Fractional Order Differentiator Research Articles

Optimal wideband digital fractional-order differentiators using gradient based optimizer

In this paper, we propose a novel optimization approach to designing wideband infinite impulse response (IIR) digital fractional order differentiators (DFODs) with improved accuracy at low frequency bands. In the new method, the objective function is formulated as an optimization problem with two tuning parameters to control the error distribution over frequencies. The gradient based optimizer (GBO) is effectively employed on the proposed objective function. A wide range of design examples are presented to illustrate the effectiveness of the proposed approach. The proposed approximations are compared to those of recent literature in terms magnitude, phase, and group delay errors. The result reveal that our method can attain approximations have a favorable low frequency performance (with about 60% of relative magnitude error reduction) and maintain a comparable accuracy at most of the Nyquist band to those of the existing ones. Thus, our approximations can be attractive for low frequency applications.

Fractional-Order Robust Control Design under parametric uncertain approach

This paper presents a novel methodology that combines fractional-order control theory with robust control under a parametric uncertainty approach to enhance the performance of linear time-invariant uncertain systems with integer or fractional order, referred to as Fractional-Order Robust Control (FORC). In contrast to traditional approaches, the proposed methodology introduces a novel formulation of inequalities-based design, thus expanding the potential for discovering improved solutions through linear programming optimization. As a result, fractional order controllers are designed to guarantee desired transient and steady-state performance in a closed-loop system. To enable the digital implementation of the designed controller, an impulse response invariant discretization of fractional-order differentiators (IRID-FOD) is employed to approximate the fractional-order controllers to an integer-order transfer function. Additionally, Hankel’s reduction order method is applied, thus making it suitable for hardware deployment. Experimental tests carried out in a thermal system and the assessment results, based on time-domain responses and robustness analysis supported by performance indices and set value analysis in a thermal system test-bed, demonstrate the improved and robust performance of the proposed FORC methodology compared to classical robust control under parametric uncertainty.

Compact active analog device for novel applications useful for sensing and measurement

The presented research introduces a novel approach to modern modular active device and provides various practical application examples tailored for industrial sensing readouts. Straightforward implementation of single active device-based topologies in frequency bands from kHz up to MHz is presented. These applications include: an electronically linearly tunable special band-pass filter that can be easily modified into a voltage-controlled oscillator, a two-port quadratic transformer, an amplitude modulator, and an equalization circuit (which serves as a fractional-order differentiator and integrator). These circuits benefit from high-impedance voltage input and low-impedance output (easily matched to 50 Ω), high processed signal levels (often reaching hundreds of mV), and simplified topologies without redundant passive elements. These features are especially valuable in tunable and configurable solutions for modern communication, sensing, audio, and consumer electronic systems, as well as in modeling various physical and biological systems. The experiments have been provided by simulations and measurements using device VCA824.

Sub-gigahertz bandwidth all-optical tunable differentiator with high-energy efficiency based on a micro-ring resonator.

We propose and demonstrate a tunable fractional-order photonic differentiator (DIFF) that can process input pulses with a sub-gigahertz bandwidth. Our scheme utilizes the self-induced optical modulation effect observed in a silicon-on-insulator micro-ring resonator. Gaussian-like pulses with varying pulse widths between 7.5 and 20 ns are employed for differentiation, achieving an energy efficiency over 45%, to the best of our knowledge, which surpasses all previously reported schemes for input pulses with a sub-gigahertz bandwidth. We simulate the temporal dynamics of pulses to gain insight into the physical mechanisms underlying the differentiated outputs and provide a method for differentiation order adjustment, which is experimentally realized using an all-optical pump-probe technique.

State estimation-based parameter identification for a class of nonlinear fractional-order systems

Parametric identification is an important part of system theory since knowledge of the parameters allows the analysis and control of the system. The aim of this paper is to propose a novel robust (against measurement noise) parameter identification method for a class of nonlinear fractional-order systems. In order to solve the parametric identification we carry out this problem to a state estimation problem, we introduce a Fractional Algebraic Identifiability (FAI) property which allows to represent the system parameters as a function of the inputs and outputs of the system, this parameter identification method provides an on-line identification process (while the system is operating), we also propose a fractional-order differentiator which allows to reduce the effect of measurement noise as well as to provide the estimation of a fractional-order derivative of the system output. Moreover, we use the Mittag–Leffler boundedness to demonstrate the convergence of this method, a different approach for this stability analysis method is given in this paper. Finally, we illustrate the accuracy and robustness of our proposed method by means of the parametric identification of two nonlinear fractional-order systems: a time-varying nonlinear fractional-order system and a nonlinear fractional-order mathematical model of a simple pendulum.

Realization of FOI and FOD of Complex Orders

This work introduces a new realization method of fractional-order integrators (FOI) and differentiators (FOD) of complex orders. The corresponding Laplace variables of the form S±α±jβ; 0 < α, β ≤ 1, are approximated and realized via minimum-phase rational transfer functions within a frequency band. The realization is adaptive since it depends on the frequency band of operation. Proper selection of the complex order and its sign play an important role in widening up and improving the phase diagram of the approximation at high frequency. Thus, making them ideal candidates to improve the leading phase of the FOI/FOD at high frequency. The main points of this work are verified via numerical simulations.

Composite iterative learning adaptive fuzzy control of fractional-order chaotic systems using robust differentiators

In conventional adaptive fuzzy control, to improve the approximation ability of fuzzy logic systems (FLSs), more fuzzy rules should be employed, which will greatly increase the computational burden. This paper investigates the adaptive fuzzy backstepping control of a specific category of incommensurate fractional-order chaotic systems afflicted by functional uncertainties and actuator faults. To address the challenging “explosion of complexity” issue, a novel modified fractional-order robust differentiator is proposed, capable of effectively suppressing noise. Importantly, an iterative learning adaptation law including parameter errors between adjacent periods and prediction errors derived from a series–parallel model is developed to improve the approximation accuracy of FLSs without using abundant fuzzy rules. Utilizing the frequency distribution model and the Lyapunov stability criterion, this approach guarantees the semi-global uniform boundedness of the closed-loop system and facilitates the convergence of tracking errors to a small region. Finally, the effectiveness of theoretical results is demonstrated through numerical simulation examples.

Control method for permanent magnet synchronous motor based on fractional active disturbance rejection

To improve the response speed and anti-interference ability of a permanent magnet synchronous motor (PMSM), a PMSM control method based on fractional-order active disturbance rejection control (FOADRC) is proposed in this study. First, a fractional-order tracking differentiator (FOTD) and fractional-order extended state observer (FOESO) are designed using the fractional calculus. According to the response speed of the FOTD and the observation effect of the FOESO, a new fastest function and new nonlinear function are designed. Second, a fractional-order active disturbance rejection (FOADR) controller is built based on the new FOTD, new FOESO, and nonlinear feedback control law, and its stability is proven. Finally, the superiority and effectiveness of the proposed strategy are verified through a simulation and experiments.

Discretization of Fractional Order Operator in Delta Domain

The fractional order operator is the backbone of the fractional order system (FOS). The fractional order operator (FOO) is generally represented as s^(±μ) (0&amp;lt;μ&amp;lt;1). Discrete time FOS can be obtained through the discretization of the fractional order operator. The FOO is the general form of either fractional order differentiator (FOD) or integrator (FOI) depending upon the values of μ. Out of the two discretization methods, direct discretization outperforms the method of indirect discretization. The mapping between the continuous time and discrete time domain is done with the development of generating function. Continuous fraction expansion (CFE) is used expand the generating function for the rational approximation of the FOO. There is an inherent problem associated with the discretization of FOO in discrete z-domain particularly at very fast sampling rate. In the other hand, discretization using delta operator parameterization provides the continuous time and discrete time results in hand to hand, when the continuous time systems are sampled at very fast sampling rate and circumventing the problem with shift operator parameterization at fast sampling rate. In this work, a new generating function is proposed to discretize the FOO using the Gauss-Legendre 3-point quadrature rule and generating function is expanded using the CFE to form rational approximation of the FOO in delta domain. The benchmark fractional order systems are considered in this work for the simulation purpose and comparison of results are made to prove the efficacy of the proposed method using MATLAB.

Parallel temporal signal processing enabled by polarization-multiplexed programmable THz metasurfaces.

Under the trends of multifunctionality, tunability, and compactness in modern wave-based signal processors, in this paper, we propose a polarization-multiplexed graphene-based metasurface to realize distinct mathematical operators on the parallel time-domain channels enabled by vertical and horizontal polarizations. The designed metasurface is composed of two perpendicularly-oriented graphene strips for each of which the chemical potential can be dynamically tuned through a DC biasing circuit. The programmable metasurface exhibits two orthogonal channels through which the time-domain input signals are elaborately processed by separate mathematical functions. Several illustrative examples are presented demonstrating that the proposed device can operate on different time-domain analog computing modes such as fractional-order differentiator and phaser at the same time. The strategy introduced in this paper will enable real-time parallel temporal analog computing and has potentially essential applications in terahertz spectroscopy architectures, communication systems, and computing technologies.

Robust Backstepping Control Combined with Fractional-Order Tracking Differentiator and Fractional-Order Nonlinear Disturbance Observer for Unknown Quadrotor UAV Systems

In this paper, we studied a fractional-order robust backstepping control (BSC) combined with a fractional-order tracking differentiator and a fractional-order nonlinear disturbance observer for a quadrotor unmanned aerial vehicle (UAV) system. A fractional-order filtered error and a fractional-order tracking differentiator were utilized in a conventional BSC system to improve the positioning control performance of a highly coupled nonlinear quadrotor UAV system and bypass the differentiation issue of the virtual control and compensation of the transformation error in the conventional BSC design. A new fractional-order disturbance observer with the sine hyperbolic function was then proposed to enhance the estimation performance of the uncertain quadrotor UAV. Sequential comparative simulations were conducted, demonstrating that the proposed positioning controller and observer utilizing fractional-order calculus outperformed those of the conventional controller and observer systems.

A Unified FPGA Realization for Fractional-Order Integrator and Differentiator

This paper proposes a generic FPGA realization of an IP core for fractional-order integration and differentiation based on the Grünwald–Letnikov approximation. All fractional-order dependent terms are approximated to simpler relations using curve fitting to enable an efficient hardware realization. Compared to previous works, the proposed design introduces enhancements in the fractional-order range covering both integration and differentiation. An error analysis between software and hardware results is presented for sine, triangle and sawtooth signals. The proposed generic design is realized on XC7A100T FPGA achieving frequency of 9.328 MHz and validated experimentally for a sine input signal on the oscilloscope. The proposed unified generic design is suitable for biomedical signal processing applications. In addition, it can be employed as a laboratory tool for fractional calculus education.

Image Enhancement Based on Rough Set and Fractional Order Differentiator

In the paper, an image enhancement algorithm based on a rough set and fractional order differentiator is proposed. By combining the rough set theory with a Gaussian mixture model, a new image segmentation algorithm with higher immunity is obtained. This image segmentation algorithm can obtain more image layers with concentrating information and preserve more image details than traditional algorithms. After preprocessing, the segmentation layers will be enhanced by a new adaptive fractional order differential mask in the Fourier domain. Experimental results and numerical analysis have verified the effectiveness of the proposed algorithm.

Recent Developments on the Realization of Fractance Device

Abstract A detailed analysis of the recent developments on the realization of fractance device is presented. A fractance device which is used to exhibit fractional order impedance properties finds applications in many branches of science and engineering. Realization of fractance device is a challenging job for the people working in this area. A term fractional order element, constant phase element, fractor, fractance, fractional order differintegrator, fractional order differentiator can be used interchangeably. In general, a fractance device can be realized in two ways. One is using rational approximations and the other is using capacitor physical realization principle. In this paper, an attempt is made to summarize the recent developments on the realization of fractance device. The various mathematical approximations are studied and a comparative analysis is also performed using MATLAB. Fourth order approximation is selected for the realization. The passive and active networks synthesized are simulated using TINA software. Various physical realizations of fractance device, their advantages and disadvantages are mentioned. Experimental results coincide with simulated results.

Intra and inter-patient arrhythmia classification using feature fusion with novel feature set based on fractional-order and fibonacci series

Cardiac ailments are increasing at an alarming rate globally due to the sedentary lifestyle and increased desk-bound activities. For decades, the ECG signal is used in aiding the analysis of human heart. Arrhythmia is a disorder that alters the normal cardiac cycle of ECG signal. The automatic classification of arrhythmia is a highly desirable and tough task. Mainly, the traditional methods of arrhythmia classification, are evaluated on the intra-patient criterion that may not befit the inter-patient criterion. A complete classification method has been proposed in the paper which performs well in intra-patient, inter-patient criterion and it is effective for minority class of MIT-BIH arrhythmia database (MIT-BIH-AD). In the proposed method, after preliminary processing based on Riesz fractional-order digital differentiator and R-peak detection, various features are extracted from ECG beat. The feature set proposed is a fusion of time-domain, time–frequency domain features and the new and novel features based on the Fibonacci series and coefficients of fractional-order Riesz based derivative signals which are proposed in this work. The results are evaluated on the MIT-BIH-AD and the results for intra-patients’ criterion have achieved an overall accuracy, sensitivity and positive predictivity values of 99.85%, 99.16%, 99.93% respectively, and for the inter-patient scheme 92.5%, 89.89% and 95.54% an average value of accuracy, sensitivity and positive predictivity of six ECG classes. The obtained results in both criterions have outperformed other methods in the literature and the proposed work has also attained better results for minority class of MIT-BIH-AD in both the scheme.

Non-asymptotic estimation for fractional integrals of noisy accelerations for fractional order vibration systems

In this paper, a class of fractional order vibration systems is considered, where the fractional differentiation orders are arbitrary real numbers between 0 and 2. The objective is to fast and robustly estimate the fractional integrals and derivatives of positions from noisy accelerations. In particular, the velocities and positions can be estimated by the proposed method. Since the proper fractional integrals of accelerations can be estimated using a numerical method, the main task of this work is to estimate unknown initial conditions. For this purpose, the generalized modulating functions method is adopted, which is recently developed to design non-asymptotic and robust fractional order differentiators. By distinguishing the cases where the stiffness matrix is invertible or not, algebraic integral formulas are provided for the unknown initial conditions in different situations. Hence, the unknowns can also be estimated even if the stiffness matrix is not invertible. Finally, the efficiency and robustness of the proposed estimators are verified by taking several numerical examples.

Design and comprehensive analysis of an ultra-fast fractional-order temporal differentiator based on a plasmonic Bragg grating microring resonator.

This paper presents the design and comprehensive analysis of an ultra-fast fractional-order temporal differentiator (DIFF) based on a plasmonic inner-wall Bragg grating microring resonator (BG-MRR). Due to the ring radius of 1.1 µm and the strong confinement of the surface plasmon polaritons in the plasmonic waveguide with very small cross-section, the overall footprint of the DIFF circuit is significantly small (approximately 4 × 2.5 µm2). By changing the coupling regimes of the microring resonator, a broad range for the differentiation order α, i.e., 0.7-1.7 and a wide 3 dB bandwidth of 3.1 THz [24.8 nm] for α = 0.7 and 3.9 THz [31.2 nm] for α = 1.7 have been realized. Comparing the outputs of the BG-MRR-based DIFF with the corresponding mathematical DIFF indicates that deviations for α > 1 are significantly larger than those of α < 1. Therefore, a fractional-order temporal DIFF circuit based on plasmonic cascaded BG-MRR has been proposed for α > 1.

Electrically tuned fractional-order temporal differentiator in silicon photonics

Here, full-wave analysis and design of an electrically tunable optical fractional-order temporal differentiator (DIFF) based on an ultra-compact microring resonator (MRR) has been presented. Electrical tunability has been realized through free carrier dispersion (FCD) effect in the resonator with P-I-N and P-N junctions. The first-order optical DIFF is realized based on the critically-coupled resonator without applying any bias voltages in the junctions. Also, fractional-order DIFF with differentiation order α > 1 is realized by over-coupled resonator with the forward-biased P-I-N and P-N junctions. While, fractional-order DIFF with α < 1 is realized by under-coupled resonator with reverse-biased P-N junction. The high performance of the designed tunable optical DIFFs and their high accuracy have been confirmed considering an ultra-fast Gaussian input pulse with short full-width at half-maximum (FWHM) of 6.3 ps.

A Weak Form Electromagnetic Transient Simulation Approach for Cole-Cole Medium

Although the finite-difference time-domain (FDTD) method is usually used to simulate the Cole-Cole model, its relatively high requirements on the grid size and time step interval limit its application in the calculation of electrically large target. Therefore, a weak form electromagnetic transient simulation method for Cole-Cole model is proposed, which is based on the discontinuous Galerkin spectral element time-domain (DG-SETD) method. However, the implementation of the Cole-Cole model using the time-domain method is difficult because of the fractional order differentiators. The z-transform (ZT) method and second order Taylor series are used to solve this difficulty in this paper. The effectiveness of this method is shown by a numerical example.

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.