228 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 6, NO. 1, JANUARY 2019 Robust Finite-time Synchronization of Non-identical Fractional-order Hyperchaotic Systems and Its Application in Secure Communication Hadi Delavari and Milad Mohadeszadeh Abstract—This paper proposes a novel adaptive sliding mode control (SMC) method for synchronization of non-identical fractional-order (FO) chaotic and hyper-chaotic systems. Under the existence of system uncertainties and external disturbances, finite-time synchronization between two FO chaotic and hyper- chaotic systems is achieved by introducing a novel adaptive sliding mode controller (ASMC). Here in this paper, a fractional sliding surface is proposed. A stability criterion for FO nonlinear dynamic systems is introduced. Sufficient conditions to guarantee stable synchronization are given in the sense of the Lyapunov stability theorem. To tackle the uncertainties and external dis- turbances, appropriate adaptation laws are introduced. Particle swarm optimization (PSO) is used for estimating the controller parameters. Finally, finite-time synchronization of the FO chaotic and hyper-chaotic systems is applied to secure communication. Index Terms—Adaptive sliding mode control (ASMC), chaos synchronization, fractional order (FO), hyper-chaotic system, Lyapunov theorem, secure communication. I. I NTRODUCTION C HAOTIC behavior is a prevalent phenomenon appearing in nonlinear systems. Chaotic systems have received more attention in the literature during the last three decades. A chaotic system is a nonlinear deterministic system that has complex and unpredictable behavior. Fractional calculus is a mathematical topic more than three centuries old, but its application to physics and engineering fields have attracted more attention only in recent years [1]− [3]. This happens because it has been recently found that several physical phenomena can be more adequately described by fractional differential equations rather than integer-order models [4], and it has been found that many FO systems can show complex dynamical behavior such as chaos. The advantages of the FO systems are that there are more degrees of freedom in the model. Also memory is included in FO systems. Many systems in interdisciplinary fields, such as Manuscript received September 22, 2015; revised December 29, 2015; accepted February 1, 2016. Recommended by Associate Editor YangQuan Chen. (Corresponding author: Hadi Delavari.) Citation: H. Delavari and M. Mohadeszadeh, “Robust finite-time syn- chronization of non-identical fractional-order hyperchaotic systems and its application in secure communication,” IEEE/CAA J. Autom. Sinica, vol. 6, no. 1, pp. 228-235, Jan. 2019. H. Delavari and M. Mohadeszadeh are with the Electrical Engineering Department, Hamedan University of Technology, Hamedan, CO 65155, Iran (e-mail: delavari@hut.ac.ir; m.mohadeszadeh@stu.hut.ac.ir) Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JAS.2016.7510145 viscoelastic materials [5] and micro-electromechanical systems [6] can be described using fractional calculus methods. Recently many researchers have recognized that many com- plex systems, such as FO Lorenz system [7], FO Chen system [8] and FO Arnodo-Coullet system [9], can be described using fractional integrals and derivatives. Since Pecora and Carroll [10] established a chaos syn- chronization scheme for two identical chaotic systems with different initial conditions, chaos synchronization has attracted a great attention. The chaotic synchronization occurs whenever the state trajectories of the slave system track the state trajec- tories of the master system in a given finite-time [11], [12]. Chaos synchronization is a contemporary topic in nonlinear science because of its broad and considerable applications in secure communication, automatic control, neural networks and etc. [13]−[15]. Due to the existence of chaos in real practical systems and many applications in physics and engineering fields, control and synchronization of FO chaotic systems have attracted many researchers attention in the past few years [16]−[23]. In [24], an active sliding mode approach for synchronization of FO chaotic system is proposed. The FO Novel and Chen hyper-chaotic systems are proposed for synchronization in [25], where the states of the FO hyper-chaotic Novel system are used to control the states of the FO hyper-chaotic Chen system. Several methods have been proposed to achieve chaos synchronization such as adaptive feedback control, adaptive impulsive control, sliding mode control, active control, back- stepping design and optimal control [26]−[36]. Most of the published papers focus on asymptotic stability which leads to infinite-time synchronization, but in practical applications, finite-time synchronization is more valuable than infinite-time synchronization. Also, most of the researches are related to synchronization between two chaotic systems without uncertainty or two identical chaotic systems, but in a real control system, due to the limitations of physical devices and the effect of interference (such as noise, temperature, etc.), uncertainties are unavoidable. Motivated by the above discussion, a novel adaptive sliding mode control approach for synchronization of a class of new FO chaotic system and a FO hyper-chaotic system is proposed. In our contribution we pursue five main research aims. First, the proposed approach is very simple and easily realized experimentally for secure communication. Second, the proposed controller can be applied for a width range of systems and is more suitable for engineering applications.