Abstract

This letter focuses on lag synchronization control analysis for memristor-based coupled neural networks with parameter mismatches. Due to the parameter mismatches, lag complete synchronization in general cannot be achieved. First, based on the -measure method, generalized Halanay inequality, together with control algorithms, some sufficient conditions are obtained to ensure that coupled memristor-based neural networks are in a state of lag synchronization with an error. Moreover, the error level is estimated. Second, we show that memristor-based coupled neural networks with parameter mismatches can reach lag complete synchronization under a discontinuous controller. Finally, two examples are given to illustrate the effectiveness of the proposed criteria and well support theoretical results.

1  Introduction

Based on the integrity of circuit relations, a fourth circuit element was first predicted by Chua (1971), who coined the term memristor (i.e., memory resistor). Unlike from the existing three circuit elements—resistor, inductor, and capacitor—a memristor (Strukov, Snider, Stewart, & Williams, 2008; Tour & He, 2008) links flux and charge (see Figure 1) and can memorize past electric charges. Thus, a memristor can remember its past dynamical history. Due to its unique properties of memory and nanoscale, the memristor is considered a promising candidate for simulating biological synapses. By replacing resistors in artificial neural networks, memristor-based neural networks (Adhikari, Yang, Kim, & Chua, 2012; Adhikari, Kim, Budhathoki, Yang, & Chua, 2015; Ascoli, Lanza, Corinto, & Tetzlaff, 2015; Wang et al., 2012; Wen, Huang, Zeng, Chen, & Li, 2015) can be constructed. In contrast to conventional neural networks, memristor-based neural networks (Wu & Zeng, 2013; Yang, Cao, & Yu, 2014; Zhang, Shen, & Wang, 2013), can have more computation capacity and information storage, which expand the application of neural networks in associative memory learning and image processing.

Figure 1:

The relationship of the four fundamental elements.

Figure 1:

The relationship of the four fundamental elements.

A memristor-based neural network model is different from traditional neural networks, which can be regarded as differential equations with discontinuous right-hand sides. Therefore, the dynamical analysis of memristor-based neural networks is more complex than dynamical analysis in general, and the criteria and methods used for traditional neural network models cannot be applied directly to memristor-based neural network models. There has been some valuable work on the dynamical behaviors of memristive neural networks (Bao & Zeng, 2013; Guo, Wang, & Yan, 2013, 2014; Velmurugan, Rakkiyappan, & Cao, 2016; Wen, Zeng, Huang, & Zhang, 2014). Based on Lyapunov functional and Filippov solutions, sufficient conditions were given to guarantee the passivity of memristor-based recurrent neural networks with discrete and distributed delays (Zhang, Shen, Yin, & Sun, 2015). Zhang, Li, Huang, and Huang (2016) considered an array of linearly coupled memristor-based neural networks with time-varying delay by using an intermittent control technique and obtained stability and synchronization criteria for these networks. Synchronization is an important collective behavior in nature, and some work has addressed synchronization problems of memristor-based neural networks—for example, the exponential synchronization and antisynchronization problem of these networks with time-varying delays. By adopting matrix measures and Halanay inequalities, Bao, Park, and Cao (2015) obtained some simple criteria for memristor-based neural networks. Guo, Wang, and Yan (2015) also obtained global exponential synchronization criteria for memristor-based recurrent neural networks with static and dynamic coupling. Wang and Shen (2015) and Yang and Ho (2016) discussed finite-time stabilizability, instabilizability, and synchronization of delayed memristive neural networks under a nonlinear discontinuous controller. By combining open loop control and linear feedback control, and using the theory of differential equations with discontinuous right-hand side, Abdurahman, Jiang, and Rahman (2015) derived function projective synchronization results for a class of memristor-based Cohen-Grossberg neural networks with time-varying delays. As a result, it could be easy to obtain complete and antisynchronization criteria from the function projective synchronization results. In addition, parameter mismatches (Han, Li, & Huang, 2010; He, Qian, & Han, 2011) are inevitable in practical applications due to noise or other artificial factors, that play an important role in the quality of synchronization. Furthermore, during signal transmission from the master system to the slave system, transmittal delay is unavoidable, so it is very important to study the lag synchronization problem of memristive neural networks with time delay. To the best of our knowledge, no literature has appeared concerning the problems of the lag synchronization of memristor-based neural networks with parameter mismatches. Almost all of the research has addressed the synchronization of memristive neural networks based on either Filippov theory or activation function requiring strong assumptions. However, since memristor neural networks’ connection weights depend on these networks’states, memristor neural networks can be seen as parameter mismatches. This motivates us to explore the synchronization of memristor-based neural networks using a parameter mismatches approach.

This letter, based on a parameter mismatches approach, analyzes the lag synchronization problem of memristor-based neural networks with two kinds of time delay. First, by introducing the -measure method and generalized Halanay inequality in designing the continuous controller, we derive simple criteria to ensure lag quasi-synchronization for memristor-based neural networks. In general, lag complete synchronization cannot be achieved with parameter mismatches so the error level is estimated accurately. Second, it is revealed that lag complete-synchronization can be achieved under the discontinuous controller: the discontinuous feedback term in a controller plays a key role in solving the synchronization of memristor-based neural networks.

The rest of this letter is organized as follows. Model formulation and some preliminaries are presented in section 2. In section 3, lag quasi-synchronization criteria for memristive neural networks with parameter mismatches are derived. Furthermore, by designing the discontinuous controller, lag complete synchronization criteria for memristive neural networks are obtained. In section 4, two numerical simulations are presented to demonstrate the validity of the proposed results. Conclusions are drawn in section 5.

Notation: Throughout this letter, denotes the -dimensional Euclidean space. denotes the identity matrix with compatible dimension. For a vector , is the 1-norm, which defines as . denotes the -norm. For scalar , denotes the family of continuous functions from to . Matrices, if their dimensions not explicitly stated, are assumed to be compatible dimensions for algebraic operations.

2  Network Models and Preliminaries

According to the property of memristors, a simplified memristor-based neural networks with time-varying delay can be described as
formula
2.1
where is the voltage of the capacitor , and are feedback functions, is time-varying delay, is the external input, and
formula
where represents the memductance between the feedback function and , and represents the memductance between the feedback function and . is the parallel resistor to capacitance . is the external input of primitive recurrent neural networks. Then, , , and are memristor-based connection weights satisfying the following conditions:
formula
2.2
The switching jumps , , , , , , , , are constants. The initial values associated with system 2.1 are , or, equivalently,
formula
2.3
where is the state vector, is a diagonal matrix, , , and are activation functions.
In a master-slave scheme, from the master system, equation 2.1, the slave system with parameter mismatches can be written as
formula
2.4
or, equivalently,
formula
2.5
where is the state vector, is a diagonal matrix, , , and the connection weight coefficients satisfy
formula
2.6
The switching jumps , , , , , , , , are constants. The initial values associated with system 2.4 are ; . is the coupling controller with slave system 2.1, which will be designed for the synchronization objective. Due to the finite transmission speed in practice, there is unavoidable transmittal delay from master system 2.1 with transfers to slave system 2.4. Hence, we adopt the following more practical coupling controller:
formula
2.7
where is the control gains and is the transmittal delay.

Throughout this letter, we need the following assumptions:

  1. : There exist positive constants , such that
    formula
  2. : The time delay satisfies: , is the positive constant.

To obtain the main result of this letter, we introduce, the following definitions and lemmas:

Definition 1.

The slave system, equation 2.4, is said to be lag quasi-synchronized to master system 2.1 with an error level . If there exists a compact set , such that for any , the error converges to as goes to infinity.

Definition 2
(Cao, 2004). The matrix -measure of a real square matrix is denoted as
formula
where is an induced -norm of matrix . Here, are any constant numbers, and the corresponding -measure is
formula
Similarly, when the matrix is reduced as a vector, the -norm of vector is denoted as .
Remark 1.

It should be noted that several results have been given by using a matrix measure with . However, there are few results in a -measure, and can be seen a special case of our results, since is degenerated as when .

Lemma 1.

The matrix measure has the following basic properties:

Lemma 2.
(generalized Halanay inequality Wen, Yu, & Wang, 2008). If the nonnegative function satisfies
formula
where the continuous function , , , if there exists constant such that the following inequality holds,
formula
then we have
formula
where , , .

3  Main Results

In this section, we denote the lag synchronization error signal as , (). Based on matrix measure theory and the generalized Halanay inequality, we will derive sufficient conditions for lag quasi-synchronization of memristive neural networks with parameter mismatches. For convenience, we define these notations:
formula

3.1  Lag Quasi-Synchronization Criteria

Theorem 1.
Under assumptions and , suppose that and . If the following equality holds,
formula
then the trajectory of the error system converges to
formula
3.1
The master system, equation 2.1, and the slave system, equation 2.4, achieve lag quasi-synchronized with an error level , where is an arbitrarily small, positive constant.
Proof.
Consider the following Lyapunov-Krasovskii functional:
formula
3.2
According to the master system, equation 2.1, and the slave system, equation 2.4, the error system can be written as
formula
3.3
where , , that is,
formula
takes the upper-right Dini derivative of equaiton 3.2. With respect to along the solution of equation 3.3. It yields
formula
3.4
Then, from assumption , we have
formula
3.5
Combining equations 3.4 and 3.5, we obtain that
formula
3.6
From the above analysis, let , , . One has
formula
which leads to . Then, according to lemma 5, we can obtain
formula
3.7
where . From equation 3.7, we conclude that the error system converges exponentially to the set as tends to infinity. According to definition 1, this implies that the slave system, equation 2.4, and the master system, equation 2.1, achieve lag quasi-synchronization with an error level .
Remark 2.

Due to parameter mismatches, the master system and slave system cannot reach complete lag synchronization under state feedback controller. However, when the parameter disturbance is small, the error bound of the lag quasi-synchronization is estimated in accordance with the parameter mismatches in theorem 6.

Remark 3.

Quasi-synchronization, different from complete synchronization, does not require activation functions satisfying , a common assumption in neural network fields.

Note that theorem 6 gave the general expression of . In the following corollary, we will get the concrete algorithm of .

Corollary 1.
Under assumptions and , suppose that and parameter mismatches satisfy . Let . The master system, equation 2.1, and the slave system, equation 2.4, achieve a lag quasi-synchronized with an error level set,
formula
3.8
if the following equality holds:
formula
Proof.
From assumptions , we have
formula
Since , one has
formula
Based on theorem 6, we can draw the conclusion that
Remark 4.

By adopting a -measure method, we get the algebraic criteria for lag quasi-synchronization of coupled memristor neural networks, which are easier than LMIs (linear matrix inequalities) to verify. Actually, the -measure extends the -measure () (Cao & Wan, 2014; He & Cao, 2009). In Li and Cao (2014), we proved that the -measure method was superior to other measurement methods.

Remark 5.

For a traditional neural networks model, complete synchronization cannot be achieved under a parameter mismatch. However, a memristor-based neural network can be regarded as a special switched system by designing a discontinuous controller, which can suppress the switching memristor parameter. Moreover, we find that the new dynamical phenomena, that is, memristor-based neural networks, can reach lag complete synchronization under parameter mismatches. In section 3.2, we will give the criteria of lag complete synchronization, can be verified easily.

Remark 6.

Unlike Li and Cao (2014), in this letter, we adopt parameter mismatches methods and consider lag quasi-synchronization of memristor neural networks with time delays. Moreover, we find that these networks with parameter mismatches can reach complete synchronization under a discontinuous controller.

3.2  Lag Complete-Synchronization Criteria

In this section, we consider the master-slave system with constant delay. Then the slave system is changed to
formula
3.9
where
formula
The discontinuous controller is considered as
formula
3.10
where and are control gain constants.

Assume that the activation functions , satisfy the following conditions:

  1. For any two different constants , there exist constants , , , , such that
    formula

Theorem 2.

Under assumption , suppose . Slave system 3.9 with a discontinuous controller, equation 3.10, can globally exponentially lag-synchronize with the master system if there exist positive constants satisfying the following conditions:

Proof.
Consider the following Lyapunov-Krasovskii functional,
formula
Based on equation 3.9, calculating the time derivative of , one has
formula
3.11
According to condition , we have
formula
For a sufficiently small positive , the continuity argument, yields
formula
Based on the above analysis, combining condition , one has
formula
Therefore,
formula
3.12
where , , and . Following equation 3.12, converges exponentially to zero with a convergence rate of .

4  Two Illustrative Examples

In this section, we set out two illustrative examples to check the validity of the lag synchronization results in theorems 6 and 13.

4.1  Example 1

Consider the following memristor-based recurrent neural networks,
formula
4.1
where
formula
The slave system with parameter mismatches is defined as
formula
4.2
where
formula
Taking time delay , and activation functions , assumption holds with . Suppose . Choose the initial condition as , , by observing that the phase trajectory (see Figure 2) of system 4.1 is contained in the set , and the parameter mismatches satisfy , , . Thus, one obtains that , , and . Set , that is, the error level is less than 0.5. Then we have , and the control gain parameters are taken as . It is obvious that all conditions in theorem 6 hold.
Figure 2:

Phase trajectories of master system 4.1.

Figure 2:

Phase trajectories of master system 4.1.

For a numerical simulation, taking transmittal delay —the state trajectories of the master system and the slave system are described in Figures 3 and 4. One can find that approximately slows by 1 s. The synchronization errors are given in Figure 5, from which we can see that the error level is less than 0.5 by using a suitable controller. Master-slave memristive neural networks with a parameter mismatch can achieve quasi-synchronization and prove that the proposed results in theorem 6 are correct.

Figure 3:

Time trajectories of states of and .

Figure 3:

Time trajectories of states of and .

Figure 4:

Time trajectories of states of and .

Figure 4:

Time trajectories of states of and .

Figure 5:

The error-state trajectories of the coupled memristive networks.

Figure 5:

The error-state trajectories of the coupled memristive networks.

4.2  Example 2

The second example verifies the results of theorem 13—that the master-slave memristor system can achieve lag complete synchronization. Consider the following two-dimensional memristor-based recurrent neural networks,
formula
4.3
where , ,
formula
Correspondingly, the slave system can be written as
formula
4.4
where , ,
formula
Let . The nonlinear activation functions are taken as , time-varying delay . It can be seen that and . Assuming initial conditions for , the phase trajectories of the memristor-based neural networks model, equation 4.3, are as shown in Figure 6. From the conditions, it is easy to obtain that . Then we have the following inequalities:
Figure 6:

Phase trajectories of the memristor-based neural network model of equation 4.3.

Figure 6:

Phase trajectories of the memristor-based neural network model of equation 4.3.

formula

Taking , and control gains , , transmittal delay , it follows from theorem 13 that the master-slave system can reach lag complete synchronization. Figures 7 and 8 show that the slave system, equation 4.4, approximately follows the master system, equaiton 4.3, with a transmit delay of 1 s, and the lag synchronization errors and of memristive coupled networks converge to zeros, which are described in Figure 9. The simulations verify the correctness of theorem 13.

Figure 7:

Time trajectories of the states of and .

Figure 7:

Time trajectories of the states of and .

Figure 8:

Time trajectories of states and .

Figure 8:

Time trajectories of states and .

Figure 9:

The error state trajectories of variable , .

Figure 9:

The error state trajectories of variable , .

5  Conclusion

In this letter, we investigate the lag synchronization of memristor-based coupled neural networks with a time delay. By adopting continuous and discontinuous feedback controllers, with an approach of parameter mismatches, not only lag quasi-synchronization criteria for memristor-based neural networks have been obtained, but also lag complete synchronization criteria for memristive neural networks have been established. The proposed criteria are easy to verify. Furthermore, the states of memristor neural networks strictly depend on a system’s initial conditions, which can easily generate chaotic behavior. Hence, our next research topic is to explore a new analysis method for more complex dynamical behaviors of memristor-based neural networks.

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China under grants 61573096, 61603125, 61640315, and 61573189; the 333 Engineering Foundation of Jiangsu Province of China under grant BRA2015286; and the Key Program of Higher Education of He’nan Province under grant 17A120001.

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