Centralized Data-Sampling Approach for Global O(t−α) Synchronization of Fractional-Order Neural Networks with Time Delays

In this paper, the global O(t−α) 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 O(t−α) synchronization design strategy. A sufficient criterion is given under which the drive-response-based coupled neural networks can achieve global O(t−α) 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 datasampling scheme.

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][21][22][23][24][25][26][27][28][29][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][32][33][34][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 datasampling 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.

Preliminaries.
In order to facilitate understanding, we first introduce some concepts of fractional calculation.
Fractional integral with order  > 0 of function F() is characterized as where  ≥  0 , Γ(⋅) is the Gamma function, which is defined as Riemann-Liouville derivative with order  > 0 of function F() is characterized as where  ≥  0 ,  − 1 <  < , and  is a positive integer.Caputo derivative with order  > 0 of function where  ≥  0 ,  − 1 <  < , and  is a positive integer.
In this paper, consider a class of fractional-order neural networks with time delays governed by  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.

Problem Formulation.
In this paper, consider system (5) as the master/drive system, and then the slave/response system is described as where 5) and ( 7); then we can obtain the error dynamics system where The initial condition of system ( 8) is () = () fl χ() − () ∈ C  .
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 () ∈ R  represents the output of ( 8) and   denotes the sampling instant satisfying lim →+∞   = +∞.
To study global ( − ) synchronization, next, we will introduce some related definitions.
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].

Main Results
Based on the discussion in preceding section, then the datasampling controller can be designed as where  = (K  ) × ∈ R × 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  = 1, 2, . . ., , Before we began to develop theoretical criterion, we first state an important lemma, which will be used in the proving process.
Proof.Consider the Lyapunov functions candidate as and set for  ≥  0 .
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.

Two Illustrative Examples
In this section, two illustrative examples are given to demonstrate the effectiveness of theoretical criterion.
Figure 1 shows simulation result of the above neural network model, which can exhibit chaotic behavior.
Obviously, we can obtain  1 =  2 = 1, then (38) are satisfied, so the designed controller is Figure 2 is utilized to show the simulation result for the master/drive system state  1 () and the slave/response system state x1 () in Example 1. Figure 3 is utilized to show the simulation result for the master/drive system state  2 () and the slave/response system state x2 () 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.
Figure 6 is utilized to show the simulation result for the master/drive system state () and the slave/response system state x() 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.

Conclusion
In this paper, the centralized data-sampling approach for global ( − ) synchronization of fractional-order neural net-