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Applications of Fractional Operator in Image Processing and Stability of...
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Applications of Fractional Operator in Image Processing and Stability of Control Systems

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Engineering".

Deadline for manuscript submissions: closed (15 January 2023) | Viewed by 40065

Special Issue Editors

Dr. Xuefeng Zhang

E-Mail Website
Guest Editor
College of Sciences, Northeastern University, Shengyang 110819, China
Interests: singular systems; fractional systems; robust control
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Driss Boutat

E-Mail
Guest Editor
Institut National des Sciences Appliquées Centre Val de Loire, Bourges, France
Interests: non-asymptotic state estimation; non-asymptotic fractional order differentiator
Special Issues, Collections and Topics in MDPI journals
Dr. Dayan Liu

E-Mail Website
Guest Editor
INSA Centre Val de Loire, Université d’Orléans, PRISME EA 4229, CEDEX, 18022 Bourges, France
Interests: non-asymptotic state estimation; non-asymptotic fractional order differentiator
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The aim of this Special Issue is to gather articles reflecting the latest developments in applied mathematics and control engineering related to the interdisciplinary topics of control, fractional calculus, and image processing, and their applications in engineering science. Fractional calculus and fractional processes, with applications in control systems and image processing, have become a hot topic. Fractional order systems are a natural generalization of classical integer order systems, and can accurately describe many real-world physical systems. Fusion and noise suppression of medical images are becoming increasingly difficult to ignore in image processing, and these techniques provide abundant information for clinical diagnosis and treatment. Image fusion is a significant factor in image processing, owing to the increase in image acquisition models. Recently, fractional operators have been playing an important role in image processing. Additionally, powerful fractional operating tools have been introduced which can be effectively applied to the analysis and design of nonlinear control systems. Singular systems are governed by so-called singular differential equations, which endow the systems with many special features that are not found in classical systems. The approaches of fractional order control systems, which borrow from those of integer order control systems, are attracting increasing attention within the control field. 

Prof. Dr. Xuefeng Zhang
Prof. Dr. Driss Boutat
Dr. Dayan Liu
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • image processing
  • image fusion
  • image denoising
  • stability of fractional systems
  • control of fractional systems
  • singular fractional systems

Published Papers (25 papers)

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Editorial

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5 pages, 207 KiB  
Editorial
Applications of Fractional Operator in Image Processing and Stability of Control Systems
by Xuefeng Zhang, Driss Boutat and Dayan Liu
Fractal Fract. 2023, 7(5), 359; https://doi.org/10.3390/fractalfract7050359 - 28 Apr 2023
Cited by 8 | Viewed by 1358
Abstract
Over recent years, a growing number of authors’ works from various science and engineering fields have dealt with dynamical systems, described by the connection between the theory of artificial intelligence and fractional differential equations, and many computational fractional intelligence systems and stability analysis [...] Read more.
Over recent years, a growing number of authors’ works from various science and engineering fields have dealt with dynamical systems, described by the connection between the theory of artificial intelligence and fractional differential equations, and many computational fractional intelligence systems and stability analysis and image processing applications have been proposed [...] Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)

Research

Jump to: Editorial

19 pages, 2732 KiB  
Article
A Study of Fractional-Order Memristive Ant Colony Algorithm: Take Fracmemristor into Swarm Intelligent Algorithm
by Wuyang Zhu and Yifei Pu
Fractal Fract. 2023, 7(3), 211; https://doi.org/10.3390/fractalfract7030211 - 23 Feb 2023
Cited by 3 | Viewed by 998
Abstract
As the fourth fundamental circuit element, the memristor may execute computations while storing data. Fracmemristor takes advantage of the fractional calculate’s long-term memory, non-locality, weak singularity, and the memristor’s storage–computational integration. Since the physical structure of the fracmemristor is similar to the topology [...] Read more.
As the fourth fundamental circuit element, the memristor may execute computations while storing data. Fracmemristor takes advantage of the fractional calculate’s long-term memory, non-locality, weak singularity, and the memristor’s storage–computational integration. Since the physical structure of the fracmemristor is similar to the topology of the ant transfer probability flow in ACO, we propose the fractional-order memristive ant colony algorithm (FMAC), which uses the fracmemristor physical system to record the probabilistic transfer information of the nodes that the ant will crawl through in the future and pass it to the current node of the ant, so that the ant acquires the ability to predict the future transfer. After instigating the optimization capabilities with TSP, we discovered that FMAC is superior to PACO-3opt, the best integer-order ant colony algorithm currently available. FMAC operates substantially more quickly than the fractional-order memristor ant colony algorithm due to the transfer probability prediction module based on the physical fracmemristor system (FACA). Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
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10 pages, 1428 KiB  
Article
Frac-Vector: Better Category Representation
by Sunfu Tan and Yifei Pu
Fractal Fract. 2023, 7(2), 132; https://doi.org/10.3390/fractalfract7020132 - 31 Jan 2023
Cited by 1 | Viewed by 929
Abstract
For this paper, we proposed the fractional category representation vector (FV) based on fractional calculus (FC), of which one-hot label is only the special case when the derivative order is 0. FV can be considered as a distributional representation when negative probability is [...] Read more.
For this paper, we proposed the fractional category representation vector (FV) based on fractional calculus (FC), of which one-hot label is only the special case when the derivative order is 0. FV can be considered as a distributional representation when negative probability is considered. FVs can be used either as a regularization method or as a distributed category representation. They gain significantly in the generalization of classification models and representability in generative adversarial networks with conditions (C-GANs). In image classification, the linear combinations of FVs correspond to the mixture of images and can be used as an independent variable of the loss function. Our experiments showed that FVs can also be used as space sampling, with fewer dimensions and less computational overhead than normal distributions. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
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26 pages, 4774 KiB  
Article
A Fractional-Order Telegraph Diffusion Model for Restoring Texture Images with Multiplicative Noise
by Xiangyu Bai, Dazhi Zhang, Shengzhu Shi, Wenjuan Yao, Zhichang Guo and Jiebao Sun
Fractal Fract. 2023, 7(1), 64; https://doi.org/10.3390/fractalfract7010064 - 05 Jan 2023
Cited by 3 | Viewed by 1662
Abstract
Multiplicative noise removal from texture images poses a significant challenge. Different from the diffusion equation-based filter, we consider the telegraph diffusion equation-based model, which can effectively preserve fine structures and edges for texture images. The fractional-order derivative is imposed due to its textural [...] Read more.
Multiplicative noise removal from texture images poses a significant challenge. Different from the diffusion equation-based filter, we consider the telegraph diffusion equation-based model, which can effectively preserve fine structures and edges for texture images. The fractional-order derivative is imposed due to its textural detail enhancing capability. We also introduce the gray level indicator, which fully considers the gray level information of multiplicative noise images, so that the model can effectively remove high level noise and protect the details of the structure. The well-posedness of the proposed fractional-order telegraph diffusion model is presented by applying the Schauder’s fixed-point theorem. To solve the model, we develop an iterative algorithm based on the discrete Fourier transform in the frequency domain. We give various numerical results on despeckling natural and real SAR images. The experiments demonstrate that the proposed method can remove multiplicative noise and preserve texture well. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
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20 pages, 701 KiB  
Article
Quadratic Admissibility for a Class of LTI Uncertain Singular Fractional-Order Systems with 0 < α < 2
by Yuying Wang, Xuefeng Zhang, Driss Boutat and Peng Shi
Fractal Fract. 2023, 7(1), 1; https://doi.org/10.3390/fractalfract7010001 - 20 Dec 2022
Cited by 6 | Viewed by 981
Abstract
This paper provides a unified framework for the admissibility of a class of singular fractional-order systems with a given fractional order in the interval (0, 2). These necessary and sufficient conditions are derived in terms of linear matrix [...] Read more.
This paper provides a unified framework for the admissibility of a class of singular fractional-order systems with a given fractional order in the interval (0, 2). These necessary and sufficient conditions are derived in terms of linear matrix inequalities (LMIs). The considered fractional orders range from 0 to 2 without separating the ranges into (0, 1) and [1, 2) to discuss the admissibility. Moreover, the uncertain system with the fractional order in the interval (0, 2) is norm-bounded. The quadratic admissibility and general quadratic stability of the system are analyzed, and the equivalence between the two is proved. All the above can be expressed in terms of strict LMIs to avoid any singularity problem in the solution. Finally, the effectiveness of the method is illustrated by three numerical examples. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
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19 pages, 1535 KiB  
Article
Adaptive Fuzzy Backstepping Control of Fractional-Order Chaotic System Synchronization Using Event-Triggered Mechanism and Disturbance Observer
by Zhiye Bai, Shenggang Li, Heng Liu and Xiulan Zhang
Fractal Fract. 2022, 6(12), 714; https://doi.org/10.3390/fractalfract6120714 - 30 Nov 2022
Cited by 4 | Viewed by 1351
Abstract
The synchronization of fractional-order chaotic systems is investigated using command-filtered adaptive fuzzy control with a disturbance observer, where an event-triggered mechanism and backstepping control technique are employed. In order to relieve the pressure of the continuous update of the controller and improve the [...] Read more.
The synchronization of fractional-order chaotic systems is investigated using command-filtered adaptive fuzzy control with a disturbance observer, where an event-triggered mechanism and backstepping control technique are employed. In order to relieve the pressure of the continuous update of the controller and improve the resource utilization, an event-triggered control strategy is constructed to reduce the amount of communication for the actuator. Under the framework of adaptive fuzzy backstepping recursive design, fuzzy logical systems and disturbance observers are proposed to estimate the unknown parametric uncertainties and external disturbances, respectively. Moreover, a tracking differentiator is introduced to eliminate the drawback of the explosion of complexity in traditional backstepping. By applying the fractional-order stability theory, all closed-loop signals are bounded and chaos synchronization is achieved. Finally, a simulation example is provided to confirm the effectiveness of the designed method. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
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18 pages, 2032 KiB  
Article
Dental X-ray Identification System Based on Association Rules Extracted by k-Symbol Fractional Haar Functions
by Mona Hmoud AlSheikh, Nadia M. G. Al-Saidi and Rabha W. Ibrahim
Fractal Fract. 2022, 6(11), 669; https://doi.org/10.3390/fractalfract6110669 - 11 Nov 2022
Cited by 6 | Viewed by 1112
Abstract
Several identification approaches have recently been employed in human identification systems for forensic purposes to decrease human efforts and to boost the accuracy of identification. Dental identification systems provide automated matching by searching photographic dental features to retrieve similar models. In this study, [...] Read more.
Several identification approaches have recently been employed in human identification systems for forensic purposes to decrease human efforts and to boost the accuracy of identification. Dental identification systems provide automated matching by searching photographic dental features to retrieve similar models. In this study, the problem of dental image identification was investigated by developing a novel dental identification scheme (DIS) utilizing a fractional wavelet feature extraction technique and rule mining with an Apriori procedure. The proposed approach extracts the most discriminating image features during the mining process to obtain strong association rules (ARs). The proposed approach is divided into two steps. The first stage is feature extraction using a wavelet transform based on a k-symbol fractional Haar filter (k-symbol FHF), while the second stage is the Apriori algorithm of AR mining, which is applied to find the frequent patterns in dental images. Each dental image’s created ARs are saved alongside the image in the rules database for use in the dental identification system’s recognition. The DIS method suggested in this study primarily enhances the Apriori-based dental identification system, which aims to address the drawbacks of dental rule mining. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
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16 pages, 1522 KiB  
Article
Adaptive Neural Fault-Tolerant Control for Nonlinear Fractional-Order Systems with Positive Odd Rational Powers
by Jiawei Ma, Huanqing Wang, Yakun Su, Cungen Liu and Ming Chen
Fractal Fract. 2022, 6(11), 622; https://doi.org/10.3390/fractalfract6110622 - 25 Oct 2022
Cited by 2 | Viewed by 921
Abstract
In this paper, the problem of adaptive neural fault-tolerant control (FTC) for the fractional-order nonlinear systems (FNSs) with positive odd rational powers (PORPs) is considered. By using the radial basis function neural networks (RBF NNs), the unknown nonlinear functions from the controlled system [...] Read more.
In this paper, the problem of adaptive neural fault-tolerant control (FTC) for the fractional-order nonlinear systems (FNSs) with positive odd rational powers (PORPs) is considered. By using the radial basis function neural networks (RBF NNs), the unknown nonlinear functions from the controlled system can be approximated. With the help of an adaptive control ideology, the unknown control rate of the actuator fault can be handled. In particular, the FNSs subject to high-order terms are studied for the first time. In addition, the designed controller can ensure the boundedness of all the signals of the closed-loop control system, and the tracking error can tend to a small neighborhood of zero in the end. Finally, the illustrative examples are shown to validate the effectiveness of the developed method. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
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16 pages, 6702 KiB  
Article
Fault Diagnosis of Hydroelectric Units Based on a Novel Multiscale Fractional-Order Weighted Permutation Entropy
by Wenjing Zhang, Yuanchen Gao, Shizhe Peng, Dongdong Zhou and Bin Wang
Fractal Fract. 2022, 6(10), 588; https://doi.org/10.3390/fractalfract6100588 - 13 Oct 2022
Cited by 3 | Viewed by 1133
Abstract
To improve the noise immunity, stability and sensitivity to different signal types in the hydroelectric unit fault diagnosis model, a hydroelectric unit fault diagnosis model based on improved multiscale fractional-order weighted permutation entropy (IMFWPE) is proposed. Firstly, the fractional order and weighting theory [...] Read more.
To improve the noise immunity, stability and sensitivity to different signal types in the hydroelectric unit fault diagnosis model, a hydroelectric unit fault diagnosis model based on improved multiscale fractional-order weighted permutation entropy (IMFWPE) is proposed. Firstly, the fractional order and weighting theory is introduced into the permutation entropy (PE) to improve the sensitivity to different fault signals while improving the defect of ignoring the signal amplitude information. Additionally, considering the problem that a single scale cannot fully reflect the timing characteristics and that the traditional coarse-grained method will shorten the timing length, a new tool for measuring the complexity of timing signals, IMFWPE, is proposed by introducing an improved multiscale method. Finally, the IMFWPE values of signals are extracted as features and input to the classifier for fault identification of hydroelectric units. The experimental results show that the proposed method has the best diagnostic effect when compared with other methods, has good noise immunity and stability, and has good diagnostic capability in the actual unit environment. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
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19 pages, 8936 KiB  
Article
A Novel Adaptive Fractional Differential Active Contour Image Segmentation Method
by Yanzhu Zhang, Lijun Yang and Yan Li
Fractal Fract. 2022, 6(10), 579; https://doi.org/10.3390/fractalfract6100579 - 10 Oct 2022
Cited by 6 | Viewed by 1301
Abstract
When the image is affected by strong noise and uneven intensity, the traditional active contour models often cannot obtain accurate results. In this paper, a novel adaptive fractional differential active contour image segmentation method is proposed to solve the above problem. At first, [...] Read more.
When the image is affected by strong noise and uneven intensity, the traditional active contour models often cannot obtain accurate results. In this paper, a novel adaptive fractional differential active contour image segmentation method is proposed to solve the above problem. At first, in order to extract more texture parts of the image, an adaptively fractional order matrix is constructed according to the gradient information of the image, varying the fractional order of each pixel. Then, the traditional edge-stopping function in the regularization term is susceptible to noise, and a new fractional-order edge-stopping function is designed to improve noise resistance. In this paper, a fitting term based on adaptive fractional differentiation is introduced to solve the problem of improper selection of the initial contour position leading to inaccurate segmentation results so that the initial contour position can be selected arbitrarily. Finally, the experimental results show that the proposed method can effectively improve the segmentation accuracy of noise images and weak-edge images and can arbitrarily select the position selection of the initial contour. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
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17 pages, 4072 KiB  
Article
A Study of Adaptive Fractional-Order Total Variational Medical Image Denoising
by Yanzhu Zhang, Tingting Liu, Fan Yang and Qi Yang
Fractal Fract. 2022, 6(9), 508; https://doi.org/10.3390/fractalfract6090508 - 11 Sep 2022
Cited by 5 | Viewed by 1627
Abstract
Following the traditional total variational denoising model in removing medical image noise with blurred image texture details, among other problems, an adaptive medical image fractional-order total variational denoising model with an improved sparrow search algorithm is proposed in this study. This algorithm combines [...] Read more.
Following the traditional total variational denoising model in removing medical image noise with blurred image texture details, among other problems, an adaptive medical image fractional-order total variational denoising model with an improved sparrow search algorithm is proposed in this study. This algorithm combines the characteristics of fractional-order differential operators and total variational models. The model preserves the weak texture region of the image improvement based on the unique amplitude-frequency characteristics of the fractional-order differential operator. The order of the fractional-order differential operator is adaptively determined by the improved sparrow search algorithm using both the sine search strategy and the diversity variation processing strategy, which can greatly improve the denoising ability of the fractional-order differential operator. The experimental results reveal that the model not only achieves the adaptivity of fractional-order total variable differential order, but also can effectively remove noise, preserve the texture structure of the image to the maximum extent, and improve the peak signal-to-noise ratio of the image; it also displays favorable prospects for applications in medical image denoising. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
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21 pages, 776 KiB  
Article
Finite-Time Synchronization of Uncertain Fractional-Order Delayed Memristive Neural Networks via Adaptive Sliding Mode Control and Its Application
by Tianyuan Jia, Xiangyong Chen, Liping He, Feng Zhao and Jianlong Qiu
Fractal Fract. 2022, 6(9), 502; https://doi.org/10.3390/fractalfract6090502 - 07 Sep 2022
Cited by 10 | Viewed by 1429
Abstract
Finite-time synchronization (FTS) of uncertain fractional-order memristive neural networks (FMNNs) with leakage and discrete delays is studied in this paper, in which the impacts of uncertain parameters as well as external disturbances are considered. First, the fractional-order adaptive terminal sliding mode control scheme [...] Read more.
Finite-time synchronization (FTS) of uncertain fractional-order memristive neural networks (FMNNs) with leakage and discrete delays is studied in this paper, in which the impacts of uncertain parameters as well as external disturbances are considered. First, the fractional-order adaptive terminal sliding mode control scheme (FATSMC) is designed, which can effectively estimate the upper bounds of unknown external disturbances. Second, the FTS of the master–slave FMNNs is realized and the corresponding synchronization criteria and the explicit expression of the settling time (ST) are obtained. Finally, a numerical example and a secure communication application are provided to demonstrate the validity of the obtained results. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
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13 pages, 488 KiB  
Article
Sampled-Data Stabilization of Fractional Linear System under Arbitrary Sampling Periods
by Kecai Cao, Juping Gu, Jingfeng Mao and Chenglin Liu
Fractal Fract. 2022, 6(8), 416; https://doi.org/10.3390/fractalfract6080416 - 29 Jul 2022
Cited by 5 | Viewed by 1233
Abstract
The sampled-data stabilization of a fractional continuous linear system under arbitrary sampling periods was first investigated in this paper wherein novel co-designed sampled-data controllers were constructed based on the compensation of scaling gains. With the help of fractional difference approximation, sufficient and necessary [...] Read more.
The sampled-data stabilization of a fractional continuous linear system under arbitrary sampling periods was first investigated in this paper wherein novel co-designed sampled-data controllers were constructed based on the compensation of scaling gains. With the help of fractional difference approximation, sufficient and necessary conditions for global asymptotic stability were first presented in the discrete-time domain, and then co-designed sampled-data controllers were constructed with only the “newest” or “oldest” state information available for controller design. Due to the compensation scheme between scaling gains and sampling periods, much more flexibility on selecting different sampling periods was provided in the sampled-data stabilization of the fractional continuous linear system which is significantly preferred for digital implementation. Numerical studies are also presented to illustrate the effectiveness of our co-designed sampled-data controllers under different sampling periods. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
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16 pages, 1995 KiB  
Article
Ensemble FARIMA Prediction with Stable Infinite Variance Innovations for Supermarket Energy Consumption
by Jing Wang, Yi Liu, Haiyan Wu, Shan Lu and Meng Zhou
Fractal Fract. 2022, 6(5), 276; https://doi.org/10.3390/fractalfract6050276 - 22 May 2022
Cited by 4 | Viewed by 1786
Abstract
This paper concerns a fractional modeling and prediction method directly oriented toward an industrial time series with obvious non-Gaussian features. The hidden long-range dependence and the multifractal property are extracted to determine the fractional order. A fractional autoregressive integrated moving average model (FARIMA) [...] Read more.
This paper concerns a fractional modeling and prediction method directly oriented toward an industrial time series with obvious non-Gaussian features. The hidden long-range dependence and the multifractal property are extracted to determine the fractional order. A fractional autoregressive integrated moving average model (FARIMA) is then proposed considering innovations with stable infinite variance. The existence and convergence of the model solutions are discussed in depth. Ensemble learning with an autoregressive moving average model (ARMA) is used to further improve upon accuracy and generalization. The proposed method is used to predict the energy consumption in a real cooling system, and superior prediction results are obtained. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
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20 pages, 6338 KiB  
Article
Image Dehazing Based on Local and Non-Local Features
by Qingliang Jiao, Ming Liu, Bu Ning, Fengfeng Zhao, Liquan Dong, Lingqin Kong, Mei Hui and Yuejin Zhao
Fractal Fract. 2022, 6(5), 262; https://doi.org/10.3390/fractalfract6050262 - 09 May 2022
Cited by 11 | Viewed by 2309
Abstract
Image dehazing is a traditional task, yet it still presents arduous problems, especially in the removal of haze from the texture and edge information of an image. The state-of-the-art dehazing methods may result in the loss of some visual informative details and a [...] Read more.
Image dehazing is a traditional task, yet it still presents arduous problems, especially in the removal of haze from the texture and edge information of an image. The state-of-the-art dehazing methods may result in the loss of some visual informative details and a decrease in visual quality. To improve dehazing quality, a novel dehazing model is proposed, based on a fractional derivative and data-driven regularization terms. In this model, the contrast constrained adaptive histogram equalization method is used as the data fidelity item; the fractional derivative is applied to avoid over-enhancement and noise amplification; and the proposed data-driven regularization terms are adopted to extract the local and non-local features of an image. Then, to solve the proposed model, half-quadratic splitting is used. Moreover, a dual-stream network based on Convolutional Neural Network (CNN) and Transformer is introduced to structure the data-driven regularization. Further, to estimate the atmospheric light, an atmospheric light model based on the fractional derivative and the atmospheric veil is proposed. Extensive experiments display the effectiveness of the proposed method, which surpasses the state-of-the-art methods for most synthetic and real-world images, quantitatively and qualitatively. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
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19 pages, 717 KiB  
Article
A New Adaptive Robust Sliding Mode Control Approach for Nonlinear Singular Fractional-Order Systems
by Shunan Chen, Wenkai Huang and Qiang Liu
Fractal Fract. 2022, 6(5), 253; https://doi.org/10.3390/fractalfract6050253 - 06 May 2022
Cited by 5 | Viewed by 1792
Abstract
This article focuses on designing an adaptive sliding mode controller via state and output feedback for nonlinear singular fractional-order systems (SFOSs) with mismatched uncertainties. Firstly, on the basis of extending the dimension of the SFOS, a new integral sliding mode surface is constructed. [...] Read more.
This article focuses on designing an adaptive sliding mode controller via state and output feedback for nonlinear singular fractional-order systems (SFOSs) with mismatched uncertainties. Firstly, on the basis of extending the dimension of the SFOS, a new integral sliding mode surface is constructed. Through this special sliding surface, the sliding mode of the descriptor system does not contain a singular matrix E. Then, the sufficient conditions that ensure the stability of sliding mode motion are given by using linear matrix inequality. Finally, the control law based on an adaptive mechanism that is used to update the nonlinear terms is designed to ensure the SFOS satisfies the reaching condition. The applicability of the proposed method is illustrated by a practical example of a fractional-order circuit system and two numerical examples. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
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13 pages, 323 KiB  
Article
Stability Analysis of the Nabla Distributed-Order Nonlinear Systems
by Cuihong Wang, Tianfen Zhu and Yangquan Chen
Fractal Fract. 2022, 6(5), 228; https://doi.org/10.3390/fractalfract6050228 - 20 Apr 2022
Cited by 3 | Viewed by 1579
Abstract
The stability of the nabla discrete distributed-order nonlinear dynamic systems is investigated in this paper. Firstly, a sufficient condition for the asymptotic stability of the nabla discrete distributed-order nonlinear systems is proposed based on Lyapunov direct method. In addition, some properties of the [...] Read more.
The stability of the nabla discrete distributed-order nonlinear dynamic systems is investigated in this paper. Firstly, a sufficient condition for the asymptotic stability of the nabla discrete distributed-order nonlinear systems is proposed based on Lyapunov direct method. In addition, some properties of the nabla distributed-order operators are derived. Based on these properties, a simpler criterion is provided to determine the stability of such systems. Finally, two examples are given to illustrate the validity of these results. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
17 pages, 10205 KiB  
Article
Image Enhancement Based on Rough Set and Fractional Order Differentiator
by Xuefeng Zhang and Lewen Dai
Fractal Fract. 2022, 6(4), 214; https://doi.org/10.3390/fractalfract6040214 - 11 Apr 2022
Cited by 24 | Viewed by 2094
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
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16 pages, 461 KiB  
Article
Robust H∞ Control of Fractional-Order Switched Systems with Order 0 < α < 1 and Uncertainty
by Bingxin Li, Xiangfei Zhao, Yaowei Liu and Xin Zhao
Fractal Fract. 2022, 6(3), 164; https://doi.org/10.3390/fractalfract6030164 - 16 Mar 2022
Cited by 6 | Viewed by 1747
Abstract
In this paper, robust H∞ control for fractional-order switched systems (FOSSs) with uncertainty is studied. Firstly, the fractional-order switching law for FOSSs is proposed. Then, H∞ control for FOSSs is proven based on the switching law and linear matrix inequalities (LMIs). [...] Read more.
In this paper, robust H∞ control for fractional-order switched systems (FOSSs) with uncertainty is studied. Firstly, the fractional-order switching law for FOSSs is proposed. Then, H∞ control for FOSSs is proven based on the switching law and linear matrix inequalities (LMIs). Moreover, H∞ control for FOSSs with a state feedback controller is extended. Furthermore, the LMI-based condition of robust H∞ control for FOSSs with uncertainty is proven. Furthermore, the condition of robust H∞ control is proposed to design the state feedback controller. Finally, four simulation examples verified the effectiveness of the proposed methods. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
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14 pages, 444 KiB  
Article
Robust Control for Variable-Order Fractional Interval Systems Subject to Actuator Saturation
by Ri Liu, Zhe Wang, Xuefeng Zhang, Jianxu Ren and Qinglong Gui
Fractal Fract. 2022, 6(3), 159; https://doi.org/10.3390/fractalfract6030159 - 15 Mar 2022
Cited by 6 | Viewed by 1614
Abstract
In this paper, a class of variable-order fractional interval systems (VO-FIS) in which the system matrices are affected by the fractional order is investigated. Firstly, the sufficient conditions for robust stability of a VO-FIS with a unified order range of [...] Read more.
In this paper, a class of variable-order fractional interval systems (VO-FIS) in which the system matrices are affected by the fractional order is investigated. Firstly, the sufficient conditions for robust stability of a VO-FIS with a unified order range of ν(σ)∈(0,2) are proposed. Secondly, the stabilization conditions of a VO-FIS subject to actuator saturation are derived in terms of linear matrix inequalities (LMIs). Then, by using the proposed algorithm through an optimization problem, the stability region is estimated. To summarize, the paper gives a stabilization criterion for VO-FIS subject to actuator saturation. Finally, three numerical examples are proposed to verify the effectiveness of our results. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
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14 pages, 1041 KiB  
Article
A Numerical Method for Simulating Viscoelastic Plates Based on Fractional Order Model
by Suhua Jin, Jiaquan Xie, Jingguo Qu and Yiming Chen
Fractal Fract. 2022, 6(3), 150; https://doi.org/10.3390/fractalfract6030150 - 10 Mar 2022
Cited by 7 | Viewed by 1807
Abstract
In this study, an efficacious method for solving viscoelastic dynamic plates in the time domain is proposed for the first time. The differential operator matrices of different orders of Bernstein polynomials algorithm are adopted to approximate the ternary displacement function. The approximate results [...] Read more.
In this study, an efficacious method for solving viscoelastic dynamic plates in the time domain is proposed for the first time. The differential operator matrices of different orders of Bernstein polynomials algorithm are adopted to approximate the ternary displacement function. The approximate results are simulated by code. In addition, it is proved that the proposed method is feasible and effective through error analysis and mathematical examples. Finally, the effects of external load, side length of plate, thickness of plate and boundary condition on the dynamic response of square plate are studied. The numerical results illustrate that displacement and stress of the plate change with the change of various parameters. It is further verified that the Bernstein polynomials algorithm can be used as a powerful tool for numerical solution and dynamic analysis of viscoelastic plates. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
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14 pages, 905 KiB  
Article
Numerical Algorithm for Calculating the Time Domain Response of Fractional Order Transfer Function
by Lu Bai and Dingyü Xue
Fractal Fract. 2022, 6(2), 122; https://doi.org/10.3390/fractalfract6020122 - 19 Feb 2022
Cited by 1 | Viewed by 1844
Abstract
This paper proposes new numerical algorithms for calculating the time domain responses of fractional order transfer functions (FOTFs). FOTFs are divided into two categories, explicit fractional order transfer functions (EFOTFs) and implicit fractional order transfer functions (IFOTFs). Transforming an EFOTF into an equivalent [...] Read more.
This paper proposes new numerical algorithms for calculating the time domain responses of fractional order transfer functions (FOTFs). FOTFs are divided into two categories, explicit fractional order transfer functions (EFOTFs) and implicit fractional order transfer functions (IFOTFs). Transforming an EFOTF into an equivalent fractional order differential equation, its time domain response can be obtained by solving the equation by the difference method. IFOTF cannot be transformed into an equivalent equation, so its time domain response cannot be calculated by existing difference methods. A new numerical algorithm is designed for calculating a convolution and its inverse operation, the time domain response of IFOTF can be calculated based on the algorithm. Error analysis shows that the proposed numerical algorithms are of first-order accuracy. Four calculation examples are presented, and the results are consistent with the theoretical analysis. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
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18 pages, 6885 KiB  
Article
Adaptive Fractional Image Enhancement Algorithm Based on Rough Set and Particle Swarm Optimization
by Xuefeng Zhang, Ri Liu, Jianxu Ren and Qinglong Gui
Fractal Fract. 2022, 6(2), 100; https://doi.org/10.3390/fractalfract6020100 - 11 Feb 2022
Cited by 16 | Viewed by 2029
Abstract
This paper proposes a new image enhancement algorithm. At first, the paper uses the combination of rough set and particle swarm optimization (PSO) algorithm to distinguish the smooth area, edge and texture area of the image. Then, according to the results of image [...] Read more.
This paper proposes a new image enhancement algorithm. At first, the paper uses the combination of rough set and particle swarm optimization (PSO) algorithm to distinguish the smooth area, edge and texture area of the image. Then, according to the results of image segmentation, an adaptive fractional differential filter is used to enhance the image. Finally, the experimental results show that the image enhanced by this algorithm has clear edge, rich texture details, and retains the information of the smooth area of the image. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
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13 pages, 312 KiB  
Article
Reduced-Order H∞ Filter Design for Singular Fractional-Order Systems
by Ying Guo, Chong Lin and Bing Chen
Fractal Fract. 2022, 6(2), 97; https://doi.org/10.3390/fractalfract6020097 - 10 Feb 2022
Cited by 1 | Viewed by 1069
Abstract
This paper investigates the problem of reduced-order H∞ filter design for singular fractional-order systems with order 0<α<1. It provides necessary and sufficient conditions for designs of both reduced-order H∞ filters and zeroth-order H∞ filters. When [...] Read more.
This paper investigates the problem of reduced-order H∞ filter design for singular fractional-order systems with order 0<α<1. It provides necessary and sufficient conditions for designs of both reduced-order H∞ filters and zeroth-order H∞ filters. When reduced to special cases, the present results are shown to include those in recent works as special cases. Illustrative examples are presented to demonstrate the effectiveness of the results. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
10 pages, 876 KiB  
Article
Robust H∞ Control for Fractional Order Systems with Order α (0 < α < 1)
by Bingxin Li, Yaowei Liu and Xin Zhao
Fractal Fract. 2022, 6(2), 86; https://doi.org/10.3390/fractalfract6020086 - 02 Feb 2022
Cited by 4 | Viewed by 1421
Abstract
In the paper, H∞ and robust H∞ control for fractional order systems (FOS) with order 0<α<1 are studied. Firstly, necessary and sufficient conditions of H∞ control and state feedback controller design are proposed. Then, robust [...] Read more.
In the paper, H∞ and robust H∞ control for fractional order systems (FOS) with order 0<α<1 are studied. Firstly, necessary and sufficient conditions of H∞ control and state feedback controller design are proposed. Then, robust H∞ control for FOS with uncertainty is studied, and state feedback controller is designed. These conditions are based on linear matrix inequalities (LMI) and can be easily solved by the LMI toolbox. Finally, the effectiveness of these conditions is verified by two numerical examples. Full article
(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)
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