Arctangent Research Articles

Inverse tangent series involving pell and pell-lucas polynomials

AbstractBy means of the telescoping method, several summation formulae are established for the arctangent function with its argument being Pell and Pell–Lucas polynomials. Numerous infinite series identities involving Fibonacci and Lucas numbers are included as particular cases.

Novel fast fixed-time sliding mode trajectory tracking control for manipulator

This paper studies the issue of high-precision trajectory tracking control for manipulator systems with uncertainty disturbances and develops a fast fixed-time control scheme. First, a new type of (FTSM) surface is constructed by combining inverse trigonometric functions, regardless of the size of the system state, which can obtain a faster convergence velocity and high accuracy. The stabilization time of this sliding mode (SM) surface is independent of the initial state. Subsequently, on this basis, an adaptive fixed-time control method is constructed by combining adaptive techniques through the new fixed-time stability lemma proposed in this paper, under which the system's SM variables and tracking errors are guaranteed to reach the region near the equilibrium point at a fixed time while the adaptive law estimates the upper bound of the disturbance and effectively weakens the chattering phenomenon. Finally, the simulation comparison is carried out under different conditions, and the simulation results show the superiority of the control scheme with high precision, fast velocity, and good robustness.

An Overview of Systolic Arrays for Forward and Inverse Discrete Sine Transforms and Their Exploitation in View of an Improved Approach

This paper aims to present a unified overview of the main Very Large-Scale Integration (VLSI) implementation solutions of forward and inverse discrete sine transforms using systolic arrays. The main features of the most important solutions to implement the forward and inverse discrete sine transform (DST) using systolic arrays are presented. One of the central ideas presented in the paper is to emphasize the advantages of using regular and modular systolic array computational structures such as cyclic convolution, circular correlation, and pseudo-band correlation in the VLSI implementation of these transforms. The use of such computational structures leads to architectures well adapted to the features of VLSI technologies, with an efficient use of the hardware structures and a reduced I/O cost that helps avoiding the so-called I/O bottleneck. With the techniques presented in this review, we have developed a new VLSI implementation of the DST using systolic arrays that allow efficient hardware implementation with reduced complexity while maintaining high-speed performances. Using a new restructuring input sequence, we have been able to efficiently reformulate the computation of the forward DST transform into a special computational structure using eight short quasi-cycle convolutions that can be computed with low complexity and where some of the coefficients are identical. This leads to a hardware structure with high throughput. The new restructuring sequence is the use of the input samples in a natural order as opposed to previous solutions, leading to a significant reduction of the hardware complexity in the pre-processing stage due to avoiding a permutation stage to reverse the order. Moreover, the proposed VLSI architecture allows an efficient incorporation of the obfuscation technique with very low overheads.

Joint STO and DFO Estimation for SEFDM in Low-Earth-Orbit Satellite Communications

In this article, a joint symbol timing offset (STO) and Doppler frequency offset (DFO) estimation algorithm is proposed for spectrally efficient frequency division multiplexing (SEFDM) system in the downlink of low-earth-orbit (LEO) satellite communications. Considering the nonorthogonality of SEFDM, the proposed algorithm is derived by maximizing likelihood function. To reduce computation complexity of STO estimator, all unnecessary terms in likelihood function are ignored. Further, the proposed CFO estimator is simplified by replacing the inverse trigonometric function with polynomial approximation. According to relevant standards issued by the International Telecommunications Union and the 3rd Generation Partnership Project, simulation parameters are set to evaluate the estimation performance. Simulation results show that the proposed algorithm is robust to different packing factors of SEFDM, and outperforms traditional estimation methods.

Arbitrary Order Fractional Derivative of Inverse Fractional Trigonometric Function

Arbitrary Order Fractional Derivative of Inverse Fractional Trigonometric Function

Fast and Accurate Approximation Methods for Trigonometric and Arctangent Calculations for Low-Performance Computers

In modern computers, complicated signal processing is highly optimized with the use of compilers and high-speed processing using floating-point units (FPUs); therefore, programmers have little opportunity to care about each process. However, a highly accurate approximation can be processed in a small number of computation cycles, which may be useful when embedded in a field-programmable gate array (FPGA) or micro controller unit (MCU), or when performing many large-scale operations on a graphics processing unit (GPU). It is necessary to devise algorithms to obtain the desired calculated values without an accelerator or compiler assistance. The residual correction method (RCM) developed here can produce simple and accurate approximations of certain nonlinear functions with minimal multiply–add operations. In this study, we designed an algorithm for the approximate computation of trigonometric and inverse trigonometric functions, which are nonlinear elementary functions, to achieve their fast and accurate computation. A fast first approximation and a more accurate second approximation of each function were created using RCM with a less than 0.001 error using multiply–add operations only. This achievement is particularly useful for MCUs, which have a low power consumption but limited computational power, and the proposed approximations are candidate algorithms that can be used to stabilize the attitude control of robots and drones, which require real-time processing.

Fault Diagnosis of Oil‐Immersed Power Transformers Using SVM and Logarithmic Arctangent Transform

A new method of dissolved gas analysis is proposed to improve the accuracy of transformer fault diagnosis. The slime mold optimized support vector machine (SMA-SVM), and logarithmic arctangent transform (LOG-ACT) are combined. On the one hand, the better global optimization performance of SMA is used to optimize SVM parameters to solve the difficulty of SVM parameter selection. On the other hand, corresponding transformations are carried out for different features: the logarithmic(LOG) transformation is carried out for the original DGA data to retain the order of magnitude information. The arctangent (ACT) transformation is carried out for the ratio features to improve the data structure. Therefore, the combination of data transformation and optimization model can improve the accuracy of diagnosis from two aspects of data structure and classification algorithm. The performance of the proposed method was compared with IEC three ratio method, artificial neural network, optimized artificial neural network, GA-SVM, and PSO-SVM. Experimental results using published data show that the proposed method can significantly improve the accuracy of transformer fault diagnosis. © 2022 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

A Method Without Stability Issue for Robust Estimation of Three-Phase Grid Frequency

The grid frequency estimation using a recursive signal or an integrator-based method may suffer from a stability issue due to an accumulation or discretization error. To avoid these shortcomings and also to obtain a robust estimation of three-phase grid frequency under unbalanced and distorted conditions, this letter proposes a demodulation-based method without using a recursive signal and integrator. The proposed demodulation-based approach relies on the Clarke transform and fixed transport delay-based relatively simple finite-impulse-response filters. It includes an open-loop structure, thus unconditionally stable. It also does not demand the real-time evaluation of the trigonometric or inverse trigonometric function. Moreover, it generates improved accuracy as compared with a frequency-locked loop and phase-looked loop.

Several Double Inequalities for Integer Powers of the Sinc and Sinhc Functions with Applications to the Neuman–Sándor Mean and the First Seiffert Mean

In the paper, the authors establish a general inequality for the hyperbolic functions, extend the newly-established inequality to trigonometric functions, obtain some new inequalities involving the inverse sine and inverse hyperbolic sine functions, and apply these inequalities to the Neuman–Sándor mean and the first Seiffert mean.

Optimizing the Operation Mode of an Autonomous Wind Power Plant with a Distributed Generation System

The article considers physical interactions and a mathematical model of a wind power plant consisting of separate wind turbines distributed over a large area, which generate electric energy to common buses and from them to a transformer and load. Induction generators are used as electromechanical converters, which differ from synchronous ones in having a simple design and featuring better operational reliability. In view of distributed location of wind turbines, each of them generally operates under different wind conditions; in addition, their interaction depends both on the length of cable routes to the busbars and on the parameters of transformers. Owing to these circumstances, it is of interest to study parallel operation of distributed induction generators and the possibility to optimize the wind power plant operation mode to ensure the required quality of the generated electricity in terms of power supply frequency and voltage. The operation mode of such wind power plant – taking as an example two induction generators operating in parallel on a common resistive-inductive load – is represented by a complex T-shaped equivalent circuit, the mathematical description of which is given by nonlinear complex homogeneous equations for nodal potentials. The magnetization inductances are nonlinear functions of the EMFs, which are analytically approximated by an inverse trigonometric function. The mechanical equilibrium equations include the wind turbine torques, the electromagnetic moments of generators, the rotating parts inertia moments, and resistance and friction moments. It should be noted that with a given wind turbine size, its torque depends on the angular rotation speed, wind velocity and blade angle, and is approximated by spline functions of the load impedance and wind turbine rotation speeds. The operating mode analysis is based on the numerical calculation of non-trivial solutions of nonlinear complex equations for the potentials of the equivalent circuit nodes, load voltage and frequency, and their subsequent approximation as a function of rotation speeds and load impedance at fixed values of load inductance, capacitances, ballast conductivity, and other parameters. On the example of a calculation performed for typical parameters of a small-capacity wind power plant with a real difference of wind velocities for individual wind turbines and variable load impedance, it is shown that by changing the blade angle alone, it is not possible to stabilize the voltage parameters on the load, while by simultaneously changing the capacitive and ballast conductivities, the frequency and voltage can be stabilized at the specified levels. In particular, with increasing the wind turbine rotation speeds, the capacitive and ballast conductivity should be increased, and with an increase in load resistance, the ballast conductivity should also increase, and capacitive conductivity should decrease. As a subsequent development of this method, ways to stabilize the operation of such wind power plant with an arbitrary number of generators can be analyzed.

Bounds for Quotients of Inverse Trigonometric and Inverse Hyperbolic Functions

We establish new simple bounds for the quotients of inverse trigonometric and inverse hyperbolic functions such as sin−1xsinh−1x and tanh−1xtan−1x. The main results provide polynomial bounds using even quadratic functions and exponential bounds under the form eax2. Graph validation is also performed.

Effect of selenium on semen quality and fertility of rams

Objective: To evaluate the effect of different selenium (Se) doses on semen quality and fertility of rams. Design/Methodology/Approach: Se was administered subcutaneously each month to nine two-month-old rams (from April 2018 to February 2020). The treatments were: T1 = without Se, T2 = 0.1 mg Se kg-1, and T3 = 0.3 mg Se kg-1. Semen evaluations were carried out using an electroejaculator; a total of 180 ejaculations were analyzed. The rams with better semen quality (one per treatment) were subjected to a mating. The following variables were evaluated: volume (Vol), masal motility (MM), progressive motility (MP), sperm concentration per mL and ejaculation, acrosome integrity percentage, living spermatozoa, normal spermatozoa, and fertility percentage. The normal distribution variables were evaluated using an analysis of variance (ANOVA) and Tukey’s comparison test (Tukey, 0.05); the percentages were raised to the inverse sine in order to perform the appropriate ANOVA. Fertility was analyzed using a χ2 test. Results: There was no difference in semen quality and fertility percentage (p > 0.05). The ejaculation volume showed differences between treatments (p ≤ 0.05). Study Limitations/Implications: A study including a higher number of rams and a previous fieldwork practice should be carried out, in order to evaluate semen parameters. These evaluations would help to accurately estimate semen quality and fertility, as well as to corroborate the results. Findings/Conclusions: Se did not improve semen quality and fertility of rams

A New Root-Finding Method for Univariate Non-Linear Transcendental Equations With Quadratic Convergence

This paper presents a new algorithm to find a non-zero real root of the non-linear transcendental equations. The proposed method is based on the combination of the inverse sine series and Newton-Raphson method. Implementation of the proposed method in MATLAB is applied to different problems to ensure the methods applicability. Error calculation has been done for available existing methods and the proposed method. The suggested method is evaluated using a number of numerical examples and the results indicate that the proposed method is effective than well-knownmethods. The proposed method’s convergence is discussed, and it is shown to be quadraticallyconvergence.

New Traveling Wave Solutions for the (2 + 1)-Dimensional Heisenberg Ferromagnetic Spin Chain Equation

The principal objective of this article is to construct new exact soliton solutions of the (2 + 1)-dimensional Heisenberg ferromagnetic spin chain equation which investigates the nonlinear dynamics of magnets. Through using the complete discrimination system method, the traveling wave solutions are obtained. As a result, we get the traveling wave solutions of the Heisenberg ferromagnetic spin chain equation, which include rational function solutions, Jacobian elliptic function solutions, hyperbolic function solutions, trigonometric function solutions, inverse trigonometric function, logarithmic function. Some graphical representations of the problems and that of comparison are also provided.

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.

A technique for improving the computation of functions of triangular matrices

ABSTRACT We propose a simple technique that, if combined with algorithms for computing functions of triangular matrices, can make them more efficient. Basically, such a technique consists in a specific scaling similarity transformation that reduces the departure from normality of a triangular matrix, thus decreasing its norm and in general its function condition number. It can easily be extended to non-triangular matrices, provided that it is combined with algorithms involving a prior Schur decomposition. Situations where the technique should be used or not will be discussed in detail. Special attention is devoted to particular algorithms like the inverse scaling and squaring to the matrix logarithm or inverse cosine and the scaling and squaring to the matrix exponential. The advantages of our proposal are supported by theoretical results and illustrated with numerical experiments, involving matrices of small, medium and large size.

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.

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.

A novel spoke‐type PM motor with asymmetric rotor for low torque pulsation under MTPA control condition

This article presents a novel asymmetric spoke-type rotor structure. It is used to reduce rotor magnetic flux leakage, torque ripple and no-load back-EMF distortion rate while increasing the motor output torque. The traditional method of optimizing torque pulsation is the inverse cosine function, which gives a sinusoidal distribution of the air-gap magnetic field under no-load conditions of the motor. The fundamental wave magnetic field of the air-gap synthetic magnetic field under the load is offset from the no-load air-gap magnetic field. This paper presents the design theory and method of the new type of the asymmetric rotor. Asymmetric spatial harmonics are introduced to the rotor so that the motor has a sinusoidal distribution of the rated load air-gap field under MTPA control. In order to verify the superiority of the new rotor, the performance of four different spoke-type rotor structure motors under the same pole-slot combination was compared. Finally, an experimental prototype is built to verify the theoretical reliability against the design of this paper.