A Numerical Method for a System of Fractional Differential-Algebraic Equations Based on Sliding Mode Control

Yongpeng Tai,Lijin Wang,Zaiyong Feng,Jun Xu,Ning Chen

doi:10.3390/math8071134

Abstract

Fractional calculus is widely used in engineering fields. In complex mechanical systems, multi-body dynamics can be modelled by fractional differential-algebraic equations when considering the fractional constitutive relations of some materials. In recent years, there have been a few works about the numerical method of the fractional differential-algebraic equations. However, most of the methods cannot be directly applied in the equations of dynamic systems. This paper presents a numerical algorithm of fractional differential-algebraic equations based on the theory of sliding mode control and the fractional calculus definition of Grünwald–Letnikov. The algebraic equation is considered as the sliding mode surface. The validity of the present method is verified by comparing with an example with exact solutions. The accuracy and efficiency of the present method are studied. It is found that the present method has very high accuracy and low time consumption. The effect of violation corrections on the accuracy is investigated for different time steps.

Highlights

Fractional calculus (FC) means non-integer order derivative or integral of the variable function, which as an extension of the classical calculus theory, has become an important branch of differential equations [1]
Fractional differential-algebraic equations (FDAEs) are used to describe many physical and engineering problems, which consist of fractional differential equations and algebraic equations
Multi-body dynamics are usually described by differential-algebraic equations (DAEs) [12,13,14]

Summary

Introduction

Fractional calculus (FC) means non-integer order derivative or integral of the variable function, which as an extension of the classical calculus theory, has become an important branch of differential equations [1]. It has been confirmed that many physical phenomena can be described more accurately by using fractional calculus than classical integer-order derivative [2]. The research and application of fractional calculus theory have become more and more popular in several fields of mathematics, physics, mechanics, control, electrochemistry and engineering, etc. If some physical characteristics of viscoelastic damping materials are modeled by the fractional constitutive relationship, a group of FDAEs are set up for the system

Methods

Results

Conclusion