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) that