An improved PC scheme for nonlinear fractional differential equations: Error and stability analysis

Asl, Mohammad Shahbazi; Javidi, Mohammad

doi:10.1016/j.cam.2017.04.026

Cited by 40 publications

(19 citation statements)

References 25 publications

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“…Here, the numerical solution of the FO differential equation is done through the modified Adams-Bashforth-Moulton algorithm (Asl and Javidi, 2017; Diethelm et al, 2002). A brief description of the algorithm aforementioned is presented in below.…”

Section: Preliminaries and Problem Statementmentioning

confidence: 99%

A switching sliding mode control technique for chaos suppression of fractional-order complex systems

Roohi

Khooban

Esfahani³

et al. 2019

Transactions of the Institute of Measurement and Control

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Switching sliding mode control (SSMC) can be utilized as a robust control technique, which is appropriate for the control of highly non-linear power systems like chaotic systems. The present study proposes a switching sliding mode control technique for control and chaos suppression of non-autonomous fractional-order (FO) nonlinear power systems with uncertainties and external disturbances. In the first step, a novel fractional switching sliding surface is introduced as well as its stability analysis to the origin is demonstrated. In the second step, based on the fractional version of the Lyapunov stability theory, a robust non-singular control law is designed to ensure the convergence of the system trajectories to the proposed sliding surface. Next, the proposed SSMC approach is utilized for designing a single input switching control technique for the stabilization of a class of 3D FO chaotic power systems. In order to evaluate the effectiveness and robustness of the suggested approach in practice, two examples including control and the stabilization of FO chaotic electric motors are illustrated.

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Section: Preliminaries and Problem Statementmentioning

confidence: 99%

A switching sliding mode control technique for chaos suppression of fractional-order complex systems

Roohi

Khooban

Esfahani³

et al. 2019

Transactions of the Institute of Measurement and Control

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“…The first one is about controlling the chaotic fractional-order Lu and the second one is about controlling the fractional-order Arneodo system. Also, it should be clear that the numerical simulations are provided based on a modification of Adams-Bashforth-Moulton algorithm [57,64], in the MATLAB software with h = 0.001 as a time step.…”

Section: Numerical Simulationmentioning

confidence: 99%

“…The continuous approximation method is more applicable to reduce chattering than other methods. This method designs the control signal discontinuity by creating a narrow boundary layer around the sliding surface and no problems have been reported so far [57].…”

Section: Introductionmentioning

confidence: 99%

A non-integer sliding mode controller to stabilize fractional-order nonlinear systems

Haghighi

Ziaratban

2020

Adv Differ Equ

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In this study, we examine the stabilization of fractional-order chaotic nonlinear dynamical systems with model uncertainties and external disturbances. We used the sliding mode controller by a new approach for controlling and stabilization of these systems. In this research, we replaced a continuous function with the sign function in the controller design and the sliding surface to suppress chattering and undesirable vibration effects. The advantages of the proposed control method are rapid convergence to the equilibrium point, the absence of chattering and unwanted oscillations, high resistance to uncertainties, and the possibility of applying this method to most fractional order chaotic systems. We applied the direct method of Lyapunov stability theory and the frequency distributed model to prove the stability of the slip surface and closed loop system. Finally, we simulated this method on two commonly used and practical chaotic systems and presented the results.

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“…Fractional derivative is a good mathematical tool to describe the memory and genetic characteristics of complex systems [13]. At present, there are more than six definitions of fractional derivative, among which Riemann-Liouville and Caputo derivatives are the most commonly used [14]. As we all know, in the case of time fractional Caputo derivative, the initial conditions are expressed by the values of the unknown function and its integer derivative with clear physical meaning [15].…”

Section: Introductionmentioning

confidence: 99%

Analysis of a fractional order mathematical model for tuberculosis with optimal control

2020

J. Nonlinear Funct. Anal.

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In this paper, a fractional-order mathematical model with control is constructed to describe the transmission of tuberculosis. Two cases are considered: the constant control and the optimal control. In the former case, the stability conditions of the disease-free equilibrium and the endemic equilibrium are obtained. In the second case, optimal control theory is applied to the corresponding model. The optimal control formula is derived by use of the Hamiltonian function and the Pontryagin's Maximum Principle. In addition, some numerical simulations are performed to support our analytic results.

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An improved PC scheme for nonlinear fractional differential equations: Error and stability analysis

Cited by 40 publications

References 25 publications

A switching sliding mode control technique for chaos suppression of fractional-order complex systems

A switching sliding mode control technique for chaos suppression of fractional-order complex systems

A non-integer sliding mode controller to stabilize fractional-order nonlinear systems

Analysis of a fractional order mathematical model for tuberculosis with optimal control

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