Abstract
The sliding mode control schemes are investigated to synchronize two spatiotemporal chaotic systems, which are two arrays of a large number of coupled chaotic oscillators. Firstly, sliding mode manifolds with the desired performance are designed. The asymptotic convergence to the origin of the synchronization errors is also proved. However, the terms from parameter fluctuations in equivalent controls are usually impossible to be measured directly. So we regard them as lumped disturbances, but, for practical application, it is difficult to obtain the upper bound of lumped disturbances in advance which often results in a conservative sliding mode control law with large control effort, causing a large amount of chattering. To reduce the chattering and improve the performance of the system, a disturbance observer is designed to estimate the lumped disturbances. A composite synchronization controller that consists of a sliding mode feedback part and a feedforward compensation part based on disturbance observer is developed. The numerical simulation results are presented to show the effectiveness of the proposed methods.
1. Introduction
Synchronization is a fundamental nonlinear phenomenon. It takes place in many physical and biological processes. Motivated by potential applications in physics, electrical engineering, communication, and other fields, synchronization of chaotic system has become a topic of great interest [1, 2]. A further and natural development in this aspect concerns the synchronization of spatiotemporal chaos, which is an array of a large number of coupled spatial functional cells or units displaying complicated nonlinear behaviors [3]. There are some great advantages of spatiotemporal chaos in comparison with low-dimensional chaos. When we apply spatiotemporal chaos to information processing, each spatial cell serves as an information operator and then performs information operation simultaneously in parallel if one can properly drive and control each coupled cell. Thus, the efficiency of information processing can be significantly improved. These potential applications of synchronization of spatiotemporal chaos are extremely attractive.
So how to control and synchronize the spatiotemporal chaotic systems should be considered. Actually, some far-sighted researchers have studied the synchronization of spatiotemporal chaos via various approaches for the last decades [4–8]. Kocarev and Parlitz studied the synchronization of spatiotemporal chaos based on discrete time coupling [4]. Codreanu presented a method of active control to synchronize spatiotemporal chaotic systems [5]. Brandt et al. studied the effects of disorder in external forces on the control and synchronization of coupled nonlinear oscillator networks [6]. dos Santos et al. investigated the spatiotemporal dynamics behaviors and the conditions for synchronization of one-dimensional lattice of chaotic piecewise linear maps [7]. Lü et al. took two coupled map lattices with different structures as examples to show a method of the synchronization of spatiotemporal chaos [8].
Most studies of synchronization usually assume that the parameters of these chaotic systems are identical. However, in physical systems, this may not be the real case. Parameter mismatch or fluctuation is a common phenomenon. Shrimali et al. investigated the effect of parameter fluctuation of coupling strength on the synchronization of coupled logistic maps and pointed out that it is of crucial importance to achieve and maintain synchronization [9]. Li et al. investigated quasi-synchronization of the three-variable autocatalator model via impulsive control approach in the presence of single-parameter mismatch [10]. Shahverdiev et al. studied the influence of parameter mismatches on the synchronization between two linearly coupled chaotic nonidentical time-delayed systems [11].
In this paper, we propose a control scheme to synchronize two arrays of coupled chaotic oscillators with parameter fluctuation based on sliding mode control. The sliding mode control (SMC) is an appropriate synchronization approach in the presence of parameter mismatch or fluctuation. It has many advantages, such as easy realization, rapid response, fine transient performance, and good robust property to disturbance. One of the most intriguing characteristics of SMC is the discontinuous nature of control behavior. Several researchers have applied sliding mode control techniques to chaotic systems [12, 13]. Dadras et al. proposed a sliding mode control strategy to stabilize a new uncertain chaotic dynamical system [12]. And Yan et al. proposed a robust control scheme to synchronize a class of uncertain drive-response chaotic systems based on the sliding mode control technique [13].
However the above designed equivalent controls always contain a term from parameter fluctuation which is similar to disturbances and is impossible to measure directly. Note that for practical applications, the upper bound of lumped disturbances is difficult to obtain in advance which often results in a conservative sliding mode control law with large control effort, causing a large amount of chattering.
As an efficient technique of disturbance estimation, disturbance observer (DOB) is a good choice to alleviate the above restriction [14]. In real applications, systems always face different disturbances, including internal disturbances and external disturbances. Conventional feedback-based control methods usually cannot react directly or fast to reject these disturbances. The composite disturbance observer-based control has been successfully applied to many practical problems in various fields [15–21]. In this paper, the lumped disturbances from parameter fluctuation are estimated. Moreover, a composite synchronization controller that consists of a sliding mode feedback part and a disturbance compensation part based on disturbance observer is also developed.
This paper is organized as follows. In Section 2, the synchronization problem formulation and control scheme of two arrays of coupled chaotic oscillators with parameter fluctuation are investigated. Next, a composite synchronization controller that consists of a sliding mode feedback part and a disturbance compensation part based on disturbance observer is presented in Section 3. In Section 4, some numerical simulation results of the synchronization of two arrays of 100 coupled Lorenz oscillators with parameter mismatch are presented to show the effectiveness of the proposed method. Finally, some conclusion remarks are given in Section 5.
2. Sliding Mode Control of Two Spatiotemporal Chaotic Systems
Lorenz system is a well-known prototype for nonlinear studies and shows typically chaotic behaviors. Such spatiotemporal chaos coupled with the following diffusively Lorenz cells is considered here: where , , and are the states of the th Lorenz cell, . is the Prandtl constant, is a parameter that depends on the temperature difference, and is a constant related to the given space. When , , and , each Lorenz cell has typical chaos behavior. And is the parameter on behalf of coupling strength. The boundary condition is arranged that when the cell subscript , then is set as 1, and similarly, when , the subscript is set as . In particular, when , this array is chaotic with a Lyapunov dimension equal to , as studied by Kocarev and Parlitz [4].
The response spatiotemporal chaos has the same structure with different initial conditions, but, for the th response Lorenz cell, replace in (1) with correspondingly, where , . As the parameter depends on temperature difference, it may fluctuate with time. So and are designated as the parameters of the drive spatiotemporal chaos and the response one, respectively, where . Here is constant bias or varies slowly with time in a limited range. However, the parameter depends on the given space and, in general, does not vary with time.
Adding synchronization control signal on each “corresponding” response Lorenz cell, one has Then the forced response spatiotemporal chaos under synchronization control can be written as where , , and are the state variables of the th cell in the response spatiotemporal chaos, .
The error state vector of the synchronization between a drive cell and its “corresponding” response one is defined as follows: , . Then, the th synchronization error subsystem can be expressed as
The synchronization problem of spatiotemporal chaos considered here is that, for different initial conditions of the th drive chaotic cell and its “corresponding” response chaotic cell , they can be synchronized by an appropriate sliding mode control such that where denotes the Euclidian norm of a vector.
The design of sliding mode control generally consists of two main steps: selecting a sliding mode manifold which induces a stable reduced-order dynamics and designing a switching control to force the error states onto the sliding mode manifold and subsequently to keep them on that manifold. For the stability of the error subsystem in (4), the sliding mode manifold should satisfy the following conditions. (i) It includes all states of ; (ii) Almost everywhere .
Then the th time-varying sliding mode manifold is chosen as ; namely, the manifold is on the plane . It satisfies the above condition (i) obviously. Besides, from , it also satisfies the condition (ii). Once the sliding mode manifold is reached, from , the equivalent control can be obtained as
When the th synchronization error subsystem (4) slides on the manifold, its corresponding reduced-order synchronization error subsystem can be written as
Theorem 1. Considering the error subsystem (4) and supposing that it is forced to slide on the manifold by , then the states of the reduced-order synchronization error subsystem 7 in (7) converge asymptotically to itself equilibrium point.
Proof. Consider an order accessorial system made up of the state variable , :
Next consider the following Lyapunov function candidate:
Then from (8), its derivative is
For the parameters and which are positive,
So according to Lyapunov stability criterion, the accessorial system (12) is asymptotically stable.
From the error variables in (7), when and , then , where .
The problem becomes how to design an appropriate control law such that the sliding mode manifold is reached and maintained. For the existence of manifold , the necessary and sufficient conditions for the occurrence of the sliding mode are and . So the sliding mode control scheme is designed as
where is the gain of switching control and is the standard signum function.
For can only choose finite values, the sliding mode manifold may only exist in a certain local region. For the ergodicity of chaotic strange attractor, the region attracting to the sliding mode manifold is certainly reached. In this way, the control law for th error subsystem in (4) is designed as
Remark 2. From (13), the equivalent control includes the term of parameter fluctuation that is hard to measure directly. Considering the robust property of sliding mode control, the term can be regarded as a lumped disturbance . Then the control in (13) becomes
Theorem 3. As the disturbance term from parameter mismatch is bounded, there exists a constant satisfying . Under the control law (14), the sliding mode manifold is reached in finite time and maintained if the switching control gain satisfies .
Proof. Choosing another Lyapunov function and taking its derivative yield So for any . The existence of the sliding mode is guaranteed. The error state reaches the terminal sliding manifold from any initial condition in finite time.
Remark 4. As discussed above, if the disturbance term from parameter fluctuation is taken off, the gain of switching control in (14) should be larger than the maximum of the disturbance amplitude. For the upper bound of lumped disturbances is difficult to obtain in advance, this often results in a conservative sliding mode control law with larger enough switching control gain. However, the higher switching control gain causes a larger amount of chattering in a control system which is commonly harmful in engineering.
3. Chattering Reduction Based on Disturbance Observer
In real applications, systems always face different disturbances, including internal disturbances and external disturbances. In most cases, it is usually impossible to measure the disturbances directly in practical applications.
As pointed out in Remark 2, the term in equivalent control is hard to measure directly. If we leave it out, the switching control gain has to become higher to guarantee the sliding mode manifest of synchronization control. And some performances become worse.
Considering the error variables under the control in (14), Next we design a disturbance observer to estimate the lumped disturbance .
The synchronization problem can be interpreted as a stability one. So the diagram of a disturbance observer based on (16) is shown in Figure 1. In the diagram, the transfer functions , , and denote the th controlled model, the nominal model, and the filter of DOB, respectively. The phase functions , , , and represent the synchronization error variable in (16), the lumped disturbance, the disturbance estimate obtained by DOB, and the total synchronization control, respectively.
Theorem 5. Consider the disturbance estimation error between the lumped disturbance and its estimate, and assuming that the filter in the disturbance observer satisfies if the closed-loop system can be stabilized by the sliding mode control in (14) and the lumped disturbance has a bounded steady-state value, then the estimation error will asymptotically approach zero.
Proof. From (16), we can get And from Figure 1, the disturbance estimate can be obtained: So the phase function of the disturbance estimate error is According to the final-value theorem, we obtain the following result: Since the steady-state gain of is 1 and is bounded in the steady-state, this means that . The theorem is thus proved.
As discussed above, the key of designing disturbance observer is to give and rationally. From (16), we take the nominal model . For the implementation of disturbance observer should have a relative degree larger than or equal to the relative degree of nominal model, a one-order low-pass filter is given as where is a time constant. From Theorem 5, we obtain that , which means the estimation of DOB could eventually converge to the lumped disturbance. So the new control law is gotten by synthesizing the sliding mode controller and disturbance compensation part: Under above control, the error variables in (16) become
Corollary 6. From Theorem 3, if the switching control gain is larger than the upper bound of , namely, , then, under the control law (22), the sliding mode manifold is reached in finite time and maintained.
Remark 7. When the design of DOB is appropriate, will be a good estimate of and the upper bound of can be much less than that of . In this case, the switching control gain required is quite smaller than . This means the discontinuous terms of the sliding mode controller in (22) will be much smaller than that in (14) and the chattering will be reduced.
Remark 8. The proposed control method in (22) can be regarded as a composite control structure that consists of a SMC part plus a disturbance compensation part. The SMC control law and the disturbance observer are designed separately. The parameters of SMC and the disturbance observer can be also chosen separately.
4. Simulation Results
In order to demonstrate the performances of the proposed scheme, some numerical simulations of synchronization between two spatiotemporal chaotic systems were presented here. Each spatiotemporal chaos has 100 coupled chaotic Lorenz cells as described in (1) and (3).
The Prandtl constant and the constant related to the given space were set as and in both drive and response Lorenz chaotic arrays. The parameter of temperature difference is set as in drive array. However, the corresponding parameter in response array fluctuates slowly in the range from 27.44 to 28.56 (total in ±2%). Each initial condition was independently given.
The composite disturbance observer-based sliding mode control in (22) was switched on at . The filter constant is chosen as , and the gain of switching control in (22) is . Some simulation results have been shown in Figure 2. It illustrates the spatiotemporal evolution of every synchronization error variable between the drive and the corresponding response cells. In the first 50 seconds, each state variable is independent on the “corresponding” drive state variable . However, after the control is switched on, their difference becomes less and less. Several seconds later, it becomes indistinguishable.
To investigate the difference between the SMC method in (14) and the composite control method () in (22), some simulations and experimental studies about the synchronization of above spatiotemporal chaos have been done. The time constant parameter of DOB is also . The parameter is set as in drive array, while the corresponding parameter in response array is set a fixed value to simulate the lumped disturbance, . And the control is added at in each case. As an example, the synchronization error variable and control under the SMC and control schemes are shown in Figure 3 for simulation results.
(a)
(b)
In Figure 3, under the composite control of with gain , the synchronization error variable becomes rapidly close to zero. However, when only SMC method is applied with equal gain , the error variable does not converge to zero. For the amplitude of disturbance is bounded in 15, the switch control gain is then updated as . In this case, the error variable also becomes close to zero rapidly.
It should be noted that the simulated term is not a simple static disturbance but a dynamical one. It depends not only on parameter but also on the variable which varies rapidly with time. The estimate error of DOB could not eventually converge to zero in this case. However, the upper bounds of are still much less than that of . So the discontinuous terms of the composite controller () are much smaller than those of the SMC controller, and the chattering reduces obviously, as shown in Figure 3(b).
5. Conclusion
In this work, a composite sliding mode control scheme is developed to synchronize two arrays of coupled chaotic cells with parameter fluctuation. A sliding mode controller is firstly presented to synchronize the two arrays of coupled Lorenz cells. The condition of switching control gain is also given when the parameter fluctuation terms in equivalent controls are taken off as lumped disturbances. In order to reduce the chattering, a disturbance observer is designed to estimate it and compensate it in control. In this composite disturbance-based sliding mode control scheme, the switching control gain is usually much smaller than the original one, and the discontinuous terms become smaller.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors gratefully acknowledge the helpful comments and suggestions of the reviewers, who have improved the presentation of the paper. This work is partially supported by the National Natural Science Foundation of China (11178008).