Wavelet analysis in linear dynamic systems control and diagnostic problems

Mathematica modeling, numerical technique and program complexes


Аuthors

Konysheva V. Y.*, Maximov N. А.**, Sharonov A. V.***

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: konysheva.vict@gmail.com
**e-mail: N-A-Maximov47@yandex.ru
***e-mail: anatoly.sharonov@yandex.ru

Abstract

The paper considers one of the possible approaches to solving the problem of linear dynamical systems control and diagnostics. Analysis of available publications [2, 4, 9] on the subject revealed that the statement of the problem of diagnostics assumes knowing of the current values of the dynamic systems state vector. However, measuring all the state vector coordinates is not always possible. Firstly, such situation may arise due to the lack of “access points”. Secondly, if the measurement results of some of the state vector coordinates contain “unacceptable” errors, which do not allow them to be used to solve problems of control and diagnostics of objects.

The proposed approach presupposes the existence of a control points shortage, but requires the observance of Kalman’s observability conditions, which allows restoring all the coordinates of the state vector. With this regard, the problem posed combines three problems: the problem of estimating the coordinates of the state vector, the problem of determining the altered parameters of the object, and the problem of determining the moments of appearance of these alterations (the moments of “discord”). Solution of the first problem is traditionally associated with the constructing the Kalman filter. To solve the second problem, the authors propose to apply the equations of parametric sensitivity. To localize the instants of the onset of such “discords”, the article proposes to apply the expansion of the parametric sensitivity functions in Fourier series with respect to the orthonormal wavelet basis.

The results of mathematical modeling of the solution of the problem of control and diagnostics of the simplest linear dynamical systems confirmed the operability of the proposed approach.

Keywords:

linear dynamical system, control points, parametric sensitivity equations, parametric sensitivity functions, wavelet analysis

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