Volume 14, no. 4Pages 120 - 125

One-Dimensional Kalman Filter in Algorithms for Numerical Solution of the Problem of Optimal Dynamic Measurement

A.L. Shestakov, A.V. Keller
The article proposes the use of a digital one-dimensional Kalman filter in the implementation of numerical algorithms for solving the problem of optimal dynamic measurements to restore a dynamically distorted signal in the presence of noise. 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. The main position of the theory of optimal dynamic measurements is the modeling 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 simulated and observed output (or observed) signal is estimated. The presence of noise at the output of the measuring device makes it necessary to use digital filters in the numerical algorithms. Smoothing filters used for unknown probabilistic parameters of interference are not effective enough for filtering peak-like signals over a short time interval. In addition, the dynamics of measurements actualizes the consideration of filters that respond to rapidly changing data. The article proposes the inclusion of the procedure for filtering the observed signal into previously developed numerical algorithms, which makes it possible to either expand their application or simplify the penalty functionality.
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Keywords
optimal dynamic measurement; Kalman filter; numerical solution algorithm; Leontief type system.
References
1. 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.
2. 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.
3. 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)
4. Shestakov A., Sagadeeva M., Sviridyuk G. Reconstruction of a Dynamically Distorted Signal with Respect to the Measuring Transducer Degradation. Applied Mathematical Sciences, 2014, vol. 8, no. 41-44, pp. 2125-2130.
5. Keller A.V., Shestakov A.L., Sviridyuk G.A., Khudyukov Y.V. The Numerical Algorithms for the Measurement of the Deterministic and Stochastic Signals. Semigroups of Operators - Theory and Applications, 2015, pp. 183-195.
6. Shestakov A.L., Zagrebina S.A., Manakova N.A., Sagadeeva M.A., Sviridyuk G.A. An Algorithm for Numerically Finding the Optimal Measurement Distorted by Inertia, Resonances and Degradation of the Measuring Device. Automation and Remote Control, 2021, no. 1, pp. 55-67. (in Russian)
7. Shestakov A.L., Keller A.V. Optimal Dynamic Measurement Method Using Digital Moving Average Filter. Journal of Physics: Conference Seriesthis, 2021, vol. 1864, article ID: 012073.
8. Keller A.V. Optimal Dynamic Measurement Method Using the Savitsky-Golay Digital Filter. Differential Equations and Control Processes, 2021, vol. 2021, no. 1, pp. 1-15.
9. Kalman R.E. A New Approach to Linear Filtering and Prediction Problems. Journal of Basic Engineering, 1960, vol. 82, pp. 35-45.
10. Jazwinski A.H. Stochastic processes and filtering theory. New York: Academic Press, 1970.
11. 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.