#### Abstract

A novel approach is presented to realize the optimal exponential synchronization of nonidentical multiple time-delay chaotic (MTDC) systems via fuzzy control scheme. A neural-network (NN) model is first constructed for 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, a delay-dependent exponential stability criterion of the error system derived in terms of Lyapunov's direct method is proposed to guarantee 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). According to 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 at the same time. Finally, a numerical example with simulations is given to demonstrate the effectiveness of our approach.

#### 1. Introduction

The stability analysis and stabilization of time-delay systems are problems of considerable theoretical and practical significance and have attracted the interest of many investigators for several years. Furthermore, time delays often appear in various engineering systems [1], such as the structure control of tall buildings, hydraulics, or electronic networks. Notably, the introduction of a time-delay factor tends to complicate the analysis. Consequently, convenient methods to check stability have long been sought later. The stability criteria of time-delay systems so far have been approached from two main directions based on the dependence on the size of delay. One method is to contrive stability conditions which do not include information on the delay, while the other method takes time delay into account. The former case is often referred to as delay-independent criterion and generally gives good algebraic conditions. Nevertheless, the abandonment of information on the size of the time delay necessarily causes conservativeness of the criteria, especially when the delay is comparatively small. Hence, delay-dependent criteria are derived to deal with the stability problem in this study.

Moreover, time delays have gained increasing interest in chaotic systems, ever since chaotic phenomenon in time-delay systems was first found by Mackey and Glass [2]. Chaotic phenomena have been observed in numerous physical systems, which can lead to irregular performance and possibly catastrophic failures [3]. Chaos is a well-known nonlinear phenomenon, and it is the seemingly random behavior of a deterministic system that is characterized by sensitive dependence on initial conditions [4]. Besides, chaos is occasionally preferable but usually intrinsically unpredictable as it can restrict the operating range of many physical devices and reduce performance. Therefore, the ability to control chaos is of much practical importance. According to these properties, chaos has received a great deal of interest among scientists from various research fields [5, 6]. One of the research fields for communication, chaotic synchronization, has been investigated extensively.

The chaotic synchronization of identical systems with different initial conditions was first introduced by Pecora and Carroll in 1990 [7]. They are intended to control one chaotic system to follow another. Since the introduction of this concept, various synchronization approaches have been widely developed in the past two decades. Chaotic synchronization can be applied in the vast areas of physics and engineering science, especially in secure communication [8]. Consequently, chaotic synchronization has become a popular study [9, 10]. However, all of them are focused on synchronizing two identical chaotic systems with different initial conditions [11]. In fact, experimental and even more real systems are often not fully identical; in particular, there are mismatches in parameters of the systems [11]. Also, in many real world applications, there are no exactly two identical chaotic systems. As a result, the problem of chaos synchronization between two different uncertain chaotic systems is an important research issue [12]. For instance, He et al. [13] investigate synchronization of two nonidentical chaotic systems with time-varying delay and parameter mismatches via impulsive control. To synchronize nonidentical chaotic systems with unknown parameters, Li et al. [14] proposed an approach based on the invariance principle of differential equations, and employing a combination of feedback control and adaptive control. Li and Ge [15] presented a new fuzzy model to simulate and synchronize two totally different and complicated chaotic systems.

In general, some noise or disturbances always exist that may cause instability. The influence of the external disturbance will worsen the performance of chaotic systems. Therefore, how to reduce the effect of external disturbances in the synchronization process for chaotic systems is an important issue [16, 17]. The control has been conferred for synchronization in chaotic systems over the last few years [16–20], and the synchronization problem has been investigated extensively for time-delay chaotic systems (e.g., see [21–23]). Accordingly, the purpose of this study is to realize the exponential synchronization of nonidentical multiple time-delay chaotic (MTDC) systems and attenuate the effect of external disturbances on the control performance to a minimum level at the same time.

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 [24–29]. Neural networks consist of simple elements operating in parallel; these elements are inspired by biological nervous systems. As a result, we can train an NN to represent a particular function by adjusting the weights between elements. As in nature, the connections between elements largely determine the network function. Individuals can train a neural network to perform a particular function by adjusting the values of the connections (weights) between elements. Hence, the nonlinear systems can be approximated as close as desired by the NN models via repetitive training. Recently, numerous reports on the success of NN applications in control systems have appeared in the literature (see [30–35]). For instance, Limanond et al. [30] applied neural networks to the optimal etch time control design for a reactive ion etching process. Enns and Si [32] advanced an NN-based approximate dynamic programming control mechanism to helicopter flight control. Despite several promising empirical results and its nonlinear mapping approximation properties, the rigorous closed-loop stability results for systems using NN-based controllers are still difficult to establish. Therefore, an LDI state-space representation was introduced to deal with the stability analysis of NN models (see [36]).

In the past few years, significant research efforts have been devoted to fuzzy control, which has attracted a great deal of attention from both the academic and industrial communities, and there have been many successful applications. For example, Wang et al. [37] presented a new measurement system that comprises a model-based fuzzy logic controller, an arterial tonometer, and a micro syringe device for the noninvasive monitoring of the continuous blood pressure wave form in the radial artery. A good tracking performance control scheme, a hybrid fuzzy neural-network control for nonlinear motor-toggle servomechanisms, was given by Wai [38]; Hwang et al. [39] developed the trajectory tracking of a car-like mobile robot using network-based fuzzy decentralized sliding-mode control; a hybrid fuzzy-PI speed controller for permanent magnet synchronous motors was proposed in Sant [40]; Spatti et al. [41] introduced a fuzzy control strategy for voltage regulation in electric power distribution systems—this real-time controller would act on power transformers equipped with under-load tap changers.

In spite of the successes of fuzzy control, many basic problems remain to be solved. Stability analysis and systematic design are certainly among the most important issues for fuzzy control systems. Recently, significant research efforts have been devoted to these issues (see [42–45] and the references therein). However, all of them have neglected the modeling errors between the fuzzy models and the nonlinear systems. In fact, 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 [46]. In recent years, novel approaches to overcome the influence of modeling errors in the field of model-based fuzzy control for nonlinear systems have been proposed by Kiriakidis [46], Chen et al. [47, 48], and Cao et al. [49, 50].

Almost all the existing research works of synchronization method made use of fuzzy models to approximate the chaotic systems (see [3, 4, 28, 42] and the references therein). Although using fuzzy models to approximate the chaotic systems is more simple than the neural-networks (NNs), the NN models will approach the chaotic systems by iterative training and adjusting the weights. In other words, the modeling errors of NN models will be much less than those of fuzzy models. With a view to the abovementioned, a novel approach is proposed via the neural-network-(NN-) based technique to realize the optimal exponential synchronization of nonidentical multiple time-delay chaotic (MTDC) systems such that the trajectories of the slave systems can approach those of the master systems and the effect of external disturbances on the control performance can be attenuated to a minimum level. First, the NN model is constructed for the chaotic systems with multiple time delays. Then, a linear differential inclusion (LDI) state-space representation is established for the dynamics of the NN model. Next, in terms of Lyapunov’s direct method, a delay-dependent criterion is derived to guarantee the exponential stability of the error system between the master system and slave system. Subsequently, the stability condition of this criterion is reformulated into a linear matrix inequality (LMI). According to 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 at the same time.

The remainder of this paper is organized as follows. The system description is arranged in Section 2. In Section 3, a robustness design of fuzzy control and a delay-dependent stability criterion are proposed to realize the optimal exponential synchronization. The design algorithm is given in Section 4. In Section 5, the effectiveness of the proposed approach is illustrated by a numerical simulation. Finally, the conclusions are drawn in Section 6.

#### 2. Problem Formulation

Consider two different 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, are the time delays, is the control input, and denotes the external disturbance. Besides, and are the state vectors of and , respectively.

In this section, a neural-network (NN) model is first constructed for the MTDC system. The dynamics of the NN model are then converted into a linear differential inclusion (LDI) state-space representation. Finally, based on the LDI state-space representation, a fuzzy controller is synthesized to realize the synchronization of nonidentical 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 identification. Specifically, the number of the layer is appended as a superscript to the names for each of these variables. Thus, the weight matrix for the th layer is written as . Furthermore, 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 handle the synchronization problem of MTDC systems, this study establishes the following LDI state-space representation for the dynamics of the NN model, described as [36, 51] where is a positive integer, is a vector signifying the dependence of on its elements, are constant matrices, and . Moreover, it is assumed that and . According to the properties of LDI, without loss of generality, can be replaced by . The following procedure represents the dynamics of the NN model (4) using the LDI state-space representation [36].

To begin with, notice that the output satisfies where and denote the minimum and 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: Besides, on the basis of the interpolation method, the transfer function can be represented as follows [36]: where the interpolation coefficients and . Equations (3) and (10) show that Hence, 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 th layer, respectively. Finally, based on (6), the dynamics of the NN model (12) 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 . Furthermore, the LDI state-space representation (14) can be rearranged as follows: where and are the partitions of corresponding to the partitions of .

From the abovementioned, the NN models of the master and slave chaotic systems are described by the following LDI state-space representations (16) and (17), respectively:

##### 2.3. Fuzzy Controller

On the basis of the state-feedback control scheme, a fuzzy controller is utilized to make the slave system synchronize with the master system. The fuzzy controller is in the following form:

Control Rule IF is and and is , where , and is the number of IF-THEN rules of the fuzzy controller and are the fuzzy sets. Therefore, the final output of this fuzzy controller can be inferred as follows: with , is the grade of membership of in .

#### 3. Stability Analysis and Chaotic Synchronization via Fuzzy Control

In this section, the synchronization of nonidentical multiple time-delay chaotic (MTDC) systems is examined under the influence of modeling error. The exponential synchronization scheme of the multiple time-delay chaotic systems is described as follows.

##### 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 (19) 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 (23) and (24) show that Namely, is bounded by the specified structured bounding matrix .

*Remark 1 (see [47]). * 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 depiction for the bounding matrix is where for . Notice that can be chosen by other forms as long as . The validity of (23) is then checked in the simulation. If it is not satisfied, we can expand the bounds for all elements in and repeat the design procedure until (23) 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 (20). Moreover, in general, some noises or disturbances always exist that may cause instability. The influence of the external disturbance will worsen the performance of chaotic systems. 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, the fuzzy controller (19) realizes exponential synchronization and at the same time achieves the optimal control performance in this study.

Before examination of the stability of the error system, some definitions and a lemma are given follows.

Lemma 2 (see [52]). * For the real matrices and with appropriate dimension,
**
where is a positive constant.*

*Definition 3 (see [51]). *The slave system (2) can exponentially synchronize with the master system (1) (i.e., the error system (20) is exponentially stable) if there exist two positive numbers and such that the synchronization error satisfies
where the positive number is called the exponential convergence rate.

*Definition 4 (see [19–23]). *The master system (1) and slave system (2) are said to be exponential synchronization if the following conditions are satisfied:(i) with zero disturbance (i.e., ), the error system (20) with the fuzzy controller (19) is exponentially stable;(ii) under the zero initial conditions (i.e., for , 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 strong as possible) to satisfy the previous conditions, the fuzzy controller (19) is an optimal synchronizer [18].

Theorem 5. *For given positive constants and , if there exist two symmetric positive definite matrices , and positive constants , so that the following inequalities hold, then the exponential synchronization with the disturbance attenuation is guaranteed via the fuzzy controller (19) consider.**where , for ; and .*

*Proof. *Let the Lyapunov function for the error system (20) be defined as
where the weighting matrices and . We then evaluate the time derivative of on the trajectories of (20) to obtain
According to Lemma 2 and (33), we have
From (36), we have
where

Integrating (37) from to , the following inequality is obtained as
With zero initial conditions (i.e., for ), we have
That is, (30) and the control performance are achieved with a prescribed attenuation .

Since

(from (32)), we can get the following inequality from (37):
Then, we can easily obtain
where .

Equations (32) and (43) show that
That is, . Therefore, we conclude that
Hence, on basis of the Definition 3**,** the error system (20) with the fuzzy controller (19) is exponentially stable for .

Corollary 6. *Equations (31a) and (31b) can be reformulated into LMIs via the following procedure. **By introducing the new variables , , and , (31a) and (31b) can be rewritten as follows:**for ; and . According to Schur’s complement [36], it is easy to show that the linear matrix inequalities in (46a) and (46b) are equivalent to the following LMIs in (47a) and (47b):**where
*

Hence, 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 7 (see [53]). * In order to verify the feasibility of solving the inequalities in (47a) and (47b) using 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, (49a) can be represented as (47a) and (47b)) and*

From the abovementioned, the LMI constraint is always strictly feasible in , and the original LMI (49a) is feasible if and only if the global minimum (the global minimum is the scalar value returned as the output argument by feasp) of (49b) satisfies . In other words, if will satisfy (47a) and (47b) then the stability conditions (31a) and (31b) in Theorem 5 can be met. Then, the obtained fuzzy controller (19) can exponentially stabilize the error system, and the control performance is achieved at the same time.

Corollary 8. *In order 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.

*Remark 9. *In order to reduce the computational burden, this study sets the positive constants and as unity.

*Remark 10. *It is an important issue to reduce the effect of external disturbances in the 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 (31a) more difficult to satisfy.

*Remark 11. *According to (25), the modeling error is assumed to be bounded by the specified structured bounding matrix , and then a larger results in a larger . Since the matrices must be negative definite to meet the stability condition (31a), a larger will make Theorem 5 more difficult to satisfy.

#### 4. Algorithm

The complete design procedure can be summarized as follows.

*Problem 1. *Given two different multiple time-delay chaotic systems with different initial conditions, the problem is centered on how to synthesize a fuzzy controller to realize the optimal *H*^{∞} exponential synchronization.

We can solve this problem based on the following steps.

*Step 1. *Construct the neural-network (NN) models of the master system (1) and the slave system (2), respectively. According to the interpolation method, the NN models are then converted into LDI state-space representations.

*Step 2. *On the basis of the state-feedback control scheme, a fuzzy controller (19) is synthesized to exponentially stabilize the error system.

*Step 3. *Define the synchronization error , and then the dynamics of the error system (20) can be obtained.

*Step 4. *Based on Corollary 8, 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 illustrates the effectiveness of the previous algorithm.

*Problem 2. *The purpose of this example is to synthesize a fuzzy controller to achieve optimal exponential synchronization. Consider the modified multiple time-delay Genesio and Lorenz 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 the external disturbance .

Figures 2(a) and 2(b) show the chaotic behaviors of the master (51) and slave (52) systems, respectively.

**(a)**

**(b)**

*Solution 1. *We can solve the previous problem based on 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 7–3, and the transfer functions of the hidden layer are chosen as follows:
On the other hand, the transfer functions of the output layer are chosen as follows:
After training, we can obtain the following connection weights (the indices in state that the weight of the layer in the NN model represents the connection to the neuron from the source):
Then, the net inputs of the th () layer are as follows (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):
Based on (8), the minimum and maximum of the derivative of each transfer function shown in (53) and (54) can be obtained as follows:

In order to simplify the notation, we let , , , and . Then, according to the interpolation method, we have On the basis of (9), let then, , .

Plugging (56a) and (56b) into (59) leads to where , , ,

Next, by renumbering the matrices shown in (61), the NN model of master system can be rewritten as the following LDI state-space representation: where ,