Fractional Order Integral Research Articles

A priori error analysis for a finite element approximation of dynamic viscoelasticity problems involving a fractional order integro-differential constitutive law

We consider a fractional order viscoelasticity problem modelled by a power-law type stress relaxation function. This viscoelastic problem is a Volterra integral equation of the second kind with a weakly singular kernel where the convolution integral corresponds to fractional order differentiation/integration. We use a spatial finite element method and a finite difference scheme in time. Due to the weak singularity, fractional order integration in time is managed approximately by linear interpolation so that we can formulate a fully discrete problem. In this paper, we present a stability bound as well as a priori error estimates. Furthermore, we carry out numerical experiments with varying regularity of exact solutions at the end.

FPAA Implementations of Fractional-Order Chaotic Systems

In this paper, fractional-order chaotic systems in an analog-based platform are realized using field programmable analog arrays (FPAA) hardware. With the help of this work, we aim to increase the complexity of chaotic systems. Approximated transfer functions in frequency domain are obtained by analyzing different values of fractional-order integrator with the Charef approximation method. In this study, fractional-order numerical calculation of Rssler and Sprott type-H chaotic systems is carried out. MATLAB Simulink model for chaotic systems that satisfy the conditions of chaos in the boundaries of fractional order value is schematically presented. Moreover, CAM designs and analysis that facilitate the realization of fractional-order transfer functions in FPAA platforms are introduced. The analog-based FPAA experimental and numerical outcomes for fractional order chaotic systems are demonstrated. The comparison of the results obtained in the numerical analysis and simulation study with the experimental results is given. This study confirms that the unpredictability of the chaos carrier signals realized by digital-based can be increased with analog-based FPAA hardware and fractional-order structures so as to provide safer transfer of information signals.

Fractionally integrated Gauss-Markov processes and applications

We investigate the stochastic processes obtained as the fractional Riemann-Liouville integral of order α∈(0,1) of Gauss-Markov processes. The general expressions of the mean, variance and covariance functions are given. Due to the central role, for the fractional integral of standard Brownian motion and of the non-stationary/stationary Ornstein-Uhlenbeck processes, the covariance functions are carried out in closed-form. In order to clarify how the fractional order parameter α affects these functions, their numerical evaluations are shown and compared also with those of the corresponding processes obtained by ordinary Riemann integral. The results are useful for fractional neuronal models with long range memory dynamics and involving correlated input processes. The simulation of these fractionally integrated processes can be performed starting from the obtained covariance functions. A suitable neuronal model is proposed. Graphical comparisons are provided and discussed.

Guaranteed cost and finite-time event-triggered control of fractional-order switched systems

This paper considers the problem of guaranteed cost and finite-time event-triggered control of fractional-order switched systems. Firstly, an event-triggered scheme including both the information of current state and an exponential decay function is proposed, and a novel cost function that adopts the characteristics of fractional-order integration is presented. Secondly, some sufficient conditions are derived to guarantee that the corresponding closed-loop system is finite-time stable with a certain cost upper bound, using multiple Lyapunov functions and average dwell time approach. Meanwhile, the event-triggered parameters and state feedback gains are simultaneously obtained via solving linear matrix inequalities. Moreover, Zeno behavior does not exist by finding a positive lower bound of the triggered interval. Finally, an example about fractional-order switched electrical circuit is provided to show the effectiveness of the proposed method.

Image Denoising of Adaptive Fractional Operator Based on Atangana–Baleanu Derivatives

A fractional integral operator can preserve an image edge and texture details as a denoising filter. Recently, a newly defined fractional-order integral, Atangana–Baleanu derivatives (ABC), has been used successfully in image denoising. However, determining the appropriate order requires numerous experiments, and different image regions using the same order may cause too much smoothing or insufficient denoising. Thus, we propose an adaptive fractional integral operator based on the Atangana–Baleanu derivatives. Edge intensity, global entropy, local entropy, and local variance weights are used to construct an adaptive order function that can adapt to changes in different regions of an image. Then, we use the adaptive order function to improve the masks based on the Grumwald–Letnikov scheme (GL_ABC) and Toufik–Atangana scheme (TA_ABC), namely, Ada_GL_ABC and Ada_TA_ABC, respectively. Finally, multiple evaluation indicators are used to assess the proposed masks. The experimental results demonstrate that the proposed adaptive operator can better preserve texture details when denoising than other similar operators. Furthermore, the image processed by the Ada_TA_ABC operator has less noise and more detail, which means the proposed adaptive function has universality.

Mathematical analysis and emulation of the fractional-order cubic flux-controlled memristor

Cubic model of flux-controlled memristor model is presented in this work and the model is analyzed at various fractional orders. Analytic computations with a sinusoidal source shows that the v-i loop can have up to two more additional intersection points aside from the origin and are located in the 1st and 3rd quadrants respectively. The locations of the intersection points and the area of the v-i loops are both dependent on the excitation signal frequency and fractional orders. Floating and Grounded emulator circuits for the cubic models are designed incorporating a fractional order integrator designed from the 2nd-order rational approximation of s-α. PSPICE simulations verify the correctness of the numerical analysis and calculations. An application example shows that the fractional-order cubic model can generate chaos in Chua’s circuit and produced an increase in the system dynamics.

Application of Fractional-Order Calculus to Improve the Mathematical Model of a Two-Mass System with a Long Shaft

Based on the general theory of fractional order derivatives and integrals, application of the Caputo–Fabrizio operator is analyzed to improve a mathematical model of a two-mass system with a long shaft and concentrated parameters. Thus, the real transmission of complex electric drives, which consist of long shafts with a sufficient degree of adequacy, is presented as a two-mass system. Such a system is described by ordinary fractional order differential equations. In addition, it is well known that an elastic mechanical wave, propagating along a drive transmission with a long stiff shaft, creates a retardation effect on distribution of the time–space angular velocity, the rotation angle of the shaft, and its elastic moment. The approach proposed in the current work helps to take in account the moving elastic wave along the shaft of electric drive mechanism. On this basis, it is demonstrated that the use of the fractional order integrator in the model for the elastic moment enables it to reproduce real transient processes in the joint coordinates of the system. It also provides an accuracy equivalent to the model with distributed parameters. The distance between the traditional model and the model in which the fractional integral is used for the elastic moment modelling in a two-mass system, with a long shaft, is analyzed.

THE CALCULUS OF BIVARIATE FRACTAL INTERPOLATION SURFACES

In this paper, we investigate partial integrals and partial derivatives of bivariate fractal interpolation functions (FIFs). We also prove that the mixed Riemann–Liouville fractional integral and derivative of order [Formula: see text], of bivariate FIFs are again bivariate interpolation functions corresponding to some iterated function system (IFS). Furthermore, we discuss the integral transforms and fractional order integral transforms of the bivariate FIFs.

Numerical solution of fractional differential equations using hybrid Bernoulli polynomials and block pulse functions

Block pulse functions are piecewise constant and not smooth enough. Therefore, they offer limited accuracy when used to approximate functions and unable to find highly accurate numerical solutions of fractional differential equations (FDEs). To overcome this problem, we present in this paper a new efficient numerical method for solving FDEs. A hybrid Bernoulli polynomials and block pulse functions operational matrix of fractional order integrals is derived and used to convert the underlying FDEs into a system of algebraic equations. The solutions of the FDEs are obtained by solving the algebraic equations. Simulation examples are given to verify the effectiveness of our proposed method, and the results show that the method is much more efficient and accurate than other known methods.

A deep learned type-2 fuzzy neural network: Singular value decomposition approach

The main objective of this study is to present a novel dynamic fractional-order deep learned type-2 fuzzy logic system (FDT2-FLS) with improved estimation capability. The proposed FDT2-FLS is constructed based on the criteria of singular value decomposition and uncertainty bounds type-reduction. The upper and the lower singular values of the set of inputs are estimated by a simple filter and the output is obtained by fractional-order integral of the uncertainty bounds type-reduction. Using stability criteria of fractional-order systems, the adaptation rules of the consequent parameters are extracted such that the globally Mittag-Leffler stability is achieved. The proposed FDT2-FLS is employed for online dynamic identification of a hyperchaotic system, online prediction of chaotic time series and online prediction of glucose level in type-1 diabetes patients and its performance is compared with other well-known methods. It is shown that the proposed mechanism results in significantly better prediction and estimation performance with less tunable parameters in just one learning epoch.

Spatial Path Tracking Controllers for Autonomous Ground Vehicles: Conventional and Nonconventional Schemes

Unlike time-based path tracking controllers, the [Formula: see text]-controller is a spatial path tracking controller. It is a purely geometric path tracking controller and essentially a P-controller to maintain the reasonable spatial distance, [Formula: see text], from the vehicle to the desired path. In this paper, we present some enhancement schemes using the non-conventional PI control laws via optimization. We propose to use a nonlinear term [Formula: see text] for the proportional controller. A fractional-order integral used to achieve a PI[Formula: see text] control. Among the schemes, an optimization search procedure applied to find optimal controller gains by meshing the regions around the values from approximate linear designs. The performance index for parametric optimization is the integration of the absolute purely spatial deviation from the desired path. Three different types of road shape were chosen and the Gazebo-ROS simulation results were presented to show the effectiveness of the proposed enhancement schemes. The results show that in some cases a smaller [Formula: see text] and [Formula: see text] can be achieved by using [Formula: see text] controller, but its disadvantage is there may be some oscillation. For PI[Formula: see text] controller, there is an additional adjustable parameter [Formula: see text], better performance can be achieved without significant disadvantages which is worth in-depth research.

Numerical solution of fractional Volterra integral equation with piecewise continuous kernel

Integral equations play the important role in applied mathematics, related to many areas of theory, especially applications. In this article, we consider the numerical method for solving Volterra integral equations for the fractional order of integration. The error of this method is O(1/N).

Fractional data-driven control for a rotary flexible joint system

As one of the most promising topics in complex control processes, data-driven techniques have been widely used in numerous industrial sectors and have developed over the past two decades. In addition, the fractional-order controller has become more attractive in applied studies. In this article, a fractional integral control is implemented for a rotary flexible joint system. Moreover, an adjusted virtual reference feedback tuning (VRFT) technique is used to tune the fractional-order integrator. In this method, fractional integral control is designed based on state feedback control. Then, VRFT is adjusted and applied to the fractional integral controller. The effectiveness of the proposed adjusted VRFT method is discussed and presented through simulation and experimental results. The tracking performance of the rotary arm and the minimization of the vibration tip is evaluated based on the proposed method. In this article, the comparison of our proposed VRFT fractional scheme is made with the classical state feedback as well as a recently developed state feedback-based fractional order integral (SF-FOI) controller. The current investigations determine the performance improvement of our proposed scheme of comparable structure to the recent SF-FOI, with the introduction of the VRFT to the SF-FOI scheme.

The best one-side approximation of classes of integrals of fractional order with regard to position of point on interval

We obtain the asymptotic estimations for the best one-sided point-wise approximation to the classes $W_{\infty}^r$, $r > 1$ (in case of fractional $r$) by algebraic polynomials.

Active Realization of Fractional-Order Integrators and Their Application in Multiscroll Chaotic Systems

This paper presents the design, simulation, and experimental verification of the fractional-order multiscroll Lü chaotic system. We base them on op-amp-based approximations of fractional-order integrators and saturated series of nonlinear functions. The integrators are first-order active realizations tuned to reduce the inaccuracy of the frequency response. By an exponential curve fitting, we got a convenient design equation for realizing fractional-order integrators of orders from 0.1 to 0.95. The results include simulations in SPICE of the mathematical description and the electronic implementation and experimental measurements that confirm them. Monte Carlo and sensitivity tests revealed a robust realization. Contrary to its passive counterparts, the suggested realizations significantly reduce design and implementation efforts by favoring resistors and capacitors with commercial values and reducing hardware requirements.

Solvability of generalized fractional order integral equations via measures of noncompactness

In this article, we work on the existence of solution of generalized fractional integral equations of two variables. To achieve our main objective, we establish a new fixed point theorem using measure of noncompactness and a new contraction operator which generalized the Darbo’s fixed point theorem (DFPT). Also we obtain the corresponding coupled fixed point theorem. Finally we apply this generalized DFPT on the generalized fractional integral equations of two variables and illustrate our findings with the help of an example.

Integrals and derivatives of fractional order based on Laplace type integral transformations with applications

Integrals and derivatives of fractional order theory is being developed. An analogue of the operational calculus is constructed for a differential operator with piecewise constant coefficients. Various constructions of the generalized Laplace transform are proposed. The transformation operators establish a connection between the Mellin – Laplace integral transformations and the generalized Laplace integral transformation. Isomorphism between the space of originals and the space of generalized originals is found. Mellin – Laplace type inversion formulas are established. Theorems on the differentiation of the generalized original and others are proved. A definition of a generalized convolution is given and a formula for its calculation is established, a connection between the generalized and classical convolution is indicated. On the basis of the concept of generalized convolution, a definition of a generalized integral and a generalized fractional derivative is given. Relations between generalized fractional integrals and Riemann-Liouville integrals of fractional order are established. For a model equation of heat conduction with a piecewise constant coefficient, the problem of calculating the heat flow transfer is solved. The heat flow is expressed as a generalized time derivative of the order of 1/2 of the measured temperature dependence at the boundary.

Social spider optimisation based identification and optimal control of fractional order system

Fractional order derivatives and integrals are infinite-dimensional operators and non-local in time. Currently, the researchers are working on the solution of the fractional optimal control problem...

Reconnection–less Reconfigurable Fractional–Order Current–Mode Integrator Design With Simple Control

A design of a fractional-order (FO) integrator is introduced for the operation of the resulting solution in the current mode (CM). The solution of the integrator is based on the utilization of RC structures, but in comparison to other RC structure based FO designs, the proposed integrator offers the electronic control of the order. Moreover, the control of the proposed integrator does not require multiple specific and accurate values of the control voltages/currents in comparison to the topologies based on the approximation of the FO Laplacian operator. The electronic control of a gain level (gain adjustment) of the proposed integrator is available. The paper offers the results of Cadence IC6 (spectre) simulations and more importantly experimental measurements to support the presented design. The proposed integrator can be used to build various FO circuits as demonstrated by the utilization of the integrator into a structure of a frequency filter in order to provide FO characteristics.

Low-speed control of PMSM based on ADRC + FOPID

ABSTRACT In this study, a new low-speed control strategy based on active disturbance rejection control (ADRC) + fractional-order proportion integration differentiation (FOPID) is proposed to realize fast and accurate control of permanent magnet synchronous motor (PMSM) at low speed. First, a new nonlinear function with enhanced smoothness and continuity around the origin is designed. On the basis of the above nonlinear function, an improved extended state observer (ESO) is developed. Then, the parameters of ESO are adjusted online by the designed fuzzy rules. Considering the improved dynamic performance of the FOPID controller, the nonlinear state error feedback control law module of ADRC is replaced by FOPID to calculate the proportion, differential, and integral of the error signal. Finally, the simulation analysis of the new low-speed control system based on ADRC + FOPID is carried out. The results show that the proposed algorithm has faster response speed and better anti-interference ability in the low-speed control of PMSM than the traditional ADRC.