Fractional Order Integral Research Articles

Modeling the Dynamics of Supercapacitors by Means of Riemann–Liouville Integral Definition

The application of fractional calculus to obtain dynamic models for supercapacitors represents an alternative approach to obtaining simpler and more accurate models. This paper presents a model for the supercapacitor in the time domain, based on the use of the fractional or non-integer order integral. This fractional model is compared with the conventional simple model, which is typically used in industrial applications. This fractional integral-based model provides satisfactory fits in relation to the number of parameters used in the model. Furthermore, an interpretation of the effect of the application of fractional integration is presented for constant current charging and discharging processes at constant current, using the Riemann–Liouville definition for the non-integer order integral. Supercapacitors are devices that exhibit non-linear behavior, with a distinct charging and discharging operation. There are several methods of dynamic analysis for the characterization of supercapacitors. The information extracted from these methods is essential for understanding the behavior of supercapacitors and, thus, ensuring that processes involving supercapacitors are as efficient as possible. This paper presents a dynamic analysis based on charge and discharge operations with constant currents. The conclusion is that the fractional model provides fairly accurate fits.

Effect of moisture in flax fiber on viscoelastic properties of the manufactured flax fiber reinforced polymer by fractional-order viscoelastic model

Flax Fiber Reinforced Polymer (FFRP) composites, known for their eco-friendliness and mechanical strength, find extensive application in construction. This study investigates the influence of moisture content in flax fiber fabrics on FFRP's viscoelastic properties. Isothermal frequency sweep tests were conducted to assess dynamic mechanical properties under varying moisture levels. The Huet-Sayegh viscoelastic model, with fractional order integration, was employed for analysis. Results indicate optimal viscoelastic properties when fabric humidity matches manufacturing conditions. However, the correlation between the loss factor and the humidity is not straightforward. In summary, FFRP's capacity to store and dissipate energy diminishes with the increased moisture content, but the reduction does not follow a specific pattern with humidity levels.

On the Boundary Functional of a Semi-Markov Process

In this paper, we consider the semi-Markov random walk process with negative drift, positive jumps. An integral equation for the Laplace transform of the conditional distribution of the boundary functional is obtained. In this work, we define the residence time of the system by generalized exponential distributions with different parameters via fractional order integral equation. The purpose of this paper is to reduce an integral equation for the Laplace transform of the conditional distribution of a boundary functional of the semi-Markov random walk processes to fractional order differential equation with constant coefficients.

FLRNN-FGA: Fractional-Order Lipschitz Recurrent Neural Network with Frequency-Domain Gated Attention Mechanism for Time Series Forecasting

Time series forecasting has played an important role in different industries, including economics, energy, weather, and healthcare. RNN-based methods have shown promising potential due to their strong ability to model the interaction of time and variables. However, they are prone to gradient issues like gradient explosion and vanishing gradients. And the prediction accuracy is not high. To address the above issues, this paper proposes a Fractional-order Lipschitz Recurrent Neural Network with a Frequency-domain Gated Attention mechanism (FLRNN-FGA). There are three major components: the Fractional-order Lipschitz Recurrent Neural Network (FLRNN), frequency module, and gated attention mechanism. In the FLRNN, fractional-order integration is employed to describe the dynamic systems accurately. It can capture long-term dependencies and improve prediction accuracy. Lipschitz weight matrices are applied to alleviate the gradient issues. In the frequency module, temporal data are transformed into the frequency domain by Fourier transform. Frequency domain processing can reduce the computational complexity of the model. In the gated attention mechanism, the gated structure can regulate attention information transmission to reduce the number of model parameters. Extensive experimental results on five real-world benchmark datasets demonstrate the effectiveness of FLRNN-FGA compared with the state-of-the-art methods.

Adaptive tracking control of underactuated AUV with historical navigation information and piecewise weighted fractional order integration

In this study, a piecewise adaptive weighted fractional order integration (PAWFOI) is introduced in cascade-backstepping control, which improves the effectiveness of autonomous underwater vehicles (AUVs) in suppressing navigation errors during long-duration operations. Additionally, PAWFOI significantly reduces the computational complexity compared to conventional fractional order integration, ensuring a consistent computational speed. The adaptive weight selector dynamically adjusts the weights based on the AUV state information, achieving enhanced tracking performance. Based on Lyapunov’s theory, the results demonstrated that AUV tracking control can be achieved by selecting appropriate control parameters. Finally, an example is presented to verify the correctness and validity of the theoretical results.

Solvability of Quadratic Integral Equations of Urysohn Type Involving Hadamard Variable-Order Operator

This study investigates the existence of solutions to integral equations in the form of quadratic Urysohn type with Hadamard fractional variable order integral operator. Due to the lack of semigroup properties in variable-order fractional integrals, it becomes challenging to get the existence and uniqueness of corresponding integral equations, hence the problem is examined by employing the concepts of piecewise constant functions and generalized intervals to address this issue. In this context, the problem is reformulated as integral equations with constant orders to obtain the main results. Both the Schauder and Banach fixed point theorems are employed to prove the uniqueness results. In addition, an illustration is included in order to verify those results in the final step.

The Generalized Fox–Wright Function: The Laplace Transform, the Erdélyi–Kober Fractional Integral and Its Role in Fractional Calculus

In this paper, we consider and study in detail the generalized Fox–Wright function Ψ˜qp introduced in our recent work as an extension of the Fox–Wright function Ψqp. This special function can be seen as an important case of the so-called I-functions of Rathie and H¯-functions of Inayat-Hussain, that in turn extend the Fox H-functions and appear to include some Feynman integrals in statistical physics, in polylogarithms, in Riemann Zeta-type functions and in other important mathematical functions. Depending on the parameters, Ψ˜qp is an entire function or is analytic in an open disc with a final radius. We derive its basic properties, such as its order and type, and its images under the Laplace transform and under classical fractional-order integrals. Particular cases of Ψ˜qp are specified, including the Mittag-Leffler and Le Roy-type functions and their multi-index analogues and many other special functions of Fractional Calculus. The corresponding results are illustrated. Finally, we emphasize the role of these new generalized hypergeometric functions as eigenfunctions of operators of new Fractional Calculus with specific I-functions as singular kernels. This paper can be considered as a natural supplement to our previous surveys “Going Next after ‘A Guide to Special Functions in Fractional Calculus’: A Discussion Survey”, and “A Guide to Special Functions of Fractional Calculus”, published recently in this journal.

Fractional Neutral Integro-Differential Equations with Nonlocal Initial Conditions

We primarily investigate the existence of solutions for fractional neutral integro-differential equations with nonlocal initial conditions, which are crucial for understanding natural phenomena. Taking into account factors such as neutral type, fractional-order integrals, and fractional-order derivatives, we employ probability density functions, Laplace transforms, and resolvent operators to formulate a well-defined concept of a mild solution for the specified equation. Following this, by using fixed-point theorems, we establish the existence of mild solutions under more relaxed conditions.

Augmenting the Stability of Automatic Voltage Regulators through Sophisticated Fractional-Order Controllers

The transition from traditional to renewable energy sources is a critical issue in current energy-generation systems, which aims to address climate change and the increased demand for energy. This shift, however, imposes additional burdens on control systems to maintain power system stability and quality within predefined limits. Addressing these challenges, this paper proposes an innovative Modified Hybrid Fractional-Order (MHFO) automatic voltage regulator (AVR) equipped with a fractional-order tilt integral and proportional derivative with a filter plus a second-order derivative with a filter FOTI-PDND2N2 controller. This advanced controller combines the benefits of a (FOTI) controller, known for enhancing dynamic performance and steady-state response, with a (PDND2N2) controller to improve system robustness and adaptability. The proposed MHFO controller stands out with its nine tunable parameters, providing more extensive control options than the conventional three-parameter PID controller and the five-parameter FOPID controller. Furthermore, a recent optimization approach using a growth optimizer (GO) has been formulated and applied to optimally adjust the MHFO controller’s parameters simultaneously. The performance of the proposed AVR based on the MHFO-GO controller is scrutinized by contrasting it with various established and developed optimization algorithms. The comparative study shows that the AVR based on the MHFO-GO controller surpasses other AVR controllers from the stability, robustness, and dynamic response speed points of view.

Compact active analog device for novel applications useful for sensing and measurement

The presented research introduces a novel approach to modern modular active device and provides various practical application examples tailored for industrial sensing readouts. Straightforward implementation of single active device-based topologies in frequency bands from kHz up to MHz is presented. These applications include: an electronically linearly tunable special band-pass filter that can be easily modified into a voltage-controlled oscillator, a two-port quadratic transformer, an amplitude modulator, and an equalization circuit (which serves as a fractional-order differentiator and integrator). These circuits benefit from high-impedance voltage input and low-impedance output (easily matched to 50 Ω), high processed signal levels (often reaching hundreds of mV), and simplified topologies without redundant passive elements. These features are especially valuable in tunable and configurable solutions for modern communication, sensing, audio, and consumer electronic systems, as well as in modeling various physical and biological systems. The experiments have been provided by simulations and measurements using device VCA824.

Robust Fractional-Order PI/PD Controllers for a Cascade Control Structure of Servo Systems

In this paper, a cascade control structure is suggested to control servo systems that normally include a servo motor in coupling with two kinds of mechanism elements, a translational or rotational movement. These kinds of systems have high demands for performance in terms of fastest response and no overshoot/oscillation to a ramp function input. The fractional-order proportional integral (FOPI) and proportional derivative (FOPD) controllers are addressed to deal with those control problems due to their flexibility in tuning rules and robustness. The tuning rules are designed in the frequency domain based on the concept of the direct synthesis method and also ensure the robust stability of controlled systems by using the maximum sensitivity function. The M-Δ structure, using multiplicative output uncertainties for both control loops simultaneously, is addressed to justify the robustness of the controlled systems. Simulation studies are considered for two kinds of plants that prove the effectiveness of the proposed method, with good tracking of the ramp function input under the effects of the disturbances. In addition, the robustness of the controlled system is illustrated by a structured singular value (µ) plot in which its value is less than 1 over the frequency range.

Discrete version of fundamental theorems of fractional order integration for nabla operator

Discrete version of fundamental theorems of fractional order integration for nabla operator

Design of fractional MOIF and MOPIF controller using PSO algorithm for the stabilization of an inverted pendulum‐cart system

AbstractThe topic of this paper is the design of two fractional order schemes, based on a state feedback for linear integer order system. In the first one of the state feedback is associated with a fractional order integral () controller. In the second structure the state feedback is associated with a fractional order proportional integral () controller. With such controllers, the closed loop system with state feedback described by the state equations splits in n‐subsystems with different fractional orders derivatives of the state variable. In order to find the optimal parameters value of both controllers () and (), a multi‐objective particle swarm optimization algorithm is used, with the integral of absolute error, the overshoot , the Buslowicz stability criterion are considered as objective functions. The multi‐objective integral fractional order controller and the multi‐objective proportional integral fractional order controller are applied to stabilize the inverted pendulum‐cart system (IP‐C), and their performance is compared to the fractional order controller. The simulation results of these innovative controllers are also compared with those obtained by conventional proportional–integral–derivative and fractional order proportional–integral–derivative controllers. The robustness of the proposed controllers against disturbances is investigated through simulation runs, considering the non‐linear model of the IP‐C system. The obtained results demonstrate that our approach not only leads to high effectiveness but also showcases remarkable robustness, supported by both simulation and experimental results.

INITIAL-BOUNDARY VALUE PROBLEMS TO THE TIME-NONLOCAL DIFFUSION EQUATION

This article investigates a fractional diffusion equation involving Caputo fractional derivative and Riemann-Liouville fractional integral. The equation is supplemented by initial and boundary conditions in the domain defined by the interval by space 0<x<1 and interval by time 0<t<T. The fractional operators are defined rigorously, utilizing the Caputo fractional derivative of order β and the Riemann-Liouville fractional integral of order α, where 0<α<β≤1. The main results include the presentation of well-known properties associated with fractional operators and the establishment of the unique solution to the given problem. The key findings are summarized through a theorem that provides the explicit form of the solution. The solution is expressed as a series involving the two-parameter Mittag-Leffler function and orthonormal eigenfunctions of the Sturm-Liouville operator. The uniqueness of the solution is proven, ensuring that the problem has a single, well-defined solution under specific conditions on the initial function. Furthermore, the article introduces and proves estimates related to the Mittag-Leffler function, providing bounds crucial for the convergence analysis. The convergence of the series is investigated, and conditions for the solution to belong to a specific function space are established. The uniqueness of the solution is demonstrated, emphasizing its singularity within the given problem. Finally, the continuity of the solution in the specified domain is confirmed through the uniform convergence of the series.

Study of fractional variable-order lymphatic filariasis infection model

Abstract Variable-order derivatives are the natural extension of ordinary as well as of fractional-order differentiations and integration, respectively. Numerous suggestions for fractional variable-order operators have been made in the literature over time. Therefore, this is the moment to shine a light on the variable-order fractional calculus, due to the fact that it accurately describes the mathematical underpinnings and emphasizing the modeling utility via using contemporary numerical techniques. This study focuses on investigating a fractional variable-order model of lymphatic filariasis infection using with Atangana–Beleanue–Caputo derivative. Our investigations have led to the development of newly refined results, focusing on both qualitative and numerical aspects of analysis. To achieve our research objectives, we employ the fixed point theorems of Banach and Krasnoselskii. These theorems serve as powerful tools, allowing us to establish results regarding the existence of solutions to the model. Additionally, for precise numerical simulations, we employ the fractional Euler’s method, a sophisticated computational technique that allows us to effectively simulate and interpret the results both numerically and graphically. These graphs illustrate distinct variable-orders, providing a comprehensive understanding of the model’s behavior under different conditions. Here, it should be kept in mind that we have select various continuous functions for variable to present our graphical illustration.

A Comparative performance evaluation of a complex-order PI controller for DC–DC converters

Switched-mode DC–DC converters are required to maintain constant output under uncertainties and variations in input voltage and load. The controller’s robustness is crucial in such systems, and hence various controllers have been proposed in the past few decades for DC–DC converter control. Fractional order PID controllers have attracted the attention of researchers due to their robustness and flexibility in control of power converters. Such controllers use fractional orders of integration and differentiation. Complex order controllers are the generalized form of the fractional order PID controllers and have complex orders of integration and differentiation. These controllers are very robust in the control of nonlinear systems with time-varying parameters. But complex order controllers have been very sparingly used in power electronic control. This paper proposes a complex order PI controller with a complex order integrator for controlling DC–DC buck and boost converters. The complex PID controller has four parameters to be tuned. The complex order PI controller is designed by optimization using the metaheuristic Cohort Intelligence algorithm. The results are compared with that of a fractional-order PID controller. It was observed that the complex PI controller gave a better response than the FOPID controller and was more robust to parameter variations.

Hamilton‐Jacobi‐Bellman equation based on fractional random impulses system

AbstractIn this talk, we consider the optimal control problem for fractional order random impulses differential equations (), and there is no corresponding Hamilton‐Jacobi‐Bellman (HJB) equation proposed for fractional order differential equations in existing literature. The lack of semi group properties in fractional order makes the original dynamic programming methods unable to directly handle fractional order problems. We have dealt with this problem by combining the properties of fractional order integrals. In solving this problem, we found that the order of the original system in HJB equation is at least 1. When the order is less than 1, our approach to fractional order ideas provides a possibility to solve the problem.

Adaptive cruise control system with fractional order ANFIS PD+I controller: optimization and validation

Designing the control structures of fractional order PID controllers has proven to be effective in providing adaptability in set point tracing the performance of a nonlinear cruise control system. Wheel rolling resistance, wind drag force, and road gradient are incorporated into the design to better describe the system under consideration and to show how the nonlinear cruise control system behaves. This study presents a comparative investigation using simulation between control structures such as fractional order proportional–integral–derivative, fractional order integral minus proportional derivative, and fractional order proportional integral minus derivative. By preserving integral error indices as the goal function, a genetic algorithm is used to improve the controller gain parameters and fractional scaling values. To prevent integral windup conflicts and derivative boost issues, both traditional fractional order structures and adaptive neuro-fuzzy-based fractional order structures were used to create the adaptive cruise control system. The FO ANFIS PD plus I controller for the cruise control system exceeds the competition in servo and regulatory difficulties.

Advanced golden jackal optimization-based fractional-order integral terminal sliding-mode (FO-ITSM) controller for level control of the conical tank system

This article examines a conical tank nonlinear system's level control. A crucial component of facilities like milk processing plants and sugarcane mills is the conical tank. The fluid drains entirely due to the conical shape of the tank. The conical form of the system causes nonlinear dynamics. The conical tank nonlinear system transfer mechanism is not easily accessible, and different operating heights require different transfer mechanisms, making controller design difficult. Developing the Fractional-Order Integral Terminal Sliding-Mode (FO-ITSM) Controller system for Advanced Golden Jackal Optimisation (GJO) based conical tank systems is the main objective of this paper. This is utilised to deliver real-time liquid level management in non-linear conical systems. The dynamics of these models make it possible to identify the conical tank system that produces control signals from liquid samples taken at reference levels with greater accuracy. This system, which is employed on multiple platforms, maintains and controls the liquid level while meeting all design specifications, including time constant, no overshoot, less settling, and rise time. For research simulation work, MATLAB software's FOMCON toolbox is utilised.

Smith predictor based fractional order controller design for improved performance and robustness of unstable FOPTD processes

Abstract Performance and robustness are essential characteristics for the application of unstable time-delayed systems. As tasks become more complex, traditional control methods cannot meet such demands for performance and robustness. The present work aims to develop fractional order-based controllers for enhanced Smith predictor-based unstable first-order plus time-delayed systems (FOPTD) with improved performance and robustness. In the current work, fractional order controllers using a Genetic Algorithm (GA) are designed with enhanced SP (Smith Predictor) structure to control unstable first-order time-delayed processes to improve performance. Furthermore, in the feedback path a fractional order (FO) filter is used to further improve robustness and performance. A systematic methodology is proposed for obtaining the optimum fractional order filter parameters based on the minimization of Integral Absolute Error (IAE). The recommended approach is beneficial to balance the necessary tradeoff between performance and robustness. Also, the proposed method provides flexibility in tuning the degree of freedom by adding a fractional order integrator, thus leading to robust performance. The efficacy of the recommended controller is analyzed by simulating numerical examples from the literature. The proposed controller provides enhanced performance and robustness compared to the literature.