scholarly journals Parallel Algorithm for Numerical Methods Applied to Fractional-order System

2020 ◽  
Vol 21 (4) ◽  
pp. 701-707
Author(s):  
Florin Rosu
Keyword(s):  
Numerical Methods ◽  
Numerical Method ◽  
Parallel Algorithm ◽  
Fractional Order ◽  
Fractional Order System ◽  
Order System ◽  
Fractional Order Systems ◽  
Predictor Corrector

A parallel algorithm is presented that approximates a solution for fractional-order systems. The algorithm isimplemented in CUDA, using the specific GPU capabilities. The numerical methods used are Adams-Bashforth-Moulton (ABM) predictor-corrector scheme and Diethelm’s numerical method. A comparison is done between these numerical methods that adapts the same algorithm for the approximation of the solution.

scholarly journals Elegant Chaos in Fractional-Order System without Equilibria

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Donato Cafagna ◽  
Giuseppe Grassi
Keyword(s):  
Numerical Methods ◽  
Fractional Order ◽  
Chaotic System ◽  
Chaotic Dynamics ◽  
Fractional Order System ◽  
Order System ◽  
System Parameters ◽  
System Equations ◽  
Predictor Corrector

A new fractional-order chaotic system with no equilibria is presented. The proposed system can be considered elegant in the sense given by Sprott (2010), since the corresponding system equations contain very few terms and the system parameters have a minimum of digits. The chaotic dynamics are analyzed using the predictor-corrector algorithm when the fractional-order of the derivative is 0.98. Finally, the presence of chaos is validated by applying different numerical methods.

scholarly journals Some Fundamental Properties on the Sampling Free Nabla Laplace Transform

2019 ◽  
Author(s):  
Yiheng Wei ◽  
Yuquan Chen ◽  
Yong Wang ◽  
YangQuan Chen
Keyword(s):  
System Theory ◽  
Laplace Transform ◽  
Fractional Order ◽  
Fractional Order System ◽  
The Other ◽  
Order System ◽  
Fractional Order Systems ◽  
Fundamental Properties ◽  
Other Hand

Abstract Discrete fractional order systems have attracted more and more attention in recent years. Nabla Laplace transform is an important tool to deal with the problem of nabla discrete fractional order systems, but there is still much room for its development. In this paper, 14 lemmas are listed to conclude the existing properties and 14 theorems are developed to describe the innovative features. On one hand, these properties make the Ntransform more effective and efficient. On the other hand, they enrich the discrete fractional order system theory.

Application of Incomplete Gamma Functions to the Initialization of Fractional-Order Systems

2007 ◽  
Author(s):  
Tom T. Hartley ◽  
Carl F. Lorenzo
Keyword(s):  
System Theory ◽  
Fractional Order ◽  
Gamma Function ◽  
Incomplete Gamma Function ◽  
Fractional Order System ◽  
Order System ◽  
Fractional Order Systems ◽  
Gamma Functions ◽  
Incomplete Gamma Functions

This paper reviews some properties of the gamma function, particularly the incomplete gamma function and its complement, as a function of the Laplace variable s. The utility of these functions in the solution of initialization problems in fractional-order system theory is demonstrated.

Toward Searching Possible Oscillatory Region in Order Space for Nonlinear Fractional-Order Systems

2013 ◽  
Vol 9 (2) ◽  
Author(s):  
Mohammad Saleh Tavazoei
Keyword(s):  
Fractional Order ◽  
Fractional Order System ◽  
Order System ◽  
Fractional Order Systems ◽  
Order Sets ◽  
Special Order

Finding the oscillatory region in the order space is one of the most challenging problems in nonlinear fractional-order systems. This paper proposes a method to find the possible oscillatory region in the order space for a nonlinear fractional-order system. The effectiveness of the proposed method in finding the oscillatory region and special order sets placed in its boundary is confirmed by presenting some examples.

On Observability of Fractional Order Systems

2009 ◽  
Author(s):  
Jocelyn Sabatier ◽  
Mathieu Merveillaut ◽  
Ludovic Fenetau ◽  
Alain Oustaloup
Keyword(s):  
Parabolic Equation ◽  
State Space ◽  
Fractional Order ◽  
The State ◽  
Fractional Order System ◽  
Order System ◽  
Fractional Order Systems ◽  
System State ◽  
Fractional Systems ◽  
Real State

In this paper, fractional order system observability is discussed. A representation of these systems that involves a classical linear integer system and a system described by a parabolic equation is used to define the system real state and to conclude that the system state cannot be observed. However, it is also shown that the state space like representation usually encountered in the literature for fractional systems, can be used to design Luenberger like observers that permit an estimation of important variables in the system.

scholarly journals Application of Selected Numerical Methods to Model the Fractional-Order System Behavior of Nonlaminated Magnetic Actuators

2021 ◽  
Author(s):  
Martin Hecht ◽  
Robert Seifert ◽  
Wilfried Hofmann
Keyword(s):  
Numerical Methods ◽  
Fractional Order ◽  
Eddy Currents ◽  
Tracking Error ◽  
Measurement Data ◽  
Analytical Approximation ◽  
Fractional Order System ◽  
Order System ◽  
Model Accuracy ◽  
Magnetic Actuators

The electromagnetic dynamics of nonlaminated magnetic actuators are highly influenced by eddy currents and minor perturbations like core saturation, hysteresis as well as fringing and leakage fluxes. In the literature, analytical high-fidelity models describing these phenomena are known, which lead to complex reluctance networks or transcendental system descriptions with fractional-order characteristics. Therefore, they are not suitable for a direct implementation within the actuator control. Previously, we provided appropriate analytical rational approximations that allow a digital real-time implementation of these models on a microcontroller. However, the inclusion of the minor perturbations, if possible, leads to impractical model orders requiring simplifications, which compromise the model accuracy. This article studies numerical methods to reduce high model orders or directly approximate the transcendental systems or empirical measurement data. The greater degree of freedom allows for a possible higher model accuracy with sufficiently low orders. We review and improve existing approaches like Levy's method and Vector Fitting and apply them to the frequency response of the underlying fractional-order system. Furthermore we propose an order reduction algorithm based on a pole-zero-cancellation with tracking error compensation. Using measurement data, a comparison shows that the numerical approaches match or excel our previously studied analytical approximation.

scholarly journals Application of Selected Numerical Methods to Model the Fractional-Order System Behavior of Nonlaminated Magnetic Actuators

2021 ◽  
Author(s):  
Martin Hecht ◽  
Robert Seifert ◽  
Wilfried Hofmann
Keyword(s):  
Numerical Methods ◽  
Fractional Order ◽  
Eddy Currents ◽  
Tracking Error ◽  
Measurement Data ◽  
Analytical Approximation ◽  
Fractional Order System ◽  
Order System ◽  
Model Accuracy ◽  
Magnetic Actuators

The electromagnetic dynamics of nonlaminated magnetic actuators are highly influenced by eddy currents and minor perturbations like core saturation, hysteresis as well as fringing and leakage fluxes. In the literature, analytical high-fidelity models describing these phenomena are known, which lead to complex reluctance networks or transcendental system descriptions with fractional-order characteristics. Therefore, they are not suitable for a direct implementation within the actuator control. Previously, we provided appropriate analytical rational approximations that allow a digital real-time implementation of these models on a microcontroller. However, the inclusion of the minor perturbations, if possible, leads to impractical model orders requiring simplifications, which compromise the model accuracy. This article studies numerical methods to reduce high model orders or directly approximate the transcendental systems or empirical measurement data. The greater degree of freedom allows for a possible higher model accuracy with sufficiently low orders. We review and improve existing approaches like Levy's method and Vector Fitting and apply them to the frequency response of the underlying fractional-order system. Furthermore we propose an order reduction algorithm based on a pole-zero-cancellation with tracking error compensation. Using measurement data, a comparison shows that the numerical approaches match or excel our previously studied analytical approximation.

Synchronization of a Fractional-Order System with Unknown Parameters

2013 ◽  
Vol 850-851 ◽  
pp. 796-799
Author(s):  
Xiao Ya Yang
Keyword(s):  
Numerical Simulation ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Stability Theory ◽  
Chaotic Attractor ◽  
Fractional Order System ◽  
Order System ◽  
Fractional Order Systems ◽  
Unknown Parameters ◽  
The Stability

In this paper, synchronization of a fractional-order system with unknown parameters is studied. The chaotic attractor of the system is got by means of numerical simulation. Then based on the stability theory of fractional-order systems, suitable synchronization controllers and parameter identification rules for the unknown parameters are designed. Numerical simulations are used to demonstrate the effectiveness of the controllers.

scholarly journals The Dynamics and Synchronization of a Fractional-Order System with Complex Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Xiaoya Yang ◽  
Xiaojun Liu ◽  
Honggang Dang ◽  
Wansheng He
Keyword(s):  
Fractional Order ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Complex Variables ◽  
Fractional Order System ◽  
Order System ◽  
Chaotic Attractors ◽  
Fractional Order Systems ◽  
System Parameters ◽  
The Stability

A fractional-order system with complex variables is proposed. Firstly, the dynamics of the system including symmetry, equilibrium points, chaotic attractors, and bifurcations with variation of system parameters and derivative order are studied. The routes leading to chaos including the period-doubling and tangent bifurcations are obtained. Then, based on the stability theory of fractional-order systems, the scheme of synchronization for the fractional-order complex system is presented. By designing appropriate controllers, the synchronization for the system is realized. Numerical simulations are carried out to demonstrate the effectiveness of the proposed scheme.

Initialization Energy in Fractional-Order Systems

2015 ◽  
Author(s):  
Tom T. Hartley ◽  
Jean-Claude Trigeassou ◽  
Carl F. Lorenzo ◽  
Nezha Maamri
Keyword(s):  
Laplace Transform ◽  
Fractional Order ◽  
Caputo Derivative ◽  
Fractional Order System ◽  
Order System ◽  
Time Varying ◽  
Fractional Order Systems ◽  
History Of ◽  
Infinite Energy ◽  
Order Element

This paper seeks a deeper understanding of the need for time-varying initialization of fractional-order systems. Specifically, the paper determines the energy stored in a fractional-order element based on the history of the element, and shows how this initialization energy is manifest into the future as an initialization function. Further, it is shown that infinite energy is required to initialize a fractional-order system when using the Caputo derivative Laplace transform.

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