THE THEORY OF OPTIMAL MEASUREMENTS

Alevtina Viktorovna Keller, Alexander Leonidovich Shestakov, Georgiy Anatolievich Sviridyuk

Abstract


The mathematical model (MM) of the measuring transducer (MT) is discussed. The MM is intended for restoration of deterministic signals distorted by mechanical inertia of the MT, resonances in MT's circuits and stochastic perturbations. The MM is represented by the Leontieff type system of equations, reflecting the change in the state of MT under useful signal, deterministic and stochastic perturbations; algebraic system of equations modelling observations of  distorted signal; and the Showalter - Sidorov initial condition. In addition the MM of the MT includes a cost functional. The minimum point of a cost functional is a required optimal measurement. Qualitative research of the MM of the MT is conducted by the methods of the degenerate operator group's theory. Namely, the existence of the unique optimal measurement is proved. This result corresponds to input signal without stochastic perturbation. To consider stochastic perturbations it is necessary to introduce so called Nelson - Gliklikh derivative for random process. In conclusion of article observations of "noises" (random perturbation, especially "white noise") are under consideration.

Keywords


mathematical model of the measuring transducer, the Leontieff type system, the Showalter - Sidorov condition, cost functional, the Nelson - Gliklikh derivative, "white noise".

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References


Da Prato G., Zabczyk J. Stochastic Equations in Infinite Dimensions. Cambridge, Cambridge University Press, 1992.

Gantmacher F.R. The Theory of Matrices. AMS Chelsea Publishing, Reprinted by American Mathematical Society, 2000.

Gliklikh Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics. London, Springer, 2011. mboxDOI: 10.1007/978-0-85729-163-9

Gliklikh Yu.E. Study of the Leontieff Type Equations with White Noise by the Methods of Mean Derivatives of Stochastic Processes. Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2012, no. 27 (286), issue 13, pp. 24-34. (in Russian)

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 & Computer Software", 2008, no. 27 (127), issue 2, pp. 50-56. (in Russian)

Keller A.V. The Leontieff type systems: classes of problems with the Showalter-Sidorov intial condition and numerical solving The Bulletin of Irkutsk State University. Series "Mathematics", 2010, vol. 3, no. 2, pp. 30-43. (in Russian)

Keller A.V. Numerical Study of Optimal Control Problems for Leontieff Type Models. Doctoral Thesis of Physicomathematical Science. Chelyabinsk, South Ural State University, 2011. (in Russian)

Keller A.V., Nazarova E.I. The Regularization Property and the Computational Solution of the Dynamic Measure Problem. Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2010, no. 16 (195), issue 5, pp. 32-38. (in Russian)

Khudyakov Yu.V. The Numerical Algorithm to Study the Shestakov-Sviridyuk Model of the Measuring Device with Inertia and Resonances. Matematicheskiye Zametki YAGU [Mathematical Note of the YASU], 2013, vol. 20, issue 2, pp. 225-236. (in Russian)

Kovacuteacs M., Larsson S. Introduction to stochastic partial differential equations. Processing of "New Directions in the Mathematical and Computer Sciences", National Universities Commission. October 8-12. 2007. Abuja. Nigeria. Publications of the ICMCS, 2008, no. 4, pp. 159-232.

Manakova N.A. On a Hypothesis of G.A. Sviridyuk The Bulletin of Irkutsk State University. Series "Mathematics", 2011, vol. 4, no. 4, pp. 87-93. (in Russian)

Melnikova I.V., Filinkov A.I., Alshansky M.A. Abstract Stochastic Equations II. Solutions in Spaces of Abstract Stochastic Distribotions. Journal of Mathematical Sciences. 2003, vol. 116, no. 5, pp. 3620-3656.

Melnikova I.V., Filinkov A.I. Generalized solutions to abstract stochastic problems. J. Integ. Transf. and Special Funct., 2009, vol. 20, no. 3-4, pp. 199-206.

Nelson E. Dynamical Theories of Brownian Motion. Princeton, Princeton University Press, 1967.

Shestakov A.L. Dynamic Accuracy of the Transmitter With a Correction Device as a Sensor Model. Metrology. 1987, no. 2, pp. 26. (in Russian)

Shestakov A.L., Keller A.V., Nazarova E.I. Numerical solution of the optimal measurement problem. Automation and Remote Control. 2012, no. 1, pp. 107-115.

Shestakov A.L., Sviridyuk G.A. A New Approach to Measurement of Dynamically Perturbed Signal. Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2010, no. 16 (192), issue 5, pp. 116-120. (in Russian)

Shestakov A.L., Sviridyuk G.A. Optimal Measurement of Dynamically Distorted Signals. Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2011, no. 17 (234), issue 8, pp. 70-75.

Shestakov A.L., Sviridyuk G.A. A New Concept of White Noise. Obozrenie prikladnoy i promyshlennoy matematiki [Survey of Applied and Industrial Mathematics]. 2012, issue 19, no. 2, pp. 287-288.

Shestakov A.L., Sviridyuk G.A. On the Measurement of the "White Noise". Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2012, no. 27 (286), issue 13, pp. 99-108.

Shestakov A.L., Sviridyuk G.A., Khudyakov Yu.V. Dynamic Measurements in Spaces of "Noise". Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2013, no. 2, issue 13, pp. 4-11. (in Russian)

Shestakov A.L., Sviridyuk G.A., Sagadeeva M.A. Reconstruction of a Dynamically Distorted Signal with Respect to the Measure Tranducer Degradation. Applied Mathematical Sciences. 2014, vol. 8, no. 41-44, pp. 2125-2130.

Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, K"oln, Tokyo, VSP, 2003.

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 & Computer Software", 2014, no. 7, issue 1, pp. 90-103. (in Russian)

Sviridyuk G.A., Zagrebina S.A. The Showalter - Sidorov Problem as Phenomena of the Sobolev-Type Equations. The Bulletin of Irkutsk State University. Series "Mathematics", 2010, vol. 3, no. 1, pp. 51-72. (in Russian)

Zagrebina, S.A., Soldatova, E.A. The linear Sobolev-type Equations With Relatively p-bounded Operators and Additive White Noise. The Bulletin of Irkutsk State University. Series "Mathematics", 2013, no. 1, pp. 20-34. (in Russian)

Zamyshlyaeva A.A. Stochastic Incomplete Linear Sobolev Type High-Ordered Equations with Additive White Noise. Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2012, no. 40, issue 14, pp. 73-82. (in Russian)


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