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Реализация итерационного метода наименьших квадратов для оценивания параметров статических объектов в среде MatLab..pdf

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ISSN 2072-9502. Вестник АГТУ. Сер.: Управление, вычислительная техника и информатика. 2017. № 1
??? 519.24
?. ?. ???????, ?. ?. ???????
РЕАЛИЗАЦИЯ ИТЕРАЦИОННОГО МЕТОДА
НАИМЕНЬШИХ КВАДРАТОВ
ДЛЯ ОЦЕНИВАНИЯ ПАРАМЕТРОВ
СТАТИЧЕСКИХ ОБЪЕКТОВ В СРЕДЕ MATLAB
??? ????????????? ?????? ?????????? ?????????? ?????????? ???????????? ?????? ???????, ?????????? ? ???????? ???????. ??? ???? ??????????? ??????? ?????????? ???????
??????, ? ????? ???????? ????????? ?????????? ???????. ? ????? MatLab ????????? ????????????? ??????? ??? ??????, ????? ?? ??????? ?? ????????? ???? ?????????, ? ????? ??? ??????, ????? ?????? ????????? ??? ????????? ???????? ????. ? ???????? ???????? ??????? ???????????? ???????????? ???? ???????? ? ?????????? ????????? ? ????????? ??????????.
? ????? MatLab ?????? ???????????? ????? ??????????? ?????????. ?? ????? ??????? ????
????????? ?????, ??????????????? ????????????? ???????? ??????? ??????????????? ??????
???????, ???? ????????????? ???????? ??????????? ? ???? ??? ?????????? ??????????. ?????? ? ?????? ???? ????????????? ????????????? ????????????? ?????? ?????????? ?????????. ??????????????? ??????? ???????? ? ????????? ???????? ??? ?????????? ? ???????
?????. ?????????? ?????????? ?????????? ?????????? ???????????? ???????. ???????? ??????????? ?????????????, ???????? ??????? ???????? ???? ??? ??????? ???????????? ?????
?????????. ??? ???????? ???????????? ?????? ????????? ????????? ??????????????? ?????????? ? ????????? ??????? ????????? ???????????? ????????, ??????? ????????????
? ????????? ?????????? ??????????. ?????????? ????????? ???????, ??????? ??????????
??? ?????? ????????????? ?????? ?????????? ?????????. ????????? ????????? ???????????????? ??????? ????????? ???????? ??????? ??? ??? ???????????? ????????????? ??? ?????????? ?????????? ?????????????? ???????????? ????????.
???????? ?????: ????? ?????????? ?????????, ?????????????, ?????????????, ??????? ??????, ?????????? ??????????
????????
? ????????? ????? ???????? ????????????? ? ?????????? ????????? ????????? ?????
???????? [1?25], ?? ??????? ????? ????? ?? ??????????? ??????? ?????? ???????? ????????????????? ??????. ??? ?????????????? ??????? ??????? ???????? ??????????? ????????? ? ??????????
??????, ?????????????? ????????? ?? ??????-?? ???????? ?????????? ???????? ?????? ??????
? ??????? ??? ?????????? ??????? ???????????? [2, 3, 17]. ??? ???? ? ??????? ?????????? ?????? ?????????? ????? ????????? ???????? ? ???????????? ?? ??????????? ????????? [1?5]. ????????? ????????????? ???????????? ????? ???????? ??????? ??????, ??????? ??????? ????????? ???
?????????? ????????? ?????????? ?????. ?????????????? ????????? ??????????????? ????????,
??? ???????, ?? ???? ??????????? ?????????? ???????????? ????????? ????????????? ??? ???????????? ??????????. ????? ???????? ????????? ? ???????? ?????? ?????????????. ??????
???????? ????????????? ????????????? ????????????? ???????? ??????? ???????????, ???????
???? ?????????????? ??????? ?????????????. ?????? ????????? ????????????? ???????? ? ?????????, ??????????? ? ???? ?????????? ???????????? ???????. ????????????? ? ???????? ????????????? ????????????? ?????? ????????????? ? ??????? ??????????????, ??????????? ? ?????????? ????????. ????????? ?????? ????????? ????? ? ??? ????????????? ???????????? ???????? ? ???????? ??????? [6?17]. ????? ??????????????? ???????????? ????? ?????????? ????????? ??? ?????????? ?????????? ??????????? ????????. ? ?????????? ???????????? ?????? ?????
?????????????? ?? ???????????? ???????.
?????????? ??????
????????????? ???????? ??????????? ?????? ? ????????? ?????? ? ??????????? ???????:
x? ? = y + v ,
28
Математическое моделирование
??? x ? ????????? ??????? ??????; y ? ????????? ???????? ??????. ??? ????????? ?????
???????, ??? x ? ?????? ??????????? ???, ?. ?. x = ( x1 , x 2 )? . ????? ?? = (?1 , ? 2 ) ? ?????? ??????????? ?????????? ???????; v ? ??????? ??? ?? ?????? ??????? ? ??????? ??????????????
?????????. ?????????? ???????????? ???????????????, ?? ???? ??????????? ?????? ????????? ???????? xi ? ????????? yi ????????. ?? ??????????? N ????????? ???????? ? ????????? ???????? xi , yi , i ?1, N ??? ?????? ?????????? ????????? ????? ????????? ???????
????????????? ?????????? [14]:
N
s (?) =
? ( y ? x ?) .
i
?
i
2
i =1
?????? T ? x?i ?????????? ????????????????. ????????? ?????????? N ????????? ????????? ???????:
XN
? x1? ? ? x11
? ? ?
? x? ? ? x 1
=? 2 ?=? 2
? M ? ? M
? x? ? ? x1
? N? ? N
? y1 ?
x12 ?
?
? ?
2
? y2 ?
x2 ?
? , YN = ? ? .
M ?
? M ?
2 ?
?y ?
xN ?
? N?
?????? ??????????? ?????????? ?? N ??????????? ? ??????? ?????? ?????????? ????????? ?? ????????? ??????? [14]:
?? N = ( X N? X N ) ?1 X N? YN .
????????????? ?????? ?????????? ?? N ?? N + 1 -????????? ?????????????? ?? ????????? ????????:
?? N +1 = ?? N + K N +1 ( y N +1 ? x?N +1?? N ) ,
(1)
K N +1 = PN x N +1 /(1 + x?N +1 PN x N +1 ) ,
(2)
x N +1 x?N +1
) PN ,
1 + x?N +1 PN x N +1
(3)
PN +1 = ( I ? PN
??? K N +1 ? ??????????? ???????? ? PN +1 ? ?????? ????????? ?????? ??????????, ???????????
?? ??????????? N + 1 -?????????.
?????????? ?????????? ??????????? ??????? ?? ??????? ????????? ???????? ??????? ??????????? ?????????? ?0 ? ???????????? ???????? K 0 ? ??????? P0 . ??? ??????? ??????? ??????? P0 ???????????? ????????? ?? ????????, ?, ?????????????, ?????? ?? N ????????????????????.
?????????? ?????? ?????????? ??????? ? ????? MatLab
???????????? ????????? ?????????? ?????????? ??????? ????????? ?? ?????, ?????????????? ?? ???. 1, ??? ??????? ????????? ?????: Gen_?1_?2 ? ????????? ???????? ???????;
Object ? ??????; Noise ? ????????? ?????? ????; Estimation ? ???? ?????????? ?????? ??????????. ???????? ?????? ???????, ???????????? ???????? ???????????? ???????? K ? ???????
P ???????? ?? ??????????.
29
ISSN 2072-9502. Вестник АГТУ. Сер.: Управление, вычислительная техника и информатика. 2017. № 1
???. 1. ???????????? ????????? ?????????? ??????????? ??????????
???? Estimation ?? ?????????? (1)?(3) ? ????? MatLab ???????? ?? ???. 2, ? ????? ?????????? ?? ?????????? (1)?(3) ? ?? ???. 3?5.
???. 2. ???? Estimation
???. 3. ?????????? K N +1
30
Математическое моделирование
???. 4. ?????????? PN +1
???. 5. ?????????? ?????? ?????????? ?? N +1
???????????? ?????????? ??????????? ?????????? ?? N +1 ? ????? MatLab
?? ????????? (1) ? ????? ??????? x = ( x1 , x 2 )? , ????? ??????????? ?? = (?1 , ? 2 ) ? ????? ??????? y . ????????????? ????????? ??? ????????? ??????? ?????????: ?1 = 1,0;
?2 = 0,5 . ?? ???. 6?10 ????????? ?????????? ????????????? ??? ??????? ??? ???? ????????
v = 0 ? ?????????? ????????? m = 0 . ???????? K1 ?? ???. 9 ????????? ?? ????????? 0,5. ???
?????????? ???????? ?????? ????????? ?????? ?????????? ??????? ????????? ????????:
? 0,5 0 ?
?0 0?
?? . ???????????, ??? ??? P0 = ??
?? ??????? ?? ????????. ? ???????? ????????
P0 = ??
? 0 0,5 ?
?0 0?
??????? ??????? ??????? ???? ???????? ?? ??????? ?????? ? ????????? T = 6 ? T = 4 ? ???????????, ??????? ???????. ??????????? ?????????? ??? ??? ?????????.
???????? ?????? y , ??? ??????? ?? ???. 8, ???????? ????????????? ???? ???????? ????
???????. ?????? ?????????? ???????? ?????? ???????? ? ????????? ????????, ???????? ??
50 ?????. ?? ?? ????? ????? ??????? ? ? ??????? ?????? ????????? PN ? ???????????? ????????
1
2
K N . ????? ??? ????????? ???????? ????????? ?????? ??????????: ? = 0,981, ? = 0,4912.
?? ???. 11?14 ????????? ?????????? ????????????? ??? ??????? ???? ???????? v
? ??????? ??????? m ? ?????????? ? = 0,005, ??? ????????????? ??????????? ?????????
? ???????? 5?7 %. ???????? K1 ?? ???. 13 ????????? ?? ????????? 0,5.
2
31
ISSN 2072-9502. Вестник АГТУ. Сер.: Управление, вычислительная техника и информатика. 2017. № 1
???. 6. ?????? ????????? ? ??? ???? ( v = 0 )
???. 7. ??????? ??????
KN
K1
K2
???. 8. ????? ??????? ??? ???? ( v = 0 )
???. 9. ???????? K N ??? ???? ( v = 0 )
???. 10. ???????? PN ??? ???? ( v = 0 )
32
Математическое моделирование
???. 11. ?????? ?? N ??? ??????? ????
???. 12. ????? ??????? Y ??? ??????? ????
PN
KN
K1
K2
???. 13. ???????? K N ??? ??????? ????
???. 14. ???????? PN ??? ??????? ????
???????? ????????? ???????? ??????????? ??????????: ? = 0,9771, ? = 0,4817. ???
??????? ?? ??????????? ?????????????, ???????? ??????? ???????? ???? ??? ??????? ???????????? ????? ?????????.
1
2
??????????
???????????? ?????????? ??? ????????? ????????? ?????????? ????? ??? ????????
? ??????? ?????? ?????????? ???????????? ????????????????? ????????? ? ??? ??????????
? ?????? MatLab. ?????????????, ??? ????????? ????? ????????????? ? ? ?????????? ????????
? ?????????????? ???????????? ??? ????????? ??????. ? ?????????? ?????????????? ????????????? ???????????? ??????? ??? ?????????? ?????????? ?????????????? ???????????? ????????.
?????? ??????????
1. ?????? ?. ???????? ? ?????????????? ?????? ??????????. ?.: ???, 1973. 320 ?.
2. ????? ?. ????????????? ??????. ?????? ??? ????????????. ?.: ?????, 1991. 432 ?.
3. ??????? ?. ?????? ????????????? ?????? ??????????. ?????????? ?????????? ? ?????????. ?.:
???, 1975. 683 ?.
4. ????? ??. ????????????? ??????????? ???????? ?????? ? ??????????. ?.: ???????, 1973. 440 ?.
33
ISSN 2072-9502. Вестник АГТУ. Сер.: Управление, вычислительная техника и информатика. 2017. № 1
5. ????? ?. ?., ????? ??. ?????? ?????????? ? ?? ?????????? ? ????? ? ??????????. ?.: ?????,
1976. 495 ?.
6. Gupta H. K., Mehra R. K. Computational aspects of maximum likelihood estimation and reduction in sensitivity function calculation // IEEE Trans. Autom. Control. 1974. Vol. 19, no. 7. P. 774?785.
7. Astrцm K. J. Maximum Likelihood and Prediction Error Methods // Automatica. 1980. Vol. 16, no. 5.
P. 551?574.
8. Mehra R. K. Optimal input signal for parameter estimation in dynamic system ? survey and new results //
IEEE Trans. Autom. Control. 1974. Vol. AC-19, no. 6. P. 753?768.
9. Mehra R. K. On the Identification of Variences and Adaptive Kalman Filtering // IEEE Trans. Autom.
Control. 1970. Vol. AC-15, no. 2. P. 175?184.
10. Mehra R. K. Optimal Input for Linear System Identification // IEEE Trans. Autom. Control. 1974.
Vol. 19, no. 3. P. 192?200.
11. Goodwin G. C., Payne R. L. Dynamic System Identification: Experiment Design and Data Analysis.
New York: Academic Press, 1977. 291 p.
12. Antsaklis P. J., Michel A. N. Linear systems. New York: McGraw-Hill, 1997. 685 p.
13. Brown R. J., Sage A. P. Error Analysis of Modeling and Bias Errorsin Continuous Time State Estimation
// Automatica. 1971. Vol. 7. P. 577?590.
14. Goodwin G. C. Optimal Input Signals for Nonlinear-system Identification // Proc. Inst. Elec. Engrs.
1971. Vol. 118, no. 7. P. 922?926.
15. ????? ?. ?., ???? ?. ?., III. ??????????? ?????????? ?????????. ?.: ????? ? ?????, 1982. 392 ?.
16. ??????? ?. ?., ??????? ?. ?. ?????????? ?????????? ??????? ???????? ? ?????????? ??? ???????? ???????????? ?????????? ?????? ? ?????????????? ?????????????? ??????? ?????? // ????.
?????. ????. 2006. ? 3 (24). ?. 199?200.
17. ??????? ?. ?. ???????? ????????????? ???????? ???????????? ?????????? ???????????? ???????? ?? ????????? ???????: ???. ? ????. ????. ????. ???????????, 2007. 171 c.
18. ??????? ?. ?. ?????????????? ??????? ?????? ?????????????? ??????? ????????? ??? ??????
? ??????? ????????? ??????????? // ??. ????. ??. ????. 2008. ???. 3 (53). ?. 25?34.
19. Voevoda A. A., Troshina G. V. Active identification of linear stationary dynamic object on base of the
Fisher information matrix: the steady state // Proc. of the XII Intern. Conf. "Actual problems of electronic instrument engineering (APEIE-2014)" (Novosibirsk, Russia, 2?4 October 2014). Novosibirsk, 2014. P. 745?749. doi:
10.1109/APEIE.2014.7040785.
20. Voevoda A. A., Troshina G. V. Active identification of the inverted pendulum control system // Proc.
of the 18th Intern. Conf. on Soft Computing and Measurements (SCM'2015). Saint-Petersburg: LETI Publ., 2015.
Vol. 1. P. 153?156.
21. Voevoda A. A., Troshina G. V., Patrin V. M., Simakina M. V The object unknown parameters estimation
for the 'inverted pendulum-Cart' system in the steady state // Proc. of the 16th Intern. Conf. of Young Specialists
on Micro/Nanotechnologies and Electron Devices (EDM-2015), Altai, Erlagol, 29 June ? 3 July 2015. IEEE,
2015. P. 186?188.
22. ??????? ?. ?., ??????? ?. ?. ? ????????? ??????? ?????????? ? ?????? ????????????? // ??.
????. ??. ????. 2014. ???. 2 (76). C. 16?25.
23. ??????? ?. ?., ??????? ?. ?. ?? ?????? ??????? ????????? ? ??????? ?????????? ? ?????? ????????????? // ??. ????. ??. ????. 2014. ???. 4 (78). C. 53?68. doi: 10.17212/2307-6879-2014-4-53-68.
24. ??????? ?. ?. ????????????? ???????????? ???????? ? ????? Simulink. ?. 1 // ??. ????. ??.
????. 2015. ???. 3 (81). C. 55?68. doi: 10.17212/2307-6879-2015-3-55-68.
25. ??????? ?. ?. ????????????? ???????????? ???????? ? ????? Simulink. ?. 2 // ??. ????. ??.
????. 2015. ???. 4 (82). C. 31?41. doi: 10.17212/2307-6879-2015-4-31-41.
?????? ????????? ? ???????? 22.11.2016
ИНФОРМАЦИЯ ОБ АВТОРАХ
Воевода Александр Александрович ? Россия, 630073, Новосибирск; Новосибирский
государственный технический университет; д-р техн. наук, профессор; профессор кафедры автоматики; ucit@ucit.ru.
Трошина Галина Васильевна ? Россия, 630073, Новосибирск; Новосибирский государственный технический университет; канд. техн. наук, доцент; доцент кафедры вычислительной техники; troshina@corp.nstu.ru.
34
Математическое моделирование
A. A. Voevoda, G. V. Troshina
THE REALIZATION
OF THE ITERATIVE METHOD OF THE LEAST SQUARES
FOR THE ESTIMATION OF STATIC OBJECT PARAMETERS
IN MATLAB ENVIRONMENT
Abstract. It was suggested to use the system model working in real time for an iterative method
of the parameter estimation. It gives the chance to select a suitable input signal, and also to carry
out the setup of the object parameters. The object modeling for a case when the system isn't affected by the measurement noises, and also for a case when an object is under the gaussian noise was
executed in the MatLab environment. The superposition of two meanders with different periods
and single amplitude is used as an input signal. The model represents the three-layer structure in the
MatLab environment. On the most upper layer there are units corresponding to the simulation of an
input signal, directly the object, the unit of the noise simulation and the unit for the parameter estimation. The second and the third layers correspond to the simulation of the iterative method of the
least squares. The diagrams of the input and the output signals in the absence of noise and in the
presence of noise are shown. The results of parameter estimation of a static object are given. According to the results of modeling, the algorithm works well even in the presence of significant
measurement noise. To verify the correctness of the work of an algorithm the auxiliary computations have been performed and the diagrams of the gain behavior amount which is used in the parameter estimation procedure have been constructed. The entry conditions which are necessary for
the work of an iterative method of the least squares are specified. The understanding of this algorithm functioning principles is a basis for its subsequent use for the parameter estimation of the
multi-channel dynamic objects.
Key words: the method of the least squares, identification, modeling, input signal, parameter
estimation.
REFERENCES
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Ostrem K. Vvedenie v stokhasticheskuiu teoriiu upravleniia. Moscow, Mir Publ., 1973. 320 p.).
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6. Gupta H. K., Mehra R. K. Computational aspects of maximum likelihood estimation and reduction in sensitivity function calculation. IEEE Trans. Autom. Control, 1974, vol. 19, no. 7, pp. 774?785.
7. Astrцm K. J. Maximum Likelihood and Prediction Error Methods. Automatica, 1980, vol. 16, no. 5,
pp. 551?574.
8. Mehra R. K. Optimal input signal for parameter estimation in dynamic system ? survey and new results.
IEEE Trans. Autom. Control, 1974, vol. AC-19, no. 6, pp. 753?768.
9. Mehra R. K. On the Identification of Variences and Adaptive Kalman Filtering. IEEE Trans. Autom. Control, 1970, vol. AC-15, no. 2, pp. 175?184.
10. Mehra R. K. Optimal Input for Linear System Identification. IEEE Trans. Autom. Control, 1974, vol. 19,
no. 3, pp. 192?200.
11. Goodwin G. C., Payne R. L. Dynamic System Identification: Experiment Design and Data Analysis.
New York, Academic Press, 1977. 291 p.
12. Antsaklis P. J., Michel A. N. Linear systems. New York, McGraw-Hill, 1997. 685 p.
13. Brown R. J., Sage A. P. Error Analysis of Modeling and Bias Errorsin Continuous Time State Estimation. Automatica, 1971, vol. 7, pp. 577?590.
14. Goodwin G. C. Optimal Input Signals for Nonlinear-system Identification. Proc. Inst. Elec. Engrs.,
1971, vol. 118, no. 7, pp. 922?926.
35
ISSN 2072-9502. Вестник АГТУ. Сер.: Управление, вычислительная техника и информатика. 2017. № 1
15. Sage A. P., White Ch. C., III. Optimum System Control. Prentice Hall, 1977. 372 p. (Russ. ed.: Seidzh E. P.,
Uait Ch. S., III. Optimal'noe upravlenie sistemami. Moscow, Radio i sviaz' Publ., 1982. 392 p.).
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statsionarnykh diskretnykh sistem s ispol'zovaniem informatsionnoi matritsy Fishera [Estimation of model parameters of dynamics and monitoring for linear time-invariant discrete systems using the Fisher information matrix]. Nauchnyi vestnik NGTU, 2006, no. 3 (24), pp. 199?200.
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vo vremennoi oblasti. Dis. ? kand. tekhn. nauk [Active identification of linear dynamic discrete stationary objects in the time domain. Abstract of dis. cand. techn. sci.]. Novosibirsk, 2007. 171 p.
18. Troshina G. V. Vychislitel'nye aspekty zadachi vosstanovleniia vektora sostoianiia dlia modeli s netochno
zadannymi parametrami [Computational aspects of the state vector recovery task for the model with inaccurately
specified parameters]. Sbornik nauchnykh trudov NGTU, 2008, iss. 3 (53), pp. 25?34.
19. Voevoda A. A., Troshina G. V. Active identification of linear stationary dynamic object on base of the
Fisher information matrix: the steady state. Proc. of the XII Intern. Conf. "Actual problems of electronic instrument engineering (APEIE-2014)" (Novosibirsk, Russia, 2?4 October 2014). Novosibirsk, 2014. P. 745?749. doi:
10.1109/APEIE.2014.7040785.
20. Voevoda A. A., Troshina G. V. Active identification of the inverted pendulum control system. Proc.
of the 18th Intern. Conf. on Soft Computing and Measurements (SCM'2015). Saint-Petersburg: LETI Publ., 2015,
vol. 1, P. 153?156.
21. Voevoda A. A., Troshina G. V., Patrin V. M., Simakina M. V. The object unknown parameters estimation for the 'inverted pendulum-Cart' system in the steady state. Proc. of the 16th Intern. Conf. of Young Specialists on Micro/Nanotechnologies and Electron Devices (EDM-2015), Altai, Erlagol, 29 June ? 3 July 2015.
IEEE, 2015. P. 186?188.
22. Voevoda A. A., Troshina G. V. O nekotorykh metodakh fil'tratsii v zadache identifikatsii [On some methods of filtration in the identification of the problem]. Sbornik nauchnykh trudov NGTU, 2014, iss. 2 (76), pp. 16?25.
23. Voevoda A. A., Troshina G. V. Ob otsenke vektora sostoianiia i vektora parametrov v zadache identifikatsii [On the estimation of the state vector and the vector parameters to identify the problem]. Sbornik nauchnykh trudov NGTU, 2014, iss. 4 (78), pp. 53?68. doi: 10.17212/2307-6879-2014-4-53-68.
24. Troshina G. V. Modelirovanie dinamicheskikh ob"ektov v srede Simulink. Ch. 1 [Modeling of dynamic
objects in the Simulink environment]. Sbornik nauchnykh trudov NGTU, 2015, pp. 3 (81), pp. 55?68. doi:
10.17212/2307-6879-2015-3-55-68.
25. Troshina G. V. Modelirovanie dinamicheskikh ob"ektov v srede Simulink. Ch. 2 [Modeling of dynamic
objects in the Simulink environment]. Sbornik nauchnykh trudov NGTU, 2015, iss. 4 (82), pp. 31?41. doi:
10.17212/2307-6879-2015-4-31-41.
The article submitted to the editors 22.11.2016
INFORMATION ABOUT THE AUTHORS
Voevoda Alexander Aleksandrovich ? Russia, 630073, Novosibirsk; Novosibirsk State
Technical University; Doctor of Technical Sciences, Professor; Professor of the Department
of the Automation; ucit@ucit.ru.
Troshina Galina Vasil?evna ? Russia, 630073, Novosibirsk; Novosibirsk State Technical
University; Candidate of Technical Sciences, Assistant Professor; Assistant Professor of the
Department of Computer Engineering; troshina@corp.nstu.ru.
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