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METHOD FOR IDENTIFICATION OF NONLINEAR NONAUTONOMOUS DYNAMICAL SYSTEM
09.06.2025 21:32
[1. Systemy i technologie informacyjne]
Автор: Viktor Gorodetskyi, PhD National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv, Ukraine; Mykola Osadchuk, PhD National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv, Ukraine
ORCID 0000-0003-4642-3060 Viktor Gorodetskyi
ORCID 0000-0002-3409-9315 Mykola Osadchuk
Identification of ordinary differential equation (ODE) system from time series of its variables is actual in many fields of science and engineering. This problem becomes more complex for nonautonomous ODE systems, which are depending not only on constant coefficients, but also depending on variable input actions. It was noted in [1] that the identification of a nonautonomous ODE system in most cases requires simultaneous knowledge of the time series of input actions as well as observed variables of the system.
Consider ODE system, which have variables , 
and all of them are observable. ODE system have polynomial right-hand sides of equations with 
constant coefficients 
-1 in each equation. Some constant coefficients 
(one or several) are replaced with unknown time series of input actions 
.
and all of them are observable. ODE system have polynomial right-hand sides of equations with constant coefficients -1 in each equation. Some constant coefficients (one or several) are replaced with unknown time series of input actions .The aim of the research is to develop nonautonomous ODE system identification method, which can identify structure of system's equations, numerical values of constant coefficients 
and time series of unknown input actions 
from time series of observable variables 
. It is assumed, that rate of change for unknown input actions 
is lower comparing to rate of change for observable variables 
.
and time series of unknown input actions from time series of observable variables . It is assumed, that rate of change for unknown input actions is lower comparing to rate of change for observable variables .To solve the problem under consideration, a nonautonomous ODE system identification method was proposed in [2], which is based on idea from [3]. Let time series 
have a time duration 
. Let also a section (a "window") of length 
, starting from a certain time
, with 
is selected from the time series. Let some well-known method for identifying an autonomous ODE system, for example, a least squares method (LSM), is applied within the window. As a result, the constant coefficients values 
will be obtained, which are related to the window position 
. When start of the window is shifted to position
, a constant coefficients values 
will be obtained and so on. Thus, repeating of window shifting and an autonomous ODE system identification allow to obtain time series 
. Article [2] demonstrate, that correct result of identification of input actions 
should be shifted in time 
on some unknown time 
. The analytic study of system with additive input action, where 
, show that time shift is 
. This means that the result of autonomous system identification 
should be positioned at the middle of the window. Basing on the significance [4] values and analysis of graphs 
, it is possible to determine, which coefficients of unknown ODE system are zero, which are constants 
and which are input actions 
.
have a time duration . Let also a section (a "window") of length , starting from a certain time , with is selected from the time series. Let some well-known method for identifying an autonomous ODE system, for example, a least squares method (LSM), is applied within the window. As a result, the constant coefficients values will be obtained, which are related to the window position . When start of the window is shifted to position , a constant coefficients values will be obtained and so on. Thus, repeating of window shifting and an autonomous ODE system identification allow to obtain time series . Article [2] demonstrate, that correct result of identification of input actions should be shifted in time on some unknown time . The analytic study of system with additive input action, where , show that time shift is . This means that the result of autonomous system identification should be positioned at the middle of the window. Basing on the significance [4] values and analysis of graphs , it is possible to determine, which coefficients of unknown ODE system are zero, which are constants and which are input actions .The proposed method was used for identification of several nonlinear ODE systems with 
observable variables. For a system with additive input action, where 
, a structure of equations was identified, a numerical values of constant coefficients 
were determined and time series of input action 
was identified. For a system with multiplicative input action, where 
, an identification was performed and a relation 
, which was analytically obtained, was experimentally confirmed. For a system with two input actions, where
, time series of input actions 
were identified simultaneously. For an ODE system, in which all 6 constant coefficients
were replaced with input actions
, time series of all input actions were identified.
observable variables. For a system with additive input action, where , a structure of equations was identified, a numerical values of constant coefficients were determined and time series of input action was identified. For a system with multiplicative input action, where , an identification was performed and a relation , which was analytically obtained, was experimentally confirmed. For a system with two input actions, where , time series of input actions were identified simultaneously. For an ODE system, in which all 6 constant coefficients were replaced with input actions , time series of all input actions were identified.The proposed method allows to identify a nonautonomous ODE system with unknown input actions, which have low rate of change. Unlike [3], this method allows to identify not only time series of input actions, but also a structure of equations and numerical values of constant coefficients in the system. Method can be generalized by using of identification methods, other that LSM, within the window. Another way to generalize the proposed method is to use it for identification of nonautonomous ODE system from a single observable variable.
References
1. L. A. Aguirre, C. Letellier, Modeling Nonlinear Dynamics and Chaos: A Review // Mathematical Problems in Engineering. – vol. 2009, article ID 238960. – 35 p. – DOI: https://doi.org/10.1155/2009/238960
2. V. Gorodetskyi, M. Osadchuk, Identification of oscillatory systems with unknown continuous input actions // International Journal of Dynamics and Control. – 2024. – vol. 12. – P. 3534-3545. – DOI: 10.1007/s40435-024-01458-9
3. V. S. Anishchenko, A. N. Pavlov, Global reconstruction in application to multichannel communication // Phys. Rev. E. – 1998. – vol. 57. – N 2. – P. 2455-2457.
4. C. Lainscsek, C. Letellier, I. Gorodnitsky, Global modeling of the Rössler system from the z-variable // Physics letter A. – 2003. – vol. 314. – P. 409-427.
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