The Pyt'ev - Chulichkov Method for Constructing a Measurement in the Shestakov - Sviridyuk Model

One of the approaches to solution of the problem on restoring a distorted input signal by the recorded output data of the sensor is the problem on optimal dynamic measurement, i.e. the Shestakov-Sviridyuk model. This model is the basis of the theory of optimal dynamic measurements and consists of the problem on minimizing the difference between the values of a virtual observation obtained using a computational model and experimental data, which are usually distorted by some noise. We consider the Shestakov-Sviridyuk model of optimal dynamic measurement in the presence of various types of noises. In the article, the main attention is paid to the preliminary stage of the study of the problem on optimal dynamic measurement. Namely, we consider the Pyt'ev-Chulichkov method of constructing observation data, i.e. transformation of the experimental data to make them free from noise in the form of ``white noise'' understood as the Nelson-Gliklikh derivative of the multidimensional Wiener process. In order to use this method, a priori information about the properties of the functions describing the observation is used.

Keywords

optimal dynamic measurement; Leontief type system; multidimensional Wiener process; Nelson-Gliklikh derivative; algorithm to solve the problem.

References

1. Granovskii V.A. [Dynamic Measurements. Fundamentals of Metrology Provision]. Leningrad, Energoatomizdat, 1984. (in Russian)
2. Shestakov A.L. [Methods of the Automatical Control Theory to Dynamical Measurements]. Chelyabinsk, Publishing center of SUSU, 2013. (in Russian)
3. Sviridyuk G.A., Efremov A.A. An Optimal Control Problem for a Class of Linear Degenerate Equations. Doklady Mathematics, 1999, vol. 59, no. 1, pp. 157-159.
4. Shestakov A.L., Sviridyuk G.A. A New Approach to Measurement of Dynamically Perturbed Signals. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2010, no. 16 (192), issue 5, pp. 116-120.
5. Keller A.V. Numerical Solution of the Optimal Control Problem for Degenerate Linear System of Equations with Showalter-Sidorov Initial Conditions. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2008, no. 27 (127), issue 2, pp. 50-56.
6. Shestakov A.L., Keller A.V., Nazarova E.I. Numerical Solution of the Optimal Measurement Problem. Automation and Remote Control, 2012, vol. 73, no. 1, pp. 97-104. DOI: 10.1134/S0005117912010079
7. Shestakov A.L., Sviridyuk G.A., Keller A.V. The Theory of Optimal Measurements. Journal of Computational and Engineering Mathematics, 2014, vol. 1, no. 1, pp. 3-15.
8. Shestakov A.L., Keller A.V., Zamyshlyaeva A.A., Manakova N.A., Zagrebina S.A., Sviridyuk G.A. The Optimal Measurements Theory as a New Paradigm in the Metrology. Journal of Computational and Engineering Mathematics, 2020, vol. 7, no. 1, pp. 3-23. DOI: 10.14529/jcem200101
9. Keller A.V., Shestakov A.L., Sviridyuk G.A., Khudyakov Y.V. The Numerical Algorithms for the Measurement of the Deterministic and Stochastic Signals. Springer Proceedings in Mathematics and Statistics, vol. 113, 2015, pp. 183-195. DOI: 10.1007/978-3-319-12145-1_11
10. Sagadeeva M. On Nonstationary Optimal Measurement Problem for the Measuring Transducer Model. 2nd International Conference on Industrial Engineering, Applications and Manufacturing (ICIEAM-2016), Chelyabinsk, Russia, 2016, article ID: 7911710. DOI: 10.1109/ICIEAM.2016.7911710
11. Keller A.V., Zagrebina S.A. Some Generalizations of the Showalter-Sidorov Problem for Sobolev-Type Models. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 2, pp. 5-23. DOI: 10.14529/mmp150201 (in Russian)
12. Keller A.V., Sagadeeeva M.A. Degenerate Matrix Groups and Degenerate Matrix Flows in Solving the Optimal Control Problem for Dynamic Balance Models of the Economy. Springer Proceedings in Mathematics and Statistics, vol. 325, 2020, pp. 263-277. DOI: 10.1007/978-3-030-46079-2_15
13. Pyt'ev Yu.P., Chulichkov A.I. [Methods of Morphological Analysis of Pictures]. FizMatLit, Moscow, 2010. (in Russian)
14. Nelson E. Dynamical Theories of Brownian Motion. Princeton, Princeton University Press, 1967.
15. Gliklikh Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics. N.Y., Springer, 2011.
16. Gliklikh Yu.E., Zheltikova O.O. On Existence of Optimal Solutions for Stochastic Differential Inclusions with Mean Derivatives. Applicable Analysis, 2014, vol. 93, no. 1, pp. 35-45. DOI: 10.1080/00036811.2012.753588
17. Sagadeeva M.A. Reconstruction of Observation from Distorted Data for the Optimal Dynamic Measurement Problem. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2019, vol. 12, no. 2, pp. 58-66. DOI: 10.14529/mmp190207 (in Russian)
18. Leontiev V. [Economic Essays. Theories, Research, Facts, Politics]. Moscow, Politizdat, 1990. (in Russian)
19. Boyarintsev Yu.E [Linear and Nonlinear Algebra-Differential Systems]. Novosibirsk, Nauka, 2000. (in Russian)
20. M"arz R. On Initial Value Problems in Differential-Algebraic Equations and Their Numerical Treatment. Computing, 1985, vol. 35, issue 1, pp. 13-37. DOI: 10.1007/BF02240144
21. Belov A.A., Kurdyukov A.P. [Descriptor Systems and Control Problems]. Moscow, Fizmatlit, 2015. (in Russian)
22. Khudyakov Yu.V. On Mathematical Modeling of the Measurement Transducers. Journal of Computational and Engineering Mathematics, 2016, vol. 3, no. 3, pp. 68-73. DOI: 10.14529/jcem160308
23. Khudyakov Yu.V. On Adequacy of the Mathematical Model of the Optimal Dynamic Measurement. Journal of Computational and Engineering Mathematics, 2017, vol. 4, no. 2, pp. 14-25. DOI: 10.14529/jcem170202
24. Einstein A., Smoluchowski M. [Brownian Motion]. Moscow, Fizmatlit, 1936. (in Russian)
25. Demin D.S., Chulichkov A.I. Filtering of Monotonic Convex Noise-Distorted Signals and Estimates of Positions of Special Points. Fundamentalnaya i prikladnaya matematika, 2009, vol. 15, no. 6, pp. 15-31. (in Russian)