Fuzzy logic embedding of fractional order sliding mode and state feedback controllers for synchronization of uncertain f
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Fuzzy logic embedding of fractional order sliding mode and state feedback controllers for synchronization of uncertain fractional chaotic systems Seyed Mehdi Abedi Pahnehkolaei1
· Alireza Alfi1
· J.A. Tenreiro Machado2
Received: 21 December 2019 / Revised: 5 May 2020 / Accepted: 21 May 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract This paper studies the synchronization of a class of uncertain fractional order (FO) chaotic systems that is applicable in secure communication. A novel hybrid FO controller, based on sliding mode and state feedback techniques combined with fuzzy logic, is developed. The algorithm, derived via the fractional Lyapunov theory, guarantees the stability of the overall system and the convergence of the synchronization errors toward a small residual set. Simulations demonstrate the capability of the proposed control algorithm in secure communications, not only in terms of speed of response, but also by reducing the chattering phenomenon. Keywords Chaotic systems · Fractional order · Synchronization · Secure communication · Sliding mode · Fuzzy logic Mathematics Subject Classification 93D05
1 Introduction Chaotic systems (CS) involve nonlinear dynamical phenomena characterized by an unpredictable behavior and being extremely sensitive to initial conditions (Sundarapandian and Suresh 2011), often called as the butterfly effect (Alligood et al. 1997). The study of CS is motivated by the fact that a variety natural and man-made systems exhibit a chaotic nature (Alfi 2012).
Communicated by Vasily E. Tarasov.
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Alireza Alfi [email protected] Seyed Mehdi Abedi Pahnehkolaei [email protected]
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Faculty of Electrical and Robotic Engineering, Shahrood University of Technology, Shahrood 36199-95161, Iran
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Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, Rua Dr. António Bernardino de Almeida 431, 4249-015 Porto, Portugal 0123456789().: V,-vol
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S. M. Abedi Pahnehkolaei et al.
Fractional calculus (FC) is the area of mathematics that generalizes integration and differentiation to arbitrary orders. The FC remained as a pure mathematical exercise for centuries, but with the development of natural sciences (Rajendran et al. 2017; Li et al. 2017; Moghaddam et al. 2018, 2019; Ding and Nieto 2018), the theory of fractional differential equations attracted recently considerable attention. The CS that emerge in interdisciplinary fields often exhibit memory characteristics. Indeed, some examples arise in the description of viscoelasticity (Zaky and Machado 2017), hydro-turbine machines (Xu et al. 2015), financial dynamics (Xin and Zhang 2015), electromagnetic waves (Baleanu et al. 2009), dielectric polarization (Bohannan 2000), single-species model (Li et al. 2017), and ethanol prices dynamic (David et al. 2018). From the control perspective, the stability and the stabilization of FO systems with different properties are an important topic (Dabiri et al. 2018; Yang et al. 2019; Badri and Sojood
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