Research

21 pages, 1509 KiB

Open AccessArticle

Fractional-Order Accumulative Generation with Discrete Convolution Transformation

by Tao Chen

Fractal Fract. 2023, 7(5), 402; https://doi.org/10.3390/fractalfract7050402 - 16 May 2023

Cited by 1 | Viewed by 776

A new fractional accumulation technique based on discrete sequence convolution transform was developed. The accumulation system, whose unit impulse response is the accumulation convolution sequence, was constructed; then, the order was extended to fractional orders. The fractional accumulative convolution grey forecasting model GM [...] Read more.

A new fractional accumulation technique based on discrete sequence convolution transform was developed. The accumulation system, whose unit impulse response is the accumulation convolution sequence, was constructed; then, the order was extended to fractional orders. The fractional accumulative convolution grey forecasting model GM

^{r *} (1, 1)

was established on the sequence convolution. From the viewpoint of sequence convolution, we can better understand the mechanism of accumulative generation. Real cases were used to verify the validity and effectiveness of the fractional accumulative convolution method. Full article

(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing)

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15 pages, 7008 KiB

Open AccessArticle

Fractional-Order Control of Fluid Composition Conductivity

by Raluca Giurgiu, Eva-H. Dulf and Levente Kovács

Fractal Fract. 2023, 7(4), 305; https://doi.org/10.3390/fractalfract7040305 - 30 Mar 2023

Viewed by 912

Abstract

Dialysis refers to the procedure of removing waste products and excess fluids from the blood stream. This is the main form of treatment for both acute and chronic renal failure. The need for hemodialysis process optimization is increasing. More than 10% of adults [...] Read more.

Dialysis refers to the procedure of removing waste products and excess fluids from the blood stream. This is the main form of treatment for both acute and chronic renal failure. The need for hemodialysis process optimization is increasing. More than 10% of adults are affected by chronic kidney disease, and it is the nineth leading cause of deaths worldwide. Critically ill patients are particularly at risk, and their mortality is significantly affected by the hemodialysis procedures. This is the reason why the design and control of the hemodialysis process is studied by many researchers. The present paper proposes a fractional-order control of the fluid composition conductivity in this process. Fractional-order PI and PID controllers are designed with different imposed performances in order to establish the best performing controller for this medical process. The proposed fractional-order controllers are compared to the classical controller’s results in different real-world scenarios, including process parameter changes, flow changes, and priming sequences. The results are compared with a classical PID controller used in current clinical practice. The simulation results show the robustness and advantages of the proposed fractional-order PID controller over other controllers. These results could improve the clinical use of the hemodialysis process. Full article

(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing)

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10 pages, 4313 KiB

Open AccessCommunication

In-Fiber All-Optical Fractional Differentiator Using an Asymmetrical Moiré Fiber Grating

by Lucas P. Tendela, Christian A. Cuadrado-Laborde and Miguel V. Andrés

Fractal Fract. 2023, 7(4), 291; https://doi.org/10.3390/fractalfract7040291 - 28 Mar 2023

Viewed by 826

Abstract

In this work, it is demonstrated numerically that an asymmetric Moiré fiber grating operated in reflection can provide the required spectral response to implement an all-optical fractional differentiator. In our case, the accumulated phase shift is not associated with a point phase shift, [...] Read more.

In this work, it is demonstrated numerically that an asymmetric Moiré fiber grating operated in reflection can provide the required spectral response to implement an all-optical fractional differentiator. In our case, the accumulated phase shift is not associated with a point phase shift, as when working with fiber Bragg gratings and long-period gratings with punctual defects, but is distributed all over the grating. The proposed device is supported by numerical simulations, and a dimensionless deviation factor is calculated to make quantitative analysis feasible. The performance of the proposed device is analyzed using numerical simulations by computing the fractional time derivatives of the complex field of an arbitrary transform-limited Gaussian pulse. A comparison with the performance given by theoretical differentiation is also presented. Full article

(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing)

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28 pages, 8884 KiB

Open AccessArticle

On Reservoir Computing Approach for Digital Image Encryption and Forecasting of Hyperchaotic Finance Model

by Amr Elsonbaty, A. A. Elsadany and Waleed Adel

Fractal Fract. 2023, 7(4), 282; https://doi.org/10.3390/fractalfract7040282 - 24 Mar 2023

Cited by 2 | Viewed by 1141

Abstract

Forecasting the dynamical behaviors of nonlinear systems over long time intervals represents a great challenge for scientists and has become a very active area of research. The employment of the well-known artificial recurrent neural networks (RNNs)-based models requires a high computational cost, and [...] Read more.

Forecasting the dynamical behaviors of nonlinear systems over long time intervals represents a great challenge for scientists and has become a very active area of research. The employment of the well-known artificial recurrent neural networks (RNNs)-based models requires a high computational cost, and they usually maintain adequate accuracy for complicated dynamics over short intervals only. In this work, an efficient reservoir-computing (RC) approach is presented to predict the time evolution of the complicated dynamics of a fractional order hyperchaotic finance model. Compared with the well-known deep learning techniques, the suggested RC-based forecasting model is faster, more accurate for long-time prediction, and has a smaller execution time. Numerical schemes for fractional order systems are generally time-consuming. The second goal of the present study is to introduce a faster, more efficient, and simpler simulator to the fractional order chaotic/hyperchaotic systems. The RC model is utilized in a proposed RC-based digital image encryption scheme. Security analysis is carried out to verify the performance of the proposed encryption scheme against different types of statistical, KPA, brute-force, CCA, and differential attacks. Full article

(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing)

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13 pages, 29753 KiB

Open AccessArticle

A Comparison Study of Time-Domain Computation Methods for Piecewise Smooth Fractional-Order Circuit Systems

by Xi Chen, Feng Zheng and Yewen Wei

Fractal Fract. 2023, 7(3), 230; https://doi.org/10.3390/fractalfract7030230 - 04 Mar 2023

Cited by 1 | Viewed by 1089

Abstract

The role of fractional calculus in circuit systems has received increased attention in recent years. In order to evaluate the effectiveness of time-domain calculation methods in the analysis of fractional-order piecewise smooth circuit systems, an experimental prototype is developed, and the effects of [...] Read more.

The role of fractional calculus in circuit systems has received increased attention in recent years. In order to evaluate the effectiveness of time-domain calculation methods in the analysis of fractional-order piecewise smooth circuit systems, an experimental prototype is developed, and the effects of three typical calculation methods in different test scenarios are compared and studied in this paper. It is proved that Oustaloup’s rational approximation method usually overestimates the peak-to-peak current and brings in the pulse–voltage phenomenon in piecewise smooth test scenarios, while the results of the two iterative recurrence-form numerical methods are in good agreement with the experimental results. The study results are dedicated to provide a reference for efficiently deploying calculation methods in fractional-order piecewise smooth circuit systems. Some quantitative analysis results are concluded in this paper. Full article

(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing)

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24 pages, 9327 KiB

Open AccessArticle

Research on Image Encryption Based on Fractional Seed Chaos Generator and Fractal Theory

by Haiping Chang, Erfu Wang and Jia Liu

Fractal Fract. 2023, 7(3), 221; https://doi.org/10.3390/fractalfract7030221 - 01 Mar 2023

Cited by 9 | Viewed by 1184

Abstract

In this paper, a new fractional-order seed chaotic generator is designed to solve the problem of the complex operations of single low-dimensional systems and simple high-dimensional systems. The fractional-order chaotic system generated is proven to have better chaotic performance using Lyapunov exponential differential [...] Read more.

In this paper, a new fractional-order seed chaotic generator is designed to solve the problem of the complex operations of single low-dimensional systems and simple high-dimensional systems. The fractional-order chaotic system generated is proven to have better chaotic performance using Lyapunov exponential differential calculus, approximate entropy, 0–1 test and other indicators. On this basis, the “multiple squares nested body scrambling (MSNBS)” model is extended from fractal theory to complete the image scrambling process, and a new algorithm is proposed to be applied to the encryption field in combination with the fractional-order coupled chaotic system. Experimental simulations show that the algorithm can resist common differential attacks and noise attacks and improve the security of the algorithm. Full article

(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing)

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37 pages, 482 KiB

Open AccessArticle

A Fractional Chemotaxis Navier–Stokes System with Matrix-Valued Sensitivities and Attractive–Repulsive Signals

by Chao Jiang, Zuhan Liu and Yuzhu Lei

Fractal Fract. 2023, 7(3), 209; https://doi.org/10.3390/fractalfract7030209 - 22 Feb 2023

Viewed by 3705

Abstract

In this paper, we considered a fractional chemotaxis fluid system with matrix-valued sensitivities and attractive–repulsive signals on a two-dimensional periodic torus

T^{2}

. This model describes the interaction between a type of cell that proliferates following a logistic law, and the diffusion [...] Read more.

In this paper, we considered a fractional chemotaxis fluid system with matrix-valued sensitivities and attractive–repulsive signals on a two-dimensional periodic torus

T^{2}

. This model describes the interaction between a type of cell that proliferates following a logistic law, and the diffusion of cells is fractional Laplace diffusion. The cells and attractive–repulsive signals are transported by a viscous incompressible fluid under the influence of a force due to the aggregation of cells. We proved the existence and uniqueness of the global classical solution on the matrix-valued sensitivities, and the initial data satisfied the regular conditions. Moreover, by using energy functionals, the stabilization of global bounded solutions of the system was proven. Full article

(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing)

14 pages, 535 KiB

Open AccessArticle

Robust Localization for Near- and Far-Field Signals with an Unknown Number of Sources

by Tao Liu, Hao Feng, Tianshuang Qiu, Shengyang Luan and Jiacheng Zhang

Fractal Fract. 2023, 7(2), 184; https://doi.org/10.3390/fractalfract7020184 - 12 Feb 2023

Cited by 1 | Viewed by 1068

Abstract

Source location is a constant issue of importance of both theoretical study and practical engineering. Many pioneers have put out the corresponding solutions for near- or far-field signals, and preferred contributions are suggested. To our best knowledge, there are currently few focused approaches [...] Read more.

Source location is a constant issue of importance of both theoretical study and practical engineering. Many pioneers have put out the corresponding solutions for near- or far-field signals, and preferred contributions are suggested. To our best knowledge, there are currently few focused approaches to the complicated situation where both near- and far-field signals exist with an unknown number of sources. Additionally, the robustness of the method must be taken into account when the additive background noise does not follow Gaussian or super-Gaussian distribution. To solve these problems, a novel method based on phased fractional lower-order moment (PFLOM) is proposed to simultaneously better preserve the signal and suppress the noise. Secondly, the whole procedure of the method containing direction of arrival (DOA) estimation, range estimation, separation of near-and far-field sources, and crucial parameter settings are studied in detail. Finally, comprehensive Monte Carlo experiments are carried out in the simulation to demonstrate the superiority of the proposed method compared to the existing competitive methods. Due to the novel method’s effectiveness with an unknown number of sources and robustness against various noises, it is believed that it could be fully utilized in more fields. Full article

(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing)

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31 pages, 1040 KiB

Open AccessArticle

Application of Empirical Bode Analysis for Delay-Margin Evaluation of Fractional-Order PI Controller in a Renewable Distributed Hybrid System

by Soumen Biswas, Shibendu Mahata, Provas Kumar Roy and Kalyan Chatterjee

Fractal Fract. 2023, 7(2), 119; https://doi.org/10.3390/fractalfract7020119 - 26 Jan 2023

Cited by 4 | Viewed by 1314

Abstract

For an uninterrupted power supply, renewable energy promises to be a suitable alternative compared to the conventional sources. System delays or communication delays may cause significant synchronization imbalances between various components in big electrical grids. Since the properties of solar and wind generation [...] Read more.

For an uninterrupted power supply, renewable energy promises to be a suitable alternative compared to the conventional sources. System delays or communication delays may cause significant synchronization imbalances between various components in big electrical grids. Since the properties of solar and wind generation constantly change with climatic circumstances, engineers encounter many difficulties when substituting sustainable power with conventional electricity. The computation delay margin may be leveraged to handle a time-delayed automatic generation control (AGC) system. In order to regulate a distributed hybrid renewable energy system in a three-area AGC configuration, this paper investigates the influence of the fractional integral order on the stable system’s delay parameter region. By changing the fractional order range, the delay margin can be increased, potentially broadening the time-delayed system’s stability region. The controller’s stability region has dependency on the order of fraction and the time delay. For this purpose, the asymptotic Bode diagram of the time-delayed fractional proportional-integral controller is determined. The gain and phase margins are used to calculate the delay margin for the application in discussion. The Honey Badger algorithm helps to adjust the controller parameters. It is also confirmed that the suggested controller is resilient to random load perturbations, nonlinearities, and parameter variations. Full article

(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing)

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20 pages, 49814 KiB

Open AccessArticle

An End-to-End Underwater-Image-Enhancement Framework Based on Fractional Integral Retinex and Unsupervised Autoencoder

by Yang Yu and Chenfeng Qin

Fractal Fract. 2023, 7(1), 70; https://doi.org/10.3390/fractalfract7010070 - 09 Jan 2023

Cited by 2 | Viewed by 2369

Abstract

As an essential low-level computer vision task for remotely operated underwater robots and unmanned underwater vehicles to detect and understand the underwater environment, underwater image enhancement is facing challenges of light scattering, absorption, and distortion. Instead of using a specific underwater imaging model [...] Read more.

As an essential low-level computer vision task for remotely operated underwater robots and unmanned underwater vehicles to detect and understand the underwater environment, underwater image enhancement is facing challenges of light scattering, absorption, and distortion. Instead of using a specific underwater imaging model to mitigate the degradation of underwater images, we propose an end-to-end underwater-image-enhancement framework that combines fractional integral-based Retinex and an encoder–decoder network. The proposed variant of Retinex aims to alleviate haze and color distortion in the input image while preserving edges to a large extent by utilizing a modified fractional integral filter. The encoder–decoder network with channel-wise attention modules trained in an unsupervised manner to overcome the lack of paired underwater image datasets is designed to refine the output of the Retinex. Our framework was evaluated under qualitative and quantitative metrics on several public underwater image datasets and yielded satisfactory enhancement results on the evaluation set. Full article

(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing)

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18 pages, 3648 KiB

Open AccessArticle

Analysis of the Charge Density Variation Caused by the Physical Properties of the Electrodes of Lithium-Ion Batteries

by Xin Lu and Ning Chen

Fractal Fract. 2022, 6(12), 701; https://doi.org/10.3390/fractalfract6120701 - 26 Nov 2022

Viewed by 1446

Abstract

The detection and characterization of electrode performance is a key problem of lithium-ion batteries. The physical properties of the electrodes affect the charge density during the life of a battery. The charge density is difficult to monitor because of the complexity of the [...] Read more.

The detection and characterization of electrode performance is a key problem of lithium-ion batteries. The physical properties of the electrodes affect the charge density during the life of a battery. The charge density is difficult to monitor because of the complexity of the charge distribution. In this paper, a visualized fractional derivative order (FDO) is used to characterize the charge distribution and to reveal variations in the charge density associated with the physical properties of the electrode. Instantaneous discharge datasets collected at different aging stages of batteries are used to identify the FDO in the fractional derivative model. The results show that the FDO has a strong correspondence with the charge density. As the charge density decreases, the charge mobility gradually increases due to changes in the charge distribution. Moreover, this paper finds that the capacity recovery effect is closely related to the mutation of the charge density and uses the FDO to explain the charge accumulation at the sharp edges of the electrodes. The analysis of the charge density variation caused by the physical properties of the electrodes provides guidance for the detection of the electrode performance and the design of the electrode microstructure. Full article

(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing)

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15 pages, 1974 KiB

Open AccessArticle

Fractional-Order Calculus-Based Data Augmentation Methods for Environmental Sound Classification with Deep Learning

by Bilgi Görkem Yazgaç and Mürvet Kırcı

Fractal Fract. 2022, 6(10), 555; https://doi.org/10.3390/fractalfract6100555 - 29 Sep 2022

Cited by 1 | Viewed by 1420

Abstract

In this paper, we propose two fractional-order calculus-based data augmentation methods for audio signals. The first approach is based on fractional differentiation of the Mel scale. By using a randomly selected fractional derivation order, we are warping the Mel scale, therefore, we aim [...] Read more.

In this paper, we propose two fractional-order calculus-based data augmentation methods for audio signals. The first approach is based on fractional differentiation of the Mel scale. By using a randomly selected fractional derivation order, we are warping the Mel scale, therefore, we aim to augment Mel-scale-based time-frequency representations of audio data. The second approach is based on previous fractional-order image edge enhancement methods. Since multiple deep learning approaches treat Mel spectrogram representations like images, a fractional-order differential-based mask is employed. The mask parameters are produced with respect to randomly selected fractional-order derivative parameters. The proposed data augmentation methods are applied to the UrbanSound8k environmental sound dataset. For the classification of the dataset and testing the methods, an arbitrary convolutional neural network is implemented. Our results show that fractional-order calculus-based methods can be employed as data augmentation methods. Increasing the dataset size to six times the original size, the classification accuracy result increased by around 8.5%. Additional tests on more complex networks also produced better accuracy results compared to a non-augmented dataset. To our knowledge, this paper is the first example of employing fractional-order calculus as an audio data augmentation tool. Full article

(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing)

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22 pages, 2715 KiB

Open AccessArticle

Optimizing the Maximum Lyapunov Exponent of Fractional Order Chaotic Spherical System by Evolutionary Algorithms

by Vincent-Ademola Adeyemi, Esteban Tlelo-Cuautle, Francisco-Javier Perez-Pinal and Jose-Cruz Nuñez-Perez

Fractal Fract. 2022, 6(8), 448; https://doi.org/10.3390/fractalfract6080448 - 18 Aug 2022

Cited by 5 | Viewed by 1827

Abstract

The main goal of this work is to optimize the chaotic behavior of a three-dimensional chaotic-spherical-attractor-generating fractional-order system and compare the results with its novel hyperchaotic counterpart. The fractional-order chaotic system is a smooth system perturbed with a hyperbolic tangent function. There are [...] Read more.

The main goal of this work is to optimize the chaotic behavior of a three-dimensional chaotic-spherical-attractor-generating fractional-order system and compare the results with its novel hyperchaotic counterpart. The fractional-order chaotic system is a smooth system perturbed with a hyperbolic tangent function. There are two major contributions in this investigation. First, the maximum Lyapunov exponent of the chaotic system was optimized by applying evolutionary algorithms, which are meta-heuristics search algorithms, namely, the differential evolution, particle swarm optimization, and invasive weed optimization. Each of the algorithms was populated with one hundred individuals, the maximum generation was five hundred, and the total number of design variables was eleven. The results show a massive increase of over 5000% in the value of the maximum Lyapunov exponent, thereby leading to an increase in the chaotic behavior of the system. Second, a hyperchaotic system of four dimensions was constructed from the inital chaotic system. The dynamics of the optimized chaotic and the new hyperchaotic systems were analyzed using phase portraits, time series, bifurcation diagrams, and Lyapunov exponent spectra. Finally, comparison between the optimized chaotic systems and the hyperchaotic states shows an evidence of more complexity, ergodicity, internal randomness, and unpredictability in the optimized systems than its hyperchaotic counterpart according to the analysis of their information entropies and prediction times. Full article

(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing)

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17 pages, 3050 KiB

Open AccessArticle

CORDIC-Based FPGA Realization of a Spatially Rotating Translational Fractional-Order Multi-Scroll Grid Chaotic System

by Wafaa S. Sayed, Merna Roshdy, Lobna A. Said, Norbert Herencsar and Ahmed G. Radwan

Fractal Fract. 2022, 6(8), 432; https://doi.org/10.3390/fractalfract6080432 - 07 Aug 2022

Cited by 4 | Viewed by 1622

Abstract

This paper proposes an algorithm and hardware realization of generalized chaotic systems using fractional calculus and rotation algorithms. Enhanced chaotic properties, flexibility, and controllability are achieved using fractional orders, a multi-scroll grid, a dynamic rotation angle(s) in two- and three-dimensional space, and translational [...] Read more.

This paper proposes an algorithm and hardware realization of generalized chaotic systems using fractional calculus and rotation algorithms. Enhanced chaotic properties, flexibility, and controllability are achieved using fractional orders, a multi-scroll grid, a dynamic rotation angle(s) in two- and three-dimensional space, and translational parameters. The rotated system is successfully utilized as a Pseudo-Random Number Generator (PRNG) in an image encryption scheme. It preserves the chaotic dynamics and exhibits continuous chaotic behavior for all values of the rotation angle. The Coordinate Rotation Digital Computer (CORDIC) algorithm is used to implement rotation and the Grünwald–Letnikov (GL) technique is used for solving the fractional-order system. CORDIC enables complete control and dynamic spatial rotation by providing real-time computation of the sine and cosine functions. The proposed hardware architectures are realized on a Field-Programmable Gate Array (FPGA) using the Xilinx ISE 14.7 on Artix 7 XC7A100T kit. The Intellectual-Property (IP)-core-based implementation generates sine and cosine functions with a one-clock-cycle latency and provides a generic framework for rotating any chaotic system given its system of differential equations. The achieved throughputs are

821.92

Mbits/s and

520.768

Mbits/s for two- and three-dimensional rotating chaotic systems, respectively. Because it is amenable to digital realization, the proposed spatially rotating translational fractional-order multi-scroll grid chaotic system can fit various secure communication and motion control applications. Full article

(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing)

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19 pages, 7637 KiB

Open AccessArticle

Grid-Connected Inverter Based on a Resonance-Free Fractional-Order LCL Filter

by Xiaogang Wang and Junhui Cai

Fractal Fract. 2022, 6(7), 374; https://doi.org/10.3390/fractalfract6070374 - 01 Jul 2022

Cited by 2 | Viewed by 1593

Abstract

The integer-order LCL (IOLCL) filter has excellent high-frequency harmonic attenuation capability but suffers from resonance, which causes system instability in grid-connected inverter applications. This paper studied a class of resonance-free fractional-order LCL (FOLCL) filters and control problems of single-phase FOLCL-type grid-connected inverters (FOGCI). [...] Read more.

The integer-order LCL (IOLCL) filter has excellent high-frequency harmonic attenuation capability but suffers from resonance, which causes system instability in grid-connected inverter applications. This paper studied a class of resonance-free fractional-order LCL (FOLCL) filters and control problems of single-phase FOLCL-type grid-connected inverters (FOGCI). The Caputo fractional calculus operator was used to describe the fractional-order inductor and capacitor. Compared with the conventional IOLCL filter, by reasonably selecting the orders of the inductor and capacitor, the resonance peak of the FOLCL filter could be effectively avoided. In this way, the FOGCI could operate stably without passive or active dampers, which simplified the design of control system. Furthermore, compared with a single-phase integer-order grid-connected inverter (IOGCI) controlled by an integer-order PI (IOPI) controller, the FOGCI, combined with a fractional-order PI (FOPI) controller, could achieve greater gain and phase margins, which improved the system performance. The correctness of the theoretical analyses was validated through both simulation and hardware-in-the-loop experiments. Full article

(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing)

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