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

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. The aim of this Special Issue is to gather articles reflecting the latest developments in applied mathematics and advanced intelligent 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, represent a hot topic. Fractional-order systems are a natural generalization of classical integer-order systems with the capacity to accurately describe many real-world physical systems. It is increasingly difficult to ignore the fusion and noise suppression of medical images 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 played an important role in image processing. Additionally, powerful fractional operating tools have been introduced, possessing extensive applications in the analysis and design of nonlinear control systems. Singular systems are governed by so-called singular differential equations, endowing the systems with many special features 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.

More than 20 high-quality papers were accepted for publication in this Special Issue. The papers were written by different authors which proves the wide scope of the Special Issue. The published papers are briefly summarized as follows.

According to [1], Li et al. formulated the necessary and sufficient LMI-based conditions of $H_{\infty }$ control, and derived the methods of solving robust $H_{\infty }$ control problems for a fractional-order system with order $0<\alpha <1$. The innovation of this article is to use two real matrices to replace the conjugate matrix or Hermitian matrix.

The study in [2] provided the necessary and sufficient conditions for singular fractional-order systems with order $0<\alpha <1$ based on the bounded real lemma without decomposing the system matrices for designs of both reduced-order $H_{\infty }$ filters and zeroth-order $H_{\infty }$ filters. In particular, the method for solving the zeroth-order $H_{\infty }$ filtering problem was in terms of two strict LMIs, and the appropriate filters can be designed such that the filtering error systems are admissible and the transfer functions from the disturbance to the filtering error output satisfy a prescribed $H_{\infty }$-norm bound constraint.

The study in [3] proposed new numerical algorithms for calculating a convolution and its inverse operation for implicit fractional-order transfer functions. For the implicit fractional-order transfer functions, which could previously only be analyzed by frequency methods, the proposed algorithm can perform frequency-domain and time-domain analyses, directly providing their time-domain characteristics.

In [4], sufficient conditions for variable-order fractional interval systems with orders (0, 2) subject to actuator saturation for stabilization criterion in terms of LMIs were given. Based on the stability conditions, this article estimated the stability domain of the system by solving an optimization problem in terms of LMIs, and the eigenvalues of the system matrix are restrained in the right half of the plane.

Article [5] proposed the fractional-order switching law and proved $H_{\infty }$ control for the robust $H_{\infty }$ control for fractional-order switched systems with uncertainty based on the switching law and linear matrix inequalities. Moreover, $H_{\infty }$ control for fractional-order switched systems with a state feedback controller was extended, and the LMI-based condition of robust $H_{\infty }$ control for fractional-order switched systems with uncertainty was proven.

In this paper [6], Wang et al. drew a sufficient condition for the asymptotic stability of nabla discrete distributed-order nonlinear systems based on the Lyapunov direct method and comparison principle. In addition, they derived some properties based on the nabla distributed-order operators and provided a simpler criterion to determine the stability of such systems.

In [7], on the basis of extending the dimension of the nonlinear singular fractional-order systems with mismatched uncertainties, Chen et al. constructed a new integral sliding mode surface, which ensures the stability of sliding mode motion by using linear matrix inequality. In addition, the control law based on an adaptive mechanism that is used to update the nonlinear terms is designed, which can deal with the nonlinear terms to ensure the singular fractional-order systems satisfy the reaching conditions.

The paper in [8] concerned a fractional modeling and prediction method directly oriented toward an industrial time series with obvious non-Gaussian features, which extracted the hidden long-range dependence and the multifractal properties to determine the fractional order. Additionally, the authors of this article proposed a fractional autoregressive integrated moving average model considering innovations with stable infinite variance and discussed in depth the existence and convergence of the model solutions, which enabled the use of ensemble learning with an autoregressive moving average model to further improve upon accuracy and generalization.

The article in [9] constructed novel co-designed sampled-data controllers with only the “newest” or “oldest” state information available for the sampled-data stabilization of a fractional continuous linear system under arbitrary sampling periods based on the compensation of scaling gains. Cao et al. first presented the sufficient and necessary conditions for global asymptotic stability in the discrete-time domain with the help of fractional difference approximation. 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 system, which is significantly preferred for digital implementation.

In this paper [10], the author’s designed a fractional-order adaptive terminal sliding mode control scheme for the finite-time synchronization of uncertain fractional-order memristive neural networks with leakage and discrete delays, displaying the impacts of uncertain parameters as well as external disturbances, which can effectively estimate the upper bounds of unknown external disturbances. Furthermore, by simulating the finite-time synchronization of the master–slave fractional-order memristive neural networks, the corresponding synchronization criteria and the explicit expression of the settling time were obtained.

The study in [11] proposed a hydroelectric unit fault diagnosis model based on improved multiscale fractional-order weighted permutation entropy in order to improve the noise immunity, stability, and sensitivity to different signal types. Additionally, a new time-series feature-extraction method was proposed for the incomprehensiveness of the single scale and the instability of the traditional coarse-granulation method, and they proposed a new fault diagnosis method for hydroelectric units based on noise resistance and feature extraction ability combined with a classifier.

In [12], the author’s approximated the unknown nonlinear functions from the adaptive neural fault-tolerant control for the fractional-order nonlinear systems with positive odd rational powers by using the radial basis function neural networks. In particular, this paper studied the fractional-order nonlinear systems subject to high-order terms for the first time and designed the controller to ensure the boundedness of all the signals of the closed-loop control system so that the tracking error can tend to a small neighborhood of zero in the end.

In [13], the author’s employed an event-triggered mechanism and backstepping control technique while also using command-filtered adaptive fuzzy control with a disturbance observer to construct an event-triggered control strategy for the synchronization of fractional-order chaotic systems, such that all closed-loop signals were bounded and chaos synchronization was achieved.Under the framework of adaptive fuzzy backstepping recursive design, fuzzy logical systems, disturbance observers, and a tracking differentiator were proposed to estimate, respectively, the unknown parametric uncertainties, external disturbances, and the drawback of the explosion of complexity in traditional backstepping.

The authors in paper [14] provided a unified framework for the admissibility of a class of singular fractional-order systems with a given fractional order in the interval (0, 2) without separating the ranges into (0, 1) and [1, 2). The authors of this paper derived the necessary and sufficient conditions in terms of linear matrix inequalities to avoid any singularity problem in the solution, and proved the equivalence between the quadratic admissible system and the general quadratic stable system.

In [15], the authors proposed a novel analytical approach and established a novel finite-time stability lemma for a class of inertial neural networks with mixed-state time-varying delays, which was entirely different from the existing finite-time stability theorems. Moreover, an improved discontinuous reliable control mechanism was developed, which is more valid and widens the application scope compared with previous results, and novel sufficient criteria were established using finite-time stability theorems to estimate the settling time with respect to a finite-time stabilization of inertial neural networks by using a novel non-reduced-order approach and the Lyapunov functional theory.

The authors in [16] drew the fractional-order memristive ant colony algorithm, which uses the fracmemristor physical system to record the probabilistic transfer of information of the nodes that the ant will crawl through in the future, passing 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, it can be discovered that FMAC is superior to PACO-3opt, the best integer-order ant colony algorithm currently available. Additionally, due to the transfer probability prediction module based on the physical fracmemristor system, the fractional-order memristive ant colony algorithm operates substantially more quickly than the fractional-order memristor ant colony algorithm.

In [17], the authors proposed the fractional category representation vector, which can be considered as a distributional representation when negative probability is considered, and it can be used either as a regularization method or as a distributed category representation. In image classification, the linear combinations of the fractional category representation vectors correspond to the mixture of images and can be used as an independent variable of the loss function, and it can also be used for space sampling, with fewer dimensions and less computational overhead than normal distributions.

In [18], the authors proposed a new image enhancement algorithm, ensuring an image has clear edges, rich texture details, and retains the information of the smooth area. Moreover, it used the combination of rough set and particle swarm optimization algorithms 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 was used to enhance the image.

In [19], an image enhancement algorithm based on a rough set and fractional-order differentiator was proposed. By combining the rough set theory with a Gaussian mixture model, a new image segmentation algorithm with higher immunity was obtained, which can obtain more image layers with concentrated information, preserving more image details than traditional algorithms.

According to [20], based on fractional-derivative and data-driven regularization terms, a novel dehazing model was proposed to improve dehazing quality from the texture and edge information of an image. Moreover, a dual-stream network based on a convolutional neural network and transformer was introduced to structure the data-driven regularization, and an atmospheric light model based on the fractional derivative and the atmospheric veil was proposed to estimate the atmospheric light. The model proposed in this paper, which surpassed the state-of-the-art methods for most synthetic and real-world images, quantitatively, qualitatively, and effectively avoided over-enhancement and noise amplification, and extracted the local and non-local features of an image.

In [21], an adaptive medical image fractional-order total variational denoising model with an improved sparrow search algorithm, which combined the characteristics of fractional-order differential operators and total variational models, was proposed, which not only achieved the adaptivity of the fractional-order total variable differential order but also effectively removed noise, preserved the texture structure of the image to the maximum extent, and improved the peak signal-to-noise ratio of the image. By using 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 order of the fractional-order differential operator was adaptively determined.

According to [22], due to images being affected by strong noise and uneven intensity, the traditional active contour models often cannot obtain accurate results; therefore, a novel adaptive fractional differential active contour image segmentation method was proposed. This paper constructed an adaptive fractional-order matrix to extract further image texture, and also designed a new fractional-order edge-stopping function to improve noise resistance and introduced a fitting term based on adaptive fractional differentiation by reason of 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.

The work in [23] developed a novel dental identification scheme by utilizing a fractional wavelet feature extraction technique and rule mining with an a priori procedure to deal with the problem of dental image identification, extract the most discriminating image features during the mining process to obtain strong association rules, and enhance the a priori-based dental identification system, which aims to address the drawbacks of dental rule mining. To achieve feature extraction, a wavelet transform based on a k-symbol fractional Haar filter was used, and the a priori algorithm of AR mining was applied to find the frequent patterns in dental images.

The paper in [24] applied Schauder’s fixed-point theorem and developed an iterative algorithm based on the discrete Fourier transform method in the frequency domain to construct a telegraph-diffusion equation-based model to effectively preserve fine structures and edges for texture images, and the fractional-order derivative was imposed due to its textural detail-enhancing capabilities. Furthermore, the gray-level indicator was introduced, 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.