Arctangent Function Research Articles

Sensorless Control of Permanent Magnet Linear Synchronous Motor Based On Double Sliding Mode

Background & Objective: This document presents a dual sliding mode vector control method to estimate speed and position of Permanent Magnet Linear Synchronous Motor (PMLSM). Methods: Firstly, the motor simulation model is built based on the mathematical model of PMLSM. For the sake of solving the problems of the jitter and position angle deviation in the traditional SMO, the inverse hyperbolic tangent function is used and the software phase-locked loop algorithm is introduced, and the gain of exponential approach rate is improved to change the real-time, so as to eliminate the problems of high-frequency jitter and angle deviation. Secondly, global fractional order integral sliding mode control (GFOISMC) is used to replace PI control to eliminate jitter and improve the robustness to load disturbance. Finally, the dynamic performance of the system under constant speed no-load and constant speed sudden load is simulated. Results: The simulation result proves a proposed dual sliding mode control strategy combining the improved SMO and GFOISMC that can effectively reduce the jitter and improve robustness. Conclusion: In this paper, two sensorless control strategies are compared and their robust performance is studied. The two control scheme is simulated in the MATLAB/Simulink environment.

Harmonic analysis of the arctangent function regarding the angular error introduced by superimposed Fourier series for application in sine/cosine angle encoders

We present a rigorous analytical method for harmonic analysis of the angular error of rotary and linear encoders with sine/cosine output signals in quadrature that are distorted by superimposed Fourier series. To calculate the angle from measured sine and cosine encoder channels in quadrature, the arctangent function is commonly used. The hence non-linear relation between raw signals and calculated angle—often thought of as a black box—complicates the estimation of the angular error and its harmonic decomposition. By means of a Taylor series expansion of the harmonic amplitudes, our method allows for quantification of the impact of harmonic signal distortions on the angular error in terms of harmonic order, magnitude and phase, including an upper bound on the remaining error term—without numerical evaluation of the arctangent function. The same approximation is achieved with an intuitive geometric approximation in the complex plane, validating the results. Interaction effects between harmonics in the signals are considered by higher-order Taylor expansion. The approximations show an excellent agreement with the exact calculation in numerical examples even in case of large distortion amplitudes, leading to practicable estimates for the angular error decomposition.

Adaptive iterative learning control for high-order nonlinear systems with random initial state shifts

Aiming at the tracking problem for a class of high-order nonlinear systems with arbitrary initial state, this paper proposes different initial state shifts rectifying schemes for systems with different orders from the perspective of solving differential equations to ensure that the systems achieve complete tracking over the specified interval. This method relaxes the requirement of initial positioning in iterative learning control, and solves the problem of iterative learning control for high-order nonlinear systems with arbitrary initial state error. The theoretical analysis shows that the proposed schemes can make all signals in the system bounded, and ensure that the tracking error in the preset interval tends to zero as the number of iterations increases. When designing the controller, the arctangent function is used to avoid flutter of the control signal and related variables. The final simulations verify the effectiveness of the proposed algorithms.

Single-shot high-precision 3D reconstruction with color fringe projection profilometry based BP neural network

In order to reduce the influence of nonlinear factors such as gamma nonlinearity and color crosstalk on the phase retrieval accuracy with color fringe projection profilometry (CFPP) in rapid measurement, a single-shot high-precision backpropagation (BP) neural network based CFPP is proposed. Firstly, neural networks are trained from samples through a training process. The neural network inputs are the RGB components of composite color fringe patterns. The numerator and denominator of the arctangent function are used as network outputs. The monochromatic fringe patterns are adopted. The numerator and denominator of the arctangent function are calculated by the 15-step phase-shift method, which serves as the network training target. Applying this neural network to the prediction of the standard phase can reduce the influence of nonlinear factors on the phase accuracy. Then, the absolute phase was unwrapped by using geometric constraints. Finally, the 3D reconstruction with high precision can be realized by using only a single-shot color fringe. The theoretical model of the proposed method and the process of neural network training are introduced. The results of 3D reconstruction are compared with other algorithms in the same condition. The experimental results demonstrate that the RMS of the proposed algorithm can reach 0.0107 mm, compared with 0.0121 mm of the traditional 3-step phase-shift method. The proposed method is applied to the online quality inspection of relay components, and the results verify the high accuracy and high efficiency of measurement.

Probability distribution arithmetic optimization algorithm based on variable order penalty functions to solve combined economic emission dispatch problem

The optimization objective of combined economic emission dispatch (CEED) problem is to minimize the total cost of power generation and the emission of harmful gases under the premise of satisfying the load demand and operation constraints. In today's world, energy and environmental problems are becoming more and more serious. Solving CEED problems can reduce costs, save energy, and reduce environmental pollution. By adopting the maximum/maximum price penalty factor, the multi-objective CEED optimization problem is transformed into a single objective optimization problem, and a probability distribution arithmetic optimization algorithm (AOA) based on variable order penalty function was proposed to solve the CEED problem. The energy attenuation strategy was proposed to replace the variation trend of Math optimizer accelerated (MOA) in the original AOA, which better balances the exploration and exploitation of AOA. Five probability distribution functions, including exponential distribution, uniform distribution, normal distribution, beta distribution and gamma distribution, are used to replace a control parameterμ, so as to enhance its searching ability, improve the convergence speed, and enhance the ability to jump out of the local optimal. Based on the CEC-BC-2017 benchmark functions, the original AOA is compared with the five improved AOA to select the best improved strategy, and the effectiveness of the improved strategy is verified by simulation comparison with other intelligent optimization algorithms. In dealing with the constraint of CEED problem, five kinds of penalty functions with varying order were proposed, which are sine function, hyperbolic tangent function, arctangent function, V-type function and hyperbolic secant function, respectively, to replace the traditional fixed penalty function. The CEED problem with 6 units are selected and solved under the total demand of 150 MW, 175 MW, 200 MW and 225 MW respectively. The experimental results show that the AOA based on probability distribution is better than other optimization algorithms in solving CEED problems, and the variable order penalty strategy improves the solution quality and convergence speed compared with the fixed penalty strategy.

An AtanTV Nonconvex Regularization Model for MRI Reconstruction

Recently, the nonconvex regularization model has attracted much research attention in magnetic resonance imaging (MRI). In comparison with the traditional total variation (TV) regularization method, a nonconvex TV regularizer can effectively improve the fitting performance and prevent bias problems. In this work, we utilize the arctangent function as nonconvex TV regularize term (AtanTV) for MRI reconstruction. With appropriate parameters, the AtanTV model can avoid the systematic underestimation characteristic and maintain the objective function still convex. To address the AtanTV model, we propose an efficient alternating direction method of multipliers (ADMM) method to solve it. Experimental results indicate the superior performance (PSNR, RE, SSIM, etc.) of the proposed MRI reconstruction method.

Study on environment modeling and electromagnetic scattering characteristics in mountainous areas

ABSTRACT This paper proposed a modeling method to describe the environmental structure of mountainous areas by using the composite rough surface model, and the electromagnetic (EM) scattering characteristics of the composite rough surfaces are studied. Based on the Monte Carlo method and Gaussian spectral function, the rough surface of the terrestrial environment is generated. And the inverse tangent function is introduced to weighted rough surfaces with different statistical properties, making rough surfaces join smoothly. On this basis, geometric models of rough surfaces of three typical mountainous environments, valley, hillside, and ridge, are established. The shooting and bouncing ray (SBR) method is used to calculate the EM scattering characteristics. Due to the multiple reflections of EM waves, the idea of ray-density normalization (RDN) is introduced to improve the EM flow on triangular facets illuminated by multiple rays. Finally, the effects of different slope angles, surface roughness, and incident wave angles on the EM scattering characteristics are discussed to provide some useful conclusions. The simulation results show that the rough surface with a certain slope will greatly change the scattering characteristics of the whole environment and make the spatial distribution of scattering energy show a new characteristic.

A rolling element bearing fault feature extraction method based on the EWT and an arctangent threshold function

A rolling element bearing fault feature extraction method based on the EWT and an arctangent threshold function

Some Notes for Two Generalized Trigonometric Families of Distributions

The paper deals with two general families of cumulative distribution functions based on arctangent function. We provide analysis of the error of the best one-sided Hausdorff approximation for some special cases of these families. We obtain precious estimates for the value of the Hausdorff distance that can be used as an additional criterion in practice. Further, the family of recurrence generated adaptive functions is constructed and investigated. All new results are illustrated with suitable numerical experiments. Simple dynamic software modules show applicability of Hausdorff approximation.

Research on the Applicability of Vibration Signals for Real-Time Train and Track Condition Monitoring.

The purpose of this research was to analyze the possibilities for the application of vibration signals in real-time train and track control. Proper experiments must be performed for the validation of the methods. Research on vibration in the context of transport must entail many of the different nonlinear dynamic forces that may occur while driving. Therefore, the paper addresses two research cases. The developed application contains the identification of movement and dynamics and the evaluation of the technical state of the rail track. The statistics and resultant vector methods are presented. The paper presents other useful metrics to describe the dynamical properties of the driving train. The angle of the resultant horizontal and vertical accelerations is defined for the evaluation of the current position of cabin. It is calculated as an inverse tangent function of current longitudinal and transverse, longitudinal and vertical, transverse, and vertical accelerations. Additionally, the resultant vectors of accelerations are calculated.

The effect of the altitude on the performance of a solar chimney

This study aimed to show the relationship between the altitude and the performance of a solar chimney based on simulation analysis and numerical calculation. Steady-state heat transfer from isothermal vertical walls of a chimney is modelled analytically. The paper introduces the basic concepts and governing equations for modelling heat transfer and air flow rates within the solar chimney at various altitudes. The relationship between altitude, temperature, and mass flow rates for the buoyancy-driven chimney flow is complex. A multidimensional equation comprising the inverse tangent and inverse hyperbolic tangent functions is developed. A Matlab program was developed to overcome the problem of the non-linearity of the equation. The results obtained showed that the altitude has modest influence but not substantial on the performance of the solar chimney if we assumed a constant ambient air temperature for all altitudes. But in real climatic conditions, the ambient temperature is altitude dependent. In these circumstances, the performance of the solar chimney is significantly influenced by the altitude. Therefore, microclimatic conditions should be used rather than large-scale area climatic data for solar chimney design.

Structural mapping of potential field sources using BHG filter

Edge detection is one of the most commonly used interpretation techniques applied to potential data for detecting subsurface susceptibility/density discontinuities. Several edge detection techniques utilize the first, second, or even third-order gradients of the magnetic field to delineate the susceptibility contrasting zones. A new filter called Balanced Horizontal Gradient (BHG) filter is designed for enhancing the edges of the subsurface bodies that overcomes the limitations of the conventional filters. The BHG filter is based on the arctangent function of derivatives of horizontal gradients. The main advantage of the proposed filter is that it seems to detect all the source edges with no false edges generated around the true edges and effectively balances the edges arising from the shallow and deep-seated sources. We applied the filter on the aeromagnetic data from central India and identified the extensions of the Godavari rift and central Indian shear underneath the Deccan volcanic province.

Improved Variable Step Size Least Mean Square Algorithm for Pipeline Noise

In this study, we employ the active noise control (ANC) method to eliminate the low-frequency part of the noise generated by the rotation of the axial fan in heating, ventilation, and air-conditioning (HVAC) pipelines. Because the traditional variable step size least mean square (VSS-LMS) algorithm has poor tracking performance, we propose a variable step size filtered-X least mean square (FXLMS) algorithm based on the arctangent function to improve the adaptive filtering method of the convergence speed and noise cancellation effect. The step size of the proposed algorithm can be adjusted according to the error. When the error signal is significant, a larger step can be obtained, and when the error is small, the step size smoothness of the algorithm can be optimized. Compared with the traditional VSS-LMS algorithm, the convergence speed of the proposed algorithm is increased by 29%, the noise reduction effect is enhanced by 19%, and the mean square error (MSE) is reduced by 23% (0.0084). In addition, we developed a hardware experimental platform based on noise characteristics. In the noise reduction test using a GB/T 5836.2-06 standard PVC pipeline, the system reduced the noise by 12–17 dB.

EVALUATING LOG-TANGENT INTEGRALS VIA EULER SUMS

An investigation into the representation of integrals involving the product of the logarithm and the arctan functions, reducing to log-tangent integrals, will be undertaken in this paper. We will show that in many cases these integrals take an explicit form involving the Riemann zeta function, the Dirichlet eta function, Dirichlet lambda function and many other special functions. Some examples illustrating the theorems will be detailed.

New BBP-Type Formulae for π Derived from New Forms of Taylor Expansions of Inverse Tangent Function

In this work, we give two new Taylor expansions of arctan(x + ω), where ω represents a finite increment of x. We discover several remarkable infinite series from these expansions by special substitutions. Some of these infinite series give BBP-type formulae.

A pseudorandom bit generator based on hyperbolic cosine function

In the literature, little attention is paid to devising and analyzing novel one dimensional chaotic mappings. In our previous efforts, we have tried fold, translation & scale on arctangent function & sigmoid function respectively, which brings good results. In this paper, we do the same to obtain a variant of Hyperbolic Cosine Function. Both Bifurcation Diagram & Lyapunov Exponent Spectrum manifest that the new mapping possesses wonderful chaotic properties. Then, a pseudorandom bit generator is designed based on it. Pseudorandom tests demonstrate that the generator is much better than our previous ones. It owns great application prospect.

Refined Disturbance Rejection-Based Composite Control of Flexible Spacecrafts for Tracking a Tumbling Non-Cooperative Target

When tracking a tumbling non-cooperative target, the position and attitude maneuver control with high-precision as well as fast-response performances is required for flexible spacecrafts. However, the unknown rigid-flexible coupling and model uncertainties will degrade the control performances significantly. Therefore, this paper proposes a refined disturbance rejection-based composite control method for spacecraft position and attitude tracking. First of all, the disturbances including the rigid-flexible coupling and model uncertainties are described by an exogenous model by fully using the partially known disturbance information (e.g., modal frequency and damping). Then, a novel exogenous model-based sliding mode disturbance observer (SMDO) is proposed to achieve the refined disturbance estimation. Next, by combining the SMDO in the feedforward loop and an adaptive terminal sliding mode control (ATSMC) in the feedback loop, a finite-time composite control law is proposed to ensure the fast disturbance rejection and position/attitude tracking simultaneously. In the composite controller, a novel nonsingular terminal sliding mode surface with arctangent function is proposed. Moreover, a time-varying boundary Lyapunov function (BLF) is designed to preassign the transient and steady-state performances of sliding mode surface. Finally, the effectiveness of proposed method is verified via numerical simulation.

MacLaurin’s series expansions for positive integer powers of inverse (hyperbolic) sine and tangent functions, closed-form formula of specific partial Bell polynomials, and series representation of generalized logsine function

In the paper, the authors find series expansions and identities for positive integer powers of inverse (hyperbolic) sine and tangent, for composite of incomplete gamma function with inverse hyperbolic sine, in terms of the first kind Stirling numbers, apply a newly established series expansion to derive a closed-form formula for specific partial Bell polynomials and to derive a series representation of generalized logsine function, and deduce combinatorial identities involving the first kind Stirling numbers.

A fast finite sample count symmetric component extraction method for use in grid side converters

The considerable increase in the number of renewable energy sources in power systems in recent years has led to the grid side converters’ connection requirements being more and more strict. In order to comply with the newly arising grid codes, the synchronization unit has to provide a fast and robust response even under unbalanced and distorted line voltage. This paper proposes a new method for very fast extraction of symmetrical components. The method uses a finite number of grid voltage samples to solve a set of algebraic equations and obtain sine and cosine factors of the direct and inverse sequence components. The corresponding amplitudes and angles are then found by using √ and arctan functions. The proposed extraction algorithm requires a derivative estimation technique to estimate the frequency from the direct sequence angle. The performance of the proposed finite sample count (FSC) method is compared to the all-pass filter (APF) based synchronous reference frame phase-locked-loop (SRF PLL) in simulations and experimentally using a DSP based platform and grid emulator with analog outputs. Under the voltage dip, FSC responds within several hundreds of µs, whereas it takes around 3 ms for the APF SRF PLL to settle.

Event-triggered intelligent critic control with input constraints applied to a nonlinear aeroelastic system

In this paper, we establish an event-triggered intelligent control scheme with a single critic network, to cope with the optimal stabilization problem of nonlinear aeroelastic systems. The main contribution lies in the design of a novel triggering condition with input constraints, avoiding the Lipschitz assumption on the inverse hyperbolic tangent function. Based on an improved weight updating criterion that eliminates the requirement of initial admissible control, the control law is obtained approximately by online training of a single critic network. The Lyapunov stability and the Zeno phenomenon of the closed-loop system are analysed. The feasibility of the established algorithm is verified by applying it to an optimal stabilization task of a nonlinear aeroelastic system. The results reveal that the developed approach can handle input-constrained optimal control problems, with performance comparable to the time-based method that updates control inputs at each instant, while reducing the computational and communication's load.