Fractional Order Dynamic Model Research Articles

Research on anti-vibration tension control based on self-coupling fractional-order PID

Constant tension control is essential for excellent winding quality. However, the system’s nonlinearity and external disturbance make it challenging to guarantee tension control accuracy with conventional control methods. Thus, a self-coupling fractional-order proportional–integral–derivative (SC-FOPID) control scheme combined with a disturbance observer is proposed to enhance the system’s anti-vibration performance. The fractional-order dynamic model of the unwinding roller and swing rod is established by analyzing the tension mechanism. Based on deliberate analysis and calculation, the vibration shock signal can be decomposed into periodic sinusoidal disturbance and bounded noise approximately. As such, an output-based anti-vibration method using a fractional-order model can be realized, where a back recursive disturbance observer is designed to estimate the periodic component. Simultaneously, the bounded noise exhibited in vibration can be attenuated by the SC-FOPID controller. The stability is guaranteed using the Lyapunov theorem, and the simulation results show the proposed method’s effectiveness in improving the tension control performance.

Fractional order model of MRSA bacterial infection with real data fitting: Computational Analysis and Modeling

Bacterial infections in the health-care sector and social environments have been linked to the Methicillin-Resistant Staphylococcus aureus (MRSA) infection, a type of bacteria that has remained an international health risk since the 1960s. From mild colonization to a deadly invasive disease with an elevated mortality rate, the illness can present in many different forms. A fractional-order dynamic model of MRSA infection developed using real data for computational and modeling analysis on the north side of Cyprus is presented in this paper. Initially, we tested that the suggested model had a positively invariant region, bounded solutions, and uniqueness for the biological feasibility of the model. We study the equilibria of the model and assess the expression for the most significant threshold parameter, called the basic reproduction number (ℛ0). The reproductive number’s parameters are also subjected to sensitivity analysis through mathematical methods and simulations. Additionally, utilizing the power law kernel and the fixed-point approach, the existence, uniqueness, and generalized Ulam–Hyers–Rassias stability are presented. Chaos Control was used to regulate the linear responses approach to bring the system to stabilize according to its points of equilibrium, taking into account a fractional-order system with a managed design where solutions are bound in the feasible domain. Finally, numerical simulations demonstrating the effects of different parameters on MRSA infection are used to investigate the impact of the fractional operator on the generalized form of the power law kernel through a two-step Newton polynomial method. The impact of fractional orders is emphasized in the study so that the numerical solutions support the importance of these orders on MRSA infection. With the application of fractional order, the significance of cognizant antibiotic usage for MRSA infection is verified.

Fractional Order Mathematical Modelling of HFMD Transmission via Caputo Derivative

This paper studies a nonlinear fractional mathematical model for hand, foot, and mouth Disease (HFMD), incorporating a vaccinated compartment. Our initial focus involves establishing the non-negativity and boundedness of the fractional order dynamical model. The existence and uniqueness of the system are discussed using the Caputo derivative operator formulation. Applying a fixed-point approach, we obtain results that confirm the presence of at least one solution. We analyze the stability behavior at the two equilibrium points (disease-free and endemic states) of the model and derive the basic reproduction number. Numerical simulations are conducted using the fractional Euler approach, and the simulation results confirm our analytical conclusions. This comprehensive approach enhances the understanding of HFMD dynamics and facilitates the policy making of health care centers to control the further spread of this disease.

A Comparative Study of Fractional-Order Models for Supercapacitors in Electric Vehicles

Fractional-order models display many advantages compared with integer-order equivalent circuit models for modeling energy storage system, such as more precision, fewer parameters. This paper carries out a comparative study of fractional-order equivalent circuit models for modeling supercapacitor. These models are fractional-order RC model, fractional-order classic model, fractional-order RRCW model, fractional-order dynamic model, fractional-order transmission line model. An experiment load platform is built to test the supercapacitor for collecting the feature data. To be fair, the parameters of all models are optimized to achieve their best fitting accuracy by using the genetic algorithm. Accuracy and complexity two indices are considered to evaluate the above five models based on different test cycles. Simulation results show that the fractional-order RRCW model has the ideally comprehensive performance.

A fractional-order dynamical model of the SEIVRS epidemic with temporary immunity and transport-related infection

We propose an epidemic model as a system of FDEs to describe transmission amongst different locations that affect the epidemiological spread and the vaccination to control the disease. Our primary target is to focus on the necessity of the lockdown period, sometimes seen as interfering with private life in mobility, and the need for vaccines to minimize the transmission in the interior and external stages. We prove that controlling social contact was necessary during the pandemic spread to stop transmissions between different locations (travel, periodic lockdowns, social distance,…) and that re-vaccinations were needed to avoid reinfections.Using Routh Hurwitz's theory, we examin the local stability of the equilibrium points—the disease-free and coexisting. Employing the Lyapunov function, our research demonstrate global asymptotic stability towards the equilibrium points. At the end of the study, we provide some numerical simulations to corroborate our theoretical findings.

Dynamical modeling and physical analysis of pipe flow in hydraulic systems based on fractional variational theory

Based on the fractional variational theory and Hamilton principle, this paper extends a semi-analytic solution and further develops a fractional-order dynamical model of pipe flow. This model facilitates the dynamical coupling of hydraulic and mechanical systems, and ensures easily have the solution for both the physical characteristics by state space equation or the damped vibration characteristics by fractional Bagley–Torvik equation. Subsequent application in an actual hydropower system reveals the potential periodic vibration of pipe flow and the effect of fractional order memory on the dashpot time constant. The comparison of these findings with the smaller alignment index (SALI) detection results reaffirms the soundness of this study.

Synchronization of the Circadian Rhythms with Memory: A Simple Fractional-Order Dynamical Model Based on Tow Coupled Oscillators

Disruptions of the circadian rhythm are associated with internal desynchronization. It affects some internal functions of our body and behavior that are important to our health. Our modern lifestyle has contributed to millions of people developing some circadian rhythm disruptions, making the subject very important clinically as well as economically. Motivated by studying a simple mathematical model that can reveal some features of internal synchronization desynchronization, in this contribution, we extend the coupling oscillator phase model proposed by Strogatz [14] in the sense that memory is considered in the modeling. Such memory is a result of the introduction of Caputo-type fractional derivatives in the coupling oscillators’ phase model dynamics, resulting in a fractional phase model. We show that the proposed fractional coupling oscillator phase model is well-posed. Furthermore, we analyzethe synchronization phenomena. We obtain the synchronized solutions explicitly when the memory is equally distributed between the oscillators. In contrast, when distinct levels of memory are imposed in the modeling, we obtain lower and upper bounds because any existing synchronized solution must be confined in between. We present numerical realizations that support the theoretical findings in great detail.

Predicting solutions of the stochastic fractional order dynamical system using machine learning

The solution of fractional-order systems has been a complex problem for our research. Traditional methods like the predictor-corrector method and other solution steps are complicated and cumbersome to derive, which makes it more difficult for our solution efficiency. The development of machine learning and nonlinear dynamics has provided us with new ideas to solve some complex problems. Therefore, this study considers how to improve the accuracy and efficiency of the solution based on traditional methods. Finally, we propose an efficient and accurate nonlinear auto-regressive neural network for the fractional order dynamic system prediction model (FODS-NAR). First, we demonstrate by example that the FODS-NAR algorithm can predict the solution of a stochastic fractional order system. Second, we compare the FODS-NAR algorithm with the famous and good reservoir computing (RC) algorithms. We find that FODS-NAR gives more accurate predictions than the traditional RC algorithm with the same system parameters, and the residuals of the FODS-NAR algorithm are closer to 0. Consequently, we conclude that the FODS-NAR algorithm is a method with higher accuracy and prediction results closer to the state of fractional-order stochastic systems. In addition, we analyze the effects of the number of neurons and the order of delays in the FODS-NAR algorithm on the prediction results and derive a range of their optimal values.

Impact of quarantine on fractional order dynamical model of Covid-19

In this paper, a Covid-19 dynamical transmission model of a coupled non-linear fractional differential equation in the Atangana-Baleanu Caputo sense is proposed. The basic dynamical transmission features of the proposed system are briefly discussed. The qualitative as well as quantitative results on the existence and uniqueness of the solutions are evaluated through the fixed point theorem. The Ulam-Hyers stability analysis of the suggested system is established. The two-step Adams-Bashforth-Moulton (ABM) numerical method is employed to find its numerical solution. The numerical simulation is performed to accesses the impact of various biological parameters on the dynamics of Covid-19 disease.

Fractional-order model and nonlinear vibration analyses of hybrid gas journal bearing–rotor system

It is important to propose a reasonable nonlinear dynamic model of gas dynamic bearing–rotor system for the stable operation of turbocharger. This paper presents a hybrid gas journal bearing (HGJBs)–rotor system with flexible supporting structure. A fractional-order dynamic model of the symmetrical flexible HGJB–rotor system is established. A tracking–alarming method is proposed to obtain the rotational speed ratio at the bifurcation point. The stable and unstable regions of the rotational speed ratio Ωω for both the rigid bearing–rotor system and flexible HGJB–rotor system are analyzed. The nonlinear dynamic behaviors of HGJB–rotor system versus the fractional order αd and rotational speed ratio Ωω are studied. Results show the fractional-order dynamic model can realize the reasonable prediction of nonlinear dynamic behaviors. An increase in the rotational speed ratio Ωω and the decrease of fractional order αd result in a growth of the vibration amplitude. The adoption of flexible supporting structure is a good strategy to increase the size of stable region.

Fractional PIλDµ controller design for non-linear PEM fuel cell for pressure control based on a genetic algorithm

ABSTRACT A fractional-order dynamic model could more accurately model many real scenarios than an integer-order model and provide a more accurate description of numerous genuine dynamical processes. A seventh-order nonlinear proton exchange membrane fuel cell (PEMFC) model is linearised in this research, taking into account correct initial conditions and equilibrium points.We consider the fluctuating load current as a disturbance parameter to affect the system. The goal is to find a control law for the MIMO system using a fractional PID controller based on a genetic algorithm. The controller is a critical part of the fuel cell which controls its functioning and efficiency. The goal is accomplished by designing a fractional controller to adjust the natural response of the fuel cell reactor and maintain the desired Power output in the face of uncertainties and disturbances. The validation results demonstrate that the fractional PIλDµ(FOPID) control method has a smaller overshoot and higher stability than the PID control method. Moreover, it is also observed that the operation efficiency of the PEMFC has risen by 2% with a response timing of less than 0.1 s using the developed fractional PIλDµ(FOPID) control technique.

Fractional order model for complex Layla and Majnun love story with chaotic behaviour

The main purpose of the present paper is to develop and analyze fractional order dynamical model describing the change in behavior exhibited by a couple in a passionate relationship of love. Model is treated with advanced fractional operators like Atangana-Baleanu Caputo sense, Atanagana-Toufik scheme, and fractal fractional for dynamical behavior of real love story. Its ability is to show constantly changed chaotic behavior of two individuals in the relationship, the model was previously expressed in love triangles. The System is analyzed by using the fixed point theory technique, the existence and uniqueness of system are proved as well as analyzed qualitatively. Also verify the local and global stability of the fractional order system. Numerical simulations are made with the Atanagana-Toufik scheme and fractal fractional operator to measure the emotions and love relationships between two peoples. Advanced approach of such applications opens the path to foresee the love relation after observing the human personality.

COVID-19 deterministic and stochastic modelling with optimized daily vaccinations in Saudi Arabia

In this paper, we investigate the stochastic nature of the COVID-19 temporal dynamics by generating a fractional-order dynamic model and a fractional-order-stochastic model. Initially, we considered the first and second vaccination doses as multiple vaccinations were initiated worldwide. The concerned models are then tested for the Saudi Arabia second virus wave, which is assumed to start on 1st March 2021. Four daily vaccination scenarios for the first and second dose are assumed for 100 days from the wave beginning. One of these scenarios is based on function optimization using the invasive weed optimization algorithm (IWO). After that, we numerically solve the established models using the fractional Euler method and the Euler-Murayama method. Finally, the obtained virus dynamics using the assumed scenarios and the real one started by the government are compared. The optimized scenario using the IWO effectively minimizes the predicted cumulative wave infections with a 4.4 % lower number of used vaccination doses.

Static and dynamic solar photovoltaic models’ parameters estimation using hybrid Rao optimization algorithm

Photovoltaic (PV) modeling research is a growing area of interest. The accuracy of PV models pertains to the efficient design and control of PV systems. Subsequently, several static and dynamic models were developed to emulate the course of current in PV cells. Moreover, a novel fractional order dynamic PV model has been proposed to enhance the accuracy of the conventional integral order one. The parameter estimation of these models is a challenging task. Most of the existing methods designed for handling this problem require either time-consuming parameter tuning process, or malfunction for systems exposed to environmental fluctuation. Besides, there has been a lack of researches into fractional dynamic PV models. In these circumstances, a hybrid Rao_1 Optimization Algorithm, denoted as HROA, is proposed for addressing this parameter extraction problem. Its exploration capacity is enhanced by introducing a logistic chaotic map and an exploration operator from the Generalized Normal Distribution Optimization algorithm (GNDO). Then a performance learning mechanism is employed to realize an adaptive transition between exploration and exploitation. HROA stands out by being parameter-tuning-free and efficient on a wide range of PV models. For the static models, HROA offers the best performance. A comprehensive performance analysis reveals that HROA exhibits a higher accuracy and less deviation than many recent algorithms. For the dynamic models, HROA is among the 3 best algorithms, it is capable of emulating the experimental datasets with low deviation error. The results also confirm that fractional dynamic model has better fitting effect than integral one. All the evidence suggests that HROA is a promising alternative for dealing with parameter identification problems in PV systems.

Non-singleton fuzzy control for multi-synchronization of chaotic systems

In this paper the problem of multi-synchronization with time-varying switching modes is introduced and a novel adaptive control method is designed. Unlike to the most studies, in addition to the unknown dynamics, the values of fractional-orders and control input nonlinearities are also considered to be unknown. A new fractional-order dynamic model on basis of non-singleton interval type-3 fuzzy logic system (NIT3-FLS) is presented to estimate the uncertainties. A fractional-order tuning rule is derived to optimize NIT3-FLS. An adaptive compensator is designed to tackle the effects of approximation errors, switching of synchronization modes and dynamic uncertainties and perturbations. The superiority of the suggested control scheme is verified through the several simulations and comparisons with the other control systems and other kind of FLSs.

Trajectory tracking control based on non-singular fractional derivatives for the PUMA 560 robot arm

In this paper, a novel hybrid fractional-order control strategy for the PUMA-560 robot manipulator is developed and presented, which combines the derivative of Caputo–Fabrizio and the integral of Atangana–Baleanu, both in the Caputo sense. The fractional-order dynamic model of the system (FODM) is also considered which consists of two models, the robot manipulator model, and the model of the induction motors which are the actuators that drive their joints. The fractional model of the manipulator is obtained using the Euler–Lagrange formulation. On the other hand, for controlling each one of the induction motors, fractional-order controllers ${{\mathop{\mathrm{PI}}\nolimits } ^{\vartheta }}$ based on Atangana–Baleanu in the Caputo sense integral were developed. And for the trajectory tracking control, fractional-order controllers ${{\mathop{\mathrm{PD}}\nolimits } ^{\xi }}$ were developed based on the fractional derivative of Caputo–Fabrizio in the Caputo sense. Also, ordinary PI and PD controllers were developed for the PUMA robot control to compare their performance with the fractional-order controllers. The results obtained demonstrated that the fractional-order controllers have a better capability for tracking trajectory tasks than the integer-order controllers, even when changes of the desired trajectory and external disturbances are considered. Additionally, an end-effector trajectory tracking task for manufacturing applications is also considered. All numerical simulations were performed by using the same orders and gains, demonstrating that the proposed fractional-order ${{\mathop{\mathrm{PI}}\nolimits } ^{\vartheta }}$ and ${{\mathop{\mathrm{PD}}\nolimits } ^{\xi }}$ controllers are robust, under different operating conditions, for tracking trajectory tasks. The fractional-order controllers and the integer-order controllers were tuned applying the cuckoo search optimization algorithm where the root-mean-square error (RMSE) was chosen as the cost function to minimize.

FRACTIONAL-ORDER PASSIVITY-BASED ADAPTIVE CONTROLLER FOR A ROBOT MANIPULATOR TYPE SCARA

In this paper, a novel fractional-order control strategy for the SCARA robot is developed. The proposed control is composed of [Formula: see text] and a fractional-order passivity-based adaptive controller, based on the Caputo–Fabrizio and Atangana–Baleanu derivatives, respectively; both controls are robust to external disturbances and change in the desired trajectory and effectively enhance the performance of robot manipulator. The fractional-order dynamic model of the robot manipulator is obtained by using the Euler–Lagrange formalism, as well as the model of the induction motors which are the actuators that drive their joints. Through simulations results, the effectiveness and robustness of the proposed control strategy have been demonstrated. The performance of the fractional-order proposed control method is compared with its integer-order counterpart, composed of the PI controller and the conventional passivity-based adaptive controller, reported in the literature. The performance comparison results demonstrate the superiority and effectiveness of the fractional-order proposed control strategy for a SCARA robot manipulator.

Fractional-order cubic nonlinear flux-controlled memristor: theoretical analysis, numerical calculation and circuit simulation

Many studies have proved that the fractional-order system is more accurate than the integer-order system. The theory of fractional calculus has received extensive attention in many fields. The research on flux-controlled memristor has been carried out for several years; however, fractional-order flux-controlled memristor has not received widespread attention. In light of the theory of fractional calculus, a generalized fractional-order dynamic model of flux-controlled memristor is proposed for the first time in this paper. The mathematical model is established and its characteristics of the pinched hysteresis loops are analyzed by changing the order and the frequency. The specific computational formulas of areas of pinched hysteresis loops are given. The equivalent circuit of the fractional-order flux-controlled memristor is constructed, and circuit simulations are given to illustrate the effectiveness of theoretical analysis and numerical calculation. In this study, the flux-controlled memristor model is extended to the fractional-order model that is closer to the real device, and the analysis method of areas of pinched hysteresis loops is further expanded.

New approaches to the fractional dynamics of schistosomiasis disease model

In this paper, schistosomiasis fractional order dynamic model is examined via exponential law kernel sense and Mittag-Leffler kernel in Liouville–Caputo sense. Some special solutions for two operators are obtained using the iterative scheme through Laplace transform and Sumudu–Picard integration technique. The uniqueness and existence of solution for both operators are established. The numerical solutions for both operators approve that the desirable results can be obtained when alpha value is less than 1.

Static and dynamic photovoltaic models’ parameters identification using Chaotic Heterogeneous Comprehensive Learning Particle Swarm Optimizer variants

Photovoltaic modeling has attracted researchers’ attention worldwide because of its importance in the photovoltaic system design. Therefore, several photovoltaic models have been introduced as static and dynamic photovoltaic models. Moreover, a novel fractional order dynamic photovoltaic model has been recently developed to enhance the accuracy and flexibility of the conventional integral order one. The unknown parameters of these models should be extracted accurately to achieve a proper photovoltaic system design and operation. In this work, novel Chaotic Heterogeneous Comprehensive Learning Particle Swarm Optimizer variants are introduced, where the Heterogeneous Comprehensive Learning Particle Swarm Optimizer is combined with ten different chaos maps to adapt its parameters. Six Chaotic Heterogeneous Comprehensive Learning Particle Swarm Optimizer variants are proposed in addition to the standard Heterogeneous Comprehensive Learning Particle Swarm Optimizer version to identify the parameters of both the static and the dynamic models based on different experimental datasets. To demonstrate the superiority of the developed variants, their results are compared to the most recent state-of-the-art algorithms with the aid of statistical analysis methods. The main outcome is that, in both of the static and the dynamic photovoltaic models, the Chaotic Heterogeneous Comprehensive Learning Particle Swarm Optimizer variants show their efficiency, accuracy and robustness not only over Heterogeneous Comprehensive Learning Particle Swarm Optimizer but also over recently published algorithms. They provide better fitting relative to the experimental datasets with the least deviation error and the fastest convergence speed as well. In the case of static models, the fourth variant of Chaotic Heterogeneous Comprehensive Learning Particle Swarm Optimizer with an iterative map for the single diode model, the third variant of Chaotic Heterogeneous Comprehensive Learning Particle Swarm Optimizer with singer map for the double diode model of solar cell. On the other hand, for the dynamic models, the second Chaotic Heterogeneous Comprehensive Learning Particle Swarm Optimizer variant with sinusoidal map for the integral order dynamic photovoltaic model and the sixth variant of Chaotic Heterogeneous Comprehensive Learning Particle Swarm Optimizer with Gauss/mouse map for the fractional order dynamic photovoltaic model offer the best performance.