New power law inequalities for fractional derivative and stability analysis of fractional order systems

Fractional-order systems have been applied in many engineering applications. A key issue with the application of such systems is the approximation of fractional-order parameters. The numerical tools for the approximation of fractional-order parameters gained attention recently. However, available toolboxes in the literature do not have a direct option to approximate higher order systems and need improvements with the graphical, numerical, and stability analysis. Therefore, this chapter proposes a MATLAB-based GUI for the approximation of fractional-order operators. The toolbox is made up of four widely used approximation techniques, namely, Oustaloup, refined Oustaloup, Matsuda, and curve fitting. The toolbox also allows numerical and stability analysis for evaluating the performance of approximated transfer function. To demonstrate the effectiveness of the developed GUI, a simulation study is conducted on fractional-order PID control of pH neutralization process. The results show that the toolbox can be effectively used to approximate and analyze the fractional-order systems.

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Dynamic analysis of a fractional order system

International Journal of Biomathematics ◽

10.1142/s1793524515500795 ◽

2015 ◽

Vol 08 (06) ◽

pp. 1550079

Author(s):

M. Javidi ◽

N. Nyamoradi

Keyword(s):

Stability Analysis ◽

Fractional Derivative ◽

Dynamic Analysis ◽

Fractional Order ◽

Local Stability ◽

Equilibrium Points ◽

Dynamical Behavior ◽

Order System ◽

Stability Conditions ◽

Hurwitz Stability

In this paper, we investigate the dynamical behavior of a fractional order phytoplankton–zooplankton system. In this paper, stability analysis of the phytoplankton–zooplankton model (PZM) is studied by using the fractional Routh–Hurwitz stability conditions. We have studied the local stability of the equilibrium points of PZM. We applied an efficient numerical method based on converting the fractional derivative to integer derivative to solve the PZM.

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Fractional Dynamics of Computer Virus Propagation

Mathematical Problems in Engineering ◽

10.1155/2014/476502 ◽

2014 ◽

Vol 2014 ◽

pp. 1-7 ◽

Cited By ~ 5

Author(s):

Carla M. A. Pinto ◽

J. A. Tenreiro Machado

Keyword(s):

Steady State ◽

Fractional Derivative ◽

Fractional Order ◽

Initial Conditions ◽

Computer Virus ◽

Fractional Dynamics ◽

Fractional Order Systems ◽

Virus Propagation ◽

Integer Order ◽

Fast Transients

We propose a fractional model for computer virus propagation. The model includes the interaction between computers and removable devices. We simulate numerically the model for distinct values of the order of the fractional derivative and for two sets of initial conditions adopted in the literature. We conclude that fractional order systems reveal richer dynamics than the classical integer order counterpart. Therefore, fractional dynamics leads to time responses with super-fast transients and super-slow evolutions towards the steady-state, effects not easily captured by the integer order models.

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Comments on "Stability Analysis of a Class of Nonlinear Fractional-Order Systems

IEEE Transactions on Circuits & Systems II Express Briefs ◽

10.1109/tcsii.2009.2020944 ◽

2009 ◽

Vol 56 (6) ◽

pp. 519-520 ◽

Cited By ~ 10

Author(s):

M.S. Tavazoei

Keyword(s):

Stability Analysis ◽

Fractional Order ◽

Fractional Order Systems

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Fractional-order-dependent global stability analysis and observer-based synthesis for a class of nonlinear fractional-order systems

International Journal of Robust and Nonlinear Control ◽

10.1002/rnc.4250 ◽

2018 ◽

Author(s):

Mohammad Tavazoei ◽

Mohammad Hassan Asemani

Keyword(s):

Stability Analysis ◽

Global Stability ◽

Fractional Order ◽

Fractional Order Systems ◽

Global Stability Analysis

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A GENERAL COMPARISON PRINCIPLE FOR CAPUTO FRACTIONAL-ORDER ORDINARY DIFFERENTIAL EQUATIONS

Fractals ◽

10.1142/s0218348x2050070x ◽

2020 ◽

Vol 28 (04) ◽

pp. 2050070 ◽

Cited By ~ 1

Author(s):

CONG WU

Keyword(s):

Differential Equations ◽

Fractional Derivative ◽

Ordinary Differential Equations ◽

Fractional Order ◽

Comparison Principle ◽

Caputo Fractional Derivative ◽

Fractional Derivatives ◽

Fractional Order Systems ◽

Caputo Fractional Derivatives ◽

Full Result

In this paper, we work on a general comparison principle for Caputo fractional-order ordinary differential equations. A full result on maximal solutions to Caputo fractional-order systems is given by using continuation of solutions and a newly proven formula of Caputo fractional derivatives. Based on this result and the formula, we prove a general fractional comparison principle under very weak conditions, in which only the Caputo fractional derivative is involved. This work makes up deficiencies of existing results.

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