Abstract
In this paper, the global synchronization problem is investigated for a class of fractional-order neural networks with time delays. Taking into account both better control performance and energy saving, we make the first attempt to introduce centralized data-sampling approach to characterize the synchronization design strategy. A sufficient criterion is given under which the drive-response-based coupled neural networks can achieve global synchronization. It is worth noting that, by using centralized data-sampling principle, fractional-order Lyapunov-like technique, and fractional-order Leibniz rule, the designed controller performs very well. Two numerical examples are presented to illustrate the efficiency of the proposed centralized data-sampling scheme.
1. Introduction
Fractional-order calculus has gained an increasing attention in physical systems and engineering systems. Fractional-order dynamic systems containing fractional derivatives and integrals have been investigated in the field of control systems [1–19]. Investigating analytical skills in fractional-order dynamic systems is one important theme. However, it is considered that many control schemes for fractional differential equation are only a mathematical concept. A lot of control methods in integer-order dynamic systems cannot be introduced to fractional-order dynamic systems [2, 3].
As a kind of the important dynamic systems, the concept of neurodynamic systems can be traced back to the early 1940s. Neurodynamic systems behave like a synthesizer evaluating the performance of system itself via the topology structure. In recent years, modeling fractional phenomenon to neurodynamic systems has been developed in an effort to improve the neurodynamic processes [2, 3, 14, 16]. Appealing feature of fractional-order neurodynamic systems is that the infinite memory property can take the past inherited information into account, which well suits describing complex dynamic processes. For control system applications, fractional-order neurodynamic systems have greatly expanded systems of conventional difference equations, which provide an adaptive control system for standard application.
Data-sampling control of systems has been studied in a number of publications. Actually, as stated in [20–30], for complex or multivariable control systems, it is unrealistic or even impossible to sample all real-time physical signals at one single rate. In such situation, one is forced to use multirate data-sampling control. Multirate data-sampling control conditions have been derived there which are less conservative [31–35]. Multirate data-sampling control can achieve what single-rate data-sampling control cannot, for instance, gain margin improvement, centralized control, and decentralized control.
Many control schemes have been established for complex control systems, such as adaptive control [36, 37] and sliding mode control [38]. In view of the feature of fractional-order systems, compared with other control strategies, centralized data-sampling scheme is more applicable for implementation in fractional-order systems. For one thing, centralized data-sampling scheme itself is relatively cheaper and simpler to operate. Unlike a lot of data-sampling designs, these schemes are usually designed for continuous sampling, and the control cost is very high. For another thing, considering that the system structures of fractional-order systems are complex and ever changing, which may have unpredictable nonlinear effects, it is more reasonable and implementable for centralized data-sampling only carried out at part of timing nodes.
In this paper, a centralized data-sampling architecture enabled by low-bandwidth communication is proposed. The centralized data-sampling approach is developed to globally synchronize the drive-response-based coupled fractional-order neural networks with time delays. And then we present an intelligent control method for designing synchronization scheme based on centralized data-sampling principle, fractional-order Lyapunov-like technique, and fractional-order Leibniz rule. The obtained results provide novel and higher performance extension for the designed controller. The use of centralized data-sampling approach facilitates utilizing low-bandwidth communication to transmit harmonic signals. The operation principle and numerical examples based on computer simulations are also presented.
2. Preliminaries and Problem Formulation
2.1. Notation
For -dimensional vector , the norm of vector is recorded as . Denote as the space of -order continuous and differentiable functions from into . is a Banach space of all continuous functions . For any , let .
2.2. Preliminaries
In order to facilitate understanding, we first introduce some concepts of fractional calculation.
Fractional integral with order of function is characterized as where , is the Gamma function, which is defined as
Riemann-Liouville derivative with order of function is characterized as where , , and is a positive integer.
Caputo derivative with order of function is characterized as where , , and is a positive integer.
In this paper, consider a class of fractional-order neural networks with time delays governed by where , is the state vector, is a positive diagonal matrix, and denote the weight matrix and the delayed weight matrix, respectively, represents the transmission delay satisfying , and denote the activation functions at times and , respectively, is the bias, is a constant matrix, and is the output vector.
Obviously, system (5) is a more general model. In [2, 16], the model is a fractional-order system without time-delay. By comparing the system models, system (5) contains some existing fractional-order neural networks.
The initial condition of system (5) is .
In addition, in (5), the activation functions and are global Lipschitz; that is, for , there exist positive constants and such that
2.3. Problem Formulation
In this paper, consider system (5) as the master/drive system, and then the slave/response system is described as where is the controller to be designed.
The initial condition of system (7) is .
Define , from (5) and (7); then we can obtain the error dynamics system where
The initial condition of system (8) is .
In our control design, the structure-dependent centralized data-sampling is used. Moreover, the measured output of error dynamics system is sampled and then the data-sampling information is sent to the controller of the response system as where represents the output of (8) and denotes the sampling instant satisfying .
To study global synchronization, next, we will introduce some related definitions.
Definition 1. For dynamic system where is defined on , the initial condition of system (11) is . System (11) is said to be globally stable if there exists a positive constant such that for any and .
Remark 2. Convergence of fractional-order systems is totally different from conventional exponential convergence or absolute convergence, which possesses abnormal convergence behavior. In addition, according to Definition 1, global stability and global Mittag-Leffler stability are “essentially the same.” On Mittag-Leffler stability, please see some publications [2, 16].
Definition 3. For the master/drive system and the slave/response system where and are defined on , the initial conditions for systems (12) and (13) are and , respectively. The coupled systems (12) and (13) are said to be globally synchronized if the zero solution of the error dynamics system is globally stable, where .
3. Main Results
Based on the discussion in preceding section, then the data-sampling controller can be designed as where is the constant gain matrix to be determined. Therefore, the error dynamics system can be transformed into the following form:
For technical convenience, we also give mathematical expression for (16) represented by components: for ,
Before we began to develop theoretical criterion, we first state an important lemma, which will be used in the proving process.
Lemma 4 (see [3]). If , then the following properties hold:(1).(2)If and and their all derivatives are continuous in , then where , ,
Theorem 5. If there exist positive constants and such that for , set as the sampling time point such that for all ; then system (17) is globally stable. That is, the drive-response-based coupled systems (5) and (7) can reach global synchronization.
Proof. Consider the Lyapunov functions candidate asand set for .
Using Lemma 4, we can obtainfor .
On the other hand, it is not difficult to followfor , , where .
Substituting (25) into (24), it is obvious to derive that where
Utilizing (22), it is obvious that there exists a such that for .
From (17), together with the data-sampling principle (21), we compute Then where such that , when .
Substituting (30) into (26), we have when .
According to (20) and (31), for all .
By the definition of Caputo derivative, from (32), we can obtain then for . Thus, for , for , where ; then where . Therefore, system (17) is globally stable, which implies that the drive-response-based coupled systems (5) and (7) can reach global synchronization.
Remark 6. In Theorem 5, (20) is an algebraic condition, which relies on only system parameters, free-weighting parameters and . Besides, (21) is used to characterize the sampling time point, which can be determined by trigger mechanism. Generally, the discrimination conditions in Theorem 5 can be easily rectified. In addition, although the results established are based on , clearly, such results can be also applied to .
Remark 7. Aiming at the realistic environment under the limited bandwidth of communication channel, it is very critical to reduce the data transmission rate for networked systems. As (21), the centralized data-sampling mechanism for the sampling time point is an effective way, which does not waste the bandwidth of network to be with needless signals, and then reduces the data transmission and power consumption.
4. Two Illustrative Examples
In this section, two illustrative examples are given to demonstrate the effectiveness of theoretical criterion.
Example 1. Consider a two-dimensional neural network model (5) described by
Figure 1 shows simulation result of the above neural network model, which can exhibit chaotic behavior.
Obviously, we can obtain , , and ; to apply Theorem 5, it requires
Select , , , andthen (38) are satisfied, so the designed controller isFigure 2 is utilized to show the simulation result for the master/drive system state and the slave/response system state in Example 1. Figure 3 is utilized to show the simulation result for the master/drive system state and the slave/response system state in Example 1. Clearly, the drive-response-based coupled systems in Example 1 can reach global synchronization. Figure 4 shows the centralized data-sampling release instants and the corresponding release intervals. These results via computer simulations nicely demonstrate that the designed controller performs very well.
Example 2. Consider a one-dimensional neural network model (5) described by
Figure 5 shows simulation result of the above neural network model without any external control, which generates disorganized behavior.
Obviously, we can obtain , , and ; to apply Theorem 5, it requires
Select andthen (42) is satisfied, so the designed controller isFigure 6 is utilized to show the simulation result for the master/drive system state and the slave/response system state in Example 2. Figure 7 shows the centralized data-sampling release instants and the corresponding release intervals. Such results via computer simulations also indicate that the designed controller performs very well.
5. Conclusion
In this paper, the centralized data-sampling approach for global synchronization of fractional-order neural networks with time delays is investigated. Some sufficient conditions for centralized data-sampling principle are derived and proved to guarantee the global synchronization for drive-response-based coupled neural networks. The proposed theoretical results are sufficient conditions for global synchronization and contain a lot of space for further improvement. Future research can be extended to more economical and efficient event-triggered mechanism and more realistic networked systems involving stochastic effect or incomplete measurement.
Competing Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The work is supported by the Research Project of Hubei Provincial Department of Education of China under Grant T201412.