STUDY OF THE AREA OF DETERMINATION OF PARAMETERS OF THE BASIS FUNCTION OF TWO ARGUMENTS IN CONSTRUCTING THE Λ-ORTHOGONAL BASIS FUNCTION
Abstract
The purpose of the research is to determine the area of setting the argument τ and its discretization
intervals when constructing an algorithm for adaptive nonlinear
optimal smoothing of multi-parameter data of trajectory measurements, which makes it possible to
jointly implement the spatial and temporal redundancy of the data obtained. The research was
carried out by constructing a Λ-orthogonal basis function in order to obtain independent estimates
for the coefficients of the smoothing polynomial. It is shown that it is advisable to solve the problem
of determining the maximum likelihood estimate of the coefficient vector of the smoothing
polynomial by the method of successive approximations. When constructing a Λ-orthogonal basis
function, the maximum likelihood estimate of the coefficient vector of the smoothing polynomial is
achieved in 2-3 iterations. It follows from the research results presented in the paper that the accuracy
index Qт as a function of two arguments ( is the smallest value of the argument and Δτ is
the discretization interval of the argument τ) in a wide range of values of these arguments changes
slightly, but increases sharply at . In this case, the values of these arguments should not
exceed, respectively, the maximum and minimum possible numbers that can be written without loss
of accuracy in the re grid of the computer used. With a uniform discretization step of the argument τ it is advisable to choose the argument in the middle part of the interval¸where and
respectively, are the minimum and maximum numbers that can be written into the bit grid of a
computer without loss of accuracy. In case of adverse conditions Approaching the edges of the
interval can lead to an increase in calculation errors in determining the secondary parameters of
the position of the aircraft due to the fact that the main matrix of the system of equations becomes
ill-conditioned.
References
traektornykh izmereniy [Method of non-linear smoothing in data processing of trajectory
measurements], Sb. nauchnykh trudov [Collection of scientific papers]. Issue 28. Donetsk:
Donetskiy institut zheleznozhdorozhnogo transporta, 2011, pp. 93-101.
2. Mil'shteyn A.V., Motylev K.I., Paslen V.V. Issledovanie struktur bazisnykh funktsiy [Investigation
of the structures of basic functions], Sb. nauchnykh trudov [Collection of scientific papers]. Issue
29. Donetsk: Donetskiy institut zheleznozhdorozhnogo transporta, 2012, pp. 23-30.
3. Ogodniychuk N.D. Obrabotka traektornoy informatsii [Processing of trajectory information].
Part 1. Kiev: KVVAIU, 1981, 141 p.
4. Ogodniychuk N.D. O prikladnykh metodakh analiza traektornoy informatsii [On applied methods
for analyzing trajectory information], Sb. materialov NTK, posvyashchennoy 25-letiyu
uchilishcha [Collection of materials of the Scientific and Technical Committee dedicated to
the 25th anniversary of the school]. Part 1. Kiev: KVVAIU, 1977. – S. 65-84.
5. Shcherbov I.L. Issledovanie algoritma adaptivnogo nelineynogo optimal'nogo sglazhivaniya
mnogoparametricheskikh dannykh izmereniy [Study of the adaptive nonlinear optimal smoothing
algorithm for multi-parameter measurement data], Informatika i kibernetika [Informatics
and cybernetics], 2020, No. 4 (22), pp. 5-12.
6. Shcherbov I.L., Paslen V.V., Motylev K.I., Mikhaylov M.V. Metody obrabotki dannykh
izmereniy, kotorye vladeyut prostranstvennoy i vremennoy izbytochnost'yu [Measurement
data processing methods that have spatial and temporal redundancy], Sistemnye tekhnologii:
Regional'nyy mezhvuzovskiy sbornik nauchnykh rabot [System Technologies. Regional
interuniversity collection of scientific papers]. Issue 5 (46). Dnepropetrovsk, 2006, pp. 95-100.
7. Zhdanyuk B.F. Osnovy statisticheskoy obrabotki traektornoy informatsii [Fundamentals of
statistical processing of trajectory measurements]. Moscow: Sov. radio, 1978, 384 p.
8. Ogodniychuk N.D. Obrabotka traektornoy informatsii [Processing of trajectory information].
Part II. Kiev: KVVAIU, 1986, 224 p.
9. Svetozarov V.V. Osnovy statisticheskoy obrabotki rezul'tatov izmereniy [Fundamentals of
statistical processing of measurement results]. Moscow: MIFI, 2005, 54 p.
10. Savchuk V.P. Obrabotka rezul'tatov izmereniy [Processing of measurement results]. Odessa:
ONPU, 2002, 54 p.
11. Algazin S.D. Chislennye algoritmy bez nasyshcheniya v klassicheskikh zadachakh
matematicheskoy fiziki [Unsaturated numerical algorithms in classical problems of mathematical
physics]. Moscow: Nauchnyy Mir, 2002, 155 p.
12. Grantmakher F.R. Teoriya matrits [Matrix Theory]. Moscow: Fizmatlit, 2004, 560 p.
13. Lankaster P. Teoriya matrits [Matrix Theory]. Moscow: Nauka. Glav. red. fiz.-mat. lit., 1962,
280 p.
14. Volkov E.A. Chislennye metody [Numerical methods]. Moscow: Nauka. Glav. red. fiz.-mat.
lit., 1987, 248 p.
15. Shcherbov I.L., Paslen V.V., Mikhaylov M.V., Motylev K.I., Lebedenko D.M., Antikuz A.G.
O postroenii ortogonal'nykh bazisnykh funktsiy [About the construction of orthogonal basis
functions], Tupolevskie chteniya: Mezhdunarodnaya molodezhnaya nauchnaya konferentsiya,
posvyashchennaya 1000-letiyu goroda Kazani: Mater. Konferentsii [Tupolev readings: International
youth scientific conference dedicated to the 1000th anniversary of the city of Kazan.
Conference materials]. Vol. II. Kazan': Kazan. gos. tekhn. un-ta, 2005, pp. 139-140.
16. Shcherbov I.L., Paslen V.V. Obrabotka dannykh traektornogo kontrolya s ispol'zovaniem
ortogonal'nykh bazisnykh funktsiy [Trajectory control data processing using orthogonal basis
functions], Vestnik Akademii grazhdanskoy zashchity [Vestnik of the Academy of Civil Protection],
2021, Issue 1 (25), pp. 48-53.
17. Shcherbov I.L. Informatsionnaya tekhnologiya obrabotki dannykh traektornogo kontrolya [Information
technology of mathematical modeling of data processing of trajectory control],
Vestnik Donetskogo natsional'nogo universiteta. Ser. G: Tekhnicheskie nauki [Vestnik of the
Donetsk National University. Series G: Technical sciences], 2021, No. 1, pp. 71-77.
18. Ogorodniychuk N.D. Otsenka parametrov dvizheniya i tochnost' ikh opredeleniya po dannym
korrelirovannykh traektornykh izmereniy [Evaluation of motion parameters and the accuracy
of their determination according to data of correlated trajectory measurements]. Kiev:
KVVAIU, 1968, 216 p.
19. Baklitskiy, V.K., Yur'ev A.N. Korrelyatsionno-ekstremal'nye metody navigatsii [Correlationextremal
methods of navigation]. Moscow: Radio i svyaz', 1982, 256 p.
20. Dubrovskiy S.A. Prikladnoy mnogomernyy statisticheskiy analiz [Applied multivariate statistical
analysis]. Moscow: Finansy i statistika, 1982, 216 p.
21. Shcherbov I.L., Paslen V.V., Fesenko D.V., Mikhaylov M.V. Prichiny vozniknoveniya sboev i
metody ikh ustraneniya [Causes of failures and methods of their elimination],
Mezhdunarodnaya molodezhnaya nauchno-prakticheskaya konferentsiya «Chelovek i kosmos»
[International Youth Scientific and Practical Conference "Man and Space"]. Dnepropetrovsk:
NTSAOMU, 2006, pp. 120.
22. Ogodniychuk N.D., Paslen V.V., Veligdan S.V. Issledovaniya na EVM svoystv sistem LNBF i
Λ-OBF kak funktsii dvukh argumentov [Computer studies of the properties of the LNBF and
Λ-OBF systems as functions of two arguments], Radioelektronnoe oborudovanie leta-tel'nykh
apparatov [Radio-electronic equipment of aircraft]. Issue 3. Kiev: KVVAIU, 1989, pp. 90-93.
