RECONSTRUCTION OF THE TWO VARIABLES DISCONTINUOUS FUNCTION BY DIFFERENT INFORMATION OPERATORS USING TRIANGULAR ELEMENTS

The paper examines methods for constructing mathematical models of two variables discontinuous functions using various information about them: one-sided values at points and one-sided traces along a given system of lines. The case is considered when the domain of the required function is triangulated by right-angled triangles. If interpolation or approximation methods are used, then for their construction the values of the function at given points must be given; if we use interlination methods, then traces of the desired function along a given system of lines. In this work, we construct a discontinuous interpolation and approximation splines for approximating a discontinuous function of two variables with given one-sided values in a given system of points (in our case, at the vertices of right-angled triangles), and prove theorems on the estimation of the approximation error by constructed discontinuous structures. In the paper a discontinuous interlination spline, which uses completely different information about the discontinuous function, namely one-sided traces along a given system of lines (in our case, along the sides of right-angled triangles) is also built. Interlination of functions can find wide application in the aircraft and automobile body design automation; when receiving and processing the results of sonar and radar, when solving problems of computed tomography, in digital signal processing and in many other areas. In the paper theorems on the integral form and an estimate of the approximation error by the constructed discontinuous interlination operator are also proved. Computational experiments that compare the results of the approximation of a discontinuous function of two variables by different information operators using triangular elements are presented. In the future, it is planned to apply the constructed operators of discontinuous approximation and interlination to solve a two-dimensional problem of computed tomography with a significant use of the inhomogeneity of the internal structure of the body, which must be reconstructed.

A new continuous integral sliding mode control algorithm for inverted pendulum and 2-DOF helicopter nonlinear systems: Theory and experiment

This paper introduces a new continuous integral sliding mode control algorithm, where the discontinuous function of the super-twisting control law is replaced with a continuous disturbance observer for the substantial chattering attenuation. In the present integral sliding mode control, the discontinuous function generates chattering that is undesirable for several real-time applications. The proposed control strategy decreases the amplitude of the controller gain compared to the existing integral sliding mode controls, and as a consequence of this, the attenuation of chattering is achieved to a great extent. The efficacy of the proposed control algorithm is validated successfully on the single-input single-output Inverted Pendulum and 2-DOF Helicopter nonlinear coupled multi-input multi-output systems. The simulation and experimental results demonstrate the successful application of the proposed control approach to follow reference inputs and acquire robustness and stabilization of the system in the presence of limited matched perturbations and nonlinearities.

Analysis of the results of a computational experiment to restore the discontinuous functions of two variables using projections

This article presents the main statements of the method of approximation of discontinuous functions of two variables, describing an image of the surface of a 2D body or an image of the internal structure of a 3D body in a certain plane, using projections that come from a computer tomograph. The method is based on the use of discontinuous splines of two variables and finite Fourier sums, in which the Fourier coefficients are found using projection data. The method is based on the following idea: an approximated discontinuous function is replaced by the sum of two functions – a discontinuous spline and a continuous or differentiable function. A method is proposed for constructing a spline function, which has on the indicated lines the same discontinuities of the first kind as the approximated discontinuous function, and a method for finding the Fourier coefficients of the indicated continuous or differentiable function. That is, the difference between the function being approximated and the specified discontinuous spline is a function that can be approximated by finite Fourier sums without the Gibbs phenomenon. In the numerical experiment, it was assumed that the approximated function has discontinuities of the first kind on a given system of circles and ellipses nested into each other. The analysis of the calculation results showed their correspondence to the theoretical statements of the work. The proposed method makes it possible to obtain a given approximation accuracy with a smaller number of projections, that is, with less irradiation.

APLIKASI INTEGRAL DALAM BIDANG EKONOMI DAN FINANSIAL

The characteristics of a function are usually investigated by looking at the continuity of the function. But what happens if a function does not have continuous properties? To what extent can the characteristics of continuous function be maintained for discontinuous cases? The stochastic function that is widely involved in solving problems in the field of average financial mathematics is a discontinuous function. This is reflected by the acquisition of a smooth curve from the modeling drawing obtained. Today, the nature of continuous functions in [a, b] has been widely studied and developed. Some properties of the continuous function can be extended to the appropriate discontinuous function. In this paper, there will be some integral reviews for discontinuous functions which are closely related to stochastic functions.

A Novel Filter Extracted Equivalent Control Based Fixed Frequency Sliding Mode Approach for Power Electronic Converters

The key issue in the implementation of the Sliding Mode Control (SMC) in analogue circuits and power electronic converters is its variable switching frequency. The drifting frequency causes electromagnetic compatibility issues and also adversely affect the efficiency of the converter, because the proper size of the inductor and the capacitor depends upon the switching frequency. Pulse Width Modulation based SMC (PWM-SMC) offers the solution, however, it uses either boundary layer approach or employs pulse width modulation of the ideal equivalent control signal. The first technique compromises the performance within the boundary layer, while the latter may not possess properties like robustness and order reduction due to the absence of the discontinuous function. In this research, a novel approach to fix the switching frequency in SMC is proposed, that employs a low pass filter to extract the equivalent control from the discontinuous function, such that the performance and robustness remains intact. To benchmark the experimental observations, a comparison with existing double integral type PWM-SMC is also presented. The results confirm that an improvement of 20% in the rise time and 25.3% in the settling time is obtained. The voltage sag during step change in load is reduced to 42.86%, indicating the increase in the robustness. The experiments prove the hypothesis that a discontinuous function based fixed frequency SMC performs better in terms of disturbances rejection as compared to its counterpart based solely on ideal equivalent control.