Article
Automation & Control Systems
R. Almeida, S. Hristova, S. Dashkovskiy
Summary: In this study, the BIBO stability of a nonlinear Caputo fractional system with time-varying bounded delay and nonlinear output was investigated. New stability criteria were derived using the Razumikhin method, Lyapunov functions, and fractional derivatives of Lyapunov functions. The effectiveness of these theoretical results was demonstrated through numerical simulations of the system's dynamic response.
INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL
(2021)
Article
Mathematics, Interdisciplinary Applications
Lin Sun, Chunhao Cai, Min Zhang
Summary: This paper focuses on the maximum likelihood estimator for the parameter of first-order autoregressive models driven by stationary Gaussian noises and input signals. By finding the optimal input that maximizes the Fisher information and using the Laplace transform method, the asymptotic properties and design problem of the maximum likelihood estimator are investigated.
FRACTAL AND FRACTIONAL
(2022)
Article
Mathematics, Applied
Gabriel Bengochea, Manuel Ortigueira, Luis Verde-Star
Summary: The operational method proposed in the study is suitable for solving fractional differential equations with two or more noncommensurate orders, and can be applied in a recursive manner by adding or removing pseudo-poles or pseudo-zeros to recalculate the output from pre-existing solutions. Through several examples, the method's application, efficiency, and simplicity are demonstrated.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Christian Engstrom, Stefano Giani, Luka Grubisic
Summary: In this paper, the numerical inverse Laplace transform for distributed order time-fractional equations is considered. The discontinuous Galerkin scheme is used to discretize the problem in space. A method is proposed to enclose the spectrum and compute resolvent estimates independent of the problem size, which is crucial for the success of Talbot's approach. The new results are applied to time-fractional wave and diffusion-wave equations of distributed order.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Alireza Ansari, Mohammad Hossein Derakhshan, Hassan Askari
Summary: In this paper, we study the powers of the Laplacian operator in axisymmetric cylindrical geometry, obtaining the fundamental solution of the corresponding distributed order time-fractional diffusion equation. We also discuss the roles of the Mittag-Leffler and Wright functions in the structures of solutions.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2022)
Article
Mathematics, Applied
N. Vieira, M. M. Rodrigues, M. Ferreira
Summary: In this work, the Cauchy problem for the time-fractional telegraph equation of distributed order in R-n x R+ is considered. By employing the technique of the Fourier, Laplace and Mellin transforms, a representation of the fundamental solution of this equation in terms of convolutions involving the Fox H-function is obtained. Some particular choices of the density functions in the form of elementary functions are studied. Fractional moments of the fundamental solution are computed in the Laplace domain. Finally, by application of the Tauberian theorems we study the asymptotic behaviour of the second-order moment (variance) in the time domain.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2021)
Article
Mathematics, Interdisciplinary Applications
Mati Ur Rahman, Ali Althobaiti, Muhammad Bilal Riaz, Fuad S. Al-Duais
Summary: This article explores a biological population model using a specific numerical method. The numerical simulations reveal a relationship between the population density and the fractional order, showing that a higher fractional order leads to a higher population density. The results demonstrate that the method is suitable and highly accurate in terms of computational cost.
FRACTAL AND FRACTIONAL
(2022)
Article
Mathematics, Applied
Yibo Wang, Wanrong Cao
Summary: In this paper, a full-discrete scheme is proposed for a Langevin equation involving Caputo fractional derivative and additive white noise. By truncating the spectral of white noise, the fractional Langevin equation is converted into an approximate equation with random parameters, and a finite difference scheme is constructed. The consistency of the approximate equation and the error estimate of the finite difference scheme are obtained. Numerical examples are provided to verify the theoretical analysis. Additionally, a scheme based on piecewise spectral approximation of white noise is developed to achieve long-time simulation.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
Jiabin Xu, Hassan Khan, Rasool Shah, A. A. Alderremy, Shaban Aly, Dumitru Baleanu
Summary: The research paper presents an efficient technique for solving fractional-order nonlinear Swift-Hohenberg equations related to fluid dynamics, showing that the Laplace Adomian decomposition method requires minimal calculations and produces solutions in close agreement with other existing methods. Numerical examples confirm the validity of the suggested method, demonstrating its almost identical solutions with various analytical methods through graphs and tables.
Article
Computer Science, Interdisciplinary Applications
Amit Prakash, Hardish Kaur
Summary: This paper investigates the fractional Biswas-Milovic model with Kerr and parabolic law nonlinearities using fractional complex transform (FCT) combined with homotopy perturbation transform technique (HPTT). The results show that the proposed technique is reliable with less computational time and high accuracy compared to the residual power series method (RPSM). Comparative simulation studies demonstrate that the proposed technique provides better approximations.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2021)
Article
Mathematics, Applied
Jing Wang, Changfeng Shao, Xiaolu Chen, YangQuan Chen
Summary: This article presents a novel fractional-order sliding mode control method based on disturbance observer for noncommensurate fractional-order systems with mismatched disturbances. By designing fractional-order disturbance observers independently to estimate mismatched disturbances, a uniform control method is proposed. Simulation results demonstrate the improved control performance of the proposed method.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Roberto Garrappa, Andrea Giusti, Francesco Mainardi
Summary: Several approaches to formulating a fractional theory of calculus of variable order have been proposed in the literature over the years, but most lack a rigorous mathematical framework. This article discusses an alternative view on the problem originally proposed by G. Scarpi in the early seventies, framed within the recent theory of General Fractional Derivatives and Integrals, and explores practical applications of variable-order Scarpi integral and derivative using numerical methods for Laplace transforms inversion.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2021)
Article
Engineering, Multidisciplinary
Muhammad Arfan, Ibrahim Mahariq, Kamal Shah, Thabet Abdeljawad, Ghaylen Laouini, Pshtiwan Othman Mohammed
Summary: This manuscript investigates the population dynamical model of non-integer order to study the recent Covid-19 pandemic. The proposed model is analyzed qualitatively using fixed-point theory and non-linear functional analysis. Semi-analytical results are obtained through the Laplace transform with Adomian polynomial and decomposition techniques. The numerical solution is also obtained using the non-standard finite difference scheme. The Matlab simulation provides the overall spectrum and dynamical behavior of each compartment of the model.
ALEXANDRIA ENGINEERING JOURNAL
(2022)
Article
Mathematics, Applied
Chunhao Cai, Min Zhang
Summary: This paper focuses on the controlled drift estimation of the mixed fractional Ornstein-Uhlenbeck process. Two models are considered: one with optimal input to maximize the Fisher information for the unknown parameter, and the other with a constant as the controlled function. Large sample asymptotic properties of the MLE are deduced using Laplace transform computations or the Cameron Martin formula with additional insights from [12]. Additionally, the paper proves that the MLE is strongly consistent as a supplement to [12].
Article
Mathematics, Applied
Rafal Stanislawski, Krzysztof J. Latawiec
Summary: This paper introduces an extension of the Mikhailov stability criterion to a class of discrete-time noncommensurate fractional-order systems, using the nabla fractional-order Grünwald-Letnikov difference. The new stability analysis methods proposed are computationally simple and effective for both commensurate and non-commensurate fractional-order systems. The proposed methodology provides the same computational complexity for noncommensurate systems as for commensurate-order ones. Simulation examples confirm the usefulness of the methodology.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2021)
Article
Automation & Control Systems
Weijia Zheng, YangQuan Chen, Xiaohong Wang, Yong Chen, Meijin Lin
Summary: In this paper, an enhanced fractional order sliding mode control (FOSMC) method is proposed to improve the control performance of fractional order uncertain systems with multiple mis-matched disturbances. The multiple disturbances and uncertainties are estimated by finite-time disturbance observers and a fractional order extended state observer. A fractional order switching law is designed for fast convergence of system states. The proposed method incorporates feedforward compensation, the fractional order switching law, and an auxiliary state for input saturation. Numerical examples and a motor speed control problem are used to demonstrate the effectiveness of the proposed method through performance comparisons with existing control methods.
Article
Automation & Control Systems
Weijia Zheng, YangQuan Chen, Xiaohong Wang, Meijin Lin, Jing Guo
Summary: Fractional order PID (FOPID) controller offers more flexibility in design options compared to integer order PID (IOPID) controller. However, in practical implementation, it is necessary to select achievable design specifications and controller parameters. This study collects and analyzes the complete achievable regions of design specifications and models the selectable integral and derivative orders of the FOPID controller. A synthesis method is proposed to achieve the desired flat-phase characteristic.
MEASUREMENT & CONTROL
(2023)
Article
Automation & Control Systems
Yiheng Wei, YangQuan Chen, Yingdong Wei, Xuan Zhao
Summary: This paper investigates the stability analysis issue of nabla tempered fractional order systems for the first time. It defines the (discrete time) tempered Mittag-Leffler stability and derives a stability criterion using the Lyapunov method. Additionally, boundedness and attractiveness are also investigated.
ASIAN JOURNAL OF CONTROL
(2023)
Article
Thermodynamics
Zhenlong Wu, Yanhong Liu, Donghai Li, YangQuan Chen
Summary: This paper proposes a multivariable active disturbance rejection control (ADRC) for a compression liquid chiller system, where the loop coupling is estimated and compensated in real-time by enhanced reduced-order extended state observer (ESO) and static decoupling. The simulation results illustrate that the proposed control strategy can guarantee satisfactory tracking performance and estimate the loop coupling completely under different operating conditions. Compared with the decentralized regular ADRC, the performance of the loop coupling rejection and the fluctuation of the control signal have improved significantly.
Article
Engineering, Electrical & Electronic
Liping Chen, Wenliang Guo, Panpan Gu, Antonio M. M. Lopes, Zhaobi Chu, YangQuan Chen
Summary: This study investigates the stability and control issues in incommensurate fractional-order (FO) nonlinear systems with time-varying bounded uncertainties. It introduces a new stability criterion in the form of linear matrix inequality by utilizing the FO comparison principle of multi-order FO systems. Based on the proposed criteria, a state feedback controller for system stabilization is derived. Numerical simulations demonstrate the effectiveness of the theoretical formulation.
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II-EXPRESS BRIEFS
(2023)
Article
Automation & Control Systems
Chuanfan Lu, Rongnian Tang, YangQuan Chen, Chuang Li
Summary: The article introduces a method for tuning the parameters of TID controller based on the synthesis of frequency and time-domain specifications, which helps to meet the requirements of systemic stability and robustness. Simulation and experimental results demonstrate the superiority of the proposed control tuning method.
INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL
(2023)
Article
Automation & Control Systems
Zhenzhen Lu, YangQuan Chen, Yongguang Yu, Guojian Ren, Conghui Xu, Weiyuan Ma, Xiangyun Meng
Summary: In this paper, a generalized SEIHRDP fractional-order epidemic model with individual migration is established. The global properties of the proposed system are studied, and the sensitivity of parameters to the basic reproduction number is analyzed both theoretically and numerically. Based on real data from India and Brazil, it is concluded that the bilinear incidence rate provides a better description of COVID-19 transmission. The proposed system can also better predict the next peak by considering multi-peak situations. Lastly, the spread of COVID-19 between cities can be effectively controlled by limiting individual movement, enhancing nucleic acid testing, and reducing individual contact.
Article
Mathematical & Computational Biology
Zhenzhen Lu, Guojian Ren, Yangquan Chen, Xiangyun Meng, Yongguang Yu
Summary: The purpose of this paper is to ensure the physical reasonableness of epidemic models with anomalous diffusion. The researchers introduced stochastic processes and fractional-order diffusion methods to successfully incorporate anomalous diffusion into epidemic models and explored different types of these models. Additionally, the paper used real data of COVID-19 to demonstrate the spatial transmission of infectious diseases.
INTERNATIONAL JOURNAL OF BIOMATHEMATICS
(2023)
Article
Automation & Control Systems
Weijia Zheng, YangQuan Chen, Xiaohong Wang, Runquan Huang, Meijin Lin, Fang Guo
Summary: This paper proposes a fractional order sliding mode control (FOSMC) method for the control problem of permanent magnet synchronous motor (PMSM) speed servo system subject to multiple disturbances. By using an improved disturbance observer (DO) and an extended state observer (ESO), the lumped exogenous disturbances and uncertainties of the PMSM speed servo are estimated. A novel FOSMC law is developed by incorporating feedforward compensation and a fractional order switching law, improving tracking performance and robustness of the PMSM servo system.
INTERNATIONAL JOURNAL OF CONTROL AUTOMATION AND SYSTEMS
(2023)
Article
Automation & Control Systems
Xuefeng Zhang, Shun-Feng Su, Yang-Quan Chen
Summary: This paper investigates trajectory tracking control for autonomous surface vehicles with unknown dynamics and environmental disturbances. A novel adaptive fuzzy constrained control approach is proposed, which utilizes a new-type performance envelop and a singularity-free continuous control solution to achieve explicitly and freely preassigned settling time and accuracy of trajectory tracking. The fuzzy logic systems are employed to address unknown nonlinearities. The resulting control system possesses inherent robustness without disturbance observers or assuming derivative information. A comparative simulation verifies the effectiveness and superiority of the approach.
INTERNATIONAL JOURNAL OF FUZZY SYSTEMS
(2023)
Article
Engineering, Multidisciplinary
Liping Chen, Min Xue, Antonio Lopes, Ranchao Wu, YangQuan Chen
Summary: This paper investigates the asymptotic behavior of systems governed by nonlinear differential equations with two fractional derivatives. A sufficient condition for asymptotic stability is derived for such systems using the Mittag-Leffler function, Laplace transform, and generalized Gronwall inequality. Numerical examples are provided to illustrate the theoretical findings.
JOURNAL OF ENGINEERING MATHEMATICS
(2023)
Article
Acoustics
Jiazhi Cai, Yifan Liu, Qingbin Gao, YangQuan Chen
Summary: This paper presents the stability analysis and controller design of a novel active vibration suppression technique, called the fractional-order delayed resonator (FODR). The parameterized control formulas and an analytical procedure are proposed for stability analysis. The fractional feedback controller is implemented using an integer-order approximate transfer function, and a control strategy called order scheduling is introduced to enhance control flexibility. Experimental results demonstrate that this technique can widen the frequency band, improve vibration control speed, and enhance system stability robustness.
JOURNAL OF SOUND AND VIBRATION
(2023)
Article
Mathematics, Interdisciplinary Applications
Yiheng Wei, Xuan Zhao, Yingdong Wei, Yangquan Chen
Summary: This paper investigates the stability analysis problem for a class of incommensurate nabla fractional order systems, considering both Caputo definition and Riemann-Liouville definition. Several elementary fractional difference inequalities on Lyapunov functions are developed under the convex assumption. By utilizing the essential features of nabla fractional calculus, sufficient conditions are given to guarantee the asymptotic stability of the incommensurate system using the direct Lyapunov method. Four examples are provided to substantiate the efficacy and effectiveness of the theoretical results.
JOURNAL OF SYSTEMS SCIENCE & COMPLEXITY
(2023)
Article
Mathematics, Applied
Weiyuan Ma, Nuri Ma, Changping Dai, YangQuan Chen, Xinwei Wang
Summary: As the COVID-19 mutates, the infection rate is increasing rapidly and the vaccine is ineffective against the mutated strain. This paper proposes a SEIR-type fractional model with reinfection and vaccine inefficacy, which successfully captures the dynamics of the mutated COVID-19 pandemic. The model's existence, uniqueness, boundedness, and nonnegativeness are derived, and the local and global stability based on the basic reproduction number R0 are analyzed. Sensitivity analysis evaluates the impact of each parameter on R0 and ranks key epidemiological parameters. Additionally, necessary conditions for implementing fractional optimal control and corresponding optimal solutions for mitigating COVID-19 transmission are obtained.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Interdisciplinary Applications
Yixiao Ding, Ying Luo, Yangquan Chen
Summary: This paper proposes a fractional order impedance controller for industrial robot manipulator control and presents a systematic FOIC parameters tuning strategy based on frequency-domain specifications. To improve performance under dynamic disturbances, a dynamic feedforward-based fractional order impedance controller is further developed. The effectiveness of the controller is verified through simulation and physical robot manipulator prototype, and the results show that the proposed FOIC achieves better control performance than an integer order impedance controller.
FRACTAL AND FRACTIONAL
(2023)
