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Remiantis sinchronizacijos sistemos struktūrine schema sudaromas jos matematinis modelis – matricinė diferencialinė lygtis su vėluojančiu argumentu. Ši lygtis yra sprendžiama nuoseklaus integravimo metodu ir gauti rezultatai lyginami su rezultatais gautais metodu, paremtu dde23 funkcijos taikymu. Sprendžiant matricinę diferencialinę lygtį nuoseklaus integravimo metodu intervalas dalijamas į vėlavimo ilgio dalinius intervalus ir kiekviename intervale lygtis sprendžiama atskirai, kaip matricinė diferencialinė lygtis be vėluojančio argumento. Ieškodami sprendinio kiekviename intervale, taikome Laplaso transformaciją. Naudojantis matricinės diferencialinės lygties su vėluojančiu argumentu sprendiniu, išvedame sinchronizacijos sistemos pereinamųjų funkcijų ir generatorių virpesių fazių skirtumų analizines išraiškas. Remiantis gautomis išraiškomis atliekame pereinamųjų procesų tyrimus., Time delay is often encountered in various technical systems, such as electric, pneumatic and hydraulic networks, chemical processes, long transmission lines, robotics, etc. The existence of pure time lag, regardless if it is present in the control or/and the state, may cause undesirable system transient response, or even instability. The aim of this work is to investigate the mutual synchronization system, composed of five oscillators. The mathematical model of the synchronization system is the matrix differential equation with delayed argument. It is solved applying the Laplase transformation method. Using this method the analytical expressions for step responses and differences between phases are found. The results of consequent integration method are compared with the results got using method with dde23 function.

Darbe nagrinėjama ryšio tinklo tarpusavio sinchronizacijos sistema, sudaryta iš penkių į grandinę sujungtų generatorių. Darbo tikslas – sudaryti sinchronizacijos sistemos matematinį modelį, rasti sistemos pereinamųjų funkcijų ir generatorių virpesių fazių skirtumų tikslias analizines išraiškas, ištirti sistemos pereinamuosius procesus ir jos darbą nusistovėjusiame režime. Nagrinėjamos tarpusavio sinchronizacijos sistemos matematinis modelis – matricinė diferencialinė lygtis su vėluojančiu argumentu. Šią lygtį sprendžiame taikant Laplaso transformaciją ir „žingsnių“ metodą. Remdamiesi surastu sprendiniu, randame sinchronizacijos sistemos pereinamųjų funkcijų, vidinių ryšių struktūrą nusakančios matricos ir generatorių virpesių fazių skirtumų tikslias analizines išraiškas. Remiantis sinchronizacijos sistemos pereinamosiomis funkcijomis, tiriami sistemos pereinamieji procesai, atsirandantys įjungus sistemos generatorių valdymą., The mathematical model of the synchronization system composed of five joined into a chain oscillators is investigated. The mathematical model of the synchronization system is the system of matrix differential equation with delayed arguments. The solution of the matrix differential equation is obtained applying the Laplace transform and the “steps“ method. The analytical expressions of the system‘s step response functions, the matrix and the phase differences of the oscillators are obtained. Steady values of the step response functions and steady values of the phase differences are obtained applying the limiting value theorem. The transition processes of the system with equal and different delays are investigated.

Darbe nagrinėjama ryšio tinklo sinchronizacijos sistema, sudaryta iš keturių tarpusavyje sinchronizuotų generatorių. Darbo tikslas - sudaryti sinchronizacijos sistemos matematinį modelį, rasti sistemos pereinamųjų funkcijų ir generatorių virpesių fazių skirtumų tikslias analizines išraiškas, ištirti sistemos pereinamuosius procesus ir jos darbą nusistovėjusiame režime. Nagrinėjamos sistemos matematinis modelis - matricinė diferencialinė lygtis su vėluojančiu argumentu. Ši lygtis sprendžiama „žingsnių“ metodu. Tuo tikslu intervalas dalijamas į vienodo ilgio dalinius intervalus. Kiekviename daliniame intervale lygtis sprendžiama atskirai, kaip paprasta matricinė diferencialinė lygtis be vėluojančio argumento. Naudojantis gautu sprendiniu, rastos tikslios sinchronizacijos sistemos pereinamųjų funkcijų analizinės išraiškos. Remiantis šiomis išraiškomis, atliktas pereinamųjų procesų sinchronizacijos sistemoje tyrimas., The mutual synchronization system of the communication network, composed of four oscillators, is investigated in the work. Following aims were raised in this work: to compose the mathematical model of the synchronization system, to find the exact expressions of the step responses of the system and of the phase differences of the oscillations of oscillators, to investigate the transients in the synchronization system. The mathematical model of the system is the matrix differential equation with delayed argument. This equation is solved applying the method of consequent integration (method of “steps”). For this purpose the interval is divided to subintervals of equal length. The equation in each subinterval is solved separately, as ordinary differential equation without delayed argument. Using obtained expression of the solution, the expressions of step responses of the system are got. The expressions of step responses are applied investigating the transients in the synchronization system.

The mutual synchronization system composed of four oscillators is analysed in the work. The matrix differential equation with delayed arguments is the mathematical model of the synchronization system. The solution of the matrix differential equation is obtained applying the method of “steps” and Laplace transform. The solution involves the powers of the matrix describing the structure of internal links of the synchronization system. The powers of this matrix are calculated taking into account the eigenvalues and eigenvectors of the matrix. Applying solution of the matrix differential equation the step responses matrix of the system is obtained and transient responses are analysed. The dependence of the phase differences of the oscillators on initial conditions is analysed.

There are numerous synchronization methodologies available in solving the problem of network synchronization. This paper presents an analytical analysis of the mathematical model of forced synchronization system composed of four oscillators. Exact analytical and graphical expressions of transition functions and exact expressions of the phase differences between signals of oscillators of this synchronization system are obtained.

SUMMARY There are numerous methodologies available in solving the problem in synchronizing timing oscillators of the communication networks: pleziosynchronization, forced synchronization or mutual synchronization. In the presented work the mutual synchronization system composed of four oscillators is analyzed. The mathematical model of the synchronization system is matrix differential equation with delayed argument. Applying the method of “steps” and Laplace transform we find the solution of the matrix differential equation and the step responses matrix of the synchronization system. Exact analytical and graphical expressions of transition functions and exact expressions of the phase differences between signals of oscillators of this synchronization system are obtained.

The mutual in the synchronization system is analyzed in this work. The mutual in the synchronization is a double – side synchronization of interdependent tactical oscillators. Every of interdependent oscillators has influence on the frequency of other oscillators. The frequency of all oscillators will be the same and equal to the certain medium frequency which depends upon, the frequency and distances among separate oscillators of all in the settled regime of them joined into the net. The objective of the work is to analyze the synchronization system which is joined of four interdependent oscillators. The mathematical model of synchronization system is formed, analytic expressions of transitional function and differences among phases of oscillation are set transitional processes of synchronization system and differences among phases of oscillation are researched. The mathematical model of synchronization system is matrixal rectilinear dif. Equation with the argument to be late. It is solved using the method of steps and the transformation of Laplace. Exact analytical expressions of the solution components are found using Jordan’s form of descriptive matrix which describes the structure of inherent relations among the blocks of the system analytical expressions of transitional functions in the system and differences of phases of oscillation are obtained on the basis of the solution.