?h(u(t))sgn s ≥ζ sgn s.

1 Introduction

Since the pioneering work of Pecora and Carroll[1], the synchronization of chaotic systems has attracted increasing interests among many researchers, due to its useful applications in secure communication, power convertors, biological systems, information processing and chemical reactions[2-9]. By now,a wide variety of control techniques have been successfully applied to synchronize chaotic systems. Zhong et al[10]has considered the synchronization control problem for fractional-order systems based on the motive sliding mode approach. Sun et al[11]has obtained the synchronization for a class of multi-scroll chaos systems, and a self-adaptive sliding mode control project has been derived. While, from a practical point of view, it is more advantageous to synchronize chaotic systems within a finite time rather than merely asymptotically. To obtain fast convergence in a control system, the finite-time control method is an effective technique. Besides, the finite-time control techniques have demonstrated better robustness and disturbance rejection properties[12]. In recent years, some researchers have applied finite-time control techniques, Yu and Zhang[13]has used finite-time control strategies to synchronize two chaotic systems with uncertainty.

On the other hand, it has been recognized that many systems in interdisciplinary fields can be elegantly described by using fractional-order differential equations. The chaos synchronization of fractional-order systems is in great need of engineering and applications. Mao and Cheng[14]has studied the self-adaptive sliding mode synchronization issue. The chaos synchronization problem of fractional-order complex network systems has been proposed in [15]. However, the synchronization problem of the fractional-order systems in finite-time has not been dissolved and it still remains as an open and challenging problem. In practice, the effect of the dead-zone nonlinearity in control inputs can not be neglected in designing and implementing controller. Tian et al[16]has considered the finite-time synchronization problem with dead-zone input and its stability and convergence in a given finite time have been mathematically proved.

Motivated by the above discussions, in this paper, the problem of finite-time synchronization is investigated for fractional-order Victor-Carmen systems with dead-zone input. A novel fractional-order nonsingular terminal sliding surface is proposed and its finite time stability is proved. Then, on the basis of the fractional-order Lyapunov stability theory, a robust sliding control law is derived to guarantee the occurrence of the sliding motion in finite time. An estimation of the convergence time is also given.Numerical simulations demonstrate the applicability and efficiency of the proposed fractional terminal sliding mode control technique and verify the theoretical results of the paper.

2 System description

In this paper, we will use the Riemann-Liouville fractional derivative. And for the reader’s convenience, we state its definition as follows.

Definition 1[17]The Riemann-Liouville fractional derivative of order α of function f(t) is defined as

where Γ(·) is the Gamma function and t0is the initial time.

For convenience, we denote0Dαtby Dαtin what follows.

Consider the following Victor-Carmen system as the master system

where q∈(0,1), x=(x1,x2,x3)T∈R3is the system state vector of the master system,a, b, α, β, γ are the parameters, and chaos occurs in the system when α = 50, β =20, γ =4.1, a=5, b=9, q =0.873.

The slave system is presented as follows

where y =(y1,y2,y3)Tis the system state vector of (2), ?fi(y):R3→R is the model uncertainty,di(t)is the external disturbance,ui(t)is the controller to be designed later,and hi(ui(t)) is the dead-zone input determined by

where h+i(·), h?i(·)(i = 1,2,3) are nonlinear functions of ui(t), u+i,u?i(i = 1,2,3)are given constants satisfying the constraint

where β+i,β?i(i=1,2,3) are given constants.

Assumption 1 Assume that ?fi(y)(i=1,2,3)and di(t)(i=1,2,3)are bounded by

where δiand ρiare given positive constants.

To solve the finite-time synchronization problem,the error between the master and slave systems is defined as e = y ?x = (e1,e2,e3)T, Therefore, the error dynamics is obtained as follows

Lemma 1[18]Assume that a continuous, positive-definite function V(t) satisfies the following differential inequality ˙V(t)≤?pVn(t), ?t ≥t0, V(t0)≥0, where p>0 and η ∈(0,1) are two positive constants. Then, for any given t0, V(t) satisfies the following inequality

and

Lemma 2[17]Assume that p>q ≥0 and 0 ≤m ?1 ≤p

Lemma 3[17]Assume that p, q ≥0 and 0 ≤m ?1 ≤p

holds in Riemann-Liouville fractional derivatives, where m and n are two integers.

3 Main results

Generally,the design of a sliding mode controller for stabilizing the fractional order error system (3) has two steps. First, an appropriate sliding surface with the desired dynamics need to be constructed. Second, a robust control law is designed to ensure the existence of the sliding motion.

In this paper, a novel nonsingular terminal sliding surface is introduced as

where λ, μ>0.

When the system trajectories arrive the sliding surface, it follows that si(t) = 0 and ˙si(t) = 0. Taking the time derivative of the sliding surface (4), the sliding mode dynamics is obtained as follows

That is, the sliding mode dynamics is obtained as

Theorem 1The terminal sliding mode dynamics (5) is stable and its state trajectories converge to zero in the finite time T1, given by

Then it follows that

Multiplying both sides by e2λt, we have

Integrating both sides of the above equality from 0 to t, it is obvious that

we get

Thus, the proof is completed.

Once the appropriate sliding function has been selected, the next step is to design a control law which can steer the state trajectories onto the sliding mode surface in a given time and remain on it forever. A finite-time control law is proposed as follows

where kiis a positive constant, and σi= δi+ρi, λ, μ are designed in (5). Then the terminal sliding mode dynamics (5) is stable and its state trajectories converge to zero in a finite time T2.

Theorem 2Consider the error systems (3) with dead-zone nonlinear inputs.Assume that the controller of the systems is chosen as(7),then the systems trajectories will converge to the sliding surface si=0 in a finite time T2, given by

?hi(ui(t))sgn si≥ζisgn2si.

Multiplying both sides by |si|, we get

When si>0, through a similar operation, the inequality (9) still holds. Substituting (9) into (8), then we can deduce that

where k = min{k1,k2,k3}. Thus, according to Lemma 1, the system trajectories will converge to the sliding surfaces si=0, in the finite time

Therefore, this proof is completed.

4 Numerical simulations

In this section, numerical examples are presented to demonstrate the effectiveness and usefulness of the proposed finite-time control technique in synchronizing two different chaotic systems with dead-zone inputs.

Assume that the systems appear chaos attractors. We choose the parameters α=50, β =20, γ =4.1, a=5, b=9, q =0.873. In addition, the following uncertainties are considered in the simulations

The constants are set to β+i= 0.4, β?i= 0.5, βi= 0.4, γi= 2.5, λ = 1, μ = 0.5.The initial values of the systems are randomly selected as x(0)=(1,?2,?2)T, y(0)=(1,1,?1)T.

We can see that the systems are out of synchronization without controller in Figure 1. It can be seen that the synchronization errors converge to zero quickly,which implies that the trajectories of the slave system reach the trajectories of the master system in a finite time, as illustrated in Figure 2.

Figure 1 State trajectories of master-slave systems without controller (q =0.873)

Figure 2 State trajectories of master-slave systems with controller (q =0.873)

In Figure 3 to Figure 5, we see that the faster q approaches 0.873, the sooner system error converges to zero. Obviously, the control inputs are feasible in practice.The simulation results indicate that the introduced sliding mode technique has finitetime convergence and stability in both reaching and sliding mode phases.

Figure 3 The system errors (q =0.873)

Figure 4 The system errors (q =0.5)

Figure 5 The system errors (q =0.75)

5 Conclusions

In this paper, the problem of finite-time chaos synchronization between two different chaotic systems with dead-zone input is solved using a novel nonsingular terminal sliding mode scheme. A robust finite-time sliding mode controller is designed to ensure the occurrence of the sliding motion in a finite time. Finite-time stability and convergence of both sliding motion and reaching phase are proved and the exact values of the convergence times are given. Numerical simulations demonstrate the fast convergent property and robustness of the introduced technique. The proposed nonsingular terminal sliding manifold can be applied for a broad range of nonlinear control problems.