Integer Order Differentiation Research Articles

An Integer-Fractional Gradient Algorithm for Back Propagation Neural Networks

This paper proposes a new optimization algorithm for backpropagation (BP) neural networks by fusing integer-order differentiation and fractional-order differentiation, while fractional-order differentiation has significant advantages in describing complex phenomena with long-term memory effects and nonlocality, its application in neural networks is often limited by a lack of physical interpretability and inconsistencies with traditional models. To address these challenges, we propose a mixed integer-fractional (MIF) gradient descent algorithm for the training of neural networks. Furthermore, a detailed convergence analysis of the proposed algorithm is provided. Finally, numerical experiments illustrate that the new gradient descent algorithm not only speeds up the convergence of the BP neural networks but also increases their classification accuracy.

Multivariable Fractional-Order Controller Design for a Nonlinear Dual-Tank Device

Fractional calculus is defined by expanded integer order integration and differentiation. In this paper, multiple mathematical models of a nonlinear dual-tank device are precisely formulated by fractional calculus. Using the accurate model, a multivariable fractional-order controller is designed for nonlinear devices. The merits of the fractional-order design include: (1) control of multivariable nonlinearities, (2) compensation of uncertainties, and (3) elimination of coupling effects. Simulations and experiments are conducted to verify the precision of the fractional order models and the effectiveness of the multivariable fractional-order control system design.

Fractional-Order Control Techniques for Renewable Energy and Energy-Storage-Integrated Power Systems: A Review

The worldwide energy revolution has accelerated the utilization of demand-side manageable energy systems such as wind turbines, photovoltaic panels, electric vehicles, and energy storage systems in order to deal with the growing energy crisis and greenhouse emissions. The control system of renewable energy units and energy storage systems has a high effect on their performance and absolutely on the efficiency of the total power network. Classical controllers are based on integer-order differentiation and integration, while the fractional-order controller has tremendous potential to change the order for better modeling and controlling the system. This paper presents a comprehensive review of the energy system of renewable energy units and energy storage devices. Various papers are evaluated, and their methods and results are presented. Moreover, the mathematical fundamentals of the fractional-order method are mentioned, and the various studies are categorized based on different parameters. Various definitions for fractional-order calculus are also explained using their mathematical formula. Different studies and numerical evaluations present appropriate efficiency and accuracy of the fractional-order techniques for estimating, controlling, and improving the performance of energy systems in various operational conditions so that the average error of the fractional-order methods is considerably lower than other ones.

Charbonnier-Marchaud Based Fractional Variational Model for Motion Estimation in Multispectral Vision System

As we are aware that motion estimation is an active and challenging area of vision system, which leads to the applications of computer vision. In general, motion detection and tracking in the image sequence (video) is carried out based on optical flow. In the recent-past, researchers have made a significant contribution to the estimation of optical flow through integer order-based variational models, but these are limited to integer order differentiation. In this paper, a nonlinear modeling of fractional order variational model in optical flow estimation is introduced using the Charbonnier norm, which can be scaled to integer order L 1-norm. In particular, the variational functional is formulated by the inclusion of a non-quadratic penalty term, regularization term and the Marchaud’s fractional derivative. This non-quadratic penalty provides effective robustness against outliers, whereas the Marchaud’s fractional derivative possesses a non-local character, and therefore is capable to deal with discontinuous information about texture and edges, and yields a dense flow field. The numerical discretization of the Marchaud’s fractional derivative is employed with the help of Grünwald–Letnikov fractional derivative. The resulting nonlinear system is converted into a linear system and solved by an efficient numerical technique. Several experiments are performed over a spectrum of datasets. The robustness and accuracy of the proposed model are shown under different amounts of noise and through detailed comparisons with the recently published works.

Modulating functions based model-free fractional order differentiators using a sliding integration window

This paper aims to design model-free fractional order differentiators to non-asymptotically and robustly estimate both the Riemann–Liouville and Caputo fractional derivatives of an unknown signal from its discrete and noisy observation. To achieve this, new fractional integration by parts formulas are first introduced. Then, by applying the generalized modulating functions method to a constructed model, algebraic integral formulas involving the integer order derivatives of the signal are provided. These formulas can be applied to analytically or numerically calculate the fractional derivatives in noise-free case. In order to apply the proposed formulas in discrete noisy cases, a time-delayed integer order differentiator is developed by applying the modulating functions method using a sliding integration window, such that the fractional derivatives can be estimated using the estimations of the integer order ones. This integer order differentiator can cope with non-zero mean noises. Moreover, a recursive algorithm is provided to reduce the computation time for on-line applications. Finally, numerical simulation results are given to illustrate the efficiency and robustness of the proposed method with an application to fractional order PID controller.

Investigation of fractional models of damping material by a neuroevolutionary approach

Abstract This research paper deals with a problem related to the damped materials contained in structural dynamics. The problem dealt with here involves a fractional-order damping coefficient in the form of fractional derivatives that present a better mathematical model of the vibration systems. Fractional derivatives are widely used to characterize the viscoelastic features in structural designs. Unlike the integer order differentiation, fractional-order derivatives consider local as well as the global evolution of the system. Therefore, fractional differential equations can be indicated as a reasonable choice for modeling certain physical phenomena, and to present more accurate mathematical solutions to real-world applications than the ordinary differential equations. We have proposed a novel unsupervised machine learning procedure that first designs general solutions, with the help of Artificial Neural Networks (ANNs), for the fractional-order differential equation containing unknown decision weights. These weights are worked out with the help of Fractional-Order Darwinian Particle Swarm Optimization (FO-DPSO) algorithm by setting a fitness function for each case. Results obtained from our simulations are better in the sense that they are overlapping with the analytical solutions available in the literature.

On Chebyshev and Riemann-Liouville Fractional Inequalities in q-Calculus

This paper presents some new inequalities on Fractional calculus in the context of q-calculus. Fractional calculus generalizes the integer order differentiation and integration to derivatives and integrals of arbitrary order. In other words, Fractional calculus explores integrals and derivatives of functions that involve non-integer order(s). Quantum calculus (q-Calculus) on the other hand focuses on investigations related to calculus without limits and in recent times, it has attracted the interest of many researchers due to its high demand of mathematics to model complex systems in nature with anomalous dynamics. This paper thus establishes some new extensions of Chebyshev and Riemann-Liouville fractional integral inequalities for positive and increasing functions via q-Calculus.

Gegenbauer wavelet operational matrix method for solving variable-order non-linear reaction–diffusion and Galilei invariant advection–diffusion equations

In this article, a variable-order operational matrix of Gegenbauer wavelet method based on Gegenbauer wavelet is applied to solve a space–time fractional variable-order non-linear reaction–diffusion equation and non-linear Galilei invariant advection diffusion equation for different particular cases. Operational matrices for integer-order differentiation and variable-order differentiation have been derived. Applying collocation method and using the said matrices, fractional-order non-linear partial differential equation is reduced to a system of non-linear algebraic equations, which have been solved using Newton iteration method. The salient feature of the article is the stability analysis of the proposed method. The efficiency, accuracy and reliability of the proposed method have been validated through a comparison between the numerical results of six illustrative examples with their existing analytical results obtained from literature. The beauty of the article is the physical interpretation of the numerical solution of the concerned variable-order reaction–diffusion equation for different particular cases to show the effect of reaction term on the pollution concentration profile.

An Efficient R-Peak Detection Using Riesz Fractional-Order Digital Differentiator

Clinically, electrocardiogram (ECG) is a powerful tool for determining the health and functioning of the human heart. Faster detection and diagnosis of heart functioning would aid cardiologists to provide appropriate treatment to the patients (subjects). In this paper, the concept of fractional-order calculus is employed for noise cancellation and artifacts removal in ECG signal as fractional-order differentiator proved to provide more peculiar details about signals than an integer-order differentiator. In the proposed method, R-peaks are detected using Riesz fractional-order digital differentiator (RFODD) based on the differencing method. The differentiation operation enhances the high-frequency components of the signal. So, QRS complex which is a high-frequency component in ECG is accentuated and the R-peaks are detected using appropriate threshold technique. The proposed method is tested on Massachusetts Institute of Technology-Beth Israel Hospital (MIT-BIH) arrhythmia database, and the experimental results of the proposed method have achieved a sensitivity of 99.95%, positive predictivity of 99.949% and an error rate of 0.095%. ECG waveforms are analyzed on various fractional orders of RFODD, and their performance parameters, i.e., sensitivity, positive predictivity and error rate, are also calculated.

Comparison of Fractional Order Derivatives Computational Accuracy - Right Hand vs Left Hand Definition

Abstract Numerical computation of fractional order derivatives and integrals can be done by applying many approaches. They include Riemann-Liouville and Caputo formulas. Both formulas include fractional order integration and integer order differentiation but in a different order: fractional order integration can be preceded by integer order differentiation or it can be done in the reverse order. In the paper we apply both formulas for FOD computation and investigate if an application of a particular order have any advantages, e.g. resulting in higher accuracy. The outcome of the experiment is that by applying adequately selected formula of FOD computation according to the shape of a particular function, the accuracy of computation can often be increased. Application of Riemann-Liouville formula enables successful FOD computation for functions which nth derivative does not exist.

Fractional-order State Observers for Integer-Order Linear Systems

To perform a robot-assisted surgery of a prosthesis implantation on a patient’s femur, we may need to get the femoral head-neck orientation for the application. We can extract that information from Computed Tomography scans, using image processing. In image processing, edge detection often makes use of integer-order differentiation operators (e.g. Canny and LoG operators). This paper shows that introducing non-integer (fractional) differentiation to edge detectors (Fractional Canny, Fractional LoG, Fractional Derivative operators) can improve automatic edge detection results

Robust speed control of hybrid electric vehicle using fractional order fuzzy PD and PI controllers in cascade control loop

In this paper, a novel scheme is proposed for speed control of highly nonlinear hybrid electric vehicle (HEV) having an electronic throttle control system (ETCS) in the cascade control loop. Fractional order fuzzy PD (FOFPD) and fractional order fuzzy PI (FOFPI) nonlinear controllers are developed and used as primary and secondary controllers, respectively in the cascade control loop. These controllers are variable gain fuzzy controllers having an adaptive nature. Their corresponding integral counterpart, fuzzy PD and fuzzy PI controllers are realized by keeping integer order differentiator and integrator in FOFPD and FOFPI controllers. Further, fractional order PD (FOPD) and fractional order PI (FOPI) controllers are implemented by using non-integer order operators in conventional PD and PI controllers. Extensive simulations have been carried out using National Instruments software LabVIEW™ and its add-on tools such as control design and simulation toolkit to perform a comparative study for FOFPD, FPD, FOPD and PD as primary controller and FOFPI, FPI, FOPI and PI as secondary controller for setpoint tracking of speed of HEV. Multi objective genetic algorithm is used to optimize the gains of primary and secondary controllers for minimization of integral of absolute error (IAE), maximum overshoot and settling time. Performances of tuned controllers are further evaluated for tracking of speed profile, disturbance rejection and model uncertainty. It has been observed that combination of FOFPD and FOFPI controllers outperformed rest of the controllers in servo, regulatory and uncertain environment and demonstrate very robust behavior for speed control of HEV.

Edge detection based on fractional order differentiation and its application to railway track images

AbstractEdge detection is one of the most fundamental necessities in image processing. Usally, edge detection algorithms are based on integer order differentiation operators. In many applications it is essential to perform a robust edge detection also to noisy input image data with low SNR as well. Thereby, integer based differentiation operators are often not leading to sufficient detection results. For this purpose an edge detector based on fractional order differentiation is introduced, which can significantly improve the detection performance to noisy images. Furthermore, a real application scenario of fractional order based edge detection is given within a modular railway track measurement system. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Haar Wavelet Based Implementation Method of the Non–integer Order Differentiation and its Application to Signal Enhancement

Abstract Non–integer order differentiation is changing application of traditional differentiation because it can achieve a continuous interpolation of the integer order differentiation. However, implementation of the non–integer order differentiation is much more complex than that of integer order differentiation. For this purpose, a Haar wavelet-based implementation method of non–integer order differentiation is proposed. The basic idea of the proposed method is to use the operational matrix to compute the non–integer order differentiation of a signal through expanding the signal by the Haar wavelets and constructing Haar wavelet operational matrix of the non–integer order differentiation. The effectiveness of the proposed method was verified by comparison of theoretical results and those obtained by another non–integer order differential filtering method. Finally, non–integer order differentiation was applied to enhance signal.

An algebraic fractional order differentiator for a class of signals satisfying a linear differential equation

This paper aims at designing a digital fractional order differentiator for a class of signals satisfying a linear differential equation to estimate fractional derivatives with an arbitrary order in noisy case, where the input can be unknown or known with noises. Firstly, an integer order differentiator for the input is constructed using a truncated Jacobi orthogonal series expansion. Then, a new algebraic formula for the Riemann–Liouville derivative is derived, which is enlightened by the algebraic parametric method. Secondly, a digital fractional order differentiator is proposed using a numerical integration method in discrete noisy case. Then, the noise error contribution is analyzed, where an error bound useful for the selection of the design parameter is provided. Finally, numerical examples illustrate the accuracy and the robustness of the proposed fractional order differentiator.

A method of approximate fractional order differentiation with noise immunity

In order to successfully apply differentiation to quantitative analytical applications, it is important to develop a method of differentiation with noise immunity to optimize the signal-to-noise ratio. Therefore, a direct method for the approximate fractional order differentiation (FOD) of a noised signal is proposed. Using the proposed method, arbitrary order differentiation of a signal can be realized by the convolution between the fractional order differentiation of the Gaussian function and the noised signal, and the integer order differentiation is just a special case of the FOD. The proposed method is verified by comparing the FOD of the noise and noise-free peak-signals. Moreover, the proposed method is used to enhance a mass spectrum signal and locate the position of each peak.

Fractional Order Differentiation by Integration and Error Analysis in Noisy Environment

The integer order differentiation by integration method based on the Jacobi orthogonal polynomials for noisy signals was originally introduced by Mboup, Join and Fliess. We propose to extend this method from the integer order to the fractional order to estimate the fractional order derivatives of noisy signals. Firstly, two fractional order differentiators are deduced from the Jacobi orthogonal polynomial filter, using the Riemann–Liouville and the Caputo fractional order derivative definitions respectively. Exact and simple formulae for these differentiators are given by integral expressions. Hence, they can be used for both continuous-time and discrete-time models in on-line or off-line applications. Secondly, some error bounds are provided for the corresponding estimation errors. These bounds allow to study the design parameters' influence. The noise error contribution due to a large class of stochastic processes is studied in discrete case. The latter shows that the differentiator based on the Caputo fractional order derivative can cope with a class of noises, whose mean value and variance functions are polynomial time-varying. Thanks to the design parameters analysis, the proposed fractional order differentiators are significantly improved by admitting a time-delay. Thirdly, in order to reduce the calculation time for on-line applications, a recursive algorithm is proposed. Finally, the proposed differentiator based on the Riemann–Liouville fractional order derivative is used to estimate the state of a fractional order system and numerical simulations illustrate the accuracy and the robustness with respect to corrupting noises.

Fractional Governing Equations of Diffusion Wave and Kinematic Wave Open-Channel Flow in Fractional Time-Space. II. Numerical Simulations

AbstractIn this study, finite difference numerical methods, first order accurate in time and second order accurate in space, are proposed to solve the governing equations of the one-dimensional unsteady kinematic and diffusion wave open-channel flow processes in fractional time and fractional space, which were derived in the accompanying paper. Advantages of modeling open-channel flow in a fractional time-space framework over integer time-space framework are threefold. First, the nonlocal phenomena in the open-channel flow process in either space or time can be considered by taking the global correlations into consideration. Second, the proposed governing equations of the open-channel flow process in the fractional order differentiation framework are generalization of the governing equations in the integer order differentiation framework. Third, the physics of the observed heavy tailed distributions of particle displacements in transport processes, as reported in the literature, may be explained by a flow...

Image enhancement based on fractional directional derivative

This paper introduces the directional derivative to fractional derivative and proposes a new mathematical method, fractional directional derivative (FDD), and gives the corresponding numerical calculation. Compared with the traditional fractional derivative, the coefficients of FDD along the eight directions in the image plane are not the same, which can reflect different fractional change rates along different directions and is benefit to enlarge the differences among the image textures. Experiments show that the capability of nonlinearly enhancing texture details by FDD is more obvious than those by the traditional fractional derivative and integer-order differentiation operators Laplacian, Butterworth high-pass filter.

Existence of Iterative Cauchy Fractional Differential Equation

Fractional calculus and its applications (that is the theory of derivatives and integrals of any arbitrary real or complex order) are important in several widely diverse areas of mathematical, physical, and engineering sciences. It generalized the ideas of integer order differentiation and n-fold integration. Fractional derivatives introduce an excellent instrument for the description of general properties of various materials and processes.This is themain advantage of fractional derivatives in comparison with classical integer-order models, in which such effects are in fact neglected.The advantages of fractional derivatives become apparent in modeling mechanical and electrical properties of real materials, as well as in the description of properties of gases, liquids, rocks, and in many other fields. Our aim in this paper is to consider the existence and uniqueness of nonlinear Cauchy problems of fractional order in sense of Riemann-Liouville operators. Also, two theorems in the analytic continuation of solutions are studied. In the fractional Cauchy problems, we replace the first-order time derivative by a fractional derivative. Fractional Cauchy problems are useful in physics. Recently, the author studied the the fractional Cauchy problems in complex domain [1]. One of the most frequently used tools in the theory of fractional calculus is furnished by the Riemann-Liouville operators. The Riemann-Liouville fractional derivative could hardly pose the physical interpretation of the initial conditions required for the initial value problems involving fractional differential equations. Moreover, this operator possesses advantages of fast convergence, higher stability, and higher accuracy to derive different types of numerical algorithms (see [2]).