Keywords: variable order, variable order fractional control, fractional operator, controller analog realization

Keywords: Fractional order control, Fractional systems, stability theory, optimal control

Keywords: Automatic Control & Stability, System Analysis & Dynamics, Mechatronics
Abstract: The control of uncertain, nonlinear systems remains a significant and persistent challenge in modern engineering. This paper introduces a comprehensive Fractional-Order Radial Basis Function Model Reference Adaptive Control (FO-RBF-MRAC) system, uniting the strengths of MRAC, RBF networks, and fractional calculus into a single, robust architecture. The theoretical contribution is two-fold: a novel, robust fractional-order adaptation law is proposed, and the Uniformly Ultimately Bounded (UUB) stability of the complete, integrated system is rigorously proven using a detailed fractional-order Lyapunov analysis. Crucially, the controller’s efficacy is validated through a comparative simulation study on an inverted pendulum. The proposed FO-RBF-MRAC is benchmarked against its integer-order equivalent and a state-of-the-art, well-tuned Model Predictive Controller (MPC). The results demonstrate that the proposed controller achieves performance comparable to the MPC under ideal nominal conditions, but exhibits vastly superior tracking accuracy and stability when faced with significant parametric uncertainties and external noise.

Keywords: Artificial Intelligence, History of Fractional-Order Calculus, System identification & Modeling
Abstract: This paper proposes a Whale Optimization Algorithm (WOA) optimized fractional-order BP (back propagation) neural network with momentum term for rolling bearing fault diagnosis. The Caputo fractional-order gradient descent incorporating momentum term enhances weight/bias updating in BP networks. WOA adaptively configures critical parameters (fractional order, learning rate, hidden neurons) to optimize network performance. Experimental validation on CWRU (Case West Reserve University) datasets demonstrates superior accuracy over VMD-SVM (variational mode decomposition support vector machines) and other decomposition/deep learning methods, confirming significant robustness improvements across multiple metrics.

Keywords: Artificial Intelligence, Signal Processing, Mathematical methods
Abstract: Machine unlearning is essential for compliance with privacy regulations such as General Data Protection Regulation (GDPR), enabling selective removal of user data from trained models while preserving utility in sensitive applications like healthcare and finance. However, existing methods fail to account for the inherent complexity of loss landscapes and the varying stability of network layers during unlearning, leading to unstable updates, catastrophic forgetting, and poor robustness under large-scale or adversarial deletions. We introduce roughness-informed machine unlearning, a unified framework that explicitly models these challenges through geometric and statistical roughness analysis. Geometric Roughness-Informed Machine Unlearning (GRIMU) stabilizes unlearning by regularizing updates to smooth rugged loss landscapes using the Roughness Index, while Statistical Roughness-Informed Machine Unlearning (SRIMU) prioritizes adjustments in well-trained, spectrally stable layers via heavy-tailed distribution analysis. Empirical evaluations on MNIST, CIFAR-10, CIFAR-100, and UCI Adult demonstrate that GRIMU and SRIMU outperform state-of-the-art baselines including AGU, ORTR, SISA, SCRUB, AmnesiacML, SalUn, and Boundary Unlearning in test accuracy retention, privacy leakage, and KL-divergence across diverse deletion strategies (random, class-specific, and adversarial).

Keywords: Artificial Intelligence, System Analysis & Dynamics, Signal Processing
Abstract: Many have attempted using fractional calculus (FC) in optimization and machine learning in the past several years. This paper offers a holistic view on how to introduce fractional calculus in gradient descent (GD) optimization methods. We identified four different entry points to add FC ideas: 1) The updating law that could be a fractional order integrator or, in general, a fractional order system; 2) The GD gain coefficient (also known as the learning coefficient) that could be a time-varying signal as an output from a fractional order system driven by an impulse; 3) The gradient that, of course, could be a fractional order gradient based on fractional order partial derivatives; 4) The noise (additive or multiplicative) introduced artificially to the gradient that could also be fractional order stochastic processes. Therefore, in combinations, we have a total of 15 ways to apply fractional calculus in gradient descent optimization methods. We surveyed the existing sample research attempts and categorized them in one of the 15 ways in a big table. It is interesting to show what are those empty spots left on the table hinting rich future research opportunities.

Keywords: Automatic Control & Stability, Artificial Intelligence, Electrical Engineering & Electromagnetism
Abstract: Power electronic systems often suffer from unmodeled dynamics, parameter variations, and nonlinearities, which can affect closed-loop stability and transient performance. This paper proposes a novel metaheuristic approach—the PID-Based Search Algorithm (PSA)—to robustly design feedback controllers, with a focus on DC-DC buck converters. A fractional-order PI–PD (FOPI–FOPD) cascade controller is introduced, leveraging fractional calculus to extend conventional control actions via non-integer orders λ and μ, thus enhancing design flexibility and robustness. PSA simultaneously tunes both gains and fractional orders by minimizing the Integral of Time-weighted Absolute Error (ITAE), optimizing both transient and steady-state behavior. Time-domain simulations demonstrate that the PSA-optimized FOPI–FOPD controller outperforms its integer-order counterpart in terms of settling time, overshoot, and disturbance rejection, confirming its correctness and effectiveness in nonlinear converter applications.

Keywords: Mathematical methods, Physics, Special Functions
Abstract: The g-subdiffusion equation is a significant generalization of the ordinary fractional subdiffusion equation. Its main purpose is to describe diffusion processes occurring in media whose structure or properties evolve over time. This advanced modeling capability is achieved by incorporating the fractional time derivative of Caputo with respect to another function. We show an application of a subdiffusion equation with a Caputo fractional time derivative with respect to another function g to describe subdiffusion in a medium having a structure evolving over time. In this case, a continuous transition from subdiffusion to another type of diffusion may occur. The process can be interpreted as ordinary subdiffusion in which the time scale is changed by the function g. We will present several examples of complex diffusion processes in which, by choosing the appropriate function g, we will obtain a smooth transition between different types of anomalous diffusion. We will also show a method for solving this type of equation using the generalized Laplace transform.

Keywords: Mathematical methods, Physics, Special Functions
Abstract: Subdiffusion occurs in systems in which random walk of molecules is very hindered. This process is described by the subdiffusion equation with the Caputo or the Riemann-Liouville fractional time derivative. In some processes molecules can be suddenly eliminated from further subdiffusion. We consider subdiffusion with two elimination processes: when molecules become permanently immobilized or when they disappear due to their decay. The first process is described by the subdiffusion equation with a modified Riemann-Liouville time derivative. The second one is described by both the ordinary subdiffusion equation and the molecule decay equation with the fractional Caputo time derivative with respect to another function. The aim is to compare both models, in particular to derive a formula allowing to assess which of the considered processes is more effective in interrupting subdiffusion of molecules in the long time limit.

Keywords: Mathematical methods, Physics, Singularities Analysis and Integral Representations
Abstract: This paper proposes an efficient numerical algorithm for the time-space fractional diffusion equation with temporal Caputo-Hadamard derivative and spatial fractional Laplacian in two dimensions. For the time discretization, the spectral deferred correction (SDC) method is utilized. In the mean while, the fractional centered finite difference formula is adopted to approximate the two-dimensional fractional Laplacian. Numerical example is presented to illustrate the propose scheme.

Keywords: Mathematical methods, Physics, Special Functions
Abstract: In this article, we employ the invariant subspace method to systematically and analytically solve the psi-Hilfer time-fractional PDEs (tFPDEs). More precisely, this work explains how the exact solutions of linear and nonlinear psi-Hilfer tFPDEs can be obtained {with the help of the} invariant solution spaces of homogeneous linear ordinary differential equations (HLODEs). Also, different kinds of exact solutions are explicitly computed for the psi-Hilfer tFPDEs. In addition, we illustrate the dependence of the derived exact solutions on the function psi(t), the arbitrary-order parameter alpha and the type beta through graphical demonstrations.

Keywords: Mathematical methods, Physics, Special Functions
Abstract: The paper presents how separable methods can be effectively useful to find separable analytical solutions for time-delay linear and nonlinear Psi-Hilfer time-fractional convection-reaction-diffusion-wave equations (TFCRDWEs). More precisely, we derive the additive and multiplicative separable analytical solutions for the initial and the Dirichlet boundary value problems of the time-delay linear and nonlinear Psi-Hilfer TFCRDWEs. Additionally, we compare the derived analytical solutions and the results obtained in our work with those in the existing studies wherever possible.

Keywords: Others
Abstract: This work proposes a new scheme for stabilizing fractional-order systems (FOSs) in which we offer an auxiliary function such that the fractional derivative for the Lyapunov function candidate could be positive or negative, rather than the conventional negative definiteness condition required for stability. Such a foundation is employed to introduce a simple control mechanism that stabilizes the FOS at its zero equilibrium point and provides global asymptotic stability. To show the applicability of our method, we provide a numerical example supported by simulations of the Newton-Leipnik chaotic system, proving its ability to address complex fractional-order chaotic systems through a simple controller.

Keywords: Mathematical methods, Automatic Control & Stability, Others
Abstract: This work introduces a novel class of weighted fractional operators constructed through the composition of differential and integral operators. In particular, we propose the operator qDμ x, which generalizes classical fractional derivatives while maintaining essential properties such as linearity. Although the semigroup property and the Leibniz rule do not hold in their traditional forms, we derive analogous formulations by combining the proposed operator with the Riemann–Liouville derivative. Furthermore, a numerical representation based on the Gr¨unwald–Letnikov method is developed, enabling efficient discretization and simulation of the weighted operator in cases where analytical solutions are intractable. The approach also considers the interplay between Laplace transforms and convolutions, which is crucial for real-world applications in control and signal processing.

Keywords: Mathematical methods, Mechanics & Viscoelasticity
Abstract: This work presents a theoretical and numerical investigation of viscoelastic fluid flow between parallel plates subjected to step strain rate deformation. The study examines the transient stress response following a sudden, nonlinear change in deformation rate that is subsequently maintained at constant value. The governing equations form a system of integro- differential equations incorporating the Caputo derivative to model the fluid’s viscoelastic behaviour. The system is solved sequentially, with the first equation determining the velocity field and the second computing the stress field. Numerical solution of these equations presents significant challenges due to the singular kernel in the integral terms and the rapid timescales of the imposed deformation, occurring within milliseconds and approximated by a logistic-type function. A finite difference scheme is developed to address these computational difficulties, and numerical results show the method’s robustness and stability when modelling complex fluids under demanding initial conditions. The proposed approach provides an effective framework for simulating transient viscoelastic flows in confined geometries subject to abrupt deformations.

Keywords: Mathematical methods, Singularities Analysis and Integral Representations, History of Fractional-Order Calculus
Abstract: By choosing appropriate norms, a fractional integral operator can be taken as a contraction mapping on the Dimovski spaces which lie between the spaces of continuous and integrable functions on a compact interval. This fact can be used to prove existence-uniqueness results for many fractional differential equations, and also to construct explicit solutions in the form of locally uniformly convergent series. Detailed proofs are given here for Riemann–Liouville integrals, but the same methodology can and should be used for many other fractional integral operators.

Keywords: Automatic Control & Stability, Mathematical methods, System Analysis & Dynamics
Abstract: A fractional system is often represented by a transfer with a numerator and a denominator that are polynomials of non-integer orders. As for the usual systems of integer order, the zeros and the poles of a transfer are respectively defined as the zeros of its numerator and denominator. We study here the number of zeros of polynomials of non-integer orders. In the case of polynomials of integer order, the number of zeros only depends on the degree of the polynomial. In the case of polynomials of no-integer order, we show that the number of zeros depends on the higher order, and of the number of terms of the polynomial that is considered. This result is simple corollary from a theorem stated by Bellman and Cook in 1963, on the zeros of exponential polynomials. We illustrate and discuss the result through examples.

Keywords: System identification & Modeling, Signal Processing
Abstract: This paper is a literature review of the stability, resonance, and frequency response identification of fractional order transfer functions of second species, without zeros.

Keywords: System identification & Modeling, Automatic Control & Stability
Abstract: Fractional order models are useful to model physical phenomena, such as batteries, fluid motion relaxation, heat transfers) and even biomedical scenarios. Most of the proposed modelings are expressed in an explicit mathematical forms, meaning that the fractional differentiation is directly defined as an exponent power of the Laplace variable in the Laplace domain. Early fractional order models such as the Cole-Cole model for dielectric relaxation belonged effectively to this type of models. However, there are also implicit transfer function models such as the Davidson-Cole ones, which prove to be useful and more accurate. One main drawback of implicit models is its difficulty related to time-domain simulation. The present study considers a hybrid transfer function model which includes both explicit and implicit terms. Frequency-domain analysis as well as time-domain simulation will be explored. Finally a practical aspect of system identification with a such explicit and implicit fractional model will be used by using time-domain data.

Keywords: System identification & Modeling, Automatic Control & Stability, System Analysis & Dynamics
Abstract: Recent advances in fractional calculus and computation have enabled the development of more accurate and flexible models for industrial process dynamics. Among these, the Fractional First-Order Plus Dead-Time (FFOPDT) and Fractional Dual-Pole Plus Dead-Time (FDPPDT) models have shown notable performance in representing systems with overdamped step responses. This work introduces a unified analytical identification procedure for both models, derived from the process reaction curve obtained through a simple open-loop step test. The proposed methodology is validated through numerical simulations, and the results demonstrate that it achieves comparable or superior performance to existing methods, with the added benefits of analytical simplicity and computational efficiency, making it suitable for industrial applications.

Keywords: Mechanics & Viscoelasticity, Physics, System Analysis & Dynamics

Keywords: Automatic Control & Stability, Robotics
Abstract: In this work, the fractional-order active disturbance rejection control (FO-ADRC) strategy is applied to achieve a 2D navigation of a spherical microrobot inside a blood vessel. The control objective is to drive the microrobot from a known initial position to a desired target along a predefined trajectory. The microrobot dynamics are decomposed into two subsystems, each corresponding to motion along one axis, with each subsystem subject to external disturbances and coupling effects from the other axis. For each subsystem, an FO-ADRC scheme is designed, consisting of a fractional-order extended state observer (FO-ESO) and a proportional control law. The observer estimates both the internal states and the generalized disturbances, allowing for their effective compensation by the designed control law. Simulation results are presented to demonstrate the capability of the FO-ADRC in ensuring accurate trajectory tracking and effectively rejecting external disturbances and parameter uncertainties.

Keywords: Automatic Control & Stability, Robotics, System Analysis & Dynamics
Abstract: To enhance the control accuracy of servo systems, this paper proposes a novel control strategy based on a dual-inertia servo model, incorporating a fractional-order approach to improve modeling precision and system performance. First, a sliding mode observer (SMO) is constructed to estimate the system’s state variables. Then, a compensation method is designed to transform the system into a triple-integrator form. Finally, a cascade controller is developed to suppress disturbances and mitigate vibrations. The effectiveness of the proposed strategy is validated through simulation results, which illustrate its superior ability to reduce system disturbances and vibrations.

Keywords: Robotics, Automatic Control & Stability, System Analysis & Dynamics
Abstract: This paper presents a novel Fractional-Order Model Reference Adaptive Control (FO-MRAC) strategy for the adaptive feedback linearization of a nonlinear Delta parallel robot. By leveraging fractional calculus, the proposed approach enhances both robustness and adaptability in the presence of model uncertainties. The fractional-order framework enables a more accurate representation of the robot's dynamics by capturing memory effects and viscoelastic behavior phenomena typically neglected in integer-order models. The adaptive control mechanism continuously adjusts the controller parameters in real time, ensuring precise trajectory tracking without requiring exact knowledge of the system model.

Keywords: Signal Processing, Automatic Control & Stability, Electrical Engineering & Electromagnetism
Abstract: Global Positioning System (GPS) technology has become essential for modern communication and location tracking, raising signicant security concerns. To protect sensitive GPS data such as coordinates and timestamps from interception, robust encryption before transmission is paramount. Through this paper, we propose the design and implementation of a secure GPS data transmission system resorting to fractional-order chaotic dynamics to efficiently cipher data. A fractional-order Hénon-type map is used to generate complex and unpredictable sequences. The encrypted locations are transmitted wirelessly and successfully decrypted at the receiver level by applying a particular synchronization technique between fractional-order chaotic maps ensuring that the response system's geometric patterns closely match those of the driving system. Both simulation and practical results con rm the effectiveness of the proposed synchronization strategy in ensuring secure transmission of collected GPS data and their full recovery at the receiving side.

Keywords: Image Processing, Signal Processing, Robotics
Abstract: This paper presents a deep learning framework that integrates fractional-order differentiation for ultrasound image denoising over an extended range of derivative orders. Building on the Charef approximation~cite{charef2006fractional}, the method reformulates parameter computation to extend fractional gradients beyond nu > 1.0, addressing singularities in classical formulations designed for lower orders. Multi-scale directional gradient features are incorporated into a modified U-Net with a gradient consistency loss to preserve structural integrity across scales. The reformulation ensures numerical stability and enables end-to-end trainable fractional operators spanning smoothing to edge enhancement. Experiments on clinical ultrasound datasets show up to 2.3~dB PSNR and 4.7% SSIM improvements over state-of-the-art methods while preserving sharp anatomical boundaries essential for diagnostic interpretation.

Keywords: Image Processing, Mathematical methods, Signal Processing
Abstract: Accurate remote sensing of methane (CH_{4}) plumes is critical for climate mitigation, but traditional matched filtering algorithms are often ineffective. Their assumption of Gaussian background noise conflicts with the heavy-tailed, non-Gaussian nature of real-world hyperspectral data, causing detection inaccuracies. To address this, we introduce a fractional-order matched filter (FOMF) that uses fractional calculus to adapt to heavy-tailed distributions and improve sensitivity. Our proposed Fractional-Order Matched Filter (FOMF), tested on AVIRIS-ng data, successfully identified methane plumes where its standard integer-order counterpart failed completely. Quantitative analysis confirmed the fractional approach's superiority, with the FOMF achieving a Receiver Operating Characteristic (ROC) Area Under the Curve (AUC) of 0.72, outperforming both the ISTA (0.68) and NMW-FOMF (0.71) algorithms. While a benchmark Mag1c algorithm performed best, our findings validate that integrating a fractional order transforms this matched filtering approach into a robust and effective tool for gas plume detection in challenging, non-Gaussian environments.

Keywords: Automatic Control & Stability, Mathematical methods, Signal Processing
Abstract: This study addresses the challenge of preserving minimum-phase behavior in fractional-order systems (FOS) of complex orders, which are vital for control and signal processing applications involving memory effects and phase asymmetry. Traditional approximation techniques often compromise phase integrity, leading to instability or poor invertibility. Building upon El-Khazali’s first-order rational approximation, we propose a frequency-domain strategy to ensure minimum-phase realization using a Routh-Hurwitz-based safe-band verification. The method identifies analytically valid regions in the (α, β) parameter space where all approximated transfer functions maintain minimum-phase behavior across desired frequency bands. Simulation results confirm the accuracy of the method and reveal key design trends via test function maps and frequency responses. An interactive MATLAB App was developed to support real-time visualization and parameter exploration. The proposed framework provides a low-order, stable, and analytically tractable solution for minimum-phase approximation of complex-order Laplace operators.

Keywords: Automatic Control & Stability, System Analysis & Dynamics, Signal Processing
Abstract: A convolution-based time-domain approach suitable for simulation of linear, stationary systems described by irrational transfer functions is developed in this paper. A method for representing a wide class of locally integrable signals by sample vectors is introduced first. Then, a convolution algebra is developed, supporting deconvolution, convolution inverse, integer convolution powers and square roots. In addition, a method for evaluating convolution exponentials and logarithms has also been discussed. The discussion has been supported by several numerical examples.

Keywords: System Analysis & Dynamics, System identification & Modeling
Abstract: This paper presents a new methodology for identifying fractional-order systems with unknown initial conditions. The proposed approach is based on the Infinite State Representation framework, which is employed to model and estimate the frequency-distributed state of the fractional system using a Luenberger observer. A combined initialization and identification procedure enables Least Squares parameter estimation for both linear and nonlinear systems. Simple illustrative examples demonstrate the effectiveness of the proposed two-stage identification procedure, characterized by fast convergence and a significant reduction in parameter bias.

Keywords: System identification & Modeling, Automatic Control & Stability
Abstract: Continuous-time system identification with fractional model has increased in interest due to its relevance in modeling physical processes with memory and diffusion effects. This paper deals with continuous-time system identification of multiple-input single-output (MISO) fractional differentiation models. Recent studies, which have extended classic methods (e.g., OE and SRIVC) to the estimation of fractional-order models, the definition of structured-commensurability has been introduced to better cope with the estimation of fractional differentiation orders, and notably to improve the algorithm convergence. This paper investigates the convergence properties of the MISO-OOSRIVCF algorithm. After a brief method review and a SISO illustration, a detailed MISO case study is presented with convergence analysis and performance metrics. The impact of initial conditions, noise, and tuning parameters are assessed and practical strategies are then proposed for algorithm configuration for reliable system identification.

Keywords: System identification & Modeling, Signal Processing
Abstract: This paper shows how a second-species commensurate fractional order transfer function can be identified from a step response. Only step responses with overshoot or with oscillations (or both) are considered. All that the methods discussed require is knowing the time and amplitude of only two points of the step response.

Keywords: System identification & Modeling, History of Fractional-Order Calculus, Automatic Control & Stability
Abstract: This paper develops a parameter identification of fractional Wiener models based on the output error approach and the Key-term separation principle. Although the linear part is a fractional linear system. To solve the difficulty of the unmeasurable inner variables in the information vector, the auxiliary model idea is combined with the robust Levenberg-Marquardt algorithm. The efficiency of the proposed method is tested using numerical simulations.

Keywords: System Analysis & Dynamics, System identification & Modeling
Abstract: Fractional-order dynamic models may allow to capture complex dynamics, as they may be interpreted as the coexistence of extremely slow and fast time constants(Oustaloup (1995) and Trigeassou and Maamri (2025)). Different synthesis methods exist for time-domain simulation of fractional-order systems. One of the most famous methods is the Grünwald-Letnikov method, which relies on a discrete-time approximation of the fractional derivative. This method may be seen as a generalization fo the well-known backward Euler method in numerical simulation. Even though a notable advantage of backward Euler's method for linear system simulation is its unconditional stability for stable system simulation, special care may be required for unstable system simulation. This study is a short note generalizing an already well-known result concerning unstable system simulation for the case of fractional-order systems. A simple mathematical proof as well as some simulation examples will be provided.