Parameter Identification of Fractional Order Systems Using a Collocation Method Based on Hybrid Functions

2020 ◽  
Vol 142 (8) ◽  
Author(s):  
Y. Lu ◽  
J. Zhang ◽  
Y. G. Tang
Keyword(s):  
Integral Operator ◽  
Collocation Method ◽  
Fractional Order ◽  
Analytical Form ◽  
Fractional Order Systems ◽  
Actual System ◽  
Hybrid Functions ◽  
Block Pulse Functions ◽  
The Laplace Transform ◽  
Fractional Order Integral

Abstract In this paper, we propose a novel collocation method based on hybrid functions to identify the parameters and differential orders of fractional order systems (FOS). The hybrid functions consist of block-pulse functions and Taylor polynomials. The analytical form of Riemann–Liouville fractional order integral operator of these hybrid functions is derived using the Laplace transform. Then the integral operator is utilized, in conjunction with collocation points, to convert the FOS into an algebraic system directly. The parameters and differential orders of the FOS are estimated by minimizing the error between the output of the actual system and that of the estimated system. The effectiveness of the proposed method is verified through four examples.

Fractional Integrative Operator and Its FPGA Implementation

2009 ◽  
Author(s):  
R. Caponetto ◽  
G. Dongola ◽  
A. Gallo
Keyword(s):  
Integral Operator ◽  
Fractional Order ◽  
Hardware Implementation ◽  
Careful Consideration ◽  
Fractional Order Systems ◽  
Hardware Cost ◽  
Field Programmable ◽  
Digital Hardware ◽  
Real Positive Number ◽  
Fractional Order Integral

In this paper the fractional order integrative operator s−m, where m is a real positive number, is approximated via a mathematical formula and then an hardware implementation of fractional integral operator is proposed using Field Programmable Gate Array (FPGA). Digital hardware implementation of fractional-order integral operator requires careful consideration of issue of system performance, hardware cost, and hardware speed. FPGA-based implementation are up to one hundred times faster than implementations based on micro-processors; this extra speed can be exploited to allow higher performance in terms of digital approximations of fractional-order systems.

scholarly journals From fractional order equations to integer order equations

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Daniel Cao Labora ◽  
Rosana Rodríguez-López
Keyword(s):  
Integral Equation ◽  
Integral Operator ◽  
Vector Space ◽  
Integral Equations ◽  
Fractional Order ◽  
Fractional Integral ◽  
State Of The Art ◽  
Constant Coefficients ◽  
Fractional Order Integral ◽  
Ordinary Integral

AbstractThe main goal of this article is to show a new method to solve some Fractional Order Integral Equations (FOIE), more precisely the ones which are linear, have constant coefficients and all the integration orders involved are rational. The method essentially turns a FOIE into an Ordinary Integral Equation (OIE) by applying a suitable fractional integral operator.After discussing the state of the art, we present the idea of our construction in a particular case (Abel integral equation). After that, we propose our method in a general case, showing that it does work when dealing with a family of “additive” operators over a vector space. Later, we show that our construction is always possible when dealing with any FOIE under the above-mentioned hypotheses. Furthermore, it is shown that our construction is “optimal” in the sense that the OIE that we obtain has the least possible order.

scholarly journals A Numerical Method for Lane-Emden Equations Using Hybrid Functions and the Collocation Method

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Changqing Yang ◽  
Jianhua Hou
Keyword(s):  
Differential Equation ◽  
Numerical Method ◽  
Collocation Method ◽  
Chebyshev Polynomials ◽  
Initial Value Problems ◽  
Algebraic Equations ◽  
Initial Value ◽  
Numerical Examples ◽  
Hybrid Functions ◽  
Block Pulse Functions

A numerical method to solve Lane-Emden equations as singular initial value problems is presented in this work. This method is based on the replacement of unknown functions through a truncated series of hybrid of block-pulse functions and Chebyshev polynomials. The collocation method transforms the differential equation into a system of algebraic equations. It also has application in a wide area of differential equations. Corresponding numerical examples are presented to demonstrate the accuracy of the proposed method.

scholarly journals Numerical solution of fractional differential equation by wavelets and hybrid functions

2018 ◽  
Vol 36 (2) ◽  
pp. 231 ◽  
Author(s):  
Amir Hosein Refahi Sheikhani ◽  
Mahamad Mashoof
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Legendre Polynomials ◽  
Linear Equations ◽  
Operational Matrix ◽  
Numerical Examples ◽  
Hybrid Functions ◽  
Block Pulse Functions ◽  
Fractional Differential

In this paper, we introduce methods based on operational matrix of fractional order integration for solving a typical n-term non-homogeneous fractional differential equation (FDE). We use Block pulse wavelets matrix of fractional order integration where a fractional derivative is defined in the Caputo sense. Also we consider Hybrid of Block-pulse functions and shifted Legendre polynomials to approximate functions. By uses these methods we translate an FDE to an algebraic linear equations which can be solve. Methods has been tested by some numerical examples.

Synchronization in Integer and Fractional Order Chaotic Systems

2011 ◽  
pp. 127-151
Author(s):  
Ahmed E. Matouk
Keyword(s):  
Lyapunov Function ◽  
Fractional Order ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Lyapunov Stability Theory ◽  
Fractional Order Systems ◽  
Chen System ◽  
Van Der Pol ◽  
Coupling Technique ◽  
The Laplace Transform

In this chapter, the author introduces the basic methods of chaos synchronization in integer order systems, such as Pecora and Carroll method and One-Way coupling technique, applying these synchronization methods to the modified autonomous Duffing-Van der Pol system (MADVP). The conditional Lyapunov exponents (CLEs) are also calculated for the drive and response MADVP systems which match with the analytical results given by Pecora and Carroll method. Based on Lyapunov stability theory, chaos synchronization is achieved for two coupled MADVP systems by finding a suitable Lyapunov function. Moreover, synchronization in fractional order chaotic systems is also introduced. The conditions of Pecora and Carroll method and One-Way coupling method in fractional order systems are also investigated. In addition, chaos synchronization is achieved for two coupled fractional order MADVP systems using One-Way coupling technique. Furthermore, synchronization between two different fractional order chaotic systems is studied; the fractional order Lü system is controlled to be the fractional order Chen system. The analytical conditions for the synchronization of this pair of different fractional order chaotic systems are derived by utilizing the Laplace transform theory. Numerical simulations are carried out to show the effectiveness of all the proposed synchronization techniques.

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