Building an "input - state space - output" model based on the properties of linear operators using Hankel matrices.

Keywords: linear operator, projection operator, Hankel matrix, matrix rank, principal minors, dynamical system


The article is devoted to solving the problem of analyzing the structure of a dynamic object: taking into account the stochastic approach to the analysis of output signals and without taking into account the random components of the output signal based on linear mappings of a set of linear spaces, that is, the set-theoretic approach.

The stated problem of finding the structure of a dynamic object from the output signal was studied by the method of factorization of the correlation matrix of the output signal [1]. The method considered earlier and the methods considered in this work refer to inverse problems of studying dynamical systems, the essence of which is that the original observed signal is a solution to the dynamic operator of the object, and the structure of the operator itself is unknown. At the same time, there are some assumptions about its class: linear differential, nonlinear differential and differential in partial derivatives, and others.

The heuristic approach is based on the fact that the input signal acts on the object, while collecting information about all degrees of freedom of the dynamic uncontrolled object. Such an input signal, which has an infinite spectrum, is white noise. The article considers a method for finding the structure of an operator and estimating its parameters for the linear case and a method for constructing an "input - state space - output" model of a multidimensional dynamical system. The sequence of constructing a model of the operator of a linear dynamic system as a solution to the inverse problem of dynamics - to determine the structure of the operator in the state space from the output signal, will allow developing information technologies for real dynamic systems in a linear approximation.


1. Marasanov V.V., Zabytovskaya O.I., Dymova A.O. (2012) Forecasting the structure of dynamical systems. Vestnik KhNTU No. 1 (44), Pp. 292-302.
2. Gametsky A.F., Solomon D.I. (1997) Mathematical modeling of macroeconomic processes. Chisinau: Eureka. 313 p.
3. Dymova Н.O. (2018) A method for finding a model of a dynamic object from an output signal. Measuring and computing equipment in technological processes: Materials XVIII International. science and technology conference (June 8-13, 2018, Odessa); Odessa national Acad. communication named after O.S.Popova. Odesa–Khmelnytskyi: KhNU. Pр. 202-204.
4. Neimark M.A. (1969) Linear differential operators. Moscow: Nauka. 526 p.
5. Demidovich B.P., Maron I.A. (1966) Fundamentals of Computational Mathematics. Moscow: Nauka. 664 p.
6. Willems Jan K. (1989) From time series to linear system. Theory of systems. Mathematical methods and modeling. Digest of articles. Moscow: Mir. 384 p.
7. Gantmakher F.R. (2004) Matrix theory. Moscow: FIZMATLIT. 560 p.
8. Lancaster P. (1978) Matrix Theory. Moscow: Nauka. 280 p.
9. Bellman R. (1969) An introduction to matrix theory. Moscow: Nauka. 368 p.
10. Dymova A. O. (2019) Projection methods for describing the structure of an operator of linear dynamical systems. Visnyk KrNU named after Mikhail Ostrogradsky. Issue 6/2019 (119). Pp. 152-160.

Abstract views: 0
PDF Downloads: 0
How to Cite
Dymova, H. (2022). Building an "input - state space - output" model based on the properties of linear operators using Hankel matrices . COMPUTER-INTEGRATED TECHNOLOGIES: EDUCATION, SCIENCE, PRODUCTION, (48), 59-63.
Computer science and computer engineering