Search Results (158)

In recent years, fractional-order controllers have garnered increasing attention due to their enhanced flexibility and superior dynamic performance in control system design. Among them, the fractional-order Proportional–Integral–Derivative (FOPID) controller offers two additional tunable parameters, $\lambda$ and $\mu$, expanding the design space and allowing for finer performance tuning. However, the increased parameter dimensionality poses significant challenges for optimisation. To address this, the present study investigates the application of FOPID controllers to a two-wheeled self-balancing robot (TWSBR), with controller parameters optimised using intelligent algorithms. A novel Multi-Strategy Improved Beluga Whale Optimization (MSBWO) algorithm is proposed, integrating chaotic mapping, elite pooling, adaptive Lévy flight, and a golden sine search mechanism to enhance global convergence and local search capability. Comparative experiments are conducted using several widely known algorithms to evaluate performance. Results demonstrate that the FOPID controller optimised via the proposed MSBWO algorithm significantly outperforms both traditional PID and FOPID controllers tuned by other optimisation strategies, achieving faster convergence, reduced overshoot, and improved stability. Full article

A distributed weighted fusion fractional order filter is proposed for multi-sensor multi-delay fractional order systems. Firstly, the time-delay system is transformed into a non-time-delay system using the state augmentation method, and the optimal augmented fractional Kalman filter is derived. Secondly, in order to reduce the computational burden, a suboptimal fractional order Kalman filter is presented. Compared with the optimal augmented method, it greatly reduces the computational complexity, which is convenient for real-time applications. Then, in order to derive the weighting coefficient for distributed fusion, the calculation formula of filtering error variance matrix between any two sensor subsystems is derived. Finally, the distributed weighted fusion fractional order filter is presented. It is local optimal and globally suboptimal: compared with each local filter, it has higher accuracy; compared with the centralized fusion filter, it has lower accuracy and more fault tolerance. In summary, it is more suitable for practical application. Simulation results verify the effectiveness of the proposed algorithm. Full article

This paper considers the multi-term subdiffusion equation with weakly singular solutions. In order to use sparser meshes near the initial time, a novel scheme (which we call the corrected L1 scheme) on graded meshes is constructed to estimate the multi-term Caputo fractional derivative by investigating a corrected term for the nonuniform L1 scheme. Combining this nonuniform corrected L1 scheme in the temporal direction and the finite element method (FEM) in the spatial direction, a fully discrete scheme for solving the multi-term subdiffusion equation is developed. The stability result of the developed scheme is given. Furthermore, the optimal pointwise-in-time error estimate of the developed scheme is derived. Finally, several numerical experiments are conducted to verify our theoretical findings. Full article

With the rapid proliferation of real-time digital communication, particularly in multimedia applications, securing transmitted image data has become a vital concern. While chaotic systems have shown strong potential for cryptographic use, most existing approaches rely on low-dimensional, integer-order architectures, limiting their complexity and resistance to attacks. Advances in fractional calculus and memristive technologies offer new avenues for enhancing security through more complex and tunable dynamics. However, the practical deployment of high-dimensional fractional-order memristive chaotic systems in hardware remains underexplored. This study addresses this gap by presenting a secure image transmission system implemented on a field-programmable gate array (FPGA) using a universal high-dimensional memristive chaotic topology with arbitrary-order dynamics. The design leverages four- and five-dimensional hyperchaotic oscillators, analyzed through bifurcation diagrams and Lyapunov exponents. To enable efficient hardware realization, the chaotic dynamics are approximated using the explicit fractional-order Runge–Kutta (EFORK) method with the Caputo fractional derivative, implemented in VHDL. Deployed on the Xilinx Artix-7 AC701 platform, synchronized master–slave chaotic generators drive a multi-stage stream cipher. This encryption process supports both RGB and grayscale images. Evaluation shows strong cryptographic properties: correlation of $-6.1081\times {10}^{-5}$, entropy of 7.9991, NPCR of 99.9776%, UACI of 33.4154%, and a key space of $2^{1344}$, confirming high security and robustness. Full article

This study introduces a novel model reference adaptive control (MRAC) framework that incorporates fractional-order gradients (FGs) to regulate the displacement of an inverted pendulum-cart system. Fractional-order gradients have been shown to significantly improve convergence rates in domains such as machine learning and neural network optimization. Nevertheless, their integration with fractional-order error models within adaptive control paradigms remains unexplored and represents a promising avenue for research. The proposed control scheme extends the classical MRAC architecture by embedding Caputo fractional derivatives into the adaptive law governing parameter updates, thereby improving both convergence dynamics and control flexibility. To ensure optimal performance across multiple criteria, the controller parameters are systematically tuned using a multi-objective Particle Swarm Optimization (PSO) algorithm. Two fractional-order error models (FOEMs) incorporating fractional gradients (FOEM2-FG, FOEM3-FG) are investigated, with their stability formally analyzed via Lyapunov-based methods under conditions of sufficient excitation. Validation is conducted through both simulation and real-time experimentation on a physical pendulum-cart setup. The results demonstrate that the proposed fractional-order MRAC (FOMRAC) outperforms conventional MRAC, proportional-integral-derivative (PID), and fractional-order PID (FOPID) controllers. Specifically, FOMRAC-FG achieved superior tracking performance, attaining the lowest Integral of Squared Error (ISE) of $2.32\times {10}^{-5}$ and the lowest Integral of Squared Input (ISI) of 6.40 in simulation studies. In real-time experiments, FOMRAC-FG maintained the lowest ISE ($5.11\times {10}^{-6}$). Under real-time experiments with disturbances, it still achieved the lowest ISE ($1.06\times {10}^{-5}$), highlighting its practical effectiveness. Full article

Integer-order models cannot characterize the dynamic behavior of the flexible two-link manipulator (FTLM) system accurately due to its viscoelastic characteristics and flexible oscillation. Hence, this paper proposes a fractional-order modeling method and identification algorithm for the FTLM system. Firstly, we exploit the memory and history-dependent properties of fractional calculus to describe the flexible link’s viscoelastic potential energy and viscous friction. Secondly, we establish a fractional-order differential equation for the flexible link based on the fractional-order Euler–Lagrange equation to characterize the flexible oscillation process accurately. Accordingly, we derive the fractional-order model of the FTLM system by analyzing the motor–link coupling as well as the symmetry of the system structure. Additionally, a system identification algorithm based on the multi-innovation integration operational matrix (MIOM) is proposed. The multi-innovation technique is combined with the least-squares algorithm to solve the operational matrix and achieve accurate system identification. Finally, experiments based on actual data are conducted to verify the effectiveness of the proposed modeling method and identification algorithm. The results show that the MIOM algorithm can improve system identification accuracy and that the fractional-order model can describe the dynamic behavior of the FTLM system more accurately than the integer-order model. Full article

The issue of multi-switch generalized projective anti-synchronization of fractional-order chaotic systems is investigated in this work. The model is constructed using Caputo–Fabrizio derivatives, which have been rarely addressed in previous research. In order to expand the symmetric and asymmetric synchronization modes of chaotic systems, we consider modeling chaotic systems under such fractional calculus definitions. Firstly, a new fractional-order differential inequality is proven, which facilitates the rapid confirmation of a suitable Lyapunov function. Secondly, an effective multi-switching controller is designed to confirm the convergence of the error system within a short moment to achieve synchronization asymptotically. Simultaneously, a multi-switching parameter adaptive principle is developed to appraise the uncertain parameters in the system. Finally, two simulation examples are presented to affirm the correctness and superiority of the introduced approach. It can be said that the symmetric properties of Caputo–Fabrizio fractional derivative are making outstanding contributions to the research on chaos synchronization. Full article

In this research, we investigate the phenomenon of multistability and complex dynamic behaviors in plasma waves by utilizing advanced mathematical techniques. We examine how fractional-order derivatives influence plasma wave stability by applying the fractional diffusion–reaction model, the framework of nonlinear dynamical systems, and the $\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)$ method. The principal direction of our work is associated with different forms of oscillations in the plasma wave: non-linear periodic, solitons, and kink waves. This leads to the study of small amplitude pulses and solitary waves, which are significant in plasma activities. Using bifurcation analysis, we discuss how these waves appear and develop under different conditions, as well as determine which conditions generate the chaotic behavior or highly complex patterns of waves. We study the details of transitions between waves and their chaotic behavior to characterize the laws that govern their plasma environment. Moreover, we have used non-linear modeling and numerical simulations to understand in detail the complex patterns and the factors of stability underlying the phenomena of plasma waves. In addition, our study also investigates the correspondence between non-linearity, multi-stability, and the birth of complex structures such as solitons and kink waves. The solutions of the dynamical system produced by the proposed nonlinear model generate different patterns of response based on system parameter variation. These patterns include oscillations and decay behaviors. Research results about system stability and solution convergence under various parameter settings provide an extended performance evaluation of the proposed method through a better understanding of system dynamics. They increase our understanding of chaotic behavior in plasma systems and pave the way for applications in plasma physics and energy systems, as well as advanced technologies. Full article

In this survey, we have included the recent results on Lyapunov-type inequalities for differential equations of fractional order associated with Dirichlet, nonlocal, multi-point, anti-periodic, and discrete boundary conditions. Our results involve a variety of fractional derivatives such as Riemann–Liouville, Caputo, Hilfer–Hadamard, $\psi$-Riemann–Liouville, Atangana–Baleanu, tempered, half-linear, and discrete fractional derivatives. Full article

This paper investigates the bifurcation dynamics and optical soliton solutions of the integrable quintic Kundu–Eckhaus (QKE) equation with an M-fractional derivative. By adding quintic nonlinearity and higher-order dispersion, this model expands on the nonlinear Schrödinger equation, which makes it especially applicable in explaining the propagation of high-power optical waves in fiber optics. To comprehend the behavior of the connected dynamical system, we categorize its equilibrium points, determine and analyze its Hamiltonian structure, and look at phase diagrams. Moreover, integrating along periodic trajectories yields soliton solutions. We achieve this by using the simplest equation approach and the modified extended Tanh method, which allow for a thorough investigation of soliton structures in the fractional QKE model. The model provides useful implications for reducing internet traffic congestion by including fractional temporal dynamics, which enables directed flow control to avoid bottlenecks. Periodic breather waves, bright and dark kinky periodic waves, periodic lump solitons, brilliant-dark double periodic waves, and multi-kink-shaped waves are among the several soliton solutions that are revealed by the analysis. The establishment of crucial parameter restrictions for soliton existence further demonstrates the usefulness of these solutions in optimizing optical communication systems. The theoretical results are confirmed by numerical simulations, highlighting their importance for practical uses. Full article

Although fractional calculus has evolved significantly since its origin in the 1695 correspondence between Leibniz and L’Hôpital, the numerical treatment of multi-order fractional differential equations remains a challenge. Existing methods are often either computationally expensive or reliant on complex operational frameworks that hinder their broader applicability. In the present study, a novel numerical algorithm is proposed based on orthogonal hybrid functions (HFs), which were constructed as linear combinations of piecewise constant sample-and-hold functions and piecewise linear triangular functions. These functions, belonging to the class of degree-1 orthogonal polynomials, were employed to obtain the numerical solution of multi-order fractional differential equations defined in the Caputo sense. A generalized one-shot operational matrix was derived to explicitly express the Riemann–Liouville fractional integral of HFs in terms of the HFs themselves. This allowed the original multi-order fractional differential equation to be transformed directly into a system of algebraic equations, thereby simplifying the solution process. The developed algorithm was then applied to a range of benchmark problems, including both linear and nonlinear multi-order FDEs with constant and variable coefficients. Numerical comparisons with well-established methods in the literature revealed that the proposed approach not only achieved higher accuracy but also significantly reduced computational effort, demonstrating its potential as a reliable and efficient numerical tool for fractional-order modeling. Full article

Base excitation sources significantly impact vehicle-body vibrations in maglev systems, with the dynamic performance of the suspension system playing a crucial role in mitigating these effects. The second-series suspension system of a maglev vehicle typically employs an air spring, which has a great impact on the stability of maglev vehicle operation. Considering that the suspension system has certain dynamic characteristics under the foundation excitation, the present study proposes the fractional-order nonlinear Nishimura model to describe the memory-restoring force characteristics of the air spring. The fractional-order derivative term is made equivalent to a term in the form of trigonometric function, the steady-state response of the system is solved by the harmonic balance method, and the results are compared with a variety of other methods. The influence of the foundation excitation source on the dynamic behavior of the vibration isolation system is discussed significantly. The variation law of the jump phenomenon and the diversity of periodic motion of the multi-value amplitude curve are summarized. The numerical simulation also revealed the presence of multi-periodic motion in the system when variations occurred in the gap of the suspension system. Combined with the cell mapping algorithm, the distribution law of different attractors on the attraction domain of periodic motion is discussed, and the rule of the transition of periodic motion stability with different fundamental excitation amplitudes is summarized with the Lyapunov exponent. Full article

This article investigates non-fragile synchronization control and circuit implementation for incommensurate fractional-order (IFO) chaotic neural networks with parameter uncertainties. In this paper, we explore three aspects of the research challenges, i.e., theoretical limitations of uncertain IFO systems, the fragility of the synchronization controller, and the lack of circuit implementation. First, we establish an IFO chaotic neural network model incorporating parametric uncertainties, extending beyond conventional commensurate-order architectures. Second, a novel, non-fragile state-error feedback controller is designed. Through the formulation of FO Lyapunov functions and the application of inequality scaling techniques, sufficient conditions for asymptotic synchronization of master–slave systems are rigorously derived via the multi-order fractional comparison principle. Third, an analog circuit implementation scheme utilizing FO impedance units is developed to experimentally validate synchronization efficacy and accurately replicate the system’s dynamic behavior. Numerical simulations and circuit experiments substantiate the theoretical findings, demonstrating both robustness against parameter perturbations and the feasibility of circuit realization. Full article

The current research study presents a comprehensive analysis of the local discontinuous Galerkin (LDG) method for solving multi-order fractional differential equations (FDEs), with an emphasis on circuit modeling applications. We investigated the existence, uniqueness, and numerical stability of LDG-based discretized formulation, leveraging the Liouville–Caputo fractional derivative and upwind numerical fluxes to discretize governing equations while preserving stability. The method was validated through benchmark test cases, including comparisons with analytical solutions and established numerical techniques (e.g., Gegenbauer wavelets and Dickson collocation). The results demonstrate that the LDG method achieves high-accuracy solutions (e.g., with a relatively large time step size) and reduced computational costs, which are attributed to its element-wise formulation. These findings position LDG as a promising tool for complex scientific and engineering applications, particularly in modeling fractional-order systems such as RL, RLC circuits, and other electrical circuit equations. Full article

This paper is dedicated to investigating a highly accurate numerical solution for a class of 2D nonlinear time-dependent partial integro-differential equations with multi-term fractional integral items. These integrals are weakly singular with respect to time, which are handled using the product integration rule on graded meshes to compensate for the influence generated by the initial weak singular nature of the exact solution. The temporal derivative is approximated by a generalized Crank–Nicolson difference scheme, while the nonlinear term is approximated by a linearized method. Furthermore, the stability and convergence of the derived time semi-discretization scheme are strictly proved by revising the finite discrete parameters. Meanwhile, the differential matrices of the spatial high-order derivatives based on barycentric rational interpolation are utilized to obtain the fully discrete scheme. Finally, the effectiveness and reliability of the proposed method are validated by means of several numerical experiments. Full article