Fuzzy Sampled-Data Control for Synchronization of T–S Fuzzy Reaction–Diffusion Neural Networks With Additive Time-Varying Delays

This article focuses on the exponential synchronization problem of T–S fuzzy reaction–diffusion neural networks (RDNNs) with additive time-varying delays (ATVDs). Two control strategies, namely, fuzzy time sampled-data control and fuzzy time–space sampled-data control are newly proposed. Compared with some existing control schemes, the two fuzzy sampled-data control schemes cannot only tolerate some uncertainties but also save the limited communication resources for the considered systems. A new fuzzy-dependent adjustable matrix inequality technique is proposed. According to different fuzzy plant and controller rules, different adjustable matrices are introduced. In comparison with some traditional estimation techniques with a determined constant matrix, the fuzzy-dependent adjustable matrix approach is more flexible. Then, by constructing a suitable Lyapunov–Krasovskii functional (LKF) and using the fuzzy-dependent adjustable matrix approach, new exponential synchronization criteria are derived for T–S fuzzy RDNNs with ATVDs. Meanwhile, the desired fuzzy time and time–space sampled-data control gains are obtained by solving a set of linear matrix inequalities (LMIs). In the end, some simulations are presented to verify the effectiveness and superiority of the obtained theoretical results.

Abstract-This paper focuses on the exponential synchronization problem of T-S fuzzy reaction-diffusion neural networks (RDNNs) with additive time-varying delays (ATVDs). Two control strategies, namely, fuzzy time sampled-data control and fuzzy time-space sampled-data control are newly proposed. Compared with some existing control schemes, the two fuzzy sampled-data control schemes can not only tolerate some uncertainties but also save the limited communication resources for the considered systems. A new fuzzy-dependent adjustable matrix inequality technique is proposed. According to different fuzzy plant and controller rules, different adjustable matrices are introduced. In comparison with some traditional estimation techniques with a determined constant matrix, the fuzzy-dependent adjustable matrix approach is more flexible. Then, by constructing a suitable Lyapunov-Krasovskii functional (LKF) and using the fuzzydependent adjustable matrix approach, new exponential synchronization criteria are derived for T-S fuzzy RDNNs with ATVDs. Meanwhile, the desired fuzzy time and time-space sampled-data control gains are obtained by solving a set of linear matrix inequalities (LMIs). In the end, some simulations are presented to verify the effectiveness and superiority of the obtained theoretical results.

I. INTRODUCTION
D URING the past decades, fuzzy control has provoked increasing interests of many researchers from various fields. Fuzzy control is regarded as a not only useful but simple method to control many nonlinear complex systems, especially for systems or control processes with uncertainties [1]- [9]. For example, fuzzy control has been used to control two-wheeled robots in [4]. In [5], fuzzy control has been applied to stabilize the Rössler chaotic systems. In [6], the determination of the optimal green period ratios and traffic light cycle times have been realized by fuzzy control. In [7], fuzzy control has been applied to nonlinear networked systems. In [8], fuzzy control has been considered to solve the guaranteed cost control problem of uncertain stochastic fuzzy systems. In [9], fuzzy control has been used to solve the output tracking problem for T-S fuzzy systems with saturating actuators. Among diverse fuzzy control models, the T-S fuzzy model is one of the most popular ways to analyze and design fuzzy systems. Based on the T-S fuzzy model method, many T-S fuzzy systems have been diffusely investigated since they have substantial applications such as a truck-trailer system [10], Mars entry vehicles [11], and so forth.
Recently, much attention has been paid to neural networks (NNs) due to their benefits in learning algorithms and handling data. As a result, extensive applications of NNs are found in a variety of areas including financial market, image decryption, fixed-point computations, and signal processing [12]- [15]. As one of the most important dynamical behaviors of NNs, synchronization is in the spotlight. Synchronization is a universal phenomenon in many real systems and has considerable engineering applications in secure communication, biological systems, and mechatronic systems [16]- [18]. Thus, it is farreaching to study the synchronization of NNs.
In the existing literature, most of the NN models are built under the hypothesis that the interests of all neurons are evenly distributed. In fact, due to the influence of environmental factors, the reaction and diffusion phenomena inevitably exist in NNs. Therefore, it is meaningful to consider the spatial evolutions of NNs. Reaction-diffusion neural networks (RDNNs), in which the neuron states are dependent on both time and space, can perfectly describe the time and spatial evolutions. In comparison with traditional NNs, RDNNs could realize better approximations of actual systems. Until now, many interesting results on RDNNs are obtainable in the literature [19]- [23]. For instance, in [19], the impulsive synchronization problem of RDNNs has been investigated by an impulse-time-dependent LKF method. In [20], by constructing a new LKF with the neuron activation function information, stochastic synchronization has been considered for Markovian RDNNs with actuator failures. In [23], by fuzzy control, the stabilization problem has been studied for T-S fuzzy RDNNs.
In the meantime, the time delay is often encountered in RDNNs because of the finite switching speeds of amplifiers and the congestions of signal transmission. The existence of time delay may cause oscillation or instability to deteriorate the performance of RDNNs. It is, therefore, important to study RDNNs with time delay. Note that, in the existing works of RDNNs [19]- [23], the time delay is considered as a single component in the state variables. In implementation, due to the different transmission conditions in the different segments of RDNNs, signals transmitted from one point to another may lead to additive time-varying delays (ATVDs) with different properties. Thus, it is necessary to consider ATVDs for RDNNs. However, to our best knowledge, synchronization of T-S fuzzy RDNNs with ATVDs has not been considered, which is the first motivation of this note.
In order to realize the synchronization of RDNNs with time delays, various control strategies have been proposed such as quantized feedback control [24], pinning impulsive control [25], and adaptive control [26]. With the development of communication and digital technologies, sampled-data control has stimulated increasing attention [27]- [29]. Based on sampled-data control, synchronization of RDNNs has been extensively investigated [30]- [33]. For example, in [30], by time sampled-data control, the exponential synchronization problem has been studied for RDNNs with sampled-data communications. In [31], by spatial sampled-data control, the exponential synchronization of RDNNs with time delays has been investigated. In [33], by proposing a time sampleddata controller and a discontinuous LKF, synchronization criteria have been established for RDNNs with time delays. Although some new results for synchronization of RDNNs with sampled-data control have been presented in [33], a mistake occurs in V 1 (t) of the constructed LKF. The matrix dimensions of V 1 (t) are not matched because of D k ∈ R n×n and ( ∂ei(t,x) ∂x k ) T ∈ R 1×n . Moreover, in the existing works of RDNNs [30]- [33], all the sampled-data control schemes are designed with the assumption that there is no uncertainty in control processes. In practice, due to the impact of environment and restrictions of equipment, uncertainties commonly exist in the control processes of RDNNs. Hence, it is profound in both theory and application to design a fuzzy sampleddata control scheme for RDNNs. However, few works have considered such a control scheme for synchronization of T-S fuzzy RDNNs with ATVDs.
Motivated by the above-mentioned discussions, by designing fuzzy time and time-space sampled-data control, we intend to study the exponential synchronization of T-S fuzzy RDNNs with ATVDs. The main contributions are highlighted below. 1) Two control strategies, which are fuzzy time sampleddata control and fuzzy time-space sampled-data control, are proposed for T-S fuzzy RDNNs. The two fuzzy sampled-data control schemes can not only tolerate some uncertainties but also save the limited communication resources of T-S fuzzy RDNNs.
2) A fuzzy-dependent adjustable matrix inequality technique is firstly proposed. Compared with some traditional estimation techniques with a determined constant matrix, the fuzzydependent adjustable matrix inequality technique is more flexible and helpful to reduce the conservatism.
3) The ATVDs are considered for T-S fuzzy RDNNs, which generalize the existing models of RDNNs with a single timevarying delay. So the present model here can satisfy broader application requirements. Notations: Let col{· · · } denote a column vector, diag{· · · } a block-diagonal matrix, R n the n-dimensional Euclidean space, and R n×n the set of n × n real matrices. I n , 0 n and 0 n,m represent n × n identity matrix, n × n and n × m zero matrices, respectively.
x is the space variable belonging to = [α,ᾱ], α andᾱ are constants. ϕ(t, x) = col{ϕ 1 (t, x), ϕ 2 (t, x), . . . , ϕ n (t, x)} ∈ R n is the state vector with ϕ i (t, x) being the ith neuron at time t and in space x. f (ϕ(t, x)) = col{f 1 (ϕ 1 (t, x)), . . . , f n (ϕ n (t, x))} ∈ R n stands for the neuron activation function. D = diag{d 1 , d 2 , . . . , d n } ∈ R n×n , in which d i ≥ 0 represents the transmission diffusion coefficient along the ith neuron. A m = diag{a m1 , a m2 , . . . , a mn } ∈ R n×n with a mi > 0. B mij ) n×n ∈ R n×n (k = 1, 2) are the connection weight matrices. (t) = col{ 1 (t), 2 (t), . . . , n (t)} ∈ R n is the external input. The second and third equations are the Dirichlet boundary condition and initial condition, respectively. 1 (t) and 2 (t) are time-varying delays and satisfy 0 ≤ 1 (t) ≤ * By employing the weighted average fuzzy blending approach, the overall T-S fuzzy RDNN with ATVDs can be described as where ς(t) = col{ς 1 (t), . . . , ς p (t)}, θ m (ς(t)) is the normalized membership function with: and ϑ m l (ς l (t)) means the membership grade of ς l (t) in ϑ m l . Viewing system (2) as the drive system, we introduce the response system as where (2) and (3), one gets the following error system as: where The following assumption and lemmas are needed to derive the main results.
Assumption 1: For any z 1 , z 2 ∈ R, there exist scalars l − i and l + i such that f i (·) in (1) satisfies Lemma 1 [35]: For appropriate dimensional matrix Y > 0 and vector g(z), the following inequality holds: where the appropriate dimensional matrix E and vector χ(t) are independent on the integral variable.

III. MAIN RESULTS
In this section, we will investigate the exponential synchronization of T-S fuzzy RDNN with ATVDs via two different control schemes. First, by proposing a fuzzy time sampleddata control scheme and a new fuzzy-dependent adjustable matrix inequality technique, a novel exponential synchronization criterion is derived for the T-S fuzzy RDNNs (2) and (3). To show the superiority of the fuzzy-dependent adjustable matrix approach, an exponential synchronization criterion by traditional method is given for comparison. Then, in order to further save the limited network communication resources, we design a fuzzy time-space sampled-data controller. Based on the fuzzy time-space sampled-data control scheme, new sufficient conditions are further derived to exponentially synchronize the T-S fuzzy RDNNs with ATVDs.

A. Fuzzy Time Sampled-Data Control for Exponential Synchronization of RDNNs with ATVDs
Let the time sampling sequence be 0 = t 0 < t 1 < · · · < t p < · · · . The time sampling interval h p satisfies the following condition: where h is positive constant. In order to save the limited communication resources of RDNNs and tolerate some uncertainties in the designing process of the controller, according to parallel-distributed compensation method [10], the fuzzy time sampled-data controller for rule j is given by: where K j ∈ R n×n (j ∈ ‫)ג‬ are the gains to be designed. Then, the overall fuzzy time sampled-data controller is inferred by Substituting (7) into (4), we find Remark 1: In implementation, due to the impact of environment and restrictions of equipment, uncertainties ubiquitously exist in the designing process of the controller. Note that the existing control methodologies in [24]- [26], [30]- [33] are designed with an ideal hypothesis that there is no uncertainty in the designing process. Fuzzy control is commonly known as a useful method to present the control processes with uncertainties. Meanwhile, sampled-data control is more effective to save the communication resources of RDNNs in comparison with the control methods in [24]- [26]. Thus, the fuzzy time sampled-data controller is designed in (7) for exponential synchronization of T-S fuzzy RDNNs (2) and (3). Compared with the existing control schemes in [24]- [26], [30]- [33], the fuzzy time sampled-data controller (7) is more effective to tolerate some uncertainties and save the limited communication resources of RDNNs.
then the following inequality holds: where , and any matrix S 2 ∈ R n×n . Matrices F mj (m, j ∈ ‫)ג‬ and vector χ(t) are with appropriate dimensions.
Proof: For symmetric matrices Y mj ∈ R n×n (m, j ∈ ‫,)ג‬ one finds the following zero equality: From (10) and Lemma 1, we have This completes the proof. Remark 2: It is worth mentioning that the fuzzy-dependent adjustable matrix inequality technique in Lemma 4 is firstly proposed. According to different fuzzy plant rule m and controller rule j, different adjustable matrices F mj and Y mj are introduced in (9). Thus, compared with the traditional estimation technique in Lemma 1 [35] with a determined constant matrix, the fuzzy-dependent adjustable matrix inequality technique is more flexible.
Next, by constructing an appropriate LKF and using the fuzzy-dependent adjustable matrix inequality technique, a new synchronization criterion is derived for T-S fuzzy RDNNs (2) and (3). For simplicity, we denote h(t) = t − t p , Theorem 1: Let scalars * i ≥ 0, µ i (i = 1, 2), h > 0, and 0 < δ 1 < 2κ < 2γ j be given. If there exist symmetric matrices , and any matrices then the T-S fuzzy RDNNs (2) and (3) can achieve exponential synchronization under the fuzzy time sampled-data controller In the meantime, the desired fuzzy time sampled-data controller gains are given as: Proof: For t ∈ [t p , t p+1 ), choose the following LKF for error T-S fuzzy RDNN (8): where It is noted that V i (t) (i = 1, 2, . . . , 8) is continuous and V i (t) (i = 9, 10) vanish before and after t p . Then, lim t→tp V (t) = V (t p ), from which one derives that V (t) is continuous in time.
Remark 3: Based on the Lyapunov stability theory, choosing an appropriate LKF is crucial for deriving stability criteria. In this paper, (18) is chosen as the LKF. (34). It is well known that delay information and sampling information is effective to reduce the conservatism of stability criteria. In line with this, V i (t) (i = 3, 4, . . . , 8) are introduced to capture the information of the time delays 1 (t) and 2 (t) (if only single delay is considered, e.g., 1 (t), then V i (t) (i = 4, 6, 8) do not needed). V i (t) (i = 9, 10) are used to capture the information of sampling.

B. Fuzzy Time-Space Sampled-Data Control for Exponential Synchronization of RDNNs with ATVDs
Dividing into N sampling intervals, we can obtain the space sampling sequence α = x 0 < x 1 < · · · < x N =ᾱ. The space sampling interval ∆ q satisfies the following condition: where∆ is a positive constant.
Remark 5: By sampling the time t, we design the time sampled-data controller (7). Note that the state vector ϕ(t, x) of RDNN (1) is related to both time t and space x. When sampling both t and x, we design the time-space sampled-data controller (48). Compared with time sampled-data controller (7), the time-space sampled-data controller (48) uses less sampling signals, which can further save the communication resources of RDNNs.
Remark 6: In LMI-based conditions, the number of decision variables (NDVs) and the dimensions of the LMIs is two key factors for computational complexity [40]. In general, NDV is used as an index of computational complexity. By computation, the NDVs of the fuzzy time sampled-data control approach in Theorem 1 and the fuzzy time-space sampleddata control approach in Theorem 2 are 11n 2 r 2 + n 2 r + 2nr 2 + 10n 2 + 6n and 11n 2 r 2 + n 2 r + 2nr 2 + 10.5n 2 + 6.5n, respectively. Note that, in order to derive less conservative synchronization criteria, the fuzzy-dependent adjustable matrix inequality technique in Lemma 4 is used to estimate the derivative of the constructed LKF (18). By introducing more adjustable matrices, the conservatism of the obtained synchronization criteria can effectively be reduced, which will be verified in the next section. The limitation of the fuzzydependent adjustable matrix inequality technique is that it reduces the conservatism but increases the NDVs, which will increase the computational complexity to some extent. How to weigh the conservatism and computational complexity will be considered in our future work.

IV. SIMULATION EXAMPLES
In this section, some simulations are presented to verify the effectiveness and superiority of the theoretical results. In order to show how the theory results from the previous sections are applied in this section, Algorithm 1 is given to find the maximum allowable sampling period (MASP) h and controller gains K j (j = 1, 2, . . . , r).
Step 2: Use MATLAB LMI Toolbox to solve LMIs in Theorem 1 with specified h.
Step 3: If there exists a feasible solution, then let h = h + ∆h, and go to Step 2. Otherwise, go to Step 4.
Step 5: Output h = h − ∆h, which is the MASP. With the output MASP h, and using MATLAB LMI Toolbox to solve the LMIs in Theorem 1, we get the corresponding feasible matrices. Then, from (17), we find the controller gains K j (j = 1, 2, . . . , r).
When U(t, x) = 0, the trajectories of states η i (t, x) (i = 1, 2) and η(t, x) are depicted in Fig. 1. From Fig. 1, we find, the synchronization of drive-response systems (2)  x) (i = 1, 2) and the corresponding fuzzy time sampled-data controller (7). The evolution of the controlled η(t, x) is plotted in Fig.  3. From Fig. 3 one finds the exponential synchronization of the drive-response systems (2) and (3) is realized, which illustrates the effectiveness of Theorem 1 and the fuzzy time sampleddata controller (7). Then, we show the superiority of the fuzzy-dependent adjustable matrix approach. For various κ, the MASPs h by Theorem 1 and Corollary 1 are given in Table I. From  Table I, we find, for various κ, the MASPs h by Theorem 1 are all bigger than those by Corollary 1. It is noted that Theorem 1 is obtained by the fuzzy-dependent adjustable matrix approach, and Corollary 1 is obtained by the traditional estimation technique in Lemma 1 [35]. Thus, compared with the traditional estimation technique in Lemma 1 [35], the fuzzy-dependent adjustable matrix approach is more effective to reduce the conservatism.

Example 2:
This example presents the application of the obtained results to image encryption, which is based on the following algorithm.
Step 3: The encrypted signals are derived as follows.
where ⊕ is the XOR operation.
The decryption process is the reverse of the encryption process, which is omitted here.
The drive system (1) exhibits chaotic behavior as shown in Fig. 6.
Then, according to Algorithm 2, the Lena grayscale original image, encrypted image, and decrypted image are shown in Fig. 8(a), and their corresponding histograms are given in Fig. 8(b). From Fig. 8, one finds that our obtained results can successfully solve the image encryption problem of secure communication.
V. CONCLUSION In this note, we have studied the exponential synchronization problem of T-S fuzzy RDNNs with ATVDs. By proposing a fuzzy time and time-space sampled-data control schemes, the fuzzy-dependent adjustable matrix inequality technique, and constructing the suitable LKF, we have obtained some new exponential synchronization criteria for T-S fuzzy RDNNs with ATVDs. The two fuzzy sampled-data control schemes are more applicable, since they can not only tolerate some uncertainties but save the limited communication resources for T-S fuzzy RDNNs with ATVDs. The fuzzy-dependent adjustable matrix inequality technique is firstly proposed. Compared with some traditional estimation techniques with a determined constant matrix, the fuzzy-dependent adjustable matrix approach is more flexible and helpful to reduce the conservatism. Finally, we have discussed some simulations to verify the effectiveness and superiority of the obtained theoretical results. It is noted that a new time-dependent fuzzy LKF approach has been proposed in [41], which can effectively capture the information of membership functions. In our future work, the time-dependent fuzzy LKF approach will be considered for T-S fuzzy RDNNs and the fuzzy-dependent adjustable matrix inequality technique can be extended to other T-S fuzzy systems.   is currently a Reader. His current research interests include intelligent control, computational intelligence and machine learning. He has served as a program committee member, international advisory board member, invited session chair and publication chair for various international conferences and a reviewer for various books, international journals and international conferences. He is an associate editor for IEEE Transactions on Fuzzy Systems, IEEE Transactions on Circuits and Systems II: Express Briefs, IET Control Theory and Applications, International Journal of Fuzzy Systems, Neurocomputing and Nonlinear Dynamics; and guest editor and on the editorial board for a number of international journals. He was named as highly cited researcher and is an IEEE fellow. He