This chapter presents a fractional adaptive interval type-2 fuzzy logic control strategy based on active fractional sliding mode controller (FAIT2FSMC) to synchronize tow chaotic fractional-order systems. The interval type-2 fuzzy logic systems (IT2FLS) are used to approximate the plant dynamics represented by unknown functions of the system, and the IT2F adaptation law adjusts the consequent parameters of the rules based on a Lyapunov synthesis approach. One of the main contributions in this work is the use of an IT2F and an adaptive fractional order PIλ control law to eliminate the chattering action in the control signal. Based on fractional order Lyapunov stability criterion, stability analysis is performed for the proposed method for an acceptable synchronization error level. The performance of the proposed scheme is demonstrated through the synchronization of two different fractional order chaotic gyro systems. Simulations are implemented using a numerical method based on Grünwald-Letnikov approach to solve the fractional differential equations.
TopIntroduction
Fractional order systems (Hilfer, 2001; Machado et al., 2011) have shown very attractive performances and properties, and there for many applications of such systems have been reported in different areas such as signal processing, image processing (Oustaloup, 1995), automatic control (Slotine, 1991), robotics (Duarte et al., 2002), and renewable energy. A great number of research works focused on fractional systems that display chaotic behavior like: Chua circuit (Hartley, 1995), Duffing system (Arena et al., 1997), Chen dynamic (Lu and Chen, 2006), characterization (Lu et al., 2006), Rössler system and Newton-Leipnik formulation (Sheu et al., 2008). Synchronization or control of these systems is a difficult task because a main characteristic of chaotic systems is their high sensitivity to initial conditions, but it is gathering more and more research effort due to several potential applications especially in cryptography (Hosseinnia et al., 2010).
Fractional adaptive control is a growing research topic gathering the interest of a great number of researchers and control engineers (Ladaci and Charef, 2012). The main argument of this community is the significant enhancement obtained with these new real-time controllers comparatively to integer order ones (Podlubny, 1999).
Since the pioneering works of Vinagre et al. (2002) and Ladaci and Charef (2002), an increasing number of works are published focusing on various fractional order adaptive schemes such as: fractional order model reference adaptive control (Chen et al., 2016; Ladaci & Charef, 2006); Wei, 2015), fractional order adaptive pole placement control (Ladaci & Bensafia, 2016), fractional high-gain adaptive control (Ladaci et al., 2008), fractional multi-model adaptive control(Ladaci and Khettab, 2012), robust fractional adaptive control (Ladaci, 2009), fractional extremum seeking control (Neçaibia et al., 2014), Fractional IMC-based adaptive control (Ladaci, 2014), fractional adaptive sliding mode control (Efe, 2008), fractional adaptive PID control (Ladaci & Charef, 2006; Neçaibia & Ladaci, 2014).