Restrictions on the inverse Laplace transform for fractional-order systems

2014 ◽  
Author(s):  
Jay L. Adams ◽  
Robert J. Veillette ◽  
Tom T. Hartley ◽  
Lynn I. Adams
Keyword(s):  
Laplace Transform ◽  
Fractional Order ◽  
Inverse Laplace Transform ◽  
Fractional Order Systems

On the approximate inverse Laplace transform of the transfer function with a single fractional order

2020 ◽  
pp. 014233122097766
Author(s):  
Ali Yüce ◽  
Nusret Tan
Keyword(s):  
Transfer Function ◽  
Laplace Transform ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Transfer Functions ◽  
Inverse Laplace Transform ◽  
Fractional Order Systems ◽  
Approximate Inverse ◽  
Classical Mathematics ◽  
Curve Fitting Method

The history of fractional calculus dates back to 1600s and it is almost as old as classical mathematics. Although many studies have been published on fractional-order control systems in recent years, there is still a lack of analytical solutions. The focus of this study is to obtain analytical solutions for fractional order transfer functions with a single fractional element and unity coefficient. Approximate inverse Laplace transformation, that is, time response of the basic transfer function, is obtained analytically for the fractional order transfer functions with single-fractional-element by curve fitting method. Obtained analytical equations are tabulated for some fractional orders of [Formula: see text]. Moreover, a single function depending on fractional order alpha has been introduced for the first time using a table for [Formula: see text]. By using this table, approximate inverse Laplace transform function is obtained in terms of any fractional order of [Formula: see text] for [Formula: see text]. Obtained analytic equations offer accurate results in computing inverse Laplace transforms. The accuracy of the method is supported by numerical examples in this study. Also, the study sets the basis for the higher fractional-order systems that can be decomposed into a single (simpler) fractional order systems.

Stability of Nonlinear Fractional-Order Time Varying Systems

2015 ◽  
Vol 11 (3) ◽  
Author(s):  
Sunhua Huang ◽  
Runfan Zhang ◽  
Diyi Chen
Keyword(s):  
Laplace Transform ◽  
Fractional Order ◽  
Caputo Derivative ◽  
Local Stability ◽  
State Feedback ◽  
Sufficient Condition ◽  
Time Varying ◽  
Fractional Order Systems ◽  
Time Varying Systems ◽  
The Stability

This paper is concerned with the stability of nonlinear fractional-order time varying systems with Caputo derivative. By using Laplace transform, Mittag-Leffler function, and the Gronwall inequality, the sufficient condition that ensures local stability of fractional-order systems with fractional order α : 0<α≤1 and 1<α<2 is proposed, respectively. Moreover, the condition of the stability of fractional-order systems with a state-feedback controller is been put forward. Finally, a numerical example is presented to show the validity and feasibility of the proposed method.

Stability analysis of fractional-order systems with double noncommensurate orders for matrix case

2011 ◽  
Vol 14 (3) ◽  
Author(s):  
Zhuang Jiao ◽  
YangQuan Chen
Keyword(s):  
Fractional Order ◽  
Linear Time ◽  
Simple Algorithm ◽  
Inverse Laplace Transform ◽  
Order System ◽  
Necessary Condition ◽  
Fractional Order Systems ◽  
Numerical Inverse Laplace Transform ◽  
Laplace Transform Technique ◽  
Matrix Case

AbstractBounded-input bounded-output stability issues for fractional-order linear time invariant (LTI) system with double noncommensurate orders for the matrix case have been established in this paper. Sufficient and necessary condition of stability is given, and a simple algorithm to test the stability for this kind of fractional-order systems is presented. Based on the numerical inverse Laplace transform technique, time-domain responses for fractional-order system with double noncommensurate orders are shown in numerical examples to illustrate the proposed results.

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.

Further Remarks on the Existence of Periodic Solutions of Linear Time Varying Periodic Fractional Order Systems

2017 ◽  
Author(s):  
XueFeng Zhang ◽  
YangQuan Chen
Keyword(s):  
Laplace Transform ◽  
Periodic Solutions ◽  
Fractional Order ◽  
Dynamic Systems ◽  
Linear Time ◽  
Periodic Systems ◽  
Time Varying ◽  
Fractional Order Systems ◽  
Order Linear ◽  
Existence Of Periodic Solutions

Existence of periodic solutions of fractional order dynamic systems is an important and difficult issue in fractional order systems field. In this paper, the non existence of completely periodic solutions and existence of partly periodic solutions of fractional order linear time varying periodic systems and fractional order nonlinear time varying periodic systems are discussed. A new property of Laplace transform of periodic function is derived. The non existences of completely periodic solutions of fractional order linear time varying periodic systems and fractional order nonlinear time varying periodic fractional order systems are presented by Laplace transform method and contradiction approach. The existence of partly periodic solutions of fractional order dynamic systems are proved by constructing numerical examples and considering Laplace transform property approaches. The examples and state figures are given to illustrate the effectiveness of conclusion presented.

scholarly journals Parameterizing Generalized Laguerre Functions to Compute the Inverse Laplace Transform of Fractional Order Transfer Functions

MENDEL ◽  
2018 ◽  
Vol 24 (1) ◽  
pp. 79-84
Author(s):  
Vilem Karsky
Keyword(s):  
Laplace Transform ◽  
Fractional Order ◽  
Transfer Functions ◽  
Operator Space ◽  
Inverse Laplace Transform ◽  
Laguerre Functions ◽  
Two Systems ◽  
Novel Method

This article concentrates on using generalized Laguerre functions to compute the inverse Laplace transform of fractional order transfer functions. A novel method for selecting the timescale parameter of generalized Laguerre functions in the operator space is introduced and demonstrated on two systems with fractional order transfer functions.

scholarly journals Numerical Approximation of Fractional-Order Volterra Integrodifferential Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Xiaoli Qiang ◽  
Kamran ◽  
Abid Mahboob ◽  
Yu-Ming Chu
Keyword(s):  
Numerical Methods ◽  
Laplace Transform ◽  
Fractional Order ◽  
Integrodifferential Equations ◽  
Inverse Laplace Transform ◽  
Laplace Domain ◽  
The Real ◽  
Volterra Integrodifferential Equation ◽  
Engineering Sciences ◽  
Real Domain

Laplace transform is a powerful tool for solving differential and integrodifferential equations in engineering sciences. The use of Laplace transform for the solution of differential or integrodifferential equations sometimes leads to the solutions in the Laplace domain that cannot be inverted to the real domain by analytic methods. Therefore, we need numerical methods to invert the solution to the real domain. In this work, we construct numerical schemes based on Laplace transform for the solution of fractional-order Volterra integrodifferential equations in the sense of the Atangana-Baleanu Caputo derivative. We propose two numerical methods for approximating the solution of fractional-order linear and nonlinear Volterra integrodifferential equations. In our scheme, the inverse Laplace transform is approximated using a contour integration method and Stehfest method. Some numerical experiments are performed to check the accuracy and efficiency of the methods. The results obtained using these methods are compared.

scholarly journals Fractional-Order Controller Design for Oscillatory Fractional Time-Delay Systems Based on the Numerical Inverse Laplace Transform Algorithms

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Lu Liu ◽  
Feng Pan ◽  
Dingyu Xue
Keyword(s):  
Time Delay ◽  
Laplace Transform ◽  
Fractional Order ◽  
Control Method ◽  
Delay System ◽  
Inverse Laplace Transform ◽  
Delay Systems ◽  
Time Delay System ◽  
Numerical Inverse Laplace Transform ◽  
Fractional Order Controller

Fractional-order time-delay system is thought to be a kind of oscillatory complex system which could not be controlled efficaciously so far because it does not have an analytical solution when using inverse Laplace transform. In this paper, a type of fractional-order controller based on numerical inverse Laplace transform algorithm INVLAP was proposed for the mentioned systems by searching for the optimal controller parameters with the objective function of ITAE index due to the verified nature that fractional-order controllers were the best means of controlling fractional-order systems. Simulations of step unit tracking and load-disturbance responses of the proposed fractional-order optimalPIλDμcontroller (FOPID) and corresponding conventional optimal PID (OPID) controller have been done on three typical kinds of fractional time-delay system with different ratio between time delay (L) and time constant (T) and a complex high-order fractional time delay system to verify the availability of the presented control method.

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