Showing papers on "Fourier series published in 2022"

Approximation-Based Nussbaum Gain Adaptive Control of Nonlinear Systems With Periodic Disturbances

Abstract: This article considers the Nussbaum gain adaptive control issue for a type of nonlinear systems, in which some sophisticated and challenging problems, such as periodic disturbances, dead zone output, and unknown control direction are addressed. The Fourier series expansion and radial basis function neural network are incorporated into a function approximator to model time-varying-disturbed function with a known period in nonlinear systems. To deal with the problems of the dead zone output and unknown control direction, the Nussbaum-type function is recommended in the design of the control algorithm. Applying the Lyapunov stability theory and backstepping technique, the proposed control strategy ensures that the tracking error is pulled back to a small neighborhood of origin and all closed-loop signals are bounded. Finally, simulation results are presented to show the availability and validity of the analysis approach.

Novel approach for fractal nonlinear oscillators with discontinuities by fourier series

Abstract: This paper argues that in the microgravity space, the fractal nonlinear oscillators’ models with discontinuities are established by the fractal calculus, and their fractal variational principles are obtained via using the fractal semi-inverse method. Finally, a novel technology is proposed to find the frequency of the fractal nonlinear oscillators’ model by using the Fourier series. The results illustrate that the proposed method is efficient and accurate.

Adaptive the Dirichlet model of mobile/immobile advection/dispersion in a time-fractional sense with the reproducing kernel computational approach: Formulations and approximations

Abstract: In this paper, we will first present the TFMIADM with its adequate Dirichlet constraints. Right after that, we will review the formation of that model under the terms and assumptions of the RKHSM computational approach. The solutions and modeling of the utilized model will be discussed based on Caputo’s connotation of the partial time derivative. We will present the scores required to construct the appropriate spaces for the method and we will present several theories such as solutions representations, convergence restriction, and order of error. With the use of the Fourier functions expansion rule, the numeric–analytic solutions are expressed by collection sets of orthonormal functions system in [Formula: see text] and [Formula: see text] spaces. Right after that, we will solve this model in both time and space domains using the algorithms of the method used. Indeed, several drawings and tables that expound on the effectiveness and strength of the approach and its adaptation to the issue reviewed are utilized. In the end, some points of view and highlights are presented side by side with the most important modern references used.

A Pseudo-Spectral Fourier Collocation Method for Inhomogeneous Elliptical Inclusions with Partial Differential Equations

Abstract: Inhomogeneous elliptical inclusions with partial differential equations have aroused appreciable concern in many disciplines. In this paper, a pseudo-spectral collocation method, based on Fourier basis functions, is proposed for the numerical solutions of two- (2D) and three-dimensional (3D) inhomogeneous elliptic boundary value problems. We describe how one can improve the numerical accuracy by making some extra “reconstruction techniques” before applying the traditional Fourier series approximation. After the particular solutions have been obtained, the resulting homogeneous equation can then be calculated using various boundary-type methods, such as the method of fundamental solutions (MFS). Using Fourier basis functions, one does not need to use large matrices, making accrual computations relatively fast. Three benchmark numerical examples involving Poisson, Helmholtz, and modified-Helmholtz equations are presented to illustrate the applicability and accuracy of the proposed method.

Dynamic stability of hybrid fiber/nanocomposite-reinforced toroidal shells subjected to the periodic axial and pressure loadings

Abstract: Abstract In this article, the dynamic instability of the hybrid fiber/nanocomposite-reinforced toroidal shells is investigated. By using the approximation of Fourier series and applying the Galerkin method, a semi-analytical solution is achieved. The proposed method provides an accurate assessment of the dynamic stability of the shell with no computational costs in comparison to the numerical methods. By comparing the obtained results for some examples with those available in the present literature, the accuracy of present formulation is approved. In order to evaluate the effects of geometrical and mechanical specifications on the dynamic instability of the proposed shells, a comprehensive parametric study is performed, as well.

A flexible grey Fourier model based on integral matching for forecasting seasonal PM2.5 time series

Abstract: The PM 2.5 in each city exhibits seasonal and trend variations, but its seasonal pattern differed regionally. Under the novel grey modelling framework, a flexible grey Fourier model is developed by introducing the Fourier series to approximate the seasonal forcing. An integral matching method is employed to estimate the structural parameters and initial value simultaneously, then a data-driven order selection approach is utilized to accommodate various seasonal features. Next, Monte-Carlo simulation is designed to verify the effectiveness of the order selection approach and the influence of noise level. Finally, this model is established for predicting the monthly PM 2.5 of four capital cities in the Yangtze River Delta of China. The results indicate that it not only reflects the different seasonal patterns of the four cities but also performs well compared to the seven competitive models.

Short-term traffic prediction based on time series decomposition

Abstract: Traffic flow decomposition is an alternative method to explore the composition of traffic flow and improve prediction accuracy. However, most of them suffer from the inability to fully utilize the character of traffic data. This paper presents a novel framework for traffic flow decomposition and modeling named Time Series Decomposition (TSD). The traffic flow is adaptively decomposed into periodic component, residual component and volatile component which are modeled respectively. Empirical Mode Decomposition (EMD) is applied to extract the intrinsic mode functions (IMFs) of traffic flow, the periodic patterns are intuitively presented via Hilbert transform in terms of frequencies. Then the periodic component can be described as a Fourier series based on obtained frequencies. Meanwhile, the residual component is presented by IMF with the lowest frequency. The remaining part is the volatile component modeled by supervised learning. The proposed hybrid model is evaluated on the real-world dataset and compared with classical baseline models. The results demonstrate that TSD can unearth the underlying periodic patterns and provide an explicable composition of the traffic flow. Furthermore, the volatile component ensures the accuracy of single-step prediction while periodic and residual components show promising abilities in improving the multi-step prediction accuracy of short-term traffic flow.

Seeing Implicit Neural Representations as Fourier Series

Abstract: Implicit Neural Representations (INR) use multilayer perceptrons to represent high-frequency functions in low-dimensional problem domains. Recently these representations achieved state-of-the-art results on tasks related to complex 3D objects and scenes. A core problem is the representation of highly detailed signals, which is tackled using networks with periodic activation functions (SIRENs) or applying Fourier mappings to the input. This work analyzes the connection between the two methods and shows that a Fourier mapped perceptron is structurally like one hidden layer SIREN. Furthermore, we identify the relationship between the previously proposed Fourier mapping and the general d-dimensional Fourier series, leading to an integer lattice mapping. Moreover, we modify a progressive training strategy to work on arbitrary Fourier mappings and show that it improves the generalization of the interpolation task. Lastly, we compare the different mappings on the image regression and novel view synthesis tasks. We confirm the previous finding that the main contributor to the mapping performance is the size of the embedding and standard deviation of its elements.

Unified series solution for thermal vibration analysis of composite laminated joined conical-cylindrical shell with general boundary conditions

Abstract: We establish a unified series solution for the free, steady-state, and transient vibration analysis of composite laminated joined conical–cylindrical shells under thermal environment. Additionally, artificial spring technology was used to deal with the general boundary conditions and continuity conditions for the common edge of the joined conical–cylindrical shell. The spectro-geometric method (SGM) and the first-order shear deformation theory (FSDT) were employed to derive the dynamic equation of motion. Subsequently, the displacement admissible functions were expressed in a general form comprising the Fourier sine and cosine terms to manipulate the arbitrary boundary conditions without pre-satisfying them. Moreover, the convergence and solution efficiency of the proposed approach was demonstrated by comparison with the finite element method solution. Furthermore, the influence of critical factors on the vibration behaviors of the coupled shells was investigated, with findings that would aid in their structural design.

Effective numerical technique for nonlinear Caputo-Fabrizio systems of fractional Volterra integro-differential equations in Hilbert space

Abstract: The point of this paper is to analyze and investigate the analytic-approximate solutions for fractional system of Volterra integro-differential equations in framework of Caputo-Fabrizio operator. The methodology relies on creating the reproducing kernel functions to gain analytical solutions in a uniform form of a rapidly convergent series in the Hilbert space. Using the Gram-Schmidt orthonomalization process, the orthonormal basis system is constructed in a dense compact domain to encompass the Fourier series expansion in view of reproducing kernel properties. Besides, convergence and error analysis of the proposed technique are discussed. For this purpose, several numerical examples are tested to demonstrate the great feasibility and efficiency of the present method and to support theoretical aspect as well. From a numerical point of view, the acquired solutions simulation indicates that the methodology used is sound, straightforward, and appropriate to deal with many physical issues in light of Caputo-Fabrizio derivatives.

A Fourier-Based Semi-Analytical Model for Foil-Wound Solid-State Transformers

Abstract: A mesh-free semi-analytical Fourier-based method is presented to evaluate the frequency-dependent losses in the windings of solid-state transformers (SSTs) with foil-wound transformers. In the employed diffusion equation, which accounts for induced eddy currents, the imposed current density and conductivity in the transformer window area are represented using spatial Fourier series and method of images. As a result, induced current density distribution in foil conductors and ac loss in medium frequency transformers (MFTs) is calculated using the semi-analytical method. The proposed method is verified using the finite element method (FEM). It is observed that with the considered approach one can estimate the ac winding loss in the frequency range of 5–25 kHz, with 2.7% maximum absolute error and approximately 2.5 times less number of degrees-of-freedom compared to FEM computations.

Observer-Based Adaptive Control for Single-Phase UPS Inverter Under Nonlinear Load

Abstract: This article proposed a current-sensorless control scheme for the single-phase uninterruptible power supply (UPS) inverter under nonlinear load. Different from the existing current-sensorless literature, the information of load current can be obtained by the designed adaptive laws. It is meaningful for the system’s security and reliability. Under the assumption that the load is a periodic ideal-independent current source, the Fourier series in trigonometric form is expanded to model the unknown disturbance of load current. The observer-based feedback control rate and update laws are derived to adjust online the Fourier coefficients of the load current modeling. Furthermore, the stability of the system is rigorously analyzed via the Lyapunov approach. Under the same limitation of not using the current sensor, comparative experiments are designed. Simulation and experimental results have verified the effectiveness of the control approaches.

A Unified Solution for Free Vibration Analysis of Beam-Plate-Shell Combined Structures with General Boundary Conditions

Abstract: A semi-analytical method is presented to analyze free vibration response of beam-plate-shell combined structures with general boundary conditions. Based on the beam-plate-shell energy theory, the coupled annular plate-conical-cylindrical-spherical shell with stiffened rings and bulkheads regarded as the theoretical model is constructed. The unified displacement admissible functions of each substructure are expanded as modified Fourier series and auxiliary convergence functions along generatrix direction and Fourier series along circumferential direction. Virtual spring technology is adopted to express the energy stored at the junction of adjacent substructures and both boundaries. The energy variational procedure and Ritz method are used to obtain the vibrational governing equation of the combined structure. The present method provides an analytical way for the vibrational response of complicated combined structures. The convergence, accuracy and reliability are validated by comparing the free vibrational response with those of the references and finite element method. Some numerical examples show effects of different boundary conditions on the free vibration. And the influence of stiffened rings and bulkheads treated as Euler-beams and annular plates is also discussed from quantity, size and spatial distribution, offering a feasible way to design the reinforced structures and optimize the bulkheads in engineering problems.