Abstract
This paper deals with the finite-time synchronization problem of a class of fuzzy neural networks with hybrid delays and uncertain nonlinear perturbations. By applying the famous finite-time stability theory, combining differential inequality techniques, and the analysis approach, several new algebraic sufficient criteria are obtained to realize finite-time synchronization between the drive system and the response system by designing a state feedback controller and an adaptive controller. Taking discrete delays, distributed delays, and uncertain nonlinear perturbations into account in fuzzy cellular neural networks makes the neural system more general than most existing cellular neural networks. Two different novel types of controllers designed to achieve finite-time synchronization can not only effectively overcome the influence of time delays and perturbations but also change their form according to the change of system state or perturbation to achieve a better control effect. Meanwhile, some algebraic sufficient criteria obtained in this paper can be proved by the parameters of the system itself, and the complex calculation of matrix inequality is avoided. Finally, the validity of our proposed results is confirmed by several examples and simulations. Furthermore, a secure communication problem is presented to further illustrate the fact of the obtained results.
1. Introduction
Since Yang and Yang [1, 2] proposed fuzzy cellular neural networks (FCNNs) on the basis of traditional cellular neural networks (CNNs) in 1996, many researchers have performed extensive work on this topic due to their application in image processing and pattern recognition, see [3–6]. However, in the realization of neural networks, the emergence of time delays is unavoidable due to the limitation of velocity information. For one thing, discrete time delays often occur due to the limited switching speed of neurons and amplifiers [7–10]. For another, neural networks usually have spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths, the propagation velocity distribution of these paths usually results in the propagation distribution delays [11–13]. The presence of time delays can lead to instability, chaos, oscillation, and other performance degradations of the nervous system. In this case, considering the combination of discrete time delays and distributed time delays is of great significance to the study of neural networks.
Synchronization plays an essential role in practical applications such as biological systems, secure communications, and image protection. Therefore, synchronization has become a hot spot studying the dynamic behavior of neural networks. So far, many types of synchronization for fuzzy neural networks have been proposed, for instance, exponential synchronization, antisynchronization, projective synchronization, and adaptive synchronization, for example, see [14–19]. However, in practical applications, we are not only interested in the synchronization performance of the system but also more concerned with the convergence time of the system. Finite-time synchronization has the best convergence time in neural networks compared with infinite-time synchronization. That is because finite-time synchronization has better robustness and anti-interference ability. Hence, many scholars are interested in the finite-time synchronization of fuzzy neural networks and have studied it extensively [13, 20–26]. Abdurahman et al. [20] and Wang [21] investigated the finite-time synchronization problem of FCNNs with time-varying delays or time-varying coefficients and proportional delays based on the finite-time stability theory, and inequality techniques, and some criteria of finite-time synchronization for the addressed network are derived. In [22], Duan et al. studied the finite-time synchronization of delayed FCNNs with discontinuous activations. Under the framework of differential inclusions, by utilizing the discontinuous state feedback control method and constructing Lyapunov functionals, new finite-time synchronization criteria for the considered networks are established. In the same year, Tang et al. [13] further considered the finite-time cluster synchronization issue for coupled FCNNs with Markovian switching topology, discontinuous activation functions, proportional leakage, and time-varying unbounded delays, and novel quantization controllers without the sign function are designed to avoid the chattering and save communication resources. Several sufficient conditions are derived to guarantee the finite-time cluster synchronization by constructing new Lyapunov–Krasovskii functionals and utilizing M-matrix methods. What’s more, Jian and Duan [23] researched the synchronization in finite time of fuzzy neutral-type inertial neural networks with time-varying coefficients and proportional delays. To end this, by choosing proper variable transformation, the original system can be rewritten as the first-order differential system. Based on finite-time stability theory and some inequality techniques, several criteria are established to ensure that the drive-response systems can achieve finite-time synchronization. Duan et al. [24] focused on the finite-time synchronization for inertial fuzzy neural networks with time delays. The Lyapunov function constructed is more general than the used Lyapunov functions or in [23] to realize the finite-time synchronization objective.
On the other hand, uncertain perturbations are inevitable in practical situations because the actual neural networks are always in a constantly changing environment, and some unknown factors in the environment changes will always interfere with them. It is of great practical significance to consider the uncertainty perturbations in the neural network models, which makes the models more realistic. Recently, many studies have concentrated on the synchronization of neural networks with external perturbations [27–32]. According to the previous literature review, although many synchronization results have been established, these results only consider the influence of hybrid delays and uncertain perturbations. However, when modeling real-world problems, we encounter other problems such as complexity, uncertainty or ambiguity, and time delays and other external perturbations. It has been proved that fuzzy logic theory has been developed into an effective method to model and deal with complex nonlinear systems [1, 3–6, 11, 13–26, 32, 33]. Therefore, considering fuzzy logic into the model will not only make the model more complete but also have more practical significance. It is worth noting that there are few papers published on the synchronization in finite time of fuzzy neural networks with hybrid delays and uncertain nonlinear perturbations.
Motivated by the above discussions, we study the finite-time synchronization issue of fuzzy neural networks with hybrid delays and uncertain nonlinear perturbations. The main contributions of this manuscript are as follows:(a)A new general neural network model is formulated, and the model assembles fuzzy cellular neural networks, discrete time delays, distributed time delays, and uncertain nonlinear perturbations. Compared with the existing literature on neural networks [20, 21, 31, 33, 34], this model is more general and extensive.(b)Some algebraic sufficient criteria are established to ensure that the drive-response system can be synchronized in finite time. Compared with linear matrix inequality, the finite-time synchronization criterion in this paper is easy to verify with the parameters of the system and easy to implement in practice.(c)The designed state feedback controller and the adaptive controller can not only eliminate the influence of time delays and uncertain nonlinear perturbations but also change their form according to the change of system state or perturbation to achieve a better control effect.
2. Preliminaries
In this paper, we concern with the fuzzy neural networks with hybrid delays and uncertain nonlinear perturbations described bywhere ; corresponds to the number of neurons in the delayed network system; denotes the state of the th unit at moment ; represents the passive decay rates to the state of th unit; and are the connection weight of the feedback template and feed-forward template; , and denote the connection weights of the elements of the discrete fuzzy feedback MIN template, the discrete fuzzy feedback MAX template, the distributed fuzzy feedback MIN template, and the distributed fuzzy feedback MAX template, respectively; and are the elements of the fuzzy feed-forward MIN template and the fuzzy feed-forward MAX template; and denote the fuzzy AND and fuzzy OR operations, respectively; presents input and bias of the th neuron; denotes the signal activation function of the th neuron at moment ; and correspond to the discrete time-varying delay and distributed time-varying delay and satisfy , where and are nonnegative constants; represents the nonlinear perturbations, where and . Moreover, let .
According to the concept of drive-response synchronization first proposed by Pecora and Carrol in [35], we take (1) as the drive system; then, the corresponding response system is described as follows:where represents the nonlinear perturbations to system (2), and is the suitable controller to be designed for realizing synchronization of the drive-response system. The other parameters are the same as those defined in system (1).
We define as the synchronization error of system (1) and (2). Then, subtracting (1) from (2) yields the following error system:where ; ; .
Let , which is the Banach space of all continuous functions mapping with the norm . The initial values associated with systems (1) and (2) are assumed to be , respectively, , where . Then, the initial value of the error system is described as , and .
The other parameters and symbols can be seen in.
For research purposes, we make some basic assumptions and introduce the definition and the lemmas that we need to use.(i) For each , there exist a nonnegative constant such that(ii) For each , there exist nonnegative constants , and such that
Definition 1. (see [36]). The neural network (2) is said to be synchronized with (1) in finite time if, for a suitable designed controller, there exists a constant ( depends on the initial state vector error value ), such that and for , where the is called the setting time.
Lemma 1 (see [37]). We assume that there exist a continuous, positive-definite function , constants , , and an open neighborhood of the origin such that
Then, the origin of system is finite-time stable. The settling time satisfies
Moreover, if , is proper and radially unbounded, then the origin is globally finite-time stable.
Lemma 2 (see [38]).. We assume that a continuous, positive-definite function and constants satisfy the following differential inequality
Then, for any given satisfies the following inequalityand for all with the settling given by
Lemma 3 (see [39]). Let be a solution of the error system, which is defined on , . Then, function is absolutely continuous andwhere , if , while can be arbitrarily chosen in , if .
Lemma 4 (see [1]). If suppose and are two states of neural networks (1), then we have
Lemma 5 (see [20]).. If suppose is a positive integer and and are real numbers, then the following inequality holds
Lemma 6 (see [40]). Let , thenwhere are some constants, , and .
3. Main Results
In this section, two different types of controllers should be designed for the finite-time stability of the zero solution of the error system (3), which is equivalent to the finite-time synchronization between system (1) and system (2). Specifically, a state feedback controller is first designed for the finite-time synchronization problem. Then, an adaptive controller is considered based on the state feedback controller so that the control strength can be adjusted automatically. At the same time, several sufficient conditions that the considered system can achieve finite-time synchronization are obtained through rigorous mathematical proofs.
3.1. State Feedback Control
State feedback is to multiply each state variable of the system by the corresponding feedback coefficient and feed back to the input end, which is added with the reference input as the control input of the controlled system. The basic structure of state feedback control system is shown in Figure 1. In order to realize the finite-time synchronization of fuzzy neural networks with hybrid delays and uncertain nonlinear perturbations, we design a new state feedback controller for the response system.
The aforementioned controller is designed in the following form:where , the positive constants are the gain coefficients to be determined, is a tunable positive constant, and the real number satisfies .
Theorem 1. suppose that the assumptions and are satisfied, then the neural network (1) and (2) can be synchronized in a finite time under controller (15) if for any , and the following conditions hold:where , , and . Moreover, the settling time can be estimated asin which , and is a positive integer.
Proof. define the following Lyapunov function:Taking the derivative of along the trajectories of system (3) and using Lemma 3, we obtain thatAccording to and Lemma 4, we haveSubstituting inequalities (21–25) into (20), it produces thatBased on Lemma 6, we can inferSimilarly, it follows from and Lemma 6 thatSubstituting (27) and (28) into (26) yieldsBy Lemma 5 and , one can derive thatIn view ofand combining (29) and (30), we can conclude thatAccording to Lemma 1 and (32), the error system (3) will converge to zero within . Therefore, the response system (2) is synchronized with the drive system (1) in finite time under the controller (15). The proof of Theorem 1 is completed.
If there is no uncertain nonlinear perturbation in the drive system (1) and the response system (2), that is, , then the Assumption 2 is no longer needed and the control parameter in the state feedback controller (15) designed can also be removed. At the same time, Theorem 1 can be simplified. Therefore, we have the following result.
Corollary 1. suppose that the assumption and are satisfied, then under the following controller:the response system (2) can be synchronized with drive system (1) in finite time if for any , and the following conditions hold:
When the drive-response system is considered without discrete time delay (or without distributed time delay), that is, (or ), in (1) and (2), the influence of discrete time delay (or distributed time delay) cannot be considered in the uncertain nonlinear perturbation. Therefore, through the similar discussion to Theorem 1, the following two corollaries can be drawn.
Corollary 2. suppose that the assumptions and are satisfied, when the drive-response system without discrete time delays is concerned, that is, , in (1) and (2); under the following controller:the response system (2) is synchronized with the drive system (1) in finite time if (16) and the following condition hold:
Corollary 3. suppose that the assumptions and are satisfied, when the drive-response system without distributed time delays is concerned, that is, , in (1) and (2); under the following controller:the response system (2) is synchronized with the drive system (1) in finite time if (16) and the following condition hold:
3.2. Adaptive Control
It is well known that adaptive controller can automatically change the control intensity and obtain satisfactory results in practical application. Therefore, by applying adaptive technology to state feedback controller, we provide an adaptive controller such that the drive-response system can be synchronized in a finite time.
The adaptive controller and the update rules be expressed by:where , , and are constants, and are constants to be determined.
Theorem 2. suppose that the assumptions and are satisfied; then, the neural network (1) and (2) can be synchronization in a finite time if for any such that (17) and the following conditions hold:where , , and . Moreover, the settling time can be estimated asin which .
Proof. define the following Lyapunov function:Taking the derivative of along the trajectories of system (3), applying Lemma 3 and inequalities (21–25), one obtainsIt can be derived from Lemma 6 and (27) thatSimilarly, according to and (28), we getSubstituting (44) and (45) into (43), we haveBy virtue ofLet and choose , it is derived from Lemma 5 thatAccording to Lemma 2 and (48), the error system (3) will converge to zero within . Therefore, the response system (2) is synchronized with the drive system (1) in finite time under the controller (39). The proof of Theorem 2 is completed.
Remark 1. Compared with traditional state feedback controller , the state feedback controller (15) includes many compensation terms. Such a novel controller can effectively deal with the finite-time synchronization problems. From Theorem 1, it can be seen that control parameters , and play an important role in the finite-time synchronization of drive system (1) and response system (2). In addition, the settling time is inversely proportional to the tunable constant , which means that a great results in short time .
Remark 2. It is obvious from Theorem 2 that the settling time of finite-time synchronization can be effectively adjusted by control parameter , and of the adaptive controller (35). Besides, adding and into the adaptive controller (35) can further eliminate the influence of time delays and perturbations upon the states of fuzzy neural networks with hybrid delays and uncertain nonlinear perturbations.
Remark 3. If we make the perturbation term in the system, then many previous neural network models are the special cases of system (1). It is also clear from Table 2 that the model of this paper is more general. Where “” means that the model contains this component, “” means that the model does not contain this component. Moreover, since there is no finite-time synchronization of the fuzzy neural network as in form (1), the two different forms of controllers designed in this paper can successfully achieve the finite-time synchronization of the considered drive-response system. This shows that the finite-time synchronization results obtained in this paper are new.
Remark 4. Since controllers (15), (33), (35), (37) and (39) contain discrete symbolic functions, some undesirable chattering may occur when synchronization is implemented. To prevent this from happening, the methods in [20] are used, and the continuous function can be used instead of the discontinuous symbolic function. For example, the control law (15) can be redesigned aswhere and .
4. Illustrative Examples
Here, three numerical examples are provided to demonstrate that our obtained theoretical results are effectiveness.
Example 1. We consider the following 2- neural network model as the drive system:for , where . Other parameters of system (50) are as follows:The corresponding response system is depicted as follows:for , in which the parameters in system (52) are the same as the drive system (50). If , in response system (52), then state trajectories and synchronization error trajectories of (50)–(52) with the initial conditions , , and are presented in Figures 2 and 3. It can be clearly seen that system (50) and system (52) are not synchronized. Therefore, to enable them to synchronize in a finite time, the controller is given bywhere for . By simple calculations, we can select the following control gainsIt is obvious that all the conditions for Theorem 1 are completely satisfied. Therefore, system (52) is synchronized with system (50) in finite time . The state trajectories of the drive-response system under the controller (53) are shown in Figure 4. The synchronization error trajectories between the drive system (50) and the corresponding response system (52) under the controller (53) are presented in Figure 5. The validity of our obtained Theorem 1 can be seen intuitively in Figures 4 and 5.
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Example 2. We consider the following 3- neural network model as the drive system:for , where and the other coefficients and functions are taken asThe corresponding response system is depicted as follows:for , in which the parameters in system (57) are the same as the drive system (55). If in response system (57), then state trajectories and error trajectories of (55)–(57) with the initial conditions , , and are presented in Figure 6. It can be clearly seen that system (55) and system (57) are not synchronized. Therefore, to enable them to synchronize in a finite time, the controller is given bywhere for . By simple calculations, we can selectIt is easy to verify that the above parameters satisfy all the conditions of Theorem 2. Hence, the neural networks (55)–(57) can be synchronized with finite time . The trajectories of the control gains of controller (58) with the initial conditions and , are shown in Figure 7. And the state trajectories and error trajectories of the drive-response system are presented in Figure 8 under the adaptive controller (58). The effectiveness of our obtained Theorem 2 can be seen intuitively in Figure 8.
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Example 3. If , , and , then system (1) is transformed into the following neural networks model:for , where . The parameters , , and of system (60) are the same as those of system (50) and . Taking system (60) as the drive system, the response system has the following form:for , in which the parameters in system (61) are the same as the drive system (60). Meanwhile, the initial conditions for the drive system (60) and response system (61) are the same as in Example 1. If in response system (61), then state trajectories and synchronization error trajectories of (60) and (61) are shown in Figures 9 and 10. It is obvious that the systems are not synchronized. Therefore, to enable them to synchronize in a finite time, we change the controller (15) to the following form:where for . The values of , and are the same as in Example 1. In the case, the condition in Theorem 1 is satisfied. Therefore, system (61) is synchronized with system (60) in finite time . The state trajectories of the drive-response system under the controller (62) are described in Figure 11. The synchronization error trajectories between the drive system (60) and the corresponding response system (61) under the controller (62) are presented in Figure 12.
The validity of Theorem 1 and Theorem 2 we have obtained are evident from Example 1 and Example 2. It can be seen from Example 3 that the controller designed in this paper can change its form according to the change of the state or disturbance of the system so that the considered system can achieve finite-time synchronization. In addition, the convergence time is shortened when the complexity of the system is reduced. Therefore, the controller designed by this paper is more general and practical.
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5. Discussion
Based on the numerical simulations given in Section 4, it can be intuitively seen that both controllers designed in this paper can achieve finite-time synchronization of the considered drive-response system. Among them, since the internal characteristics of the system can be fully reflected by the state variables, the state feedback is beneficial to improving the control performance of the system. However, in the actual system, the introduced unknown feedback gain needs to be solved, but most of the obtained feedback gains are large and difficult to achieve. The adaptive controller can avoid large control gains and is also suitable when the system parameters are unknown or the model is uncertain, but its cost is very high. Therefore, in practical applications, the designer should select the corresponding controller according to their needs to achieve better control effects.
5.1. Comparative Analysis
In Section 3, finite-time synchronization of system (1) is realized by designing state feedback controller and adaptive controller. In the existing research, many scholars also use feedback control to realize finite-time synchronization of neural networks. Abdurahman et al. [20] realized the finite-time synchronization of the following neural networks based on the theory of finite-time convergence via controller .
If we make and replace the perturbation term with in system (1), and make in the above system, then the two systems are identical. At the same time, finite-time synchronization of the transformed system can be achieved by changing the form of the controller (15). It is worth noting that the FCNNs considered in [20] have no distributed delay, and its results cannot be adopted in the fuzzy neural networks with distributed delays. Therefore, the results obtained in this paper are more general. Besides, in [42–44], the authors put forward some criteria to ensure the finite-time synchronization of complex networks and coupled neural networks according to matrix inequalities. Compared with the sufficient criterion of the matrix inequalities obtained by them, several criteria in this paper are easier to verify with parameters and avoid the complicated calculation of the matrix inequalities. Compared with [20, 21, 27, 31, 36, 45, 46], the delay-independent feedback controllers they designed are not suitable for model (1), while the feedback controller (15) can change its form according to the state of the system or the change of perturbation to achieve finite-time synchronization of the models studied by them. In addition, for some uncertain systems and systems with unknown parameters, the adaptive controller (35) can more effectively achieve finite-time synchronization of these systems. Overall, the controller designed in this paper is more flexible and suitable for more complex systems.
5.2. Application in Secure Communication
Secure communication using synchronization between chaotic systems (chaotic secure communication, for short) is a new concept of secure communication [47]. In secure communication, message encryption is performed when two people want to communicate with each other over an insecure communication channel, but a third person cannot identify the message by listening. Therefore, one of the main objectives of chaotic secure communication is to protect information from eavesdropping and interception. It is worth noting that we can regard the communication network as a chaotic neural network, in which the transmitter and receiver correspond to the drive system and the response system, respectively. We apply the feedback controller synchronization designed in Theorem 1 to make the receiver track the sender accurately in finite time. An information signal containing the message to be transmitted can be masked by a chaotic signal and recovered using finite-time synchronization. Different strategies can be used to make the actual transmitted signal as broadband as possible. In general, three strategies are involved in secure communications with chaos [48]:(1)Signal masking, where ;(2)Modulation, where ;(3)Masking and modulation, where .
For the sake of illustration, we only use signal masking here, which is . For another, we can extract messages from receivers in a communication network through finite-time synchronization. The proposed communication system consisting of a transmitter and a receiver is shown in Figure 13. Masking techniques are proposed as follows. The transmitter is described aswhere the parameters are the same as in Example 1. is the information message, and we set makes the energy of information signal much less than chaotic carrier signal. In order not to lose generality, we make . Accordingly, the receiver is stated aswhere is the transmission signal of the system, . The transmitted information signal can be recovered by transformation . We use the same functions and parameters as in Example 1, and the information signal is arbitrarily selected as . The information signal , transmitted signal , recovered signal , and error between the information signal and the recovered signal are depicted in Figure 14. The simulation results show that the information signal can be recovered in finite time under the feedback controller (15).
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6. Conclusion
In this paper, we discuss the finite-time synchronization of a class of fuzzy neural networks with hybrid delays and uncertain nonlinear perturbations. Through the application of famous finite-time stability theory, differential inequality techniques, and the analysis approach, useful state feedback controller and adaptive controller are involved, and several new algebraic sufficient criteria are derived to realize finite-time synchronization. In addition, the system studied in this paper can contain many previous neural networks, so the system in this paper is more general. Finally, the main results obtained are as follows:(a)Some new algebraic sufficient criteria are proposed to ensure finite-time synchronization between the drive system and the response system, and the settling time is also estimated.(b)The designed state feedback controller and adaptive controller not only realize the finite-time synchronization of the drive-response system but also can change their forms according to the change of the state or perturbation of the system to obtain better a control effect.(c)The studied model and the designed controller can be effectively used in other practical applications such as secure communication.
It is well known that the settling time for finite-time synchronization depends essentially on the initial conditions of the system. However, in many practical applications, it is difficult or impossible to obtain accurate values for the initial conditions of systems such as industrial systems, robots, and biological models, which greatly limits practical applications. While in [49], Polyakov proposed the concept of fixed-time stability and pointed out that the time of fixed-time synchronization is independent of the initial synchronization error, and the settling time can be estimated in advance. From this perspective, fixed-time synchronization has more advantages. In some circumstances, there are several abrupt changes at certain moments in physical systems since instantaneous disturbances, which are called impulsive effects [50]. The influence of impulsive phenomena may appear in some other fields, such as automatic control system and artificial intelligence. Therefore, for future research, we will consider fixed-time synchronization of fuzzy neural networks with time delays and impulsive effects.
Data Availability
No data were used to support this research.
Conflicts of Interest
The authors declared that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China, under Grant 61573010, and the Opening Project of Sichuan Province University Key Laboratory of Bridge Non-Destruction Detecting and Engineering Computing, under Grant 2021QYJ06.