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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 742821, 19 pages
Optimal Exponential Synchronization of Chaotic Systems with Multiple Time Delays via Fuzzy Control
Department of Electrical Engineering, National University of Tainan, No. 33, Section 2, Shu Lin Street, Tainan 700, Taiwan
Received 15 January 2013; Revised 26 April 2013; Accepted 28 April 2013
Academic Editor: Ryan Loxton
Copyright © 2013 Feng-Hsiag Hsiao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This study presents an effective approach to realize the optimal exponential synchronization of multiple time-delay chaotic (MTDC) systems. First, a neural network (NN) model is employed to approximate the MTDC system. Then, a linear differential inclusion (LDI) state-space representation is established for the dynamics of the NN model. Based on this LDI state-space representation, this study proposes a delay-dependent exponential stability criterion of the error system derived in terms of Lyapunov’s direct method to ensure that the trajectories of the slave system can approach those of the master system. Subsequently, the stability condition of this criterion is reformulated into a linear matrix inequality (LMI). Based on the LMI, a fuzzy controller is synthesized not only to realize the exponential synchronization but also to achieve the optimal performance by minimizing the disturbance attenuation level. Finally, a numerical example with simulations is provided to illustrate the concepts discussed throughout this work.
In practice, due to information transmission, time delays naturally exist in many systems. The existence of time delay is frequently a source of instability and is encountered in various engineering systems [1–5]. Consequently, the problem of stability analysis in time-delay systems remains a major focus of researchers wishing to inspect the properties of such systems.
Since chaotic phenomenon in time-delay systems was first found by Mackey and Glass , there has been increasing interest in time-delay chaotic systems. Chaos is a well-known nonlinear phenomenon, and it is irregular, seemingly random and extremely sensitive to initial conditions . Based on these properties, chaos has received a great deal of interest among scientists from various research fields [8–12]. One of its research fields for communication, chaotic synchronization, has been investigated extensively.
The chaotic synchronization proposed by Pecora and Carroll in 1990  is intended to control one chaotic system to follow another. Since the introduction of this concept, various synchronization approaches, such as nonlinear feedback control  and adaptive control , have been widely developed in the past two decades. Chaos synchronization can be applied in the vast areas of physics and engineering science, especially in secure communication . Therefore, chaotic synchronization has become a popular study [14–23].
In real physical systems, some noises or disturbances always exist that may cause instability and thereby destroy the synchronization performance. Hence, how to reduce the effect of external disturbances in synchronization process for chaotic systems is an important issue [24, 25]. The control has been conferred for synchronization in chaotic systems over the last few years [24–28]. And the synchronization problem was also investigated extensively for time-delay chaotic systems (see, e.g., [25, 29–31]). Accordingly, the objective of this study is to realize the exponential synchronization of multiple time-delay chaotic (MTDC) systems, and at the same time the effect of external disturbance on control performance is attenuated to a minimum level.
Neural-network- (NN-) based modeling has become an active research field in the past few years due to its unique merits in solving complex nonlinear system identification and control problems [32–37]. Over the past decade, fuzzy control has rapidly developed in both the academic and industrial communities and there have been many successful applications. Despite the successes of fuzzy control, it has become evident that many basic problems remain to be solved. Stability analysis and systematic design are certainly among the most important issues for fuzzy control systems. Lately, there have been significant research efforts devoted to these issues (see [38–41]). However, all of them neglect the modeling errors between nonlinear systems and fuzzy models. The existence of modeling errors may be a potential source of instability for control designs based on the assumption that the fuzzy model exactly matches the nonlinear plant . Recently, Kiriakidis , Chen et al. [43, 44], and Cao et al. [45, 46] proposed novel approaches to overcome the influence of modeling errors in the field of model-based fuzzy control for nonlinear systems.
Consequently, an effective method is proposed via neural-network- (NN-) based technique to realize the optimal exponential synchronization of multiple time-delay chaotic (MTDC) systems in this study. Based on the above, the trajectories of slave systems can approach those of master systems and the effect of external disturbance on control performance is attenuated to a minimum level.
This study is organized as follows. The system description is arranged in Section 2. In Section 3, a robustness design of fuzzy controllers is proposed to realize the optimal exponential synchronization. The design algorithm is shown in Section 4. In Section 5, the effectiveness of the proposed approach is illustrated by a numerical example. Finally, the conclusions are drawn in Section 6.
2. System Description
Consider a pair of multiple time-delay chaotic (MTDC) systems in master-slave configuration. The dynamics of the master system () and slave system () are described as follows: where and are the nonlinear vector-valued functions, denotes the external disturbance, ) are the time delays, and is the control input. Moreover, and are the state vectors of and , respectively.
In this section, a neural network (NN) model is first established to approximate the MTDC system. The dynamics of the NN model are then converted into a linear differential inclusion (LDI) state-space representation. Finally, on the basis of the LDI state-space representation, a fuzzy controller is synthesized to realize the synchronization of MTDC systems.
2.1. Neural Network (NN) Model
The MTDC system can be approximated by an NN model, as shown in Figure 1, that has layers with neurons for each layer, in which are the state variables and are the state variables with delays.
To distinguish among these layers, the superscripts are used for identifying the layers. Specifically, we append the number of the layer as a superscript to the names for each of these variables. Thus, the weight matrix for the th layer is written as . Moreover, it is assumed that is the net input and is the transfer function of the neuron. Subsequently, the transfer function vector of the th layer is defined as where is the transfer function of the th neuron. The final output of NN model can then be inferred as follows: where with
2.2. Linear Differential Inclusion (LDI)
To deal with the synchronization problem of MTDC systems, this study establishes the following LDI state-space representation for the dynamics of the NN model, described as [47, 48]: where is a positive integer, is a vector signifying the dependence of on its elements, are constant matrices, and . Furthermore, it is assumed that and . Based on the properties of LDI, without loss of generality, we can use instead of . The following procedure represents the dynamics of the NN model (4) by LDI state-space representation .
To begin with, notice that the output satisfies where and denote the minimum and the maximum of the derivative of , respectively, and are given in the following: Subsequently, the min-max matrix of the th layer is defined as follows: Moreover, based on the interpolation method, the transfer function can be represented as follows : where the interpolation coefficients and . Equations (3) and (11) show that Therefore, the final output of the NN model (4) can be reformulated as follows: where and , , () represent the variables of the th neuron of the first, second, and the Sth layer, respectively. Finally, according to (7), the dynamics of the NN model (13) can be rewritten as the following LDI state-space representation: where , , is a positive integer, and is a constant matrix with appropriate dimension associated with . Moreover, the LDI state-space representation (15) can be rearranged as follows: where and are the partitions of corresponding to the partitions of .
2.3. Fuzzy Controller
According to the control scheme, a fuzzy controller is utilized to make the slave system synchronize with the master system.
The fuzzy controller takes the following form:
Control Rule : where , and is the number of IF-THEN rules of the fuzzy controller and are the fuzzy sets. Hence, the final output of this fuzzy controller can be inferred as follows: with , , in which is the grade of membership of in . In this study, it is also assumed that and for all . Therefore, and for all .
3. Robustness Design of Chaotic Synchronization and Stability Analysis
In this section, the synchronization of multiple time-delay chaotic (MTDC) systems is examined under the influence of modeling error.
3.1. Error Systems
From (1) and (2), the synchronization error is defined as and then the dynamics of the error system under the fuzzy control (20) can be described as follows: where , with Suppose that there exists a bounding matrix such that for the trajectory , and the bounding matrix can be described as follows: where is the specified structured bounding matrix and for ; . Equations (24) and (25) show that Namely, is bounded by the specified structured bounding matrix .
Remark 1 (see ). The following simple example describes the procedures for determining and . First, assume that the possible bounds for all elements in are
where for some with , ; ; and .
A possible description for the bounding matrix is where for . Notice that can be chosen by other forms as long as . Then, we check the validity of (24) in the simulation. If it is not satisfied, we can expand the bounds for all elements in and repeat the design procedure until (24) holds.
3.2. Delay-Dependent Stability Criterion for Exponential Synchronization
In this subsection, a delay-dependent criterion is proposed to guarantee the exponential stability of the error system described in (21). Moreover, in real physical systems, some noises or disturbances always exist that may cause instability and thereby destroy the synchronization performance. To reduce the effect of the external disturbance, an optimal scheme is used to design the fuzzy control so that the effect of external disturbance on control performance can be attenuated to a minimum level. In other words, in this study, the fuzzy controller (20) not only realizes exponential synchronization but also achieves the optimal control performance.
Prior to the examination of the stability of the error system, some definitions and a lemma are given next.
Definition 2 (see ). The slave system (2) can exponentially synchronize with the master system (1) (i.e., the error system (21) is exponentially stable) if there exist two positive numbers and such that the synchronization error satisfies The positive number is called the exponential convergence rate.
Definition 3 (see [24–28]). The master system (1) and slave system (2) are said to be exponential synchronization if the following conditions are satisfied:(i)in the case of , the error system (21) is exponentially stable,(ii)under the zero initial conditions (i.e., , in which is the maximal value of ’s) and a given constant , the following condition holds:
where the parameter is called the -norm bound or the disturbance attenuation level. If the minimum is found (i.e., the error system can reject the external disturbance as strongly as possible) to satisfy the above conditions, the fuzzy controller (20) is an optimal synchronizer .
Lemma 4 (see ). For the real matrices A and B with appropriate dimension, one has: where is a positive constant.
Theorem 5. For given positive constants a and n, if there exist symmetric positive definite matrices , and positive constant c, such that the following inequalities hold, then the exponential synchronization with the disturbance attenuation is guaranteed via the fuzzy controller (20): where for ; and .
Proof. See the appendix.
Remark 6. Based on (24), is assumed to be bounded by the specified structured bounding matrix and then the larger results in larger . Since the matrices must be negative definite to meet the stability condition (32b), the larger will make Theorem 5 more difficult to satisfy.
Corollary 7. Equations (32b) and (32c) can be reformulated into LMIs via the following procedure.
By introducing the new variables , , and , (32b) and (32c) can be rewritten as follows: for ; , and . Based on Schur’s complement , it is easy to show that the linear matrix inequalities in (33a) and (33b) are equivalent to the following LMIs in (34a) and (34b): where
Therefore, Theorem 5 can be transformed into an LMI problem, and efficient interior-point algorithms are now available in MATLAB LMI Solver to solve this problem.
Corollary 8 (see ). To verify the feasibility of solving the inequalities in (34a), (34b) by LMI Solver (MATLAB), the interior-point optimization techniques are utilized to compute feasible solutions. Such techniques require that the system of LMI is constrained to be strictly feasible; that is, the feasible set has a nonempty interior. For feasibility problems, the LMI Solver by feasp (feasp is the syntax used to test feasibility of a system of LMIs in MATLAB) is shown as follows:
(in this study, (35) can be represented as (34a), (34b))
From the above, the LMI constraint is always strictly feasible in , and the original LMI (35) is feasible if and only if the global minimum min of (36) satisfies min. In other words, if min will make (34a) and (34b) satisfied, then the stability conditions (32b) and (32c) in Theorem 5 can be met. Then, the obtained fuzzy controller (20) can exponentially stabilize the error system, and the control performance is achieved at the same time.
Remark 9. To reduce the computational burden, this study sets the positive constants a and n as unity.
Remark 10. It is an important issue to reduce the effect of external disturbances in synchronization process. The -norm bound is generally chosen as a positive small value less than unity for attenuation of disturbance. A smaller is desirable as this yields better performance. However, a smaller will result in a smaller , making the stability conditions (32b) more difficult to satisfy.
Corollary 11. To achieve optimal exponential synchronization, the fuzzy control design is formulated as the following constrained optimization problem:
More details to search the minimum are given as follows.
The positive constant is minimized by the mincx function of MATLAB LMI Toolbox. Therefore, the minimum disturbance attenuation level can be obtained.
The complete design procedure can be summarized in the following section.
Problem 1. Given two multiple time-delay chaotic systems with different initial conditions, how can a fuzzy controller be synthesized to realize the optimal exponential synchronization?
This problem can be solved according to the following steps.
Step 1. Construct the neural network (NN) models of the master system (1) and the slave system (2), respectively. On the basis of the interpolation method, the NN models are then converted into LDI state-space representations.
Step 2. According to the state-feedback control scheme, a fuzzy controller (20) is synthesized to exponentially stabilize the error system.
Step 3. Define the synchronization error , and then the dynamics of the error system (21) can be obtained.
Step 4. Based on Corollary 11, the positive constant is minimized by the mincx function of MATLAB LMI Toolbox and then we have the minimum disturbance attenuation level.
Step 5. The matrices , , and can be obtained with the minimum disturbance attenuation .
5. Numerical Example
The following example is given to illustrate the effectiveness of the proposed algorithm.
Problem 2. The purpose of this example is to synthesize a fuzzy controller to achieve optimal exponential synchronization. Consider a pair of modified multiple time-delay Chen’s chaotic systems in master-slave configuration, described as follows: where and are the state vectors of master and slave systems, respectively. Let the different initial conditions of master and slave systems be and , and let the external disturbance .
Solution. We can solve the above problem according to the following steps.
Step 1. Establish the NN models for master and slave systems via back propagation algorithm, respectively. First, the NN model to approximate the master chaotic system is constructed by 4–5–3, and the transfer functions of all hidden neurons are chosen as follows:
On the other hand, the transfer functions of all output neurons are chosen as follows:
After training, we can obtain the following the connection weights (the indices in state that the weight of the th layer in the NN model represents the connection to the th neuron from the th source):
Then, the net inputs of the th () layer are
(the symbol denotes the net input of the th neuron of the th layer in the NN model, and the indices and shown in ) indicate the same thing).
According to (9), the minimum and the maximum of the derivative of each transfer function shown in (40) and (41) can be obtained as follows:
To simplify the notation, we let , , , , , and . Then, based on the interpolation method, we have
Based on (10), let Then,
Plugging (43a)–(43c) into (45) leads towhere
Next, by renumbering the matrices shown in (48), the NN model of master system can be rewritten as the following LDI state-space representation: where ,
Similarly, the connection weights of the NN model for the slave system are obtained as follows:
Step 2. The procedures of constructing the NN model for the slave system are similar to those for the master system, and then we have the NN model of the slave system: with , and is a identity matrix. The responses of and