A non-integer sliding mode controller to stabilize fractional-order nonlinear systems
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RESEARCH
(2020) 2020:503
Open Access
A non-integer sliding mode controller to stabilize fractional-order nonlinear systems Ahmadreza Haghighi1* and Roveida Ziaratban2 *
Correspondence: [email protected] 1 Department of Mathematics, Technical and Vocational University, Tehran, Iran Full list of author information is available at the end of the article
Abstract In this study, we examine the stabilization of fractional-order chaotic nonlinear dynamical systems with model uncertainties and external disturbances. We used the sliding mode controller by a new approach for controlling and stabilization of these systems. In this research, we replaced a continuous function with the sign function in the controller design and the sliding surface to suppress chattering and undesirable vibration effects. The advantages of the proposed control method are rapid convergence to the equilibrium point, the absence of chattering and unwanted oscillations, high resistance to uncertainties, and the possibility of applying this method to most fractional order chaotic systems. We applied the direct method of Lyapunov stability theory and the frequency distributed model to prove the stability of the slip surface and closed loop system. Finally, we simulated this method on two commonly used and practical chaotic systems and presented the results. Keywords: Fractional-order system; Uncertainty; Chattering; Lyapunov theory; Sliding mode control; Frequency distributed model
1 Introduction Fractional-order calculations play an important role in various scientific fields. Recently the application of fractional-order is known as an important topic in engineering [1]. The problem of fractional-order equations was first raised by Leibniz in a letter in September 1695 on the fractional-order derivative, and has become an issue for research that is still under investigation [2]. This branch of science had long been a theoretical subject, but there was no application to it. Deficit computing have attracted the interest of many scholars in recent decades [3]. Scientists have recently shown that fractional order equations are capable of modeling different phenomena more accurately than the integer-order equations and are a powerful tool for describing the structures of a system with complex dynamics. Most systems in nature obey fractional dynamics, and their approximations are considered integer. For example, Brownian fractional motion [4], porous media dynamics [5], time-lapse random walk theory [6], heat transfer process [7], electrochemical processes and flexible structures [8] and chaos theory are among these. The ability of fractional order calculators to improve the performance of controllers has been demonstrated [9]. In 1988, Oustaloup introduced a robust fractional-order controller called Crown, and created a starting point for entering fractional-order relationships into control. Then in © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distrib
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