Development of the Theory of Optimal Dynamic Measurements

The paper presents an overview of the results of both an analytical study of optimal dynamic measurement problems and results in the development of algorithms for numerical methods for solving problems of the theory of optimal dynamic measurements. The main position of the theory of optimal dynamic measurements is the modelling of the desired input signal as a solution to the optimal control problem with minimization of the penalty functional, in which the discrepancy between the output simulated and observed signals is estimated. This theory emerged as a new approach for restoring dynamically distorted signals. The mathematical model of a complex measuring device is constructed as a Leontief-type system, the initial state of which reflects the Showalter-Sidorov condition. Initially, the mathematical model took into account only the inertia of the measuring device, later the mathematical model began to take into account the resonances that arise in the measuring device and the degradation of the device over time. The latest results take into account random noise and several approaches were developed here: the first approach is based on the Nelson-Gliklikh derivative, the second one is based on the purification of the observed signal using the Pytiev-Chulichkov method, the third one uses the purification of the observed signal by digital filters, for example, Savitsky-Golay or one-dimensional Kalman filter.

Keywords

Leontief type system; Showalter-Sidorov conditions; derivative of Nelson - Gliklich; Wiener process; optimal dynamic measurement; observation; Pytiev-Chulichkov method.

References

1. Belov A.A., Andrianova O.G., Kurdyukov A.P. Control of Discrete-Time Descriptor Systems. Cham, Springer, 2018. DOI: 10.1007/978-3-319-78479-3
2. Boyarintsev Yu.E., Chistyakov V.F. Algebro-differencial'nye sistemy: metody resheniya i issledovaniya [Differential-Algebraic Systems: Methods of Solution and Research]. Novosibirsk, Nauka, 1998. (in Russian)
3. 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)
4. DeRusso P.M., Roy R.J., Close C.M. State Variables for Engineers. New York, Chichester, Brisbane, Toronto, John Wiley & Sons, 1997.
5. Einstein A., Smoluchowski M. Brounovskoe dvizhenie [Brownian Motion]. Moscow, Fizmatlit, 1936. (in Russian)
6. Favini A., Sviridyuk G.A., Manakova N.A. Linear Sobolev Type Equations with Relatively p-Sectorial Operators in Space of "Noises". Abstract and Applied Analysis, 2016, vol. 13, pp. 1-8. DOI: 10.1155/2015/697410
7. 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
8. Gliklikh Yu.E., Makarova A.V. On Existence of Solutions to Stochastic Differential Inclusions with Current Velocities II. Journal of Computational and Engineering Mathematics, 2016, vol. 3, no. 1, pp. 48-60. DOI: 10.14529/jcem160106
9. Granovskii V.A. Dinamicheskie izmereniya. Osnovy metrologicheskogo obespecheniya [Dynamic Measurements. Fundamentals of Metrology Provision]. Leningrad, Energoatomizdat, 1984. (in Russian)
10. Keller A.V. [Leontief Type Systems: Classes of Problems with Showalter-Sidorov Initial Condition and Numerical Solutions]. The Bulletin of Irkutsk State University. Series: Mathematics, 2010, vol. 3, no. 2, pp. 30-43. (in Russian)
11. Keller A.V. Leontief-type Systems and Applied Problems. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2022, vol. 15, no. 1. pp. 23-42. (in Russian) DOI: 10.14529/mmp220102
12. 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.
13. 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
14. Keller A.V. Optimal Dynamic Measurement Method Using the Savitsky-Golay Digital Filter. Differential Equations and Control Processes, 2021, no. 1, pp. 1-15.
15. Keller A.V., Sagadeeva M.A. Convergence of the Spline Method for Solving the Optimal Dynamic Measurement Problem. Journal of Physics: Conference Series, 2021, article ID: 012074. DOI: 10.1088/1742-6596/1864/1/012074
16. Keller A.V., Sagadeeva M.A. The Optimal Measurement Problem for the Measurement Transducer Model with a Deterministic Multiplicative Effect and Inertia. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 1, pp. 134-138. (in Russian) DOI: 10.14529/mmp140111
17. 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, 2015, vol. 113, pp. 183-195. DOI: 10.1007/978-3-319-12145-1_11
18. Khudyakov Y.V. Parallelization of Algorithms for the Solution of Optimal Measurements in View of Resonances. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2013, vol. 6, no. 4, pp. 122-127. (in Russian)
19. 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
20. Nelson E. Dynamical Theory of Brownian Motion. Princeton, Princeton University Press, 1967.
21. Pyt’ev Yu.P., Chulichkov A.I. Metody morfologicheskogo analiza izobrazheniy [Methods of Morphological Analysis of Pictures]. Moscow, FizMatLit, 2010. (in Russian)
22. Sagadeeva M.A. Construction an Observation in the Shestakov-Sviridyuk Model in Terms of Multidimensional "White Noises" Distortion. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2020, vol. 12, no. 4, pp. 41-50. (in Russian) DOI: 10.14529/mmph200405
23. Sagadeeva M.A. Mathematical Bases of Optimal Measurements Theory in Nonstationary Case. Journal of Computational and Engineering Mathematics, 2016, vol. 3, no. 3, pp. 19-32. DOI: 10.14529/jcem160303
24. 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. 82-96. (in Russian) DOI: 10.14529/mmp190207
25. Sagadeeva M.A., Bychkov E.V., Tsyplenkova O.N. The Pyt'ev-Chulichkov Method for Constructing a Measurement in the Shestakov-Sviridyuk Model. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2020, vol. 13, no. 4, pp. 81-93. DOI: 10.14529/mmp200407
26. Savitzky A., Golay M.J.E. Smoothing and Differentiation of Data by Simplified Least Squares Procedures. Analytical Chemistry, 1964, vol. 36, no. 8, pp. 1627-1639. DOI: 10.1021/ac60214a047
27. Shestakov A.L. [Dynamic Precision of the Measuring Transducer with a Corrector in the Form of Sensor Model]. Metrologiya [Metrology], 1987, no. 2, pp. 26-34. (in Russian)
28. Shestakov A.L., Keller A.V. One-Dimensional Kalman Filter in Algorithms for Numerical Solution of the Problem of Optimal Dynamic Measurement. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2021, vol. 14, no. 4, pp. 120-125. (in Russian) DOI: 10.14529/mmp210411
29. Shestakov A.L., Keller A.V. Optimal Dynamic Measurement Method Using Digital Moving Average Filter. Journal of Physics: Conference Series, 2021, vol. 1864, article ID: 012073. DOI: 10.1088/1742-6596/1864/1/012073
30. 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
31. Shestakov A.L., Sviridyuk G.A. A New Approach to Measurement Dynamically Perturbed Signals. Bulletin of the South Ural State University. Series: Mathematical Modeling and Programming, 2010, no. 16 (192), pp. 116-120. (in Russian)
32. 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. (in Russian)
33. Shestakov A.L., Sviridyuk G.A., Keller A.V. Optimal Measurements. tXXI IMEKO World Congress "Measurement in Research and Industry", 2015, pp. 2072-2076.
34. Shestakov A.L., Sviridyuk G.A., Khudyukov Y.V. Dinamic Measurement in Spaces of "Noise". Bulletin of the South Ural State University. Series: Computer Technologies. Automatic Control. Radioelectronics, 2013, vol. 13, no. 2, pp. 4-11. (in Russian)
35. Shestakov A.L., Zagrebina S.A., Manakova N.A., Sagadeeva M.A., Sviridyuk G.A. Numerical Optimal Measurement Algorithm under Distortions Caused by Inertia, Resonances, and Sensor Degradation. Automation and Remote Control, 2021, vol. 82, no. 1, pp. 41-50. DOI: 10.1134/S0005117921010021
36. Shestakov A.L., Zamyshlyaeva A.A., Manakova N.A., Sviridyuk G.A., Keller A.V. Reconstruction of a Dynamically Distorted Signal Based on the Theory of Optimal Dynamic Measurements. Automation and Remote Control, 2021, vol. 82, no. 12, pp. 2143-2154. DOI: 10.1134/S0005117921120067
37. Shestakov A.L., Sagadeeva M.A., Manakova N.A., Keller A.V., Zagrebina S.A., Zamyshlyaeva A.A., Sviridyuk G.A. Optimal Dynamic Measurements in Presence of the Random Interference. Journal of Physics: Conference Series, 2018, vol. 1065, no. 21, article ID: 212012. DOI: 10.1088/1742-6596/1065/21/212012
38. 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), pp. 70-75.
39. Shestakov A.L., Keller A.V., Zamyshlyaeva A.A., Manakova N.A., Tsyplenkova O.N., Gavrilova O.V., Perevozchikova K.V. Restoration of Dynamically Distorted Signal Using the Theory of Optimal Dynamic Measurements and Digital Filtering. Measurement: Sensors, 2021, vol. 18, article ID: 100178. DOI: 10.1016/j.measen.2021.100178
40. 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
41. 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.
42. Shestakov A.L., Sviridyuk G.A., Keller A.V., Zamyshlyaeva A.A., Khudyakov Y.V. Numerical Investigation of Optimal Dynamic Measurements. Acta IMEKO, 2018, vol. 7, no. 2, pp. 65-72. DOI: 10.21014/acta_imeko.v7i2.529
43. Shestakov A.L., Sviridyuk G.A., Khudyakov Y.V. Dynamical Measurements in the View of the Group Operators Theory. Springer Proceedings in Mathematics and Statistics, 2015, vol. 113, pp. 273-286. DOI: 10.1007/978-3-319-12145-1_17
44. Shestakov A.L., Zagrebina S.A., Sagadeeva M.A., Bychkov E.V., Solovyova N.N., Goncharov N.S., Sviridyuk G.A. A New Method for Studying the Problem of Optimal Dynamic Measurement in the Presence of Observation Interference. Measurement: Sensors, 2021, vol. 18, article ID: 100266. DOI: 10.1016/j.measen.2021.100266
45. Sviridyuk G.A., Efremov A.A. Optimal Control for a Class of Degenerate Linear Equations. Doklady Akademii nauk, 1999, vol. 364, no. 3, pp. 323-325.
46. Sviridyuk G.A., Keller A.V. On the the Numerical Solution Convergence of Optimal Control Problems for Leontief Type System. Journal of Samara State Technical University. Series: Physical and Mathematical Sciences, 2011, no. 2 (23), pp. 24-33. (in Russian)
47. Zamyshlyaeva A.A., Keller A.V., Syropiatov M.B. Stochastic Model of Optimal Dynamic Measurements. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2018, vol. 11, no. 2, pp. 147-153. DOI: 10.14529/mmp180212