж Петрі Petri nets system with controlled structure discrete-continuous network automatic synthesis of Petri nets |
| url | https://journal.iasa.kpi.ua/article/view/309718 |
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