Synchronization of an Uncertain Fractional-Order Chaotic System via Backstepping Sliding Mode Control

Abstract

Backstepping control approach combined with sliding mode control (SMC) technique is utilized to realize synchronization of uncertain fractional-order strict-feedback chaotic system. A backstepping SMC method is presented to compensate the uncertainty which occurs in the slave system. Moreover, the newly proposed control scheme is applied to implement synchronization of fractional-order Duffing-Holmes system. The simulation results demonstrate that the backstepping SMC method is robust against the modeling uncertainties and external disturbances.

1. Introduction

Fractional calculus is a generalization of the ordinary integration and differentiation to noninteger-order case. In recent years, it has found widespread application in the fields of physics, applied mathematics, and engineering [1–3]. Moreover, it has been generally realized that fractional integrals and derivatives are more suitable to describe memory and hereditary properties of various materials and processes than its integer-order counterpart [4].

Chaotic behavior has been observed in the laboratory in a variety of systems, including electrical circuits, lasers, chemical reactions, fluid dynamics, and analog computers [5]. Over the past few years, more and more researchers have turned their attention to chaotic dynamics of fractional-order system. It has been shown that chaotic behavior of an integer-order nonlinear system is preserved when the order becomes fractional [6]. As a consequence, a number of fractional-order chaotic systems have been proposed, including fractional-order variants of Chua’s system [7], fractional-order Chen system [8], fractional-order Lorenz, Rössler, and Liu systems [9–11].

Synchronization control is one of the important research areas in chaos theory. During the past three decades, chaos synchronization has been attracting interest from researchers in various fields [12–14]. It should be pointed out that synchronization problem of fractional-order chaotic systems was first reported by Deng and Li [15]. Afterwards, different synchronization control methods have been successfully applied to chaos synchronization of factional order systems, such as sliding mode control [16, 17], adaptive fuzzy control [18, 19], nonlinear feedback control [20, 21], the open-plus-closed-loop control [22], observer-based control [23], and some other methods [24–26].

Backstepping approach was first proposed in [27] for designing stabilizing controls for a special type of nonlinear dynamical systems, which is referred to as the strict-feedback system. The technique consists in a recursive procedure that skillfully constructs the Lyapunov function and designs the virtual control input [28–30]. When this recursive procedure terminates, a feedback design for the real control input results [31]. On the other hand, sliding mode control (SMC) is one of the popular strategies to deal with uncertain systems. The main feature of SMC is the robustness against parameter uncertainties and external disturbances. In this paper, combining the merits of backstepping control and SMC, a backstepping sliding mode controller is devised to implement chaos synchronization of uncertain fractional-order strict feedback chaotic system.

The rest of this paper is organized as follows. In Section 2, definition and lemma in fractional calculus are presented. In Section 3, the backstepping sliding mode controller is systematically designed. In Section 4, synchronization problem of uncertain fractional-order Duffing-Holmes system is handled by the proposed method. Some concluding remarks are made in Section 5.

2. Preliminaries

There are three most frequently used definitions for fractional derivatives [1], that is, Riemann-Liouville, Grünwald-Letnikov, and Caputo definitions. Caputo definition is widely used in engineering applications since it takes on the same form as for integer-order differential equations in the initial conditions. Therefore, the following sections are based on Caputo derivative.

Definition 1 (see [1]). The Caputo fractional derivative of order 𝛼 of a continuous function 𝑓∶𝑅⁺→𝑅 is defined as 𝐷^𝛼_𝑡𝑓(𝑡)=1Γ(𝑚−𝛼)^𝑡₀𝑓^(𝑚)(𝜏)(𝑡−𝜏)^{𝛼−𝑚+1}𝑑𝑑𝜏,𝑚−1<𝛼<𝑚,_𝑚𝑑𝑡^𝑚𝑓(𝑡),𝛼=𝑚, (1) where Γ is Γ-function, and Γ(𝑧)=^∞₀𝑒^−𝑡𝑡^𝑧−1𝑑𝑡,Γ(𝑧+1)=𝑧Γ(𝑧). (2)

It should be noticed that the fractional integral of order 𝛼>0 is denoted by 𝐷_𝑡^−𝛼.

In what follows, we list several basic facts of fractional derivatives and integrals which will be used in the stability and stabilization analysis. The readers can refer to [1] for more details.

Fact 1. For 𝛼=𝑛, where 𝑛 is an integer, the operation 𝐷^𝑛_𝑡𝑓(𝑡) gives the same result as classical calculus of integer-order 𝑛. In particular, when 𝛼=1, the operation 𝐷¹_𝑡𝑓(𝑡) coincides with the ordinary derivative 𝑑𝑓(𝑡)/𝑑𝑡.

Fact 2. For 𝛼=0, the operation 𝐷^𝛼_𝑡𝑓(𝑡) is the identity operation 𝐷⁰_𝑡𝑓(𝑡)=𝑓(𝑡). (3)

Fact 3. Similary to integer-order calculus, fractional differentiation and fractional integration are both linear operations 𝐷^𝛼_𝑡𝑎𝑓(𝑡)+𝑏𝑔(𝑡)=𝑎𝐷^𝛼_𝑡𝑓(𝑡)+𝑏𝐷^𝛼_𝑡𝑔(𝑡). (4) where 𝑎 and 𝑏 are constants.

Fact 4. The additive law of exponents (semigroup property) holds 𝐷^𝛼_𝑡𝐷^𝛽_𝑡𝑓(𝑡)=𝐷^𝛽_𝑡𝐷^𝛼_𝑡𝑓(𝑡)=𝐷_𝑡^𝛼+𝛽𝑓(𝑡). (5)

Fact 5. For 𝛼>0, the following equation holds: 𝐷^𝛼_𝑡𝐷_𝑡^−𝛼𝑓(𝑡)=𝐷⁰_𝑡𝑓(𝑡)=𝑓(𝑡) (6) which means that the fractional differentiation operator is a left inverse to the fractional integration operator of the same order 𝛼.

Lemma 2 (see [32]). Let 𝑥=0 be an equilibrium point for either Caputo or RL fractional nonautonomous system 𝐷^𝑞_𝑡𝑥(𝑡)=𝑓(𝑥,𝑡), (7) where 𝑞∈(0,1) and 𝑓(𝑥,𝑡) satisfies the Lipschitz condition with Lipschitz constant 𝑙>0. Assume that there exits a Lyapunov function 𝑉(𝑡,𝑥(𝑡)) satisfying 𝛼₁‖𝑥‖^𝑎≤𝑉(𝑡,𝑥)≤𝛼₂‖𝑥‖,𝑉̇(𝑡,𝑥)≤−𝛼₃‖𝑥‖, (8) where 𝛼₁,𝛼₂,𝛼₃, and 𝑎 are positive constants and ‖⋅‖ denotes an arbitrary norm. Then the equilibrium point of system (7) is Mittag-Leffler (asymptotically) stable.

3. Main Results

In synchronization task, there are two dynamical systems, which are, respectively, called master system and slave system. From the view point of control, the task is to design a controller which obtains signals from the master system to tune the behavior of the slave system. In this paper, we consider synchronization of fractional-order strict feedback system described by 𝐷^𝑞_𝑡𝑥₁=𝑥₂,𝐷^𝑞_𝑡𝑥₂=𝑓𝑡,𝑥₁,𝑥₂, (9) which is regarded as master system and the slave system is defined as 𝐷^𝑞_𝑡𝑦₁=𝑦₂,𝐷^𝑞_𝑡𝑦₂=𝑓𝑡,𝑦₁,𝑦₂+Δ𝑓𝑡,𝑦₁,𝑦₂+𝑑(𝑡)+𝑢(𝑡), (10) where 0<𝑞≤1, 𝑋=(𝑥₁,𝑥₂)^𝑇 and 𝑌=(𝑦₁,𝑦₂)^𝑇 are the states of systems (9) and (10), Δ𝑓(𝑡,𝑦₁,𝑦₂) denotes uncertainty, 𝑑(𝑡) is the external disturbance, and 𝑢(𝑡) is the control input.

Defining the synchronization error as 𝑒_𝑖=𝑦_𝑖−𝑥_𝑖(𝑖=1,2), then the error system is given by 𝐷^𝑞_𝑡𝑒₁=𝑒₂,𝐷^𝑞_𝑡𝑒₂=𝑓𝑡,𝑦₁,𝑦₂−𝑓𝑡,𝑥₁,𝑥₂+𝐹(𝑡,𝑌)+𝑢(𝑡), (11) where 𝐹(𝑡,𝑌)=Δ𝑓(𝑡,𝑦₁,𝑦₂)+𝑑(𝑡). Suppose 𝛾 is the upper bound of function 𝐹; that is, ‖𝐹(𝑡,𝑌)‖≤𝛾.

Step 1. We start with the scalar system 𝐷^𝑞_𝑡𝑧₁=𝑒₂, (12) where 𝑧₁=𝑒₁, and 𝑒₂ is a virtual controller selected as 𝑒₂=𝛼(𝑧₁).
Choose the candidate Lyapunov function as 𝑉₁=12𝑧²₁>0; (13) then the time derivative of 𝑉₁ along the trajectories of system (12) is 𝑉̇=𝑧₁𝑧̇₁=𝑧₁𝐷_𝑡^1−𝑞𝐷^𝑞_𝑡𝑧₁=𝑧₁𝐷_𝑡^1−𝑞𝑒₂. (14) Choose 𝐷_𝑡^1−𝑞𝑒₂=−𝑘₁𝑧₁; that is, 𝛼(𝑧₁)=−𝑘₁𝐷_𝑡^𝑞−1𝑧₁, where 𝑘₁ is a positive constant. Thus, we have 𝑉̇=−𝑘₁𝑧²₁≤0.

Step 2. Let 𝑧₂=𝑒₂−𝛼(𝑧₁), so 𝐷^𝑞_𝑡𝑧₂=𝐷^𝑞_𝑡𝑒₂−𝐷^𝑞_𝑡𝛼𝑧₁=𝑓𝑡,𝑦₁,𝑦₂−𝑓𝑡,𝑥₁,𝑥₂()(𝑡)+𝐹𝑡,𝑌+𝑢+𝑘₁𝐷_𝑡^2𝑞−1𝑧₁. (15)
Design the switching surface as 𝑆=𝑘₂𝐷_𝑡⁻¹𝑧₁+𝐷_𝑡^𝑞−1𝑧₂, (16) where 𝑘₂ is the sliding surface parameter to be designed later.
According to SMC method, the condition which guarantees the trajectory of the system arrives at the sliding surface is 𝑆𝑆̇<0, and when in the sliding mode, the switching surface and its derivative must satisfy the following conditions: 𝑆(𝑡)=0,𝑆̇(𝑡)=0. (17) In view of (15), it means that 𝑆̇=𝑘₂𝑧₁+𝐷^𝑞_𝑡𝑧₂=𝑘₂𝑧₁+𝑓(𝑡,𝑌)−𝑓(𝑡,𝑋)+𝐹(𝑡,𝑌)+𝑘₁𝐷_𝑡^2𝑞−1𝑧₁+𝑢(𝑡)=0. (18) Thus, the equivalent control 𝑢_eq(𝑡) in the case of uncertainty-free is calculated by 𝑢_eq(𝑡)=𝑓(𝑡,𝑋)−𝑓(𝑡,𝑌)−𝑘₂𝑧₁−𝑘₁𝐷_𝑡^2𝑞−1𝑧₁, (19)
To satisfy the sliding mode condition, the reaching law is chosen as 𝑢_𝑟=−𝑘₃𝑆−𝛾sgn(𝑆), (20) where 𝑘₃ is a positive constant. Hence, the total control law is 𝑢=𝑢_eq+𝑢_𝑟=𝑓(𝑡,𝑋)−𝑓(𝑡,𝑌)−𝑘₂𝑧₁−𝑘₁𝐷_𝑡^2𝑞−1𝑧₁−𝑘₃𝑆−𝛾sgn(𝑆). (21)
Selecting 𝑉₂=|𝑆| (22) as a composite Lyapunov function, we have 𝑉̇₂(𝑆)𝑆̇(𝑆)𝑘=sgn=sgn₂𝑧₁+𝐷^𝑞_𝑡𝑧₂=sgn(𝑆)𝑘₂𝑧₁()()()+𝑓𝑡,𝑌−𝑓𝑡,𝑋+𝐹𝑡,𝑌+𝑘₁𝐷_𝑡^2𝑞−1𝑧₁+𝑢(𝑡). (23) Substituting the control law (21) into the right hand side of (23), we have 𝑉̇₂=sgn(𝑆)𝑘₂𝑧₁()()()+𝑓𝑡,𝑌−𝑓𝑡,𝑋+𝐹𝑡,𝑌+𝑘₁𝐷_𝑡^2𝑞−1𝑧₁+𝑓(𝑡,𝑋)−𝑓(𝑡,𝑌)−𝑘₂𝑧₁−𝑘₁𝐷_𝑡^2𝑞−1𝑧₁−𝑘₃(𝑆)(𝑆𝐹()𝑆−𝛾sgn=sgn)𝑡,𝑌−𝑘₃𝑆−𝛾sgn(𝑆)≤𝛾−𝑘₃sgn(𝑆)𝑆−𝛾sgn(𝑆)sgn(𝑆)=−𝑘₃|𝑆|. (24) Hence, according to Lemma 2, system (15) is asymptotically stable.

From the above two steps, we obtain lim_𝑡→∞𝑧_𝑖=0(𝑖=1,2). The properties 𝑧₁=𝑒₁, 𝑧₂=𝑒₂−𝛼(𝑧₁)=𝑒₂+𝑘₁𝐷_𝑡^𝑞−1𝑧₁ imply that lim_𝑡→∞𝑒_𝑖=0(𝑖=1,2). Thus, the global synchronization of systems (9) and (10) is achieved.

4. Applications

In order to demonstrate the effectiveness of the proposed control scheme, numerical simulations are made for synchronization of fractional-order Duffing-Holmes system. In the simulation, Adams-Bashforth-Moulton predictor-corrector algorithm is used, and the detailed descriptions of this algorithm are available in [33].

4.1. System Description

In [20], an integer-order chaotic Duffing-Holmes system has been studied. Let us consider the fractional-order case described by 𝐷^𝑞_𝑡𝑥₁=𝑥₂,𝐷^𝑞_𝑡𝑥₂=𝑥₁−𝑎𝑥₂−𝑥³₁+𝑏cos𝑡. (25) Initial conditions of master system and slave system are set as 𝑥₁(0)=0.2, 𝑥₂(0)=0.2 and 𝑦₁(0)=0.1, 𝑦₂(0)=−0.2, respectively.

Phase portrait of the chaotic dynamics in system (25) when 𝑞=0.98 are shown in Figure 1. Parameters 𝑎 and 𝑏 are set as 0.25 and 0.3, respectively.

Figure 1 

Phase portrait of Duffing-Holmes system (25) with fractional-order 𝑞=0.98.

The slave system is perturbed by the uncertainty and disturbance, which is described by 𝐷^𝑞_𝑡𝑦₁=𝑦₂,𝐷^𝑞_𝑡𝑦₂=𝑦₁−𝑎𝑦₂−𝑦³₁+𝑏cos𝑡+Δ𝑓(𝑡,𝑌)+𝑑(𝑡)+𝑢, (26) where Δ𝑓(𝑡,𝑌) and 𝑑(𝑡) are chosen as 0.1sin(𝑡)𝑦²₁+𝑦²₂ and 0.1sin(𝑡), respectively.

According to the scheme proposed in Section 3, the control law is designed as follows: 𝑢=𝑎𝑒₂−1+𝑘₂𝑒₁−𝑘₁𝐷_𝑡^2𝑞−1𝑒₁+𝑦³₁−𝑥³₁−𝛾sgn(𝑆)−𝑘₃𝑆, (27) in which 𝑆=𝑘₂𝐷_𝑡⁻¹𝑧₁+𝐷_𝑡^𝑞−1𝑧₂=𝑘₂𝐷_𝑡⁻¹𝑒₁+𝐷_𝑡^𝑞−1𝑒₂+𝑘₁𝐷_𝑡^2𝑞−2𝑒₁. When the parameters are chosen as 𝑞=0.98, 𝑘₁=𝑘₂=𝑘₃=2, 𝛾=0.2, the simulation results are shown in Figures 2–5. Figures 2 and 3 show the synchronization performance between drive-response system (25)-(26) and the time response of the error system, respectively. The sliding surface and the control input 𝑢 are displayed in Figures 4 and 5.

(a)

(b)

Figure 2 

Synchronization performance of fractional-order Duffing-Holmes systems (25) and (26).

(a)

(b)

Figure 3 

Synchronization error between fractional-order Duffing-Holmes systems (25) and (26).

Figure 4 

Switching surface 𝑆 of the sliding mode controller (27).

Figure 5 

Control input 𝑢 of the slave system (26).

5. Conclusions

In this paper, the backstepping SMC method is developed for fractional-order strict-feedback chaotic system. The Lyapunov function, the virtual control, the switching surface, and the actual control are systematically designed, respectively. The proposed approach, which combines the merits of backstepping control and SMC, is robust against the modeling uncertainties and external disturbances. Numerical results demonstrate that the convergence speed of the synchronized error system is satisfactory. In addition, the strategy can be generalized to the investigation of synchronization of other fractional-order strict feedback chaotic or hyperchaotic systems.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 61004078, 61104117, the Foundation of State Key Laboratory for Novel Software Technology of Nanjing University under Grant KFKT2013B11, the China Postdoctoral Science Foundation funded project, and Special Funds for Postdoctoral Innovative Projects of Shandong Province.