Limit cycle oscillation in aeroelastic systems and its adaptive fractional-order fuzzy control
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ORIGINAL ARTICLE
Limit cycle oscillation in aeroelastic systems and its adaptive fractional‑order fuzzy control Guanjun Li1,2 · Jinde Cao2,3 · Ahmed Alsaedi4 · Bashir Ahmad4
Received: 14 August 2016 / Accepted: 18 January 2017 © Springer-Verlag Berlin Heidelberg 2017
Abstract Alaeroelastic system is a complex system which can produce limit cycles oscillation. In this paper, an adaptive fractional-order fuzzy controller is presented to suppress flutter in an alaeroelastic system. The studied system is a kind of nonlinear system with two freedoms (the plunge displacement and the pitch angle). A terminal sliding mode control is proposed, the fuzzy system parameters are updated by fractional-order differential equations and the stability of the closed-loop system is discussed by means of Lyapunov stability theory. Finally, numerical simulations are demonstrated to verify the effectiveness of proposed method. Keywords Alaeroelastic system · Limit cycle oscillation · Terminal sliding mode control · Adaptive fractional-order fuzzy control
1 Introduction In recent years, fractional-order systems and controllers, which are described by fractional-order differential * Jinde Cao [email protected] 1
Department of Applied Mathematics, Huainan Normal University, Huainan 232038, China
2
School of Mathematics, Southeast University, Nanjing 210096, China
3
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
4
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
or integral equations, have received more and more attention [1–6]. Fractional calculus is an extended concept of the classical cases. In [7–15], the dynamical properties of fractional-order systems were discussed. The superiority of a fractional-order system has been more accurately shown in modeling and controlling systems to reach more satisfying specifications. It is well known, in the stability analysis of integer-order systems, the quadratic Lyapunov function is usually used. However, the fractional-order derivative of a quadratic function has a very complicated form [16–19]. So, how to use quadratic Lyapunov function in the stability analysis of fractional-order systems is a challenging work. Limit cycle oscillations (LCO) widely exists in many mechanical and physical systems [20–22]. In these complex systems, the frequency of the LCOs may be very different from eigenfrequencies near the unstable states of equilibrium. For some structures, the LCOs can become extremely dangerous, mitigation of the LCOs were investigated in [23–27]. In [28], limit cycle calculation of nonlinear aeroelastic systems was studied by using a reduced order aeroelastic model. Literature [29] introduces a novel nonlinear method by using limit cycle oscillations that shows greater sensitivity for damage detection versus linear damage detection techniques. Many researchers have focused on the stabilization of aeroelast
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