Fractional Order Differentiator Research Articles

Modelling, identification, and compensation of fractional‐order creep in frequency domain for piezoactuators

Unlike traditionally studied in time domain, the creep in a piezoelectric actuator is studied in frequency domain. It is modelled as a fractional-order integrator, identified in frequency domain, and compensated for by connecting with a fractional-order differentiator in serial. It is demonstrated that identification and compensation in frequency domain are much simpler to execute than that in time domain. Experimental results validate the effectiveness of the proposed algorithm and the creep non-linearity reduces by greater than 80%.

A Rational Approximation Method of Fractional Order Differentiators and Integrators Based On Joint Optimal Criterion

Huimin Zhao, Chen Guo, Wu Deng, Dongyan Li, Xinhua Yang and Li Zou Information Science and Technology College of Dalian Maritime University, Dalian 116026, China 2 Software Institute,Dalian Jiaotong University, Dalian 116028, China 3 Sichuan Provincial Key Lab of Process Equipment andControl (Sichuan University of Science and Engineering), Zigong 64300, China Artificial Intelligence Key Laboratory of Sichuan Province(Sichuan University of Science and Engineering) ,Zigong 64300, China E-mail: dw7689@163.com

A Novel Non-Invasive Method for Extraction of Geometrical and Texture Features of Wood

ABSTRACTNon invasive feature detection in wood based application requires exact discrimination between geometrical edges and texture. It has been found that traditional edge detection algorithms are highly sensitive to noise and texture and produces inferior results with wood. The present work encompasses a micro level reconstruction of Palash and Rosewood by using micro X-rays CT scanner. It also encompasses a new edge detection algorithm using newly constructed Chebyshev polynomial based fractional order differentiator. Transform based method has been used for reconstruction purpose. Newly designed fractional order filter has been applied on these reconstructed images. Chebyshev polynomial based fractional order differentiator has been used for filtering operation. Quadrature Mirror Filter (QMF) concept has been used for design of high pass filter and low pass filter. Preprocessing has been performed by using this filter. Canny edge detection algorithm has been used on this preprocessed image. The algorithm has been tested on two different test cases of wood images a) Palash and b) Rosewood. The effect of relaxation coefficient has also been presented and discussed.

Design of Wideband Fractional-Order Differentiator Using Interlaced Sampling Method

In this paper, the design of a wideband digital fractional-order differentiator (FOD) is investigated. First, conventional FOD designs are reviewed, and the reconstruction formula of the interlaced sampling method is used to design the proposed wideband FOD by index substitution and the Grunwald---Letnikov fractional derivative. Because a closed-form window design is obtained, the filter coefficients are easily computed. Then, the weighted least squares and convex optimization methods are applied to design non-sparse digital FODs that are optimal in the least squares or min---max sense. Next, the iterative hard thresholding and orthogonal matching pursuit methods are used to design sparse digital FODs to reduce the implementation complexity. Finally, several numerical examples are presented to show that the proposed FODs have smaller design errors in the high-frequency band than conventional digital FODs that do not use the auxiliary interlaced sampling signal.

Tunable fractional-order photonic differentiator using a distributed feedback semiconductor optical amplifier

We propose a tunable fractional-order photonic differentiator based on a distributed feedback semiconductor optical amplifier (SOA) working in reflection mode. The phase shift at the resonant wavelength can be adjusted by controlling the current injected into the DFB-SOA, which can implement the fractional-order differentiation. A 60 ps Gaussian pulse is temporally differentiated with a tunable order range from 0.7 to 1.3.

Efficient Design of Discrete Fractional-Order Differentiators Using Nelder–Mead Simplex Algorithm

Applications of fractional-order operators are growing rapidly in various branches of science and engineering as fractional-order calculus realistically represents the complex real-world phenomena in contrast to the integer-order calculus. This paper presents a novel method to design fractional-order differentiator (FOD) operators through optimization using Nelder---Mead simplex algorithm (NMSA). For direct discretization, Al-Alaoui operator has been used. The numerator and the denominator terms of the resulting transfer function are further expanded using binomial expansion to a required order. The coefficients of z-terms in the binomial expansions are used as the starting solutions for the NMSA, and optimization is performed for a minimum magnitude root-mean-square error between the ideal and the proposed operator magnitude responses. To demonstrate the performance of the proposed technique, six simulation examples for fractional orders of half, one-third, and one-fourth, each approximated to third and fifth orders, have been presented. Significantly improved magnitude responses have been obtained as compared to the published literature, thereby making the proposed method a promising candidate for the design of discrete FOD operators.

Comments on “Discretization of fractional order differentiator over Paley–Wiener space”

It is shown that the definitions underlying the algorithm proposed in Wu (2014) are not suitable for supporting the algorithm. Suitable definitions are presented.

Designs of Discrete-Time Generalized Fractional Order Differentiator, Integrator and Hilbert Transformer

In this paper, the design of a generalized fractional order differentiator (FOD) whose magnitude and phase responses can be controlled independently is investigated. First, a relation between conventional FOD and generalized FOD is studied such that the design tools of conventional FOD in the literature can be used to design variable generalized FOD directly. Then, the similar method is applied to design generalized fractional order integrator (FOI). Next, the proposed generalized FOD and FOI are used to generate a secure single side band (SSB) signal for saving the transmission bandwidth. The parameters of variable generalized FOD and FOI can be used as the secure keys for construction and reconstruction. Finally, the relation between fractional Hilbert transformer and generalized FOD is studied and the edge detection application is demonstrated to show the flexibility and effectiveness of the proposed generalized FOD.

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.

Tunable fractional-order photonic differentiator based on the inverse Raman scattering in a silicon microring resonator

A novel photonic fractional-order temporal differentiator is proposed based on the inverse Raman scattering (IRS) in the side-coupled silicon microring resonator. By controlling the power of the pump light-wave, the intracavity loss is adjusted and the coupling state of the microring resonator can be changed, so the continuously tunable differentiation order is achieved. The influences of input pulse width on the differentiation order and the output deviation are discussed. Due to the narrow bandwidth of IRS in silicon, the intracavity loss can be adjusted on a specific resonance while keeping the adjacent resonances undisturbed. It can be expected that the proposed scheme has the potential to realize different differentiation orders simultaneously at different resonant wavelengths.

A Novel Sol–Gel Thin-Film Constant Phase Sensor for High Humidity Measurement in the Range of 50%–100% RH

This paper proposes a new type constant phase sensor (CPS) for solving the problem of high humidity measurement. It is based on change in phase angle of the CPS with variation of humidity. The CPS is having interdigitated electrode sandwiched between two identical thin films of γ-Al <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> O <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> fabricated by sol-gel dipcoating method. In the presence of relative humidity, the device shows a fairly constant phase behavior over wide signal frequency and its fractional exponent reduces to nearly 0.7 value at 90% RH from the initial value of 1 at dry humidity. Results show that the CPS is effective in measuring humidity in the range of 50%-100% RH at the signal frequency of 1-5 MHz. Finally, the device has been interfaced with a simple fractional-order differentiator circuit to measure the phase angle change with change in relative humidity.

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.

A NOVEL APPROACH FOR EDGE DETECTION OF LOW CONTRAST SATELLITE IMAGES

Abstract. Discrimination between texture edges and geometrical edges is very difficult in low contrast images. Satellite images are low contrast images. It is important to extract the edges that are not clearly visible in case of Satellite images. The present work encompasses a new edge detection algorithm using newly constructed differentiator. Chebyshev polynomial based fractional order differentiator has been used for filtering operation on an image. High pass and Low pass filters are designed with the concept of Quadrature Mirror Filter (QMF). Pre-processing has been performed by using this filter. Sobel edge detection method has been applied on this pre-processed image. The algorithm has been tested with two different satellite images.

Approximations of higher-order fractional differentiators and integrators using indirect discretization

This paper describes new approximations of fractional order integrators (FOIs) and fractional order differentiators (FODs) by using a continued fraction expansion-based indirect discretization scheme. Different tenth-order fractional blocks have been derived by applying three different s-to-z transforms described earlier by Al-Alaoui, namely new two-segment, four-segment, and new optimized four-segment operators. A new addition has been done in the new optimized four-segment operator by modifying it by the zero reflection method. All proposed half (s$^{\pm 1/2})$ and one-fourth (s$^{\pm 1/4})$ differentiator and integrator models fulfill the stability criterion. The tenth-order fractional differ-integrators (s$^{\pm \alpha })$ based on the modified new optimized four-segment rule show tremendously improved results with relative magnitude errors (dB) of $\le $ -15 dB for $\alpha $ = 1/2 and $\le $ -20 dB for $\alpha $ = 1/4 in the full range of Nyquist frequency so these have been further analyzed. The main contribution of this paper lies in the reduction of these tenth-order blocks into four new fifth-order blocks of half and one-fourth order models of FODs and FOIs. The analyses of magnitude and phase responses show that the proposed new fifth-order half and one-fourth differ-integrators closely approximate their ideal counterparts and outperform the existing ones.

Wideband Differentiator and Integrator With Ideal Phase Response

In this paper, we consider the design of a digital differentiator and integrator using the natural logarithm series. The proposed class of differentiators and integrators matches the ideal phase response and the magnitude response as closely as possible at low frequencies for orders lower than those reported in the literature. Furthermore, the bandwidth of the digital differentiator and integrator is extended to cover the entire Nyquist range using a simple multirate technique. In principle, the approach yields the results that are similar to those obtained by the design of fractional order differentiators and integrators.

Optimal design of FIR fractional order differentiator using cuckoo search algorithm

In this paper, a new meta-heuristic optimization algorithm, called cuckoo search algorithm (CSA) is applied to determine the optimal coefficients of the finite impulse response-fractional order differentiator (FIR-FOD) problem. CSA is based on lifestyle and unique parasitic behavior in egg laying and breeding of some cuckoo species along with Lévy flight behavior of some birds and fruit flies. The CSA is capable of solving linear and nonlinear optimization problems. The proposed CSA method prevents the local minima problem encountered in conventional FIR-FOD design method. A novel weighted least square (WLS) fitness function is adopted to improve the response of the FOD to a great extent. The proposed CSA based method has alleviated from inherent drawbacks of premature convergence and stagnation unlike genetic algorithm (GA). To verify the effectiveness of the proposed FIR-FOD based on the cuckoo search algorithm, different set of initial population is tested by simulation. Simulation results affirm that the proposed fractional order differentiator design approach using CSA outperforms the genetic algorithm in terms design accuracy (magnitude and phase error), fast convergence rate and optimal solution. The simulation results confirmed that the proposed FOD using CSA outperforms the FOD designed using evolutionary algorithm like GA and conventional FOD design methods such as radial basis function (RBF) interpolation method and DCT interpolation method.

Designs of matrix fractional order differentiators

In this paper, the designs of matrix fractional order differentiator (MFOD) for differentiating digital signals are presented. First, the definitions of fractional derivatives are reviewed briefly and design problem of MFOD is stated. Then, three kinds of methods for designing MFOD are described including the conventional FIR and IIR filter methods, the discrete sine transform (DST) and discrete cosine transform (DCT) methods, and optimization methods. Next, numerical examples are demonstrated to compare the performances of these three design methods and the variable MFOD design is also studied. Finally, the image sharpening application and signal de-nosing application are used to show the effectiveness of the proposed matrix fractional order differentiators.

FIR Linear Phase Fractional Order Digital Differentiator Design using Convex Optimization

In this paper, design of linear phase FIR digital differentiators is investigated using convex optimization. The problem of differentiator design is first described in terms of convex optimization with different optimization variables’ options, taken one at a time. The method is then used to design first order low pass differentiators and results are compared with Salesnick’s technique and Parks McClellan algorithm. The designed FIR low pass differentiator has improvement in transition width and flexibility to optimize different parameters. The concept of low pass differentiation is further generalized to fractional order differentiators. Fractional order differentiators are designed by using minmax technique on mean square error. Design examples demonstrate easy design procedure and flexibility in the process as well as improvement over existing fractional order differentiators in terms of mean square error in passband. Finally, fractional order differentiators are designed and used for texture enhancement of color images. Better texture enhancement than existing filtering approaches is established based on average gradient and entropy values. General Terms FIR Filter, Fractional order differentiator, Low pass differentiator, Full band differentiator, Image Enhancement.

Independently Tunable Multichannel Fractional-Order Temporal Differentiator Based on a Silicon-Photonic Symmetric Mach–Zehnder Interferometer Incorporating Cascaded Microring Resonators

A multichannel fractional-order temporal differentiator with independently tunable differentiation order based on an integrated silicon-photonic symmetric Mach-Zehnder interferometer consisting of cascaded microring resonators (MRRs) is designed, fabricated, and experimentally demonstrated. By controlling the radii of the MRRs, a multichannel spectral response with uniform channel spacing is obtained, which is used to function as a multichannel temporal differentiator with multiple subdifferentiators. The differentiation order of each subdifferentiator is independently tunable by optically pumping the corresponding MRR, which leads to the phase change in the spectral response due to the two-photon absorption induced nonlinear effect. A five-channel temporal differentiator with a channel spacing of 0.49 nm is fabricated on a silicon-on-insulator chip using a CMOS-compatible process with 193-nm-deep ultraviolet lithography. Independent tuning of the differentiation orders of the subdifferentiators is experimentally demonstrated.

Fractional-order photonic differentiator using an on-chip microring resonator.

A tunable temporal photonic fractional differentiator using a silicon-on-isolator (SOI) electrically tuned microring resonator (MRR) is proposed and experimentally demonstrated. Through changing the voltage applied on the MRR, the fractional order of the photonic differentiator can be continuously tuned. The proposed fractional-order differentiator is demonstrated experimentally with Gaussian pulse injection and rectangular pulse injection, respectively. The small deviation shows the feasibility of our photonic differentiator with an integrated silicon MRR.