scholarly journals Some Numerical Examples on the Stability of Fractional Linear Dynamical Systems

2018 ◽  
Vol 17 (3) ◽  
pp. 51-66
Author(s):  
S Priyadharsini
Keyword(s):  
Fractional Order ◽  
Fractional Derivatives ◽  
Linear Models ◽  
Sufficient Conditions ◽  
Characteristic Polynomials ◽  
Caputo Fractional Derivatives ◽  
Linear Dynamical Systems ◽  
Numerical Examples ◽  
Eigen Values ◽  
The Stability

The concept of stability of a class of fractional-order linear system is considered in this paper. Existing sufficient conditions are assumed to guarantee the stability of linear models with the Caputo fractional derivatives. The results have been developed by using the concept of Laplace transform, and approximations of Mittag-Leffler.  Furthermore, results concerning asymptotical stability of linear fractional-order models are also achieved. The proposed method is based upon Eigen values and the characteristic polynomials. Numerical illustrations are specified to exhibit effectiveness of the proposed method.

scholarly journals Robust Nonfragile Controllers Design for Fractional Order Large-Scale Uncertain Systems with a Commensurate Order1<α<2

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Jianyu Lin
Keyword(s):  
Fractional Order ◽  
Uncertain Systems ◽  
Large Scale ◽  
Sufficient Conditions ◽  
Order System ◽  
Lyapunov Inequality ◽  
Numerical Examples ◽  
Decentralized Stabilization ◽  
Controller Gain ◽  
The Stability

The paper concerns the problem of stabilization of large-scale fractional order uncertain systems with a commensurate order1<α<2under controller gain uncertainties. The uncertainties are of norm-bounded type. Based on the stability criterion of fractional order system, sufficient conditions on the decentralized stabilization of fractional order large-scale uncertain systems in both cases of additive and multiplicative gain perturbations are established by using the complex Lyapunov inequality. Moreover, the decentralized nonfragile controllers are designed. Finally, some numerical examples are given to validate the proposed method.

Robust stability and stabilization of uncertain fractional order systems subject to input saturation

2017 ◽  
Vol 24 (16) ◽  
pp. 3676-3683 ◽  
Author(s):  
Esmat Sadat Alaviyan Shahri ◽  
Alireza Alfi ◽  
JA Tenreiro Machado
Keyword(s):  
Fractional Order ◽  
Robust Stability ◽  
State Feedback ◽  
Input Saturation ◽  
Sufficient Conditions ◽  
Fractional Order Systems ◽  
Numerical Examples ◽  
Stability And Stabilization ◽  
The Stability ◽  
Robust Stability And Stabilization

This paper studies the stability and the stabilization for a class of uncertain fractional order (FO) systems subject to input saturation. The Lipschitz condition and the Gronwall–Bellman lemma are adopted and sufficient conditions are derived to stabilize systems by designing a state feedback controller. Numerical examples demonstrate the effectiveness of the proposed method.

scholarly journals MATHEMATICAL ANALYSIS OF COUPLED SYSTEMS WITH FRACTIONAL ORDER BOUNDARY CONDITIONS

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040012 ◽  
Author(s):  
ZEESHAN ALI ◽  
KAMAL SHAH ◽  
AKBAR ZADA ◽  
POOM KUMAM
Keyword(s):  
Boundary Conditions ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Sufficient Conditions ◽  
Coupled Systems ◽  
Existence Theory ◽  
Third Order ◽  
Banach Contraction ◽  
The Stability ◽  
Theoretical Results

In this paper, we prove the existence, uniqueness and various kinds of Ulam stability for fractional order coupled systems with fractional order boundary conditions involving Riemann–Liouville fractional derivatives. The standard fixed point theorem like Leray–Schauder alternative and Banach contraction are applied to establish the existence theory and uniqueness. Furthermore, we build sufficient conditions for the stability mentioned above by two methods. Also, an example is given to illustrate our theoretical results. The proposed problem is the generalization of third-order ordinary differential equations with classical, initial and anti-periodic boundary conditions.

scholarly journals Efficient and Stable Numerical Methods for Multi-Term Time Fractional Sub-Diffusion Equations

2014 ◽  
Vol 4 (3) ◽  
pp. 242-266 ◽  
Author(s):  
Jincheng Ren ◽  
Zhi-zhong Sun
Keyword(s):  
Fractional Derivatives ◽  
Diffusion Equations ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Caputo Fractional Derivatives ◽  
Numerical Examples ◽  
One Dimensional ◽  
Compact Difference ◽  
Spatial Discretisation ◽  
The Stability

AbstractSome efficient numerical schemes are proposed for solving one-dimensional (1D) and two-dimensional (2D) multi-term time fractional sub-diffusion equations, combining the compact difference approach for the spatial discretisation and L1 approximation for the multi-term time Caputo fractional derivatives. The stability and convergence of these difference schemes are theoretically established. Several numerical examples are implemented, testifying to their efficiency and confirming their convergence order.

scholarly journals Ulam–Hyers–Mittag-Leffler stability for tripled system of weighted fractional operator with TIME delay

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohammed A. Almalahi ◽  
Satish K. Panchal ◽  
Fahd Jarad ◽  
Thabet Abdeljawad
Keyword(s):  
Fixed Point ◽  
Time Delay ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Sufficient Conditions ◽  
Fractional Operator ◽  
Caputo Fractional Derivatives ◽  
Picard Operator

AbstractThis study is aimed to investigate the sufficient conditions of the existence of unique solutions and the Ulam–Hyers–Mittag-Leffler (UHML) stability for a tripled system of weighted generalized Caputo fractional derivatives investigated by Jarad et al. (Fractals 28:2040011 2020) in the frame of Chebyshev and Bielecki norms with time delay. The acquired results are obtained by using Banach fixed point theorems and the Picard operator (PO) method. Finally, a pertinent example of the results obtained is demonstrated.

Dynamical Behaviors of a Fractional-Order Predator–Prey Model with Holling Type IV Functional Response and Its Discretization

2019 ◽  
Vol 20 (2) ◽  
pp. 125-136
Author(s):  
A. M. Yousef ◽  
S. Z. Rida ◽  
Y. Gh. Gouda ◽  
A. S. Zaki
Keyword(s):  
Functional Response ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Dynamical Behaviors ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Theoretical Results

AbstractIn this paper, we investigate the dynamical behaviors of a fractional-order predator–prey with Holling type IV functional response and its discretized counterpart. First, we seek the local stability of equilibria for the fractional-order model. Also, the necessary and sufficient conditions of the stability of the discretized model are achieved. Bifurcation types (include transcritical, flip and Neimark–Sacker) and chaos are discussed in the discretized system. Finally, numerical simulations are executed to assure the validity of the obtained theoretical results.

scholarly journals Admissibility of Fractional Order Descriptor Systems Based on Complex Variables: An LMI Approach

2020 ◽  
Vol 4 (1) ◽  
pp. 8
Author(s):  
Xuefeng Zhang ◽  
Yuqing Yan
Keyword(s):  
Fractional Order ◽  
Complex Plane ◽  
Sufficient Conditions ◽  
Complex Variables ◽  
Necessary And Sufficient Conditions ◽  
Descriptor Systems ◽  
Fractional Order Systems ◽  
Lmi Approach ◽  
Numerical Examples ◽  
Necessary And Sufficient

This paper is devoted to the admissibility issue of singular fractional order systems with order α ∈ ( 0 , 1 ) based on complex variables. Firstly, with regard to admissibility, necessary and sufficient conditions are obtained by strict LMI in complex plane. Then, an observer-based controller is designed to ensure system admissible. Finally, numerical examples are given to reveal the validity of the theoretical conclusions.

scholarly journals Observation of a Class of Disturbance in Time Series Expansion for Fractional Order Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Yiheng Wei ◽  
Hamid Reza Karimi ◽  
Jinwen Pan ◽  
Qing Gao ◽  
Yong Wang
Keyword(s):  
Time Series ◽  
Series Expansion ◽  
Fractional Order ◽  
Disturbance Observer ◽  
Higher Order ◽  
Transient Dynamics ◽  
Fractional Order Systems ◽  
Weighted Function ◽  
Numerical Examples ◽  
The Stability

This paper is concerned with the problem of designing disturbance observer for fractional order systems, of which the disturbance is in time series expansion. The stability of a special observer with the selected nonlinear weighted function and transient dynamics function is rigorously analyzed for slowly varying disturbance. In addition, the result is also extended to estimate slope forms disturbance and higher order disturbance of fractional order systems. The efficacy of the proposed method is validated through numerical examples.

scholarly journals Dynamic Analysis of a Fractional-Order Model for Hepatitis B Virus with Holling II Functional Response

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Ruiqing Shi ◽  
Ting Lu ◽  
Cuihong Wang
Keyword(s):  
Hepatitis B Virus ◽  
Hepatitis B ◽  
Fractional Order ◽  
Reproduction Number ◽  
Sufficient Conditions ◽  
Order Model ◽  
Fractional Order Model ◽  
B Virus ◽  
The Stability ◽  
Holling Ii Functional Response

In this paper, a fractional-order model is constructed to describe the transmission of Hepatitis B Virus (HBV). Firstly, the existence and uniqueness of positive solutions are proved. Secondly, the basic reproduction number and the sufficient conditions for the existence of two equilibriums are obtained. Thirdly, the stability of equilibriums are analyzed. After that, some numerical simulations are performed to verify the theoretical prediction. Finally, a brief discussion is presented.

scholarly journals The Robust Control and Synchronization of a Class of Fractional-Order Chaotic Systems with External Disturbances via a Single Output

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Runzi Luo ◽  
Haipeng Su
Keyword(s):  
Robust Control ◽  
Fractional Order ◽  
Chaos Control ◽  
Chaotic Systems ◽  
Sufficient Conditions ◽  
External Disturbances ◽  
Numerical Examples ◽  
Single Output ◽  
Lorenz Chaotic System ◽  
Control And Synchronization

This paper investigates the stabilization and synchronization of a class of fractional-order chaotic systems which are affected by external disturbances. The chaotic systems are assumed that only a single output can be used to design the controller. In order to design the proper controller, some observer systems are proposed. By using the observer systems some sufficient conditions for achieving chaos control and synchronization of fractional-order chaotic systems are derived. Numerical examples are presented by taking the fractional-order generalized Lorenz chaotic system as an example to show the feasibility and validity of the proposed method.

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