This chapter addresses the fuzzy adaptive controller design for the generalized projective synchronization (GPS) of incommensurate fractional-order chaotic systems with actuator nonlinearities. The considered master-slave systems are with different fractional-orders, uncertain models, unknown bounded disturbances, and non-identical form. The suggested controller includes two keys terms, namely a fuzzy adaptive control and a fractional-order variable structure control. The fuzzy logic systems are exploited for approximating the system uncertainties. A Lyapunov approach is employed for determining the parameter adaptation laws and proving the stability of the closed-loop system. At last, simulation results are given to demonstrate the validity of the proposed synchronization approach.
TopIntroduction
Throughout the last decades, fractional-order (nonlinear or linear) plants (i.e. the systems with fractional integrals or derivatives) have been studied by several works in many branches of engineering and sciences (Podlubny,1999; Hilfer,2000). It turned out that several plants, in interdisciplinary research areas, may present fractional-order dynamics including: fluid mechanics, spectral densities of music, transmission lines, cardiac rhythm, electromagnetic waves, viscoelastic systems, dielectric polarization, heat diffusion systems, electrode-electrolyte polarization, and many others (Baleanu et al., 2010; Sabatier et al., 2007; Bouzeriba et al.,2016a; Boulkroune et al.,2016).
Chaotic systems are deterministic and nonlinear dynamical plants. They are also characterized by the self similarity of the strange attractor and extreme sensitivity to initial conditions (IC) quantified respectively by fractal dimension and the existence of a positive Lyapunov exponent (Bouzeriba et al.,2016a ; Boulkroune et al.,2016). In recent works, it was made known that several fractional-order systems may perform chaotically, e.g. fractional-order Lü system (Deng & Li, 2005), fractional-order Arneodo system (Lu,2005), fractional-order Lorenz system (Grigorenk & Grigorenko,2003), fractional-order Rössler system (Li & Chen,2004), and so on.
The sliding mode control technique is an effective tool to construct robust adaptive controllers for nonlinear systems with bounded external disturbances and uncertainties. The later has several attractive features, including finite and fast time convergence, strong robustness with respect to unmodelled dynamics, parameters variations and external disturbances. The purpose of the sliding mode is extremely simple: it consists to oblige the system states to arrive at a suitably desired sliding surface based on a discontinuous control. Recently, the fractional-order calculus is employed within the sliding mode control methodologies, in order to seek better performances. Many works have coped with control problems of nonlinear systems with fractional-orders (Calderon et al.,2006; Efe & Kasnakoglu,2008; Si-Ammour et al.,2009; Pisano et al.,2010). It is worthy to note that the selection of the sliding surface for this class of systems is not an easy task in general. In numerous recent researches, the fuzzy logic system is combined with the sliding mode control in order to remove the main issues of the sliding mode control, including the high-gain authority and chattering in the system. This hybridization can smoothen the sliding mode control in diverse ways, and can also successfully approximate online the model, uncertainties and disturbances present in the system (Bouzeriba et al.,2016a; Boulkroune et al.,2016a).