The robust adaptive exponential synchronization problem of stochastic chaotic systems with structural perturbations is investigated in mean square. The stochastic disturbances are assumed to be Brownian motions that act on the slave system and the norm-bounded uncertainties exist in all parameters after decoupling. The stochastic disturbances could reflect more realistic dynamical behaviors of the coupled chaotic system presented within a noisy environment. By using a combination of the Lyapunov functional method, the robust analysis tool, the stochastic analysis techniques, and adaptive control laws, we derive several sufficient conditions that ensure the coupled chaotic systems to be robustly exponentially synchronized in the mean square for all admissible parameter uncertainties. This approach cannot only make the outputs of both master and slave systems reach synchronization with the passage of time between both systems but also attenuate the effects of the perturbation on the overall error system to a prescribed level. The main results are shown to be general enough to cover many existing ones reported in the literature.

1. Introduction

Synchronization is a fundamental phenomenon that enables coherent behavior in coupled dynamical systems. Since Pecora and Carroll [1] proposed a method to synchronize two identical chaotic systems with differential initial conditions, chaos synchronization has attracted much attention from various fields during the last decades [2, 3]. Many methods and experimental techniques have been presented to realize the synchronization of two identical or different chaotic systems [4], such as adaptive control [5], sliding mode control [6, 7], nonlinear feedback control [8, 9], and fuzzy system based control [10]. Among all these methods, adaptive control method has been used widely to treat the unknown parameters and uncertainties [11, 12]. For example, a strategy for adaptive synchronization of some electrical chaotic circuit based nonlinear control and robust adaptive generalized projective synchronization of Genesio-Tesi chaotic system with uncertain parameters are discussed in [13, 14], respectively. In [15], the stability of adaptive synchronization of chaotic systems was proposed by Sorrentino et al. A robust adaptive synchronization for a class of uncertain chaotic systems with application to Chua’s circuit was presented by Koofigar et al. [16]. Hu et al. developed a robust adaptive finite-time synchronization of two different chaotic systems with parameter uncertainties [17]. Bowong and Tewa investigated the unknown inputs’ adaptive observer for a class of chaotic systems with uncertainties [18]. However, we have noted that, in all of the above mentioned papers, the chaotic systems are deterministic differential equations without any random parameters. In addition, chaos synchronization is far from being straightforward because different aspects affect significantly the ability of systems to synchronize. A particularly important aspect is to consider the presence of many disturbances and uncertainties sources, which are unavoidable parts of any practical implementation. Therefore, robustness is a very desirable characteristic of a synchronization approach. In general, it is worth pointing out that the stochastic factors like uncertainty of the structure parameter, perturbation of external noise, and stochastic input are ubiquitous in nature, society, economy, and realistic engineering. Under normal circumstances, those stochastic factors just play a minor influence. However, when the designer of the system needs to make a choice, it will become a dominant factor which could affect the trend of deterministic system [19].

Synchronization of chaotic systems affected by both structural and stochastic disturbances poses new challenges for the understanding of stability, sensitivity and robustness, bifurcations and chaos, and so forth. When analyzing the dynamical behaviors of chaotic systems, stochastic disturbances and modeling errors are probably two of the main sources that result in uncertainties. To overcome these difficulties, various adaptive synchronization schemes have been proposed and investigated (see an excellent text in [20]). An adaptive almost surely asymptotically synchronization for stochastic delayed neural networks with Markovian switching was discussed by Ding et al. [21] while Li and Fu proposed the synchronization of chaotic delayed neural networks with impulsive and stochastic perturbations [22]. M. Chen and W.-H. Chen proposed robust adaptive neural network synchronization controller design for a class of time delay uncertain chaotic systems in [23] and a robust stability and -control of uncertain systems with impulsive perturbations [24]. Recently, Fang et al. developed a robust adaptive exponential synchronization of stochastic perturbed chaotic delayed neural networks with parametric uncertainties [25]. It appears clearly from the above works that most did not consider structural perturbations. Some previous research such as [24, 25] took into account the structured uncertainties with the stochastic perturbations but supposed that the parametric uncertainties are time varying and norm-bounded and satisfy certain elementary conditions such that the uncertain matrices can be decomposed in product of Lebesque measurable function in order to develop the problem in terms of linear matrix inequality. However, this assumption seems bulky in engineering applications and also leads to the unpredictability of the LMI. According to the best of our knowledge, there are still few results about the synchronization of chaotic systems with both structural perturbations and stochastic disturbances. Therefore, robust adaptive synchronization analysis for uncertain stochastic systems has emerged as a challenging research issue.

In this paper, we discuss the asymptotical synchronization and almost surely synchronization for two different chaotic systems with the consideration of both structural and stochastic perturbations. With the help from the Lyapunov functional method and adaption method, we employ the robust analysis tool and the stochastic analysis techniques to derive some relevant conditions under which the coupled systems is globally robustly synchronized in the mean square for all admissible parameter uncertainties. These conditions guarantee the robustness of the controller against the effect of exoteric perturbations. The rest of the paper is organized as follows: Section 2 presents the Problem’s formulation and preliminaries. In Section 3 we present our main result that consists of a new controller that is robust enough against the effect of exoteric perturbation of modified Colpitts oscillator and Chua’s circuit. The computer simulation is given in Section 4. Finally, conclusions are presented in Section 5.

2. Problem’s Formulation and Preliminaries

Let two classes of nonlinear systems be given in the following form:where and are the systems states and and are the output vectors of systems (1) and (2), respectively. ; are known matrices such that and are observable. are nonlinear continuously differentiable vector function.

Considering the effect of structural and exoteric perturbation on system’s parameters, the master system is given byBased on the observer method, the slave is constructed with nonidentical parametric uncertainties and stochastic perturbation given bywhere and are bounded structural variations of the system which satisfy the condition ( being a positive constant) and is the nonlinear controller. is measurable output. is -dimensional Brownian motion defined on complete probability space with a natural filtration generated by , where is associated with the canonical space generated by with the probability measure . is the noise intensity matrix.

Define the synchronization error vector as So the error dynamical system iswhere It is clear that the synchronization problem can be transformed to be the equivalent problem of stabilizing the error system (6).

Definition 1. Systems (3) and (4) are said to be globally stable in mean squares if for any given initial condition the following equality is verified: where is the mathematical expectation.

Assumption 2. Let us define and assume ; that is, .
Given a scalar , the error system is said to be robustly stochastically stable with noise attenuation if it is robustly stochastically stable and under zero initial conditions: For nonzero , In this paper, the following performance index for a prescribed is considered: The problem to be tackled is formulated as follows. For the stochastic systems (3) and (4) and a given scalar , we like to design an adaptive feedback controller such that the resulting master-slave trajectories robustly stochastically synchronize with noise attenuation level which is the disturbance attenuation level [26] defined by The control requirement is that the response of error to the disturbances with the stabilization controller should be brought to less than which is the suboptimal limits. For the standard problem, a controller exists if and only if a unique positive definite solution to the two Riccati equations exists [27]. State space formulas can be derived for all controllers such that the -norm of the closed loop transfer function is less than .

Assumption 3. Assume that the noise intensity function matrix is uniformly Lipschitz continuous in terms of the norm induced by the trace inner product on the matrices as follows:for all . is a real positive matrix.

Definition 4. There exists continuous function such that an operator from along the trajectory of the error system (6) is defined as where

Definition 5. Systems (3) and (4) are said to be exponentially synchronized in mean square if there exist constants and , such thatwith and is called the decay rate of exponential synchronization.

Lemma 6. Suppose two continuous functions and a nonzero decreasing function such that and then there exists a suitable positive constant satisfying the following inequality:

3. Main Results

Assumption 7. We can choose the controller in this following form:where is the matrix with appropriate dimension and is the estimated feedback gain which is updated according to the following adaptation algorithm: where is a positive constant to be specified by the designer.
The control problem with a feedback (18) is to find an admissible internally stabilizing control which would be attenuating disturbances such that the norm of the stable closed loop system from the disturbances to the controlled outputs is less than ( is equal to 1 for optimal and slightly greater than 1 for suboptimal control). The control objective is stated in the mathematical form in (9). One advantage of this type of controller is that it can be easily constructed through time varying resistors, capacitors, or operational amplifiers and their combinations or using a digital signal processor together with the appropriate converters.

Assumption 8. There exist two known positive functions and such that . We assume the following conditions: (1)For bounded and , is also bounded.(2)The function enters the closed ball and remains there with and are positive constants.

Assumption 9. The “robust performant” gain scheduled system is assumed to be bounded in response to certain added inputs. Moreover, the performance level must not degrade discontinuously for arbitrarily small errors in controller (18).

Lemma 10 (Gronwall inequality (see [28])). Let us consider three continuous positive functions , , and on satisfying the inequality , , and then , .

Lemma 11. For two integrable functions and , there exist two positive constants and such that

Lemma 12. For any matrices , , , and we have the following inequalities:

Assumption 13. We can choose a constant matrix and two positive definite matrices and such thatwhere

Theorem 14. System (3) exponentially synchronizes system (4) under controller (18) in mean square.

Proof. The closed loop error system with control (18) can be written in the following form: Consider the Lyapunov function: where is a constant to be defined later.
Relation (25) yields by differentiating with respect to time It is clear from Lemma 12 that For any time , the system state is bounded (as the system is chaotic); hence we may write . Here , , , , and are positive constants.
Relation (28) leads toLet one recall relations (23): Remark 1. There exist a lot of papers that study synchronization problems of master-slave chaotic systems, in which the choice of the performance index which satisfies condition (9) remains a hard task in practice and leads sometimes to the poor performance of the control. Our method does not need the arbitrary choice of this quantity. Besides, the control objective is obtained by the selection of the initial value of the feedback gain which enables approaching the synchronization. This partly shows the less conservativeness of our control strategy.
Using inequality (31), we obtain Finally, by letting one obtainsThen, if the inequality holds, we haveFor all nonzero , under zero initial conditions, we have . Integrating both sides of (35) from 0 to and taking the expectation, we haveWe derive the cost function asSubstituting (35) into (37) yieldswhereFor a positive constant , if the following condition holds:then we have which leads to the following conclusion: . This implies that the error system (20) via control inputs with the update law (19) is asymptotically stable. Then, (9) follows immediately from (37) and the proof of Definition 1 is completed.
From Lemma 6, we getIt follows from Ito’s formula that From Lemma 11, (43) becomes Using Gronwall’s inequality (Lemma 10), we have where and
Relation (45) guarantees the exponential synchronization. This completes the proof.

Remark 2. It is important to show that the trajectories of the slave system are bounded under the feedback in the presence of perturbations: (R-1)Since the systems are assumed to operate in chaotic mode without feedback, their trajectories converge to compact invariant set. Let one assume that is the spectral radius of the perturbation on the system dynamics. We set , and let the closed ball strictly contains such a compact set (since the chaotic trajectories are bounded, we assume that they are contained in ); let be the smallest number such that . The previous development shows that is positive definite Lyapunov function with a negative definite derivative for any values of the states contained in a compact set. Hence, there always exist control gains such that the system under feedback is forward complete.(R-2)It may be shown that the system under control is forward complete; that is, if there exists a set of initial conditions and gains such that, together, they generate solutions that tend to infinity, these solutions may unboundedly grow only in infinite time. From this, it follows that for each and , there exists such that , where is in general a nondecreasing function of its arguments. We assume that is the synchronization time since, by assumption, the system operates in open loop for all for any for all for any . That is, the solutions are bounded. Note that the bound is independent of the gains, and for , we can safely assume that and depends only on ; hence, (R-2) of the proposition follows. If and are given and , then depends on and and so does ; hence, (R-1) of the proposition follows. In either case, does not depend on the initial conditions , nor is it required that .

4. Simulation Investigation

In this section, we will provide simulation results for a system where the master oscillator is the perturbed modified Colpitts oscillators and the observer is a perturbed and stochastic Chua system.

From (1), the modified Colpitts systems whose dimensionless equations are given in [29] is represented by the values as follows: The slave based on the observer method is the Chua’s circuit and can be generalized in form (2) [30], where and are continuous-time functions defined, respectively, by and with .

If the parameters are taken as , , , , , , , , , and , the systems specified by (46) and (47) exhibit chaotic dynamics. Figures 1 and 2 give the time history and phase portrait of both systems.

Remark 3. Systems (1) and (2) are quite different to make our illustration general. Most previous studies in the literature have predominantly concentrated on standard systems such as the Lorentz, the classic Colpitts oscillator, the Chua system, the Chen system, the Lu system, or the Rössler system either in the studies of their stability analysis and periodic oscillations or in the studies of their synchronization (the general form of these dynamical systems is provided in (1)). It has been shown that the modified Colpitts oscillator presents different topology and can exhibit complicated dynamics with reference to circuits previously mentioned. If , the problem becomes complex and requires some supplementary conditions by comparison to those employed in the literature; this aspect makes the present result more general with respect to those encountered in the literature.

For convenience, we select the following matrices:where .

If we let , then the estimated feedback gain is updated according to the following adaptation algorithm: Since structural variations may take any forms in theory depending on the sources of disturbances, we illustrate the proposed control scheme with a more global example where many perturbation sources are taken into consideration [2931] under the assumption that . Hence we choose an arbitrary structural variation as follows:where .

Let us select the perturbed matrix simply in the following form: We adopt as an additive Gaussian noise such that and is estimated as . The noise perturbation can be taken as and is a 3-dimensional Brownian motion satisfying and . The initial values of master system and the slave system are adopted as and , respectively. In addition, we select and which lead to .

Figure 3 shows the time response of the synchronization errors without the adaptive robust feedback controller designed in (18). From these figures, one can see that the synchronization errors diverge. Figure 4 depicts the output errors with the robust adaptive feedback controller , from which we can see that the master system (3) and slave system (4) can be exponentially synchronized in mean square. From the theoretical analysis, it appears that the complexity of perturbations cannot give any information about the performance of the system. It is hard to obtain an optimal value of feedback control gain. In order to explore the synchronization behavior of the scheme, is taken as a variable of the output error (we recall that the initial value of the system variables does not affect the evolution of the feedback gain) to see how the solution of the output error evolves with the variation of the parameter. From Figure 5, it can be observed that and converge to zero at about which means that the synchronization can be achieved when .

5. Conclusion

Because the adaptive control has some interesting features such as low sensitivity to external disturbances, robustness to the plant uncertainties, and easy realization, in this paper, we use this method to realize exponential synchronization of two different uncertain chaotic systems in which the slave system is noise perturbed. By employing the Lyapunov functional method and adaptive control, several sufficient conditions have been obtained which ensure the coupled chaotic systems to be exponentially robustly synchronized in the mean square. Furthermore, the theoretical analysis is easily verified by using the standard numerical software. We have selected two perturbed systems consisting of modified Colpitts oscillator and Chua’s oscillator. It was found that the controller maintains robust stable synchronization in the presence of exoteric perturbations, structural uncertainties, and noises.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.