Design of Nonlinear Conformable Fractional-Order Sliding Mode Controller for a Class of Nonlinear Systems
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Design of Nonlinear Conformable Fractional-Order Sliding Mode Controller for a Class of Nonlinear Systems Sara Haghighatnia1 · Heydar Toossian Shandiz2 Received: 6 January 2019 / Revised: 19 April 2019 / Accepted: 26 April 2019 © Brazilian Society for Automatics--SBA 2019
Abstract The paper aims to develop a new fractional-order nonlinear sliding mode controller with conformable fractional-order (CFO) derivative for a class of uncertain non-autonomous systems considering disturbance. The control method involves a novel CFO nonlinear surface as well as a new CFO switching function. The stability of the CFO controller is proved by means of the Lyapunov stability theorem. The simulation results show the improvement in the developed controller in terms of faster convergence speed, chattering reduction and lower control effort. Moreover, it needs simple calculations. Keywords Conformable fractional-order derivative · Fractional Lyapunov stability · Fractional-order sliding mode control · Nonlinear systems · Robust control
1 Introduction Nowadays, because of the benefits of fractional-order (FO) calculus, FO controller design has become an interesting issue in control engineering. However, it has some problems such as complexity in calculations. In 2014, conformable fractional-order (CFO) derivative was presented by Khali et al. CFO has turned out to be superior over other definitions because of simplicity in calculations and some properties of integer operators such as chain rule, Leibniz rule, the product rule and the formula of the quotient which cannot be satisfied by other fractional definitions (Khalil et al. 2014; Abdeljawad 2015). CFO operators soon attracted a lot of attention. For instance, the solutions of some types of CFO equation were obtained in Ilie et al. (2018), Khalil and Abu-Shaab (2015) and Chung (2015a). Some features of conformable derivative were presented in Abdeljawad (2015). FO differential equations with three-point boundary and initial values were investigated based on CFO in Batarfi et al. (2015). In Çenesiz et al. (2016), some CFO nonlinear partial differential equations were solved. The advantages of CFO derivative were
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Heydar Toossian Shandiz [email protected]
1
Faculty of Electrical and Robotic Engineering, Shahrood University of Technology, Shahrood, Iran
2
Electrical Department, Engineering Faculty, Ferdowsi University of Mashhad, Mashhad, Iran
shown in Avci et al. (2017). Some CFO properties were developed under interval uncertainty in Salahshour et al. (2015). In Chung (2015b), fractional Newtonian mechanics were investigated using CFO calculus. Stability of CFO systems was analysed in Rezazadeh et al. (2017). In He et al. (2017), the behaviour of CFO chaos systems was studied by means of the numerical solution of their equations. In addition, conformable Adomian decomposition method was introduced as an efficient way for numerical solution of CFO differential equations. CFO chaotic systems were discussed in Wang (2018). FO regularised long-wave equations were investigated
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