Reconstruction of Observation from Distorted Data for The Optimal Dynamic Measurement Problem

The theory of optimal dynamic measurements is based on the minimization of the difference between the values of virtual observation, i.e. an observation obtained with the help of a computational model, and experimental data, which are usually distorted by some noise. The article describes a mathematical model of optimal dynamic measurement in the presence of various types of interference. In addition, the article proposes an algorithm to reconstruct the values of observation from the values obtained during the experiment, which are assumed to be distorted by some random influences. It is assumed that the experimental data are influenced by ' white noise', which is understood as a derivative of Nelson-Gliklich from the Wiener process. In order to reconstruct observation values, we use a priori information about the form of the function describing the observation values. The reconstruction procedure consists of two stages. At the first stage, we formulate the criterion for determining the position of the extreme point of the signal using a special type of statistics. At the second stage, we describe the procedure to reconstruct the signal values on the basis of information about the position of the extreme point and the shape of the signal convexity.

Keywords

Wiener process; Brownian motion; Nelson-Gliklikh derivative; statistical hypothesis.

References

1. Shestakov A.L. Metody teorii avtomaticheskogo upravleniya v dinamicheskih izmereniyah [Methods of the Automatical Control Theory to Dynamical Measurements]. Chelyabinsk, Publishing center of SUSU, 2013. (in Russian)
2. Pyt'ev Yu.P., Chulichkov A.I. Metody morfologicheskogo analiza izobrazheniy [Methods of Morphological Analysis of Pictures]. Moscow, FizMatLit, 2010. (in Russian)
3. Granovskii V.A. Dinamicheskie izmereniya. Osnovy metrologicheskogo obespecheniya [Dynamic Measurements. Fundamentals of Metrology Provision]. Leningrad, Energoatomizdat, 1984. (in Russian)
4. Tikhonov A.N, Arsenin A.Ia. tSolutions of Ill-Posed Problems. Winston, Halsted Press, 1977.
5. Ivanov V.K., Vasin V.V., Tanana V.P. tTheory of Linear Ill-posed Problems and Its Applications. Utrecht, Boston, VSP, 2002.
6. Lavrentiev M.M., Romanov V.G., Shishatskii S.P. Ill-Posed Problems of Mathematical Physics and Analysis. Providence, AMS, 1986.
7. Shestakov A.L. Modal Synthesis of a Measurement Transducer. Problemy Upravleniya i Informatiki (Avtomatika), 1995, no. 4, pp. 67-75. (in Russian)
8. Derusso R.M., Roy R.J., Close C.M. State Variables for Engineers. N.Y., London, Sydney, Wiley, 1965.
9. Kurzhanski A.B. Upravlenie i nablyudenie v usloviyah neopredelennosti [Control and Observation under Conditions of Uncertainty]. Moscow, Nauka, 1977. (in Russian)
10. Kuzovkov N.T. Modal'noe upravlenie i nablyudeniya ustrojstva [Modal Management and the Observing Devices]. Moscow, Mashinostroenie, 1976. (in Russian)
11. Shestakov A.L., Sviridyuk G.A. Optimal Measurement of Dynamically Distorted Signals. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2011, no. 17 (234), issue 8, pp. 70-75.
12. 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.
13. Shestakov A.L., Sviridyuk G.A., Keller A.V. Optimal Measurements. XXI IMEKO World Congress 'Measurement in Research and Industry', 2015, pp. 2072-2076.
14. Keller A.V. On the Computational Efficiency of the Algorithm of the Numerical Solution of Optimal Control Problems for Models of Leontieff Type. Journal of Computational and Engineering Mathematics, 2015, vol. 2, no. 2, pp. 39-59. DOI: 10.14529/jcem150205
15. 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
16. 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
17. Sagadeeva M. On Nonstationary Optimal Measurement Problem for the Measuring Transducer Model. 2nd International Conference on Industrial Engineering, Applications and Manufacturing (ICIEAM-2016), 2016, article ID: 7911710, 3 p. DOI: 10.1109/ICIEAM.2016.7911710
18. Einstein A. Zur theorie der brownschen bewegung. tAnnalen der Physik (ser. 4), 1905, vol. 19, pp. 371-381. (in German)
19. Krylov N.V. Introduction to the Theory of Diffusion Processes. Providence, American Mathematical Society, 1994 .
20. Melnikova I.V., Filinkov A.I. Abstract Cauchy Problem in Spaces of Stochastic Distributions. Journal of Mathematical Sciences, 2008, vol. 149, no. 5, pp. 1567-1579. DOI: 10.1007/s10958-008-0082-4
21. Melnikova I.V., Alshanskiy M.A. White Noise Calculus in Applications to Stochastic Equations in Hilbert Spaces. tJournal of Mathematical Sciences, 2016, vol. 218, no. 4, pp. 395-429. DOI: 10.1007/s10958-016-3038-0
22. Nelson E. Dynamical Theories of Brownian Motion. Princeton, Princeton University Press, 1967.
23. Gliklikh Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics. N.Y., London, Dordrecht, Heidelberg, Springer, 2011.
24. 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
25. Shestakov A.L., Sviridyuk G.A. On the Measurement of the ' White Noise'. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2012, no. 27 (286), issue 13, pp. 99-108.
26. Sviridyuk G. A., Manakova N.A. The Dynamical Models of Sobolev Type with Showalter-Sidorov Condition and Additive 'Noise'. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 1, pp. 90-103. DOI: 10.14529/mmp140108 (in Russian)
27. Favini A., Sviridyuk G.A., Manakova N.A. Linear Sobolev Type Equations with Relatively tp-Sectorial Operators in Space of 'Noises'. Abstract and Applied Analysis, 2015, vol. 2015, article ID: 697410, 8 p. DOI: 10.1155/2015/697410
28. Favini A., Sviridyuk G.A., Sagadeeva M.A. Linear Sobolev Type Equations with Relatively p-Radial Operators in Space of 'Noises', Mediterranean Journal of Mathematics, 2016, vol. 13, no. 6, pp. 4607-4621. DOI: 10.1007/s00009-016-0765-x
29. Favini A., Sviridyuk G.A., Zamyshlyaeva A.A. One Class of Sobolev Type Equations of Higher Order with Additive 'White Noise'. Communications on Pure and Applied Analysis, 2016, vol. 15, no. 1, pp. 185-196. DOI: 10.3934/cpaa.2016.15.185
30. Favini A., Zagrebina S.A., Sviridyuk G.A. Multipoint Initial-Final Value Problems for Dynamical Sobolev-Type Equations in the Space of Noises. Electronic Journal of Differential Equations, 2018, article ID: 128, 10 p. Aviable at: https://ejde.math.txstate.edu/Volumes/2018/128/favini.pdf
31. Zagrebina S.A., Sukacheva T.G., Sviridyuk G.A. The Multipoint Initial-Final Value Problems for Linear Sobolev-Type Equations with Relatively p-Sectorial Operator and Additive 'Noise'. Global and Stochastic Analysis, 2018, vol. 5, no. 2, pp. 129-143.
32. Kovacs M., Larsson S. Introduction to Stochastic Partial Differential Equations. Proceedings of 'New Directions in the Mathematical and Computer Sciences', National Universities Commission, Abuja, Nigeria, October 8-12, 2007. V. 4. Publications of the ICMCS, Lagos, 2008, pp. 159-232.
33. Zagrebina S.A., Soldatova E.A., Sviridyuk G.A. The Stochastic Linear Oskolkov Model of the Oil Transportation by the Pipeline.tSpringer Proceedings in Mathematics and Statistics, 2015, vol. 113, pp. 317-326. DOI: 10.1007/978-3-319-12145-1_20
34. Zamyshlyaeva A.A., Sviridyuk G.A. The Linearized Benney-Luke Mathematical Model with Additive White Noise. Springer Proceedings in Mathematics and Statistics, 2015, vol. 113, pp. 327-336. DOI: 10.1007/978-3-319-12145-1_21
35. 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)
36. Sviridyuk G.A., Keller A.V. On the Numerical Solution Convergence of Optimal Control Problems for Leontief Type System. Vestnik of Samara State Technical University. Series: Physical and Mathematical Sciences, 2011, no. 2 (23), pp. 24-33. (in Russian)
37. Belov A.A., Kurdyukov A.P. Deskriptornye sistemy i zadachi upravleniya [Descriptor Systems and Control Problems]. Moscow, Fizmthlit, 2015. (in Russian)
38. Shestakov A.L., Sviridyuk G.A., Hudyakov Yu.V. Dinamic Measurement in Spaces of 'Noise'. Bulletin of the South Ural State University. Series: Computer Technologies, Automatic Control, Radio Electronics, 2013, vol. 13, no. 2, pp. 4-11. (in Russian)
39. Banasiak J., Lachowicz M., Moszynski M. Chaotic Behavior of Semigroups Related to the Process of Gene Amplification-Deamplification with Cell Proliferation. Mathematical Biosciences, 2007, vol. 206, no. 2, pp. 200-2015. DOI: 10.1016/j.mbs.2005.08.004
40. Banasiak J., Pichor K., Rudnicki R. Asynchronous Exponential Growth of a General Structured Population Model. Acta Applicandae Mathematicae, 2012, vol. 119, no. 1, pp. 149-166. DOI: 10.1007/s10440-011-9666-y