Synthesis of a smoothing bicubic spline for differentiating experimental data of nonparametric identification

Over the past two decades, so-called Voltaire series have been used to describe the dynamics of nonlinear systems in terms of input-output. Nonparametric identification of models using Voltaire series consists in constructing estimates for impulse transition functions (IPFs) depending on two or more arguments, which naturally makes identification algorithms much more complicated than in the one-dimensional case. So, in order to identify the two-dimensional IPF (corresponding to the quadratic term of the Voltaire series), it is necessary to calculate the second-order mixed derivatives of the output two-dimensional signal of the system, when a series of rectangular pulses of different amplitudes at different times are fed to its input. Everyone knows, the problem of differentiation is an ill-posed problem and one of the manifestations of incorrectness is poor resistance to errors in the initial data. It is proposed to use two-dimensional smoothing cubic (bicubic) spline (abbreviated SBS) to overcome this problem. The two tasks that constitute SBS synthesis: assignment and implementation of different types of boundary conditions at the border of the rectangular region where SBS is determined; optimal values estimation of two smoothing parameters due to the different “smoothness” of IPF for different two arguments. An acceptable solution to this synthesis problem is proposed in the paper. Our performed computational experiment showed the efficiency of the proposed algorithm for calculating second-order mixed derivative from noisy initial data.

BUILDING 3D SURFACES OF LAND STORAGE VERTICAL CYLINDRICAL STEEL TANK USING BICUBIC SPLINE INTERPOLATION

The purpose of this work is to increase the accuracy, quality and information content of geodetic surveys of vertical steel tanks by using modern geodetic equipment and creating algorithms for data processing of these observations. Method. In order to increase the information content of data for straightening, it is proposed to calculate the geometric parameters of vertical steel tanks not only in places where data are directly obtained through instrumental observations, but also at any point of the 3D surface of the tank. The paper describes an algorithm for creating a 3D surface of a tank by bicubic spline interpolation (BSI). Results on the basis of the conducted research, it was established that the developed algorithm could be used and the 3D-surface spatial coordinates were determined. The method of determining the geometric parameters of vertical steel tanks by using BSI is improved. Scientific novelty and practical significance. Bicubic spline interpolation (BSI) was used for the first time. It greatly increases the accuracy and informality of the results of the control. The practical significance is confirmed by the control of the geometric parameters of a vertical cylindrical steel tank with a nominal capacity of 75.000 m3 with a floating roof and a double wall of the LODS “Brody” company.