Fractional-Order Complex Systems: Advanced Control, Intelligent Estimation and Reinforcement Learning Image-Processing Algorithms

In this Special Issue on “Applications of Fractional Operators in Image Processing and Stability of Control Systems”, more than 20 high-quality papers have been published. An escalating body of research across various scientific and engineering disciplines has concentrated on dynamical systems, which are distinguished by the interplay between artificial intelligence theory and fractional differential equations. As a result, a multitude of computational fractional intelligent systems have emerged, each accompanied by a stability analysis and potential applications in image processing.

This current Special Issue builds upon and extends the work presented in the preceding edition, with the aim of compiling articles that reflect the most recent advancements in applied mathematics and advanced intelligent control engineering. The contributions encompass interdisciplinary topics such as control theory, fractional calculus, and their applications in image processing, in addition to practical implications within the realm of engineering science. This collection not only underscores the depth of research in these areas, but also signifies the broadening scope and theoretical underpinnings of fractional intelligent systems.

This Special Issue compiles cutting-edge research in applied mathematics and advanced intelligent control engineering, focusing on interdisciplinary topics such as control theory, fractional calculus, and image processing. The articles highlight the significant attention that fractional calculus has received, particularly its applications in control systems and image processing. Fractional-order systems extend classical integer-order models, providing a more accurate description of real-world physical phenomena. In image processing, critical techniques like noise suppression and image fusion in medical imaging are essential for clinical diagnosis and treatment. The increasing diversity of image acquisition models has further emphasized the importance of these techniques. Recent advancements have seen fractional operators play a pivotal role in image processing, serving as powerful tools for noise reduction and feature enhancement. Additionally, new fractional operating tools have been developed, enhancing the analysis and design of nonlinear control systems. Singular systems, characterized by singular differential equations, exhibit unique properties distinct from classical systems. Methodologies for fractional-order control systems, inspired by integer-order approaches, are gaining traction within the control community due to their enhanced capabilities. Overall, this Special Issue showcases the latest developments in fractional calculus and its transformative impact on multiple disciplines, setting the stage for future innovations in applied mathematics and engineering.

Seventeen high-quality papers were accepted for publication in this Special Issue. The papers were written by different authors, showcasing the wide scope of the Special Issue. The published papers will be briefly summarized.

In the work of [1], a unified framework based on the linear matrix inequality region was proposed to analyze the stability and tolerability of fractional-order systems and singular fractional-order systems in the range of 0 < $\alpha$ < 2, and to solve the related stability analysis and controller design problems, making the analysis and design process simpler and more unified.

The study in [2] proposed a fractional differentiation-based depth image enhancement method, which solved the problems of blurring, aliasing, and other issues that were prone to occur during the processing of depth images. The paper enhanced the textural details and clarity of the images using fractional differential operations, effectively improving the quality of the depth images and providing a new and effective means for processing and analyzing depth images.

The study in [3] proposed a hierarchical security control framework for non-linear cyber-physical systems under denial-of-service attacks. Using fractional-order calculus theory and incorporating system state feedback and attack information, the method ensured system stability and security, effectively mitigating the disruptions and damage caused by DoS attacks. The framework allowed for the precise control and adjustment of system states, enhancing the system’s resistance to interference and providing new insights and methodologies to ensure the security of cyber-physical systems.

In [4], a novel admissibility criteria was proposed for descriptor fractional-order systems (DFOSs) with the order of (0,2) without separating the order into (0,1) and [1,2). The methods included the use of alternate admissibility criteria based on non-strict and strict linear matrix inequalities, with a focus on reducing the decision variables. These criteria ensured the stability and admissibility of DFOSs, enriching the theoretical research and allowing for its wider application, particularly in handling systems with uncertain derivative matrices.

In [5], a performance control approach is proposed for lower-triangular systems with unknown fractional powers and uncertainties. This method uses barrier functions to limit tracking errors and remove previous restrictions, enhancing its applicability. The control law ensures accurate and fast tracking without the need for function approximation, parameter identification, or additional techniques. Overall, this paper contributed a robust and flexible solution to the tracking control problem for lower-triangular systems with unknown fractional powers and uncertainties.

The article in [6] introduced a backstepping control strategy for unmanned surface vehicle trajectory tracking, incorporating a fractional-order finite-time command filter to estimate the derivatives of intermediate controls and reduce chatter, as well as a fractional-order finite-time disturbance observer to approximate and compensate for model uncertainties and external disturbances. The proposed method ensured the global asymptotic stability of the closed-loop system.

The study in [7] proposed an adaptive fractional multi-scale optimization optical flow algorithm that balances global features and local textures to mitigate over-smoothing in total variation models. By constructing a fractional-order discrete L1-regularization total variational optical flow model and utilizing the ant lion algorithm for iterative calculation, the algorithm dynamically adjusted the fractional order to ensure convergence and efficiency. This enhanced the flexibility of optical flow estimations using weak gradient textures and significantly improved the multi-scale feature extraction rates.

The paper in [8] focused on crop and weed segmentation in heterogeneous data environments for smart farming. By minimizing pixel variability and using just one training sample, it achieved real-world applicability. It also integrated fractal dimension estimation for distributional characteristics, pioneering the use of heterogeneous data to address the reductions in accuracy following segmentation.

In [9], the authors present numerical simulations of anti-symmetric matrices in the stability criteria for fractional-order systems. They then studied the admissibility criteria for descriptor fractional-order systems with orders of (0,2), focusing on systems without boundary axis eigenvalues. A unified admissibility criterion using minimal linear matrix inequalities was also provided.

The authors in paper [10] proposed a T-S fuzzy modeling approach for fuzzy fractional-order, singular perturbation, and multi-agent systems of the orders (0,2). A fuzzy observer-based controller was designed to achieve consensus and decompose the error system into fuzzy singular fractional-order systems. Consensus conditions were derived using linear matrix inequalities without equality constraints, addressing issues of uncertainty and nonlinearity. The effectiveness of the approach was verified through an RLC circuit model and numerical example.

The study in [11] applied a fractional-order system to model an industrial process with inertia and a large time delay, extending the traditional integer-order model. An output-error identification algorithm was used to determine the fractional-order model’s parameters. Compared to integer-order models, the fractional-order model provided a better fit for selective catalytic reduction denitrification process data. A PI controller was designed based on both models, and validation tests showed the fractional-order model’s advantages in such industrial processes.

In [12], a rough body segmentation-based gender recognition network was proposed to address the challenges regarding the low recognition performance obtained in long-distance gender recognition using only IR cameras. The method emphasizes the silhouette of a person through a body segmentation network, integrating anthropometric loss, and uses an adaptive body attention module to effectively combine segmentation and classification. Fractal dimension estimation was introduced to analyze the complexity and irregularity of the body region, enhancing the framework’s analytic capabilities.

The authors in paper [13] presented a novel approach to multi-focus image fusion by integrating fractal dimension and coupled neural P (CNP) systems within a nonsubsampled contourlet transform framework. Addressing the limitations of camera lenses and depth-of-field effects, this method used a local topology-based CNP model to merge low-frequency components and a spatial frequency and fractal dimension-based focus measure to merge high-frequency components. This multi-focus image-fusion method significantly improved image clarity throughout the entire scene.

The study in [14] addressed the limitations of limited camera viewing angles in image-based plant classification by introducing a method that incorporates both shallow segmentation and classification networks. The proposed shallow plant segmentation network used adversarial learning with a discriminator network, while the shallow plant classification network applied residual connections. Additionally, fractal dimension estimation was utilized to analyze the segmentation results, aiming to improve the overall classification performance.

The work in [15] addressed the emerging threat that generator adversarial networks (GANs) pose to finger-vein recognition systems by developing a new, densely updated, contrastive learning-based self-attention GAN to create elaborate fake finger-vein images for training spoof detectors. Additionally, an enhanced ConvNeXt-Small model with a large kernel attention module was proposed as a new spoof detector. To improve spoof detection performance, fractal dimension estimation was introduced to analyze the complexity and irregularity of class activation maps from real and fake finger-vein images, facilitating the generation of more realistic and sophisticated fake finger-vein images.

The authors of [16] proposed an maximum apple diameter estimation model for smart agriculture and precision agriculture using RGB-D camera fusion depth information. The model was developed by collecting and statistically analyzing the diameter of a Red Fuji apple, obtaining depth and two-dimensional size information of apple images using an Intel RealSense D435 RGB-D camera and LabelImg software, and exploring the relationship between these variables and the maximum apple diameter. Multiple regression analyses and nonlinear surface fitting were used to construct the model. The proposed model can provide a theoretical basis and technical support for selective apple-picking operations using intelligent robots based on apple size grading.

The paper in [17] examined the leader-following $H_{\infty }$ consensus of fractional-order multi-agent systems (FOMASs) under conditions of input saturation using output feedback. The study provided sufficient conditions for a $H_{\infty }$ consensus for FOMASs with $\alpha$ in the ranges (0,1) and [1,2). It used iterative linear matrix inequalities to solve the quadratic matrix inequalities and transformed the input saturation issue into optimal solutions of linear matrix inequalities to estimate stable regions. The approach avoided disassembling the entire multi-agent system and reduced conservatism.