Robust Synchronization of Fractional-Order Hyperchaotic Systems Subjected to Input Nonlinearity and Unmatched External Perturbations

Abstract

This paper investigates the robust synchronization problem for a class of fractional-order hyperchaotic systems subjected to unmatched uncertainties and input nonlinearity. Based on the sliding mode control (SMC) technique, this approach only uses a single controller to achieve chaos synchronization, which reduces the cost and complexity for synchronization control implementation. As expected, the error states can be driven to zero or into predictable bounds for matched and unmatched perturbations, respectively, even with input nonlinearity.

1. Introduction

Synchronization, which means “designing a system whose behavior mimics that of another chaotic system,” has become more and more interesting topic to engineering and science communities [1]. Fractional calculus as an extension of ordinary calculus has a 300-year-old mathematical topic; the applications of the fractional calculus to physics and engineering are just a recent focus of interest [2, 3]. It has been recognized that many dynamical systems can be more precisely modeled by using the means of the fractional calculus, such as mechanics [4, 5], image processing [6], viscoelastic materials [7], electrical circuits [8], and population models [9]. Meanwhile, it has been demonstrated that some dynamics of fractional-order systems can behave chaotically or hyperchaotically [10, 11]. Due to the potential applications in physics and engineering, many methods have been presented to achieve synchronization for fractional-order chaotic systems such as sliding mode control [12, 13], 𝐻^∞ control method [14], and active control [15], among many others [16, 17]. Unfortunately, all synchronization schemes in the above-mentioned papers for fractional-order chaotic systems are derived on the basis of the ideal assumption of control input or matched external perturbations. As well known, the control schemes for robust chaos synchronization can be realized by electronic components such as operational amplifier (OPA), resistor, and capacitor. However, in practice, there always exists nonlinearity in the control input including saturation, backlash, and dead zone in OPA or electromechanical devices. Therefore the implementation of control inputs of practical systems is frequently subjected to nonlinearity as a result of physical limitations. It has been shown that input nonlinearity might cause a serious degradation of the system performance, a reduced rate of response, and, in a worst-case scenario, system failure if the controller is not well designed [18, 19]. Therefore, its effect cannot be ignored in analysis of control design and realization for chaos synchronization. On the other hand, for designing a robust control, sliding mode control is frequently adopted due to its inherent advantages of easy realization, fast response, good transient performance, and being insensitive to variation in plant parameters or external disturbances [20, 21]. However, the property of robustness to external perturbations is just for the case of matched condition. The dynamics of controlled systems in the sliding manifold is still influenced by unmatched perturbations. Therefore, it still needs to discuss the effect of unmatched external perturbations for fractional-order chaotic systems in the sliding mode.

Motivated by the above discussions, this paper considers the robust synchronization problem for robust synchronization of fractional-order hyperchaotic systems subjected to input nonlinearity and unmatched external perturbations. To achieve this goal, a new fraction-integer integral (FII) switching surface is newly proposed such that it becomes easy to analyze the stability of the closed-loop nonlinear chaotic systems. Having established the fractional switching surface, a sliding mode controller is designed. This controller is robust to the nonlinear input and guarantees the occurrence of sliding motion of the controlled fractional-order chaotic system. In our design, a single controller is used enough to realize synchronization, which reduces the cost and complexity for synchronization control implementation. As expected, the synchronization error states can be driven to zero with the matched perturbations or into predictable bounds with unmatched perturbations.

This paper is organized as follows. Section 2 describes the problem formulation, FII switching surface, and the sliding mode controller design; a numerical example to demonstrate the effectiveness of the proposed method is included in Section 3. In Section 4, we draw conclusions on the new results. Throughout this paper, it is noted that notation 𝑀^𝑇 is used to denote the transpose for a square matrix 𝑀, while, for 𝑥∈ℜ^𝑛, ‖𝑥‖=(𝑥^𝑇𝑥)^1/2 denotes the Euclidean norm of the vector. Note that, for a scalar 𝜎, sign(𝜎) is the sign function of 𝜎; when 𝜎>0, sign(𝜎)=1; when 𝜎=0, sign(𝜎)=0; when 𝜎<0, sign(𝜎)=−1.

2. System Description and Problem Formulation

Consider a four-dimensional fractional-order hyperchaotic system; the dynamics is described by the following equations [22]: 𝐷^𝑞𝐷𝑥=𝑎𝑥−𝑦,_𝑞𝑦=𝑥−𝑦𝑧²,𝐷^𝑞𝑧=−𝑏₁𝑦−𝑏₂𝑧−𝑏₃𝐷𝑤,_𝑞𝑤=𝑧+𝑐𝑤, (1) where 𝑎, 𝑏₁, 𝑏₂, 𝑏₃, 𝑐 are system parameters. 𝐷^𝑞 denotes the Riemann-Liouville fractional derivative of order 𝑞∈𝑅 defined as follows [23]: 𝐷^𝑞𝑓(𝑡)=1Γ𝑑1−𝑞𝑑𝑡^𝑡₀𝑓(𝜏)(𝑡−𝜏)^𝑞𝑑𝜏. (2) Also Γ(𝑞) is the Euler Gamma function given as Γ𝑞=^∞₀𝑣^𝑞−1𝑒^𝑣𝑑𝑣. (3)

The order denoted by 𝑞 is subject to 0<𝑞<1.

In this paper, we focus on system (1) since it is a hyperchaotic system with more complicated dynamical behavior. Also, methods developed herein are also applicable to other fractional chaotic systems. System (1) generates chaotic oscillations when the system parameters and initial condition are set as 𝑎=0.56, 𝑏₁=1.0, 𝑏₂=1.0, 𝑏₃=6.0, 𝑐=0.8, and 𝑞=0.95 and initial condition[𝑥(0)𝑦(0)𝑧(0)𝑤(0)] =[0.50.3−0.10.1]. Figure 1 shows the typical chaotic attractors. This paper aims to design a robust synchronization controller such that the response system, even with unmatched external perturbations and input nonlinearity, is able to mimic the behavior of the drive chaotic system. Let the drive system and response system be defined below, respectively.

(a)

(b)

(c)

Figure 1 

Attractors of the considered fractional-order system: (a) three-dimensional view (𝑥−𝑦−𝑧); (b) three-dimensional view (𝑥−𝑦−𝑤); (c) three-dimensional view (𝑥−𝑧−𝑤).

Drive system is 𝐷^𝑞𝑥_𝑚=𝑎𝑥_𝑚−𝑦_𝑚,𝐷^𝑞𝑦_𝑚=𝑥_𝑚−𝑦_𝑚𝑧²_𝑚,𝐷^𝑞𝑧_𝑚=−𝑏₁𝑦_𝑚−𝑏₂𝑧_𝑚−𝑏₃𝑤_𝑚,𝐷^𝑞𝑤_𝑚=𝑧_𝑚+𝑐𝑤_𝑚, (4)and response system is 𝐷^𝑞𝑥_𝑠=𝑎𝑥_𝑠−𝑦_𝑠+𝑑₁,𝐷^𝑞𝑦_𝑠=𝑥_𝑠−𝑦_𝑠𝑧²_𝑠+𝑑₂(𝑢),𝐷+𝜙_𝑞𝑧_𝑠=−𝑏₁𝑦_𝑠−𝑏₂𝑧_𝑠−𝑏₃𝑤_𝑠+𝑑₃,𝐷^𝑞𝑤_𝑠=𝑧_𝑠+𝑐𝑤_𝑠+𝑑₄, (5) where 𝑢(𝑡)∈𝑅 is the control input. 𝜙(𝑢(𝑡)) is a continuous nonlinear function and 𝜙(0)=0, where 𝜙∶𝑅→𝑅 with the law 𝑢(𝑡)→𝜙(𝑢(𝑡)) and inside sector [𝛽₁𝛽₂]; that is, 𝛽₂𝑢²(𝑡)≥𝑢(𝑡)𝜙(𝑢(𝑡))≥𝛽₁𝑢²(𝑡), (6) where 𝛽₁ and 𝛽₂ are nonzero positive constants [19]. A nonlinear function 𝜙(𝑢(𝑡)) is illustrated in Figure 2. Also 𝑑_𝑖(𝑡), 𝑖=1,2,3,4, are the unavoidable external perturbations in practical systems and assumed bounded; that is, 𝑑_𝑖(𝑡)≤𝛼_𝑖,𝑖=1,2,3,4, (7) where 𝛼_𝑖>0 are given. Generally, 𝑑₂ is called the matched perturbation and 𝑑_𝑖, 𝑖=1,3,4, are the unmatched perturbations. Now define the synchronization error as 𝑒₁=𝑥_𝑠−𝑥_𝑚, 𝑒₂=𝑦_𝑠−𝑦_𝑚, 𝑒₃=𝑧_𝑠−𝑧_𝑚, 𝑒₄=𝑤_𝑠−𝑤_𝑚, respectively. Then yield the following error system: 𝐷^𝑞𝑒₁=𝑎𝑒₁−𝑒₂+𝑑₁,𝐷^𝑞𝑒₂=𝑒₁−𝑦_𝑠𝑧²_𝑠+𝑦_𝑚𝑧²_𝑚+𝑑₂(𝑢),𝐷+𝜙_𝑞𝑒₃=−𝑏₁𝑒₂−𝑏₂𝑒₃−𝑏₃𝑒₄+𝑑₃,𝐷^𝑞𝑒₄=𝑒₃+𝑐𝑒₄+𝑑₄. (8)

Figure 2 

A scalar nonlinear function 𝜙(𝑢) inside sector [𝛽1𝛽2].

Obviously, the aim of this work is to propose a sliding mode control law 𝑢(𝑡) subjected to input nonlinearity specified by (6), such that the resulting tracking error state vector 𝐸=[𝑒₁𝑒₂𝑒₃𝑒₄] can be forced to zero or into a predictable bound when unmatched external perturbations are present. Accordingly, to achieve the control goal by using the SMC technique, there exist two basic steps for the design procedure. The first step is to construct an appropriate switching surface such that the sliding motion can result in lim_𝑡→∞‖𝐸(𝑡)‖≤𝜌 and 𝜌≥0 are a predictable constant depending on the external perturbations, which will be explained later. The second step is to establish a SMC law which can guarantee the attraction of the sliding manifold even with the input nonlinearity (6).

2.1. Switching Surface Design of Chaos Synchronization

To complete the design steps above, we firstly propose a novel type of FII switching surface as 𝜎(𝑡)=𝐼^1−𝑞𝑒₂(𝑡)+^𝑡₀𝐾𝐸(𝜏)𝑑𝜏, (9) where 𝐼^1−𝑞𝑒₂(𝑡) is the Riemann-Liouville fractional integral of order 1−𝑞 given by 𝐼^1−𝑞𝑒₂(𝑡)=1Γ1−𝑞^𝑡₀𝑒₂(𝜏)(𝑡−𝜏)^𝑞𝑑𝜏. (10)

Obviously, when the system operates in the sliding mode, the controlled system satisfies the following conditions [20, 21]: 𝜎(𝑡)=0;𝜎̇(𝑡)=0. (11) Then, based on (8)–(11), one can deduce the following result: 𝐷^𝑞𝐸=(𝐴−𝐵𝐾)𝐸+𝐷, (12) where 𝑒𝐸=¹𝑒₂𝑒₃𝑒₄;𝐴=𝑎−10000000−𝑏₁−𝑏₂−𝑏₃;0100𝑘001𝑐𝐵=;𝐾=₁𝑘₂𝑘₃𝑘₄𝑑;𝐷=¹0𝑑₃𝑑₄. (13) When the system enters into the sliding mode, the system dynamics is governed by (12). It has been shown that system (12) without external perturbations is asymptotically stable if the eigenvalues of the matrix 𝐴−𝐵𝐾 satisfy the following argument stability criterion [24]: min_𝑖arg𝜆_𝑖(𝜋𝐴−𝐵𝐾)>𝑞2,𝑖=1,2,…,𝑛. (14)

By (13), obviously (𝐴,𝐵) is controllable. Therefore, a parameter vector 𝐾 does exist such that the maximum real part eigenvalue of 𝐴−𝐵𝐾 is negative and (14) is satisfied. Furthermore, we can easily assign the system performance in the sliding mode just by selecting an appropriate matrix 𝐾 using any pole assignment method.

The solution of the dynamics (12) can be obtained as follows [23]: 𝐸(𝑡)=𝑒^{(𝐴−𝐵𝐾)(𝑡−𝑡1)}_𝑞𝐸𝑡₁+^𝑡−𝑡1₀𝑒^{(𝐴−𝐵𝐾)(𝑡−𝑡1}_𝑞^−𝑠)𝐷(𝑠)𝑑𝑠, (15) where 𝑒^{(𝐴−𝐵𝐾)(𝑡−𝑡1)}_𝑞=𝑡−𝑡₁^∞𝑞−1_𝑘=0(𝐴−𝐵𝐾)^𝑘𝑡−𝑡₁^𝑘𝑞Γ(𝑘+1)𝑞 (16)

is the q-exponential function and represents the transition matrix of system (12).

From (15), we have ‖𝐸(𝑡=𝑒)‖_{(𝐴−𝐵𝐾)(𝑡−𝑡1)𝑞}𝐸𝑡₁+^𝑡−𝑡1₀𝑒^{(𝐴−𝐵𝐾)(𝑡−𝑡1}_𝑞^−𝑠)𝐷(𝑠)≤𝑒𝑑𝑠_{(𝐴−𝐵𝐾)(𝑡−𝑡1)𝑞}𝐸𝑡₁+‖𝐷(𝑠)‖^𝑡−𝑡1₀𝑒^{(𝐴−𝐵𝐾)(𝑡−𝑡1}_𝑞^−𝑠)≤𝑒𝑑𝑠_{(𝐴−𝐵𝐾)(𝑡−𝑡1)𝑞}𝐸𝑡₁+_𝑖=1,3,4𝛼_𝑖^𝑡−𝑡1₀𝑒^{(𝐴−𝐵𝐾)(𝑡−𝑡1}_𝑞^−𝑠)𝑑𝑠. (17) Furthermore, ^𝑡−𝑡1₀𝑒^{(𝐴−𝐵𝐾)(𝑡−𝑡1}_𝑞^−𝑠)=()𝑑𝑠𝐴−𝐵𝐾⁻¹𝐸_𝑞,1(𝐴−𝐵𝐾)𝑡−𝑡₁^𝑞−𝐼_𝑛, (18) where 𝐸_𝑞,1((𝐴−𝐵𝐾)(𝑡−𝑡₁)^𝑞) denotes the Mittag-Leffler function defined as [23] 𝐸_𝑞,1(𝐴−𝐵𝐾)𝑡−𝑡₁^𝑞=^∞_𝑘=0(𝐴−𝐵𝐾)^𝑘𝑡−𝑡₁^𝑘𝑞Γ.𝑘𝑞+1 (19) Since we assign an appropriate matrix 𝐾 such that the argument stability criterion (14) is satisfied, then lim_𝑡→∞𝐸_𝑞,1((𝐴−𝐵𝐾)(𝑡−𝑡₁)^𝑞)=0 and lim_𝑡→∞𝑒_𝑞^{(𝐴−𝐵𝐾)𝑡}=0. Therefore, from (17), (18), and (19), we have lim_𝑡→∞‖𝐸(𝑡)‖≤lim_𝑡→∞𝑒^{(𝐴−𝐵𝐾)(𝑡−𝑡1)}_𝑞𝐸𝑡₁+_𝑖=1,3,4𝛼_𝑖lim_𝑡→∞^𝑡−𝑡1₀𝑒^{(𝐴−𝐵𝐾)(𝑡−𝑡1}_𝑞^−𝑠)𝑑𝑠≤𝜌=_𝑖=1,3,4𝛼_𝑖(𝐴−𝐵𝐾)⁻¹ (20)

According to the discussion above, we can conclude that when the fractional-order system is in the sliding manifold, the tracking error ‖𝐸‖ can converge to a predictable bound 𝜌 relative to ∑_𝑖=1,3,4𝛼_𝑖 and parameter matrix 𝐾 chosen in the switching surface (9).

2.2. Design of Sliding Mode Controller with Input Nonlinearity

In order to guarantee the occurrence of sliding manifold even with the input nonlinearity, we choose a sliding mode control of the form 𝑢(𝑡)(𝜎(𝑡1=−𝜁𝜂sign)),𝜁>𝛽₁, (21) where 𝜂=|𝑒₁−𝑦_𝑠𝑧²_𝑠+𝑦_𝑚𝑧²_𝑚+𝐾𝐸|+𝛼₂.

In the following, the proposed scheme (21) will be proved to be able to derive the uncertain error dynamics (8) onto the sliding mode 𝜎(𝑡)=0.

Theorem 1. If the control 𝑢(𝑡) is given by (21), the reaching condition of expression 𝜎(𝑡)𝜎̇(𝑡)<0 of the sliding mode is satisfied in spite of the input nonlinearity.

Proof. Substituting (9), (10), and (21) into 𝜎(𝑡)𝜎̇(𝑡), we obtain 𝜎(𝑡)𝜎̇(𝑡)=𝜎(𝑡)𝐷^𝑞𝑒₂(𝑡)(𝑡𝑒+𝐾𝐸=𝜎)₁(𝑡)−𝑦_𝑠(𝑡)𝑧²_𝑠(𝑡)+𝑦_𝑚(𝑡)𝑧²_𝑚(𝑡)+𝑑₂(𝑢(𝑡(𝑡𝑒+𝜙))+𝐾𝐸=𝜎)₁(𝑡)−𝑦_𝑠(𝑡)𝑧²_𝑠(𝑡)+𝑦_𝑚(𝑡)𝑧²_𝑚(𝑡)+𝐾𝐸+𝑑₂(𝑡)𝜙(𝑢(𝑡≤𝑒+𝜎))₁(𝑡)−𝑦_𝑠(𝑡)𝑧²_𝑠(𝑡)+𝑦_𝑚(𝑡)𝑧²_𝑚(𝑡)+𝐾𝐸|𝜎(𝑡)|+𝛼₂|𝜎(𝑡(𝑡)𝜙(𝑢(𝑡|𝜎(𝑡(𝑡)𝜙(𝑢(𝑡.)|+𝜎))=𝜂)|+𝜎)) (22) Furthermore, from (6), we have 𝛽₂𝜁²𝜂²sign(𝜎(𝑡))²≥−𝜁𝜂sign(𝜎(𝑡))𝜙(𝑢(𝑡))≥𝛽₁𝜁²𝜂²sign(𝜎(𝑡))². (23) Since 𝜎²(𝑡)≥0, we get 𝛽₂𝜁²𝜂²sign(𝜎(𝑡))²𝜎²(𝑡)≥−𝜁𝜂sign(𝜎(𝑡))𝜙(𝑢(𝑡))𝜎²(𝑡)≥𝛽₁𝜁²𝜂²sign(𝜎(𝑡))²𝜎²(𝑡)⟹𝛽₂𝜁²𝜂²|𝜎(𝑡)|²≥−𝜁𝜂|𝜎(𝑡)|²𝜙(𝑢(𝑡))≥𝛽₁𝜁²𝜂²|𝜎(𝑡)|²⟹𝛽₂𝜁𝜂|𝜎(𝑡)|≥−|𝜎(𝑡)|𝜙(𝑢(𝑡))≥𝛽₁𝜁𝜂|𝜎(𝑡)|⟹−𝛽₂𝜁𝜂|𝜎(𝑡)|≤𝜎(𝑡)𝜙(𝑢(𝑡))≤−𝛽₁𝜁𝜂|𝜎(𝑡)|. (24) By placing (24) into (22), we get 𝜎(𝑡)𝜎̇(𝑡)≤−𝛽₁|𝜎(𝑡|𝜎(𝑡≤𝜁𝜂)|+𝜂)|1−𝛽₁𝜁𝜂|𝜎(𝑡)|. (25) Since 𝜁>1/𝛽₁ has been selected in (21), it can be concluded that the hitting condition 𝜎(𝑡)𝜎̇(𝑡)<0 is satisfied. Thus, the proof is achieved completely.

Remark 2. The controller in (21) demonstrates a discontinuous control law and the phenomenon of chattering will appear. In order to eliminate the chattering, controller (21) can be modified as 𝑢(𝑡)𝜎=−𝜁𝜂|𝜎|1+𝜀,𝜁>𝛽₁, (26) where 𝜀 is a sufficiently small positive constant. From the works [21, 25], the solution of system (8) with (21) can be made arbitrarily close to solution (8) with (26), if one chooses 𝜀 sufficiently small.

Remark 3. Obviously, for the case of 𝑞=1, the considered system (1) degenerates to an integer-order chaotic system and the design method developed in this paper is also available just by some minor modifications.

3. An Illustrative Example

In this section, to verify the validity of the proposed synchronization scheme, we numerically examine the synchronization. Here, the drive system and response system accord with (4) and (5), respectively. The input nonlinearity is defined as 𝜙(𝑢(𝑡))=[1+0.05sin(𝑢(𝑡))]𝑢(𝑡). (27) According to (6), 𝛽₁=0.95, 𝛽₂=1.05 can be obtained. In numerical simulation, all system’s parameters are chosen as 𝜁=2>1/𝛽₁. And the parameter 𝜀=0.01 in (26) is selected.

Case 1. Consider the case of a nominal system with 𝑑₁(𝑡)=𝑑₂(𝑡)=𝑑₃(𝑡)=𝑑₄(𝑡)=0. As mentioned in Section 2, the proposed design procedure can be summarized as follows.

Step 1. According to (9), the switching surface, with 𝐾=[−12.48246.8483−7.8920−16.8563] satisfying (14), is given by 𝜎(𝑡)=𝐼^1−𝑞𝑒₂(𝑡)+^𝑡₀𝐾𝐸(𝜏)𝑑𝜏. (28)

Step 2. According to (26), the sliding mode control law is obtained as follows: 𝑢(𝑡)𝜎=−2𝜂|𝜎|1+0.01,2>𝛽₁, (29) where 𝜂=|𝑒₁−𝑦_𝑠𝑧²_𝑠+𝑦_𝑚𝑧²_𝑚+𝐾𝐸|.

In numerical simulations, the simulations are all performed by setting 𝑞=0.95 and the initial values of the master and slave systems are given, respectively, as [𝑥_𝑚(0)^𝑦_𝑚(0)𝑧_𝑚(0)𝑤_𝑚(0)]^𝑇=[0.5−0.20.20.5]^𝑇 and [𝑥_𝑠(0)^𝑦_𝑠(0)𝑧_𝑠(0)𝑤_𝑠(0)]^𝑇=[0.10.10.10.1]^𝑇. The simulation results are shown in Figures 3, 4, 5, 6, and 7. Figure 3 shows the corresponding 𝜎(𝑡) for the controlled fractional-order hyperchaotic systems under the proposed sliding mode control (29). Figures 4–6 represent, respectively, the state responses, error states’ responses, and the control input. From the simulation results, it is shown that the proposed controller (29) can drive the resulting tracking errors lim_𝑡→∞|𝑒_𝑖(𝑡)|=0, 𝑖=1,2,3,4, which fully coincide with theoretical results in this paper.

Figure 3 

The time response of switching surface 𝜎(𝑡).

(a)

(b)

(c)

(d)

Figure 4 

State responses of controlled drive and response systems: (a) state response of 𝑥𝑚, 𝑥𝑠; (b) state response of 𝑦𝑚, 𝑦𝑠; (c) state response of 𝑧𝑚, 𝑧𝑠; (d) state response of 𝑤𝑚, 𝑤𝑠.

(a)

(b)

(c)

(d)

Figure 5 

Error states of controlled drive and response systems: (a) error state of 𝑒1; (b) error state of 𝑒2; (c) error state of 𝑒3; (d) error state of 𝑒4.

Figure 6 

The time response of control input 𝑢(𝑡).

Figure 7 

The error bound with matched perturbations.

Case 2. Consider the case of a non-nominal system with unmatched external perturbations of 𝑑₁(𝑡)=0.1sin(4𝑡), 𝑑₂(𝑡)=0.3sin(4𝑡), 𝑑₃(𝑡)=0, 𝑑₄(𝑡)=0. Under the same simulation conditions as in Case 1, the switching surface with 𝐾=[−12.48246.8483−7.8920−16.8563] is given by 𝜎(𝑡)=𝐼^1−𝑞𝑒₂(𝑡)+^𝑡₀𝐾𝐸(𝜏)𝑑𝜏 (30) and the sliding mode control law is given as in (29).

The time response of the error states, under the proposed sliding mode controller, is shown in Figure 8. It also shows that the error norm ‖𝐸(𝑡)‖ is bounded in the estimated error bound 𝜌=0.2439 as predicted.

Figure 8 

The error bound with unmatched perturbations.

4. Conclusion

This paper presents a method to design a sliding mode controller for the fractional-order hyperchaotic system subjected to unmatched perturbations and input nonlinearity. A new switching surface of fraction-integer integral (FII) type has been proposed such that the stability of the fractional chaotic system dynamics in the sliding mode is easily ensured. Illustrative examples, including nominal (matched) and nonnominal (unmatched) cases, have been presented to demonstrate the validity of the proposed synchronization scheme.

Conflict of Interests

The authors, Teh-Lu Liao, Jun-Juh Yan, and Jen-Fuh Chang, declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This paper was supported by the National Science Council of Taiwan under Contracts NSC101-2221-E-006-190-MY2 and NSC102-2221-E-366-003.