Hyperbolic Secant Function Research Articles

Several Derivative Formulas of Two Exponential Functions and Real Power of Hyperbolic Secant Function with a Generalization of a Formula for Specific Partial Bell Polynomials

In the paper, by virtue of some identities for the partial Bell polynomials and with the aid of the Faá di Bruno formula, the author presents several derivative formulas of two exponential functions and the real power of the hyperbolic secant function, and generalizes a formula for specific partial Bell polynomials.

A Hybrid Fuzzy Deep Belief Network Extreme Learning Machine Framework With Hyperbolic Secant Activation Function for Robust Semi‐Supervised Sentiment Classification

ABSTRACTSentiment classification deals with extracting and classifying the text sentiment. Fuzzy Deep Belief Network (DBN) has proved its efficiency in dealing with sentiment analysis and suitability for classifying unlabeled or semi‐labeled data. Previous structures of deep belief networks are mostly made of traditional activation functions such as sigmoid. In this paper, a new activation function, which is referred to as hyperbolic secant function, is proposed. The new activation function not only solves gradient zeroing problem but also increases the accuracy and efficiency. Besides, extreme learning machine (ELM) is proposed as the decision layer to increase the accuracy and improve the generalizability through solving gradient‐based learning problem. The efficiency of the proposed method has been experimented on “IMDB” movie critic dataset, 20‐newspaper dataset and Sentiment Analysis dataset. The results of the proposed method are more accurate and precise as compared with the previous approaches.

Effective regulation of the interaction process among three optical solitons

Abstract The interaction between three optical solitons is a complex and valuable research direction, which is of practical application for promoting the development of optical communication and all-optical information processing technology. In this paper, we start from the study of the variable-coefficient coupled higher-order nonlinear Schrödinger equation (VCHNLSE), and obtain an analytical three-soliton solution of this equation. Based on the obtained solution, the interaction of the three optical solitons is explored when they are incident from different initial velocities and phases. When the higher-order dispersion and nonlinear functions are sinusoidal, hyperbolic secant, and hyperbolic tangent functions, the transmission properties of three optical solitons before and after interactions are discussed. Besides, this paper achieves effective regulation of amplitude and velocity of optical solitons as well as of the local state of interaction process, and interaction-free transmission of the three optical solitons is obtained with a small spacing. The relevant conclusions of the paper are of great significance in promoting the development of high-speed and large-capacity optical communication, optical signal processing, and optical computing.

Research on active vibration control of blade using variable-step-size filtered-x least mean square algorithm

The engine blade is one of the important parts of the aero engine, vibration can lead to engine blade off or fracture. Active Vibration Control (AVC) has a higher flexibility is increasingly used to suppress the vibration of the blade. The filtered-x least mean square (FxLMS) algorithm is an adaptive algorithm commonly used in active vibration control, but its fixed step size has a contradiction between convergence speed and steady-state error. A new variable-step-size FxLMS (VSSFxLMS) algorithm based on hyperbolic secant function was proposed, and the sech function ensures the control signal is bounded. In the early stage, a larger step size enables the error to decrease rapidly. The step size decreases as the error decreases, and a smaller step size ensures a smaller steady-state error. Firstly, this paper summarizes three VSSFxLMS algorithms and proposes an improved algorithm; Secondly, it compares and analyses the mentioned algorithms; Then, a joint simulation platform using ADAMS and Simulink is built to verify the effectiveness of the proposed algorithms; Finally, an experimental platform was built to complete the blade vibration active control experiments using several VSSFxLMS algorithms, and the results of the experiment verify the better vibration suppression effect of the proposed algorithm.

A NEW PERSPECTIVE ON THE NONLINEAR DATE–JIMBO–KASHIWARA–MIWA EQUATION IN FRACTAL MEDIA

In this paper, we first created a fractal Date–Jimbo–Kashiwara–Miwa (FDJKM) long ripple wave model in a non-smooth boundary or microgravity space recorded. Using fractal semi-inverse skill (FSIS) and fractal traveling wave transformation (FTWT), the fractal variational principle (FVP) was derived, and the strong minimum necessary circumstance was attested with the He Wierstrass function. We have discovered two distinct solitary wave solutions, the square form of the hyperbolic secant function and the hyperbolic secant function form. Then, soliton solutions are cultivated through FVP and the minimum steady state condition. Finally, the influences of non-smooth boundaries on solitons were tackled, and the properties of the solution were demonstrated through three-dimensional contour lines. Fractal dimension can impact waveforms, but cannot affect their vertex values. The presentation of soliton solutions (SWS) using techniques is not only laudable but also noteworthy. The technique employed can also be used to investigate solitary wave solutions of other local fractional calculus partial differential equations.

Adaptive optimal control of under-actuated robotic systems using a self-regulating nonlinear weight-adjustment scheme: Formulation and experimental verification.

This paper formulates an innovative model-free self-organizing weight adaptation that strengthens the robustness of a Linear Quadratic Regulator (LQR) for inverted pendulum-like mechatronic systems against perturbations and parametric uncertainties. The proposed control procedure is devised by using an online adaptation law to dynamically adjust the state weighting factors of LQR's quadratic performance index via pre-calibrated state-error-dependent hyperbolic secant functions (HSFs). The updated state-weighting factors re-compute the optimal control problem to modify the state-compensator gains online. The novelty of the proposed article lies in adaptively adjusting the variation rates of the said HSFs via an auxiliary model-free online self-regulation law that uses dissipative and anti-dissipative terms to flexibly re-calibrate the nonlinear function's waveforms as the state errors vary. This augmentation increases the controller's design flexibility and enhances the system's disturbance rejection capacity while economizing control energy expenditure under every operating condition. The proposed self-organizing LQR is analyzed via customized hardware-in-loop (HIL) experiments conducted on the Quanser's single-link rotational inverted pendulum. As compared to the fixed-gain LQR, the proposed SR-EM-STC delivers an improvement of 52.2%, 16.4%, 55.2%, and 42.7% in the pendulum's position regulation behavior, control energy expenditure, transient recovery duration, and peak overshoot, respectively. The experimental outcomes validate the superior robustness of the proposed scheme against exogenous disturbances.

Hyperbolic-Secant-Function-Based Fast Sliding Mode Control for Pantograph Robots

Technological advancements in control systems are indispensable and require a robust controller with fast convergence in numerous sectors like robotics, space engineering, etc. A sliding mode controller (SMC) is a robust controller when subjected to nonlinearities, external disturbances, and uncertainties in system dynamics. However, SMC is prone to chattering, which can damage the plant in long term and results in low tracking accuracy. In this research paper, a novel hyperbolic secant function (HSF)-based SMC is proposed that achieves fast convergence, reduces chattering, and improves tracking performance. Stability analysis is conducted based on Lyapunov stability criteria. Comparative simulations and experiments on a pantograph robot (PR) show the effectiveness of the proposed algorithm.

An improved reptile search algorithm based on mathematical optimization accelerator and elementary functions

The reptile search algorithm (RSA) is a dynamic and effective meta-heuristic algorithm inspired by the behavior of crocodiles in nature and the way of hunting prey. Unlike other crawler search algorithms, it uses four novel mechanisms to update the location of the solutions, such as walking at high or on the belly, and hunting in a coordinated or cooperative manner. In this algorithm, the total number of iterations is divided into four intervals, and different position-updating strategies are used to make the algorithm easily fall into the local optimum. Therefore, an improved reptile search algorithm based on a mathematical optimization accelerator (MOA) and elementary functions is proposed to improve its search efficiency and make it not easily fall into local optimum. MOA was used to realize the switching of RSA’s four searching modes by introducing random perturbations of six elementary functions (sine function, cosine function, tangent function, arccosine function, hyperbolic secant function and hyperbolic cosecant function), four mechanisms are distinguished by random number instead of the original RSA algorithm’s inherent four mechanisms by iteration number, which increases the randomness of the algorithm and avoids falling into local optimum. The random perturbations generated by elementary functions are added to the variation trend of parameter MOA to improve the optimization accuracy of the algorithm. To verify the effectiveness of the proposed algorithm, 30 benchmark functions in CEC2017 were used for carrying out simulation experiments, and the optimization performance was compared with BAT, PSO, ChOA, MRA and SSA. Finally, two practical engineering design problems are optimized. Simulation results show that the proposed sechRSA has strong global optimization ability.

Image Enhancement via Special Functions and Its Application for Near Infrared Imaging.

Image enhancement is important given that it can be used to highlight the area of interest in the images. This article designs four filters via special function for realizing image enhancement. Firstly, a filter based on the exponential function is designed. When the value of the progression is even, the edge feature can be extracted. When the value of the progression is odd, sharp contrast can be obtained. Secondly, a filter is built using hyperbolic cosine and its inverse function, where a printmaking feature can be extracted. Thirdly, a filter is made via a hyperbolic secant function and its inverse. It can lead to the extraction of image edge. When the progression value is increasing, marginal effect can be found and the brightness is decreasing. Ripple morphology can be found. Fourthly, a filter is constructed through a hyperbolic sine function and its inverse, where marginal features can be extracted. Furthermore, these filters are useful for extracting the marginal features even when a high noise density of 0.9 is added to the original images. They are useful for highlighting the images acquired from near infrared imaging.

Robust generalized hyperbolic secant algorithm for nonlinear active noise control

In nonlinear active noise control (ANC) system where the primary noise obeys non-Gaussian distribution, the traditional filter-s least mean square (FsLMS) algorithm is hardly satisfactory. In order to surmount this deficiency, a novel generalized hyperbolic secant function with lp norm (FsGHSFP) algorithm is proposed for nonlinear ANC system, which designs a robust optimization criterion utilizing the generalized hyperbolic secant distribution function. The FsGHSFP algorithm has robust noise attenuation capability for non-Gaussian noise, but suffers from the problem of tuning the parameter. To address this problem, an improved FsGHSFP (IFsGHSFP) algorithm is developed by taking advantage of the fact that continuous mixed norm does not need to adjust norm parameter. Moreover, in order to strike a balance between convergence and misalignment, thereby further enhancing the robustness of the ANC system, a convex combination algorithm of IFsGHSFP is suggested. The robustness mechanism of the proposed algorithms is analyzed, as well as stability conditions and computational complexity. It is verified by simulations that the proposed algorithms accelerate the convergence speed and improve the noise attenuation performance. In addition, the experimental results further validate the effectiveness of the proposed algorithms.

Design the arbitrary order calculus operator by a simulated hyperbolic function for analytical applications

The existing problem of data processing by derivative, especially the higher order derivative, is that current operators lack robustness and accuracy for data with noise. The difference between the frequency distribution of the signal and the noise, as well as the low-frequency suppression from derivatives, is an obvious contrast that can be visually observed in the frequency domain. This contrast must be harmonious to become the better operator and requires an accurate description of the distribution of the processed data. Therefore, this paper proposed the profiles of the three basic types of spectrum peaks – Lorenz, Voigt, or Gaussian are unified in the frequency domain, for actual spectra which are single peaks or composed of multiple peaks, the envelope can be approximated with an exponential expression. Since the distribution of noise is flat, a spectrum with noise can be enveloped by a function that is the sum of an exponential function and a constant. A more precise response function for separating noise can be constructed on the envelope, which can be well simulated by the hyperbolic secant (Sech) function. The simulated hyperbolic function (SH) switched by the inverse Fourier transform is an available convolution kernel in the time domain. Around this kernel, the numerical calculus (Grünwald-Letnikov) and the corresponding adjustment can realize an arbitrary order calculus for the raw spectrum. Based on that, we presented an executable MATLAB script, and verified it interactively in the mathematical analytics, with or without noise. The results showed that the computation's accuracy and precision fully satisfied the operators' requirements. Moreover, the method has been applied in two cases of analytical chemistry to show its potential for a wide range of applications. A higher-order derivative that is free from noise interference aids the peak fit to obtain more specific results. The fractional order derivative spectroscopy yields less uncertainty for direct quantification in a multi-component system.

Model Construction and Parameters Acquisition of the Predicted Surface Movement Deformation under Thick Loose Layer Mining Area

In China, gas and oil reserves are very scarce, but coal resources are abundant in the energy architecture, which decides that coal will remain the dominant energy source for a long time in the future. The accurate prediction of the size and extent of surface movement after coal seam mining is of great significance for the safe promotion of production activities in the mine area and the safety of people’s lives and properties in the mine area. The surface movement deformation under thick loose seam conditions indicates the phenomenon of a large subsidence value and influence range. To predict the size and range of surface movement deformation under thick loose layer conditions accurately, a hyperbolic secant model is constructed based on the hyperbolic secant function. For high nonlinearity of the model parameters, the adaptive step fruit fly algorithm (ASFOA) is introduced into the process of solving the model parameters. Simulation experiments are conducted in three aspects: monitoring point antideficiency, antigross error, and parameter stability. The simulation results show that the ASFOA algorithm achieves high accuracy in finding the parameters of the hyperbolic secant model. The hyperbolic secant model was applied to the 11111 working face under the mining conditions of thick loose layer geology in the Huainan mine. The engineering application results indicate that the hyperbolic secant model performs well on the prediction of surface movement deformation under thick loose layer conditions.

Soliton formation and dynamics in the quintic nonlinear media with [formula omitted]-invariant harmonic-Gaussian potential

In this paper, we introduce a PT-symmetric harmonic-hyperbolic-Gaussian (HHG) potential in the nonlinear Schrödinger (NLS) equation with quintic nonlinearity. Firstly, we present the critical thresholdless PT-phase symmetry breaking in the linear regime. Secondly, we give both exact and numerical single-/double-hump solitons of the quintic NLS equations with the PT-symmetric HHG potential, and show their stabilities by studying the spectrum stability analysis and direct wave propagations. Finally, we investigate the interactions between the single-/double-hump solitons and the exotic hyperbolic secant function, as well as the adiabatic excitations of the solitons with the operation of an adiabatic switch function. All these obtained results provide a theoretical guidance for the applications of the PT-symmetric theory into diverse areas of physics.

Time delay estimation method based on generalized logarithmic hyperbolic secant function in impulsive noise

In this paper, a generalized logarithmic hyperbolic secant (GLHS) function is introduced that can effectively suppress impulsive noise while guarding the signal of interest from damage. Also, an analysis of the optimal scaling parameter choices for the GLHS function was studied. Then, in order to address the performance drawbacks of the traditional time delay estimation methods based on correlation under an impulsive noise environment, a novel GLHS-based correlation (GLHSC) is further developed, and the reliable time delay estimation result is obtained by finding the peak of GLHSC. The comprehensive Monte Carlo simulation results demonstrate that the performance of the method based on GLHSC is better than other robust competitive methods based on correlation in terms of probability of resolution and estimation accuracy, especially in a heavy-tailed noise environment.

Observer-based independent joint control for a coupled rigid-flexible manipulator with actuator saturation based on distributed parameter model

This study is concerned with an observer-based independent joint control scheme for a coupled two-link rigid-flexible robotic manipulator considering the effects of actuator saturation and external disturbances. A distributed parameter model of the robotic manipulator system described by ordinary and partial differential equation (ODE-PDE) is derived by using Hamilton’s Principle. An anti-saturation controller is developed to cope with the input saturation problem by using hyperbolic tangent and hyperbolic secant function. Considering the external disturbance, a disturbance observer is designed to estimate the external disturbance. The observer-based independent joint controller can adjust the joint angle and suppress the vibration simultaneously without the need for end-mounted actuators. The extended LaSalle’s Invariance Principle is adopted to strictly prove the asymptotic stability of the robotic manipulator system. The simulation comparisons validate the effectivity of the proposed controllers and observers.

An effective Reinforcement Learning method for preventing the overfitting of Convolutional Neural Networks

Convolutional Neural Networks are machine learning models that have proven abilities in many variants of tasks. This powerful machine learning model sometimes suffers from overfitting. This paper proposes a method based on Reinforcement Learning for addressing this problem. In this research, the parameters of a target layer in the Convolutional Neural Network take as a state for the Agent of the Reinforcement Learning section. Then the Agent gives some actions as forming parameters of a hyperbolic secant function. This function’s form is changed gradually and implicitly by the proposed method. The inputs of the function are the weights of the layer, and its outputs multiply by the same weights to updating them. In this study, the proposed method is inspired by the Deep Deterministic Policy Gradient model because the actions of the Agent are into a continuous domain. To show the proposed method’s effectiveness, the classification task is considered using Convolutional Neural Networks. In this study, 7 datasets have been used for evaluating the model; MNIST, Extended MNIST, small-notMNIST, Fashion-MNIST, sign language MNIST, CIFAR-10, and CIFAR-100.

Improved Robust Total Least Squares Adaptive Filter Algorithms Using Hyperbolic Secant Function

In the errors-in-variables (EIV) model, the total least squares (TLS) algorithm has shown good convergence performance. In this model, the input and output signals are simultaneously infected with noise. However, when the output signal is contaminated with impulse noise, the convergence performance of TLS will decrease or even diverge. In order to settle this problem, we propose a hyperbolic secant total least squares (HSTLS) algorithm through adopting the hyperbolic secant function as the cost function. The HSTLS algorithm shows better performance in non-Gaussian noise environment. At the same time, in order to deal with the contradiction between the algorithm’s convergence speed and steady-state deviation with the fixed step size, this brief also provides a new variable step size strategy. The improved variable step size HSTLS (VHSTLS) algorithm shows better performance in simulation. Furthermore, to reduce steady-state deviation of the HSTLS algorithm in sparse systems, a series of improved HSTLS algorithms based on the sparse criterion have also been proposed, and they also have excellent convergence performance.

Hyperbolic Secant Subsidence Prediction Model under Thick Loose Layer Mining Area

In China, as a major resource, coal has made great contributions to national energy security and social development. The mining of coal resources can cause surface subsidence damage, and in particular, the mining of coal resources in thick loose layer mines is the most serious. How to accurately predict the surface subsidence caused by coal mining in thick loose layer mines has become an urgent problem to be solved. To solve this problem, numerical simulations based on the measured data were used to reveal that the thickness of the loose layer is the intrinsic mechanism that affects the value of the surface subsidence and the large range of subsidence. On this basis, the hyperbolic secant function is used as the influence function of unit mining to derive the expected model of subsidence under thick loose layer conditions: the hyperbolic secant subsidence prediction model. Compared with the probability integral method, the hyperbolic secant subsidence prediction model’s prediction accuracy RMSE value is improved by 38%. The hyperbolic secant subsidence prediction model can realize accurate estimation of the subsidence value in the thick loose layer mine area. This greatly enriches the mining subsidence prediction theory and provides a scientific basis for the assessment of surface damage and ecological environment restoration after coal seam mining under a thick loose seam mining area.

A new class of out-gap discrete solitons in binary waveguide arrays.

We analytically and numerically investigate beyond-band discrete solitons, which present a completely new class of stable localized out-gap solitons with detunings being located beyond the two bands of the linear plane waves in a periodic binary waveguide array. Each of the even and odd components of these discrete solitons does not change its sign across the transverse direction of the binary waveguide array. The even and odd components of these newly found discrete solitons can be approximately presented by two hyperbolic secant functions with the only difference in their peaks. This approximation is especially good in the low-intensity regime in which the detuning of these solitons can asymptotically reach the two limits of a linear spectrum. These distinguishing features altogether make the newly found discrete solitons different from all other classes of discrete solitons investigated earlier in binary waveguide arrays. Two transformation rules for constructing even and odd components of these discrete solitons are also found for various combinations of signs of the propagation mismatch σ and nonlinear coefficient γ.

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.