Fixed-Time Synchronization of Delayed Fractional-Order Memristor-Based Fuzzy Cellular Neural Networks

In this article, we discussed the fixed-time synchronization of fractional-order memristor-based fuzzy cellular neural networks (FMFCNNs) with time-varying delays. Based on the differential inclusion theory and discontinuous state feedback control technique, some useful novel criteria of the fixed-time synchronization for FMFCNNs are derived by constructing appropriate Lyapunov function. The results in our paper improve and generalize current literature research results significantly. Finally, the effectiveness of our control schemes for the fixed-time synchronization is demonstrated through numerical simulations.


I. INTRODUCTION
As an extension of integer derivative and integral to arbitrary order, fractional-order calculus can date from 300 years ago [1]. There are many kinds of definitions for fractional integration and fractional differentiation such as Caputo derivative, Riemann-Liouville derivative and etc. Fractional differential equation has been deemed to be a powerful tool for the modeling of practical problems in biology, chemistry, physics, medicine, economics and other sciences [2]. Contrasting with classical integer-order systems, the reality can be better described by fractional-order systems for the reason that fractional-order differentiation takes into account the present state and all the history of its previous states [3], [4]. In other words, fractional-order systems have memory and heredity. Therefore, many scholars have applied fractional operators to neural networks to build fractional models [5].
On the other hand, since Chua and Yang first proposed cellular neural networks in 1988 [6], [7], many scholars have conducted extensive research on cellular neural networks owing to the widespread application of cellular neural networks in many fields such as quadratic optimization, image The associate editor coordinating the review of this manuscript and approving it for publication was Norbert Herencsar . processing, pattern recognition, associative memories and etc.. In 1996, based on the traditional cellular neural network, T. Yang and L. B. Yang first proposed the fuzzy cell neural network. Compared with the traditional cellular neural network, the fuzzy cellular neural network adds fuzzy logic (fuzzy AND and fuzzy OR) to its structure, and maintains the local connection between cells. The research results show that fuzzy cell neural networks have better applications in pattern recognition and image processing [8]- [11].
Among the neural network cluster behaviors, synchronization is the most important one and its manifestations are various, such as asymptotic synchronization [12], [13], exponential synchronization [14], [15], robust synchronization [16], [17], finite time synchronization [18]- [20], and etc. Among the existing references related to the synchronization of fuzzy cellular neural networks, [21] studied the asymptotic synchronization of non-identical chaotic fuzzy cellular neural networks with time-varying delay based on sliding mode control. Based on Lyapunov functional theory and inequality techniques, [22] discussed the exponential lag synchronization of delayed fuzzy cellular neural networks by periodic intermittent control methods. Reference [23] discussed the exponential lag synchronization of neural networks with time-varying delays. Based on the linear matrix VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ inequality method and the Lyapunov-Krasovskii stability theory, [24] studied the synchronization of bidirectional associative memory fuzzy neural networks with different delays. Reference [25] discussed the finite-time synchronization of fuzzy cellular neural networks with time-varying delays. Reference [26] studied the finite-time stability and synchronization of fractional-order memristive fuzzy cellular neural networks. Reference [27] studied the problem of finite-time synchronization of delayed fuzzy cellular neural networks with discontinuous activation functions. Reference [28] studied the finite-time synchronization of a class of fuzzy cellular neural networks with proportional delay and time-varying coefficients. Reference [29] considered the finite-time stability of a class of fractional-order fuzzy neural networks with proportional delay, and the sufficient conditions for the finite-time stability of the fuzzy neural network were derived based on differential inequalities and finite-time qualitative theory. Reference [30] considered the finite-time cluster synchronization of coupled fuzzy cellular neural networks with Markov switching topology, discontinuous activation function, proportional leakage and time-varying unbounded delay. By careful summarization and analysis, it is not hard to find that the current synchronization of fuzzy cellular neural networks is either asymptotic synchronization or finite-time synchronization, which makes it a big drawback in practical applications. First of all, the life of the controlled plant is limited, asymptotic synchronization is obviously unreasonable. On the other hand, although finite-time synchronization is achieved in a limited time, the synchronization time depends on the initial state of the system. In other words, the initial conditions of the system must be given first. However, the initial conditions of the actual system are often difficult to estimate or even unavailable.
Recently, in the framework of Filippov's solution theory, [31] discussed the fixed-time stability and fixed-time synchronization of coupled neural networks with nonlinear and discontinuous activation functions. Based on Lyapunov's stability theory, [32] studied the fixed-time synchronization problem of memristive fuzzy bidirectional associative memory cellular neural networks with time-varying delay by using nonlinear feedback control, and obtained some new criteria that can make the drive-response memristive fuzzy bidirectional associative memory cellular neural network achieve fixed time synchronization. Reference [33] discussed fixed-time synchronization of delayed memristive fuzzy cellular neural networks. Reference [34] mainly studied the finite-time and fixed-time robust synchronization of fuzzy Cohen-Grossberg neural networks with discontinuous activation functions, and derived some algebraic criteria that enable the drive-response system to achieve synchronization within a fixed time. In the framework of Filippov's solution theory, [35] studied the fixed-time synchronization problem of fuzzy cellular neural networks with discontinuous activation functions and time-varying delays, and established some useful new criteria for fixed-time synchronization.
To the best of our knowledge, fixed-time synchronization control of FMFCNNs with time-varying delay is not yet completely studied, which motivated our research. This article aims to establish some novel fixed-time synchronization criteria for FMFCNNs with time-varying delay. The main contributions of this article are summarized as follows.
• For the first time, the fixed-time synchronization problem for FMFCNNs with time-varying is studied. In practical applications, fixed-time synchronization is more general and practical than finite-time synchronization and asymptotic synchronization.
• By constructing a nonlinear feedback controller and choosing a simple Lyapunov function, some sufficient conditions which are easy to verify are obtained to ensure the fixed-time stability of FMFCNNs and the fixed-time synchronization of the drive-response FMFCNNs systems.
• The theoretical results obtained are more general and can improve or supplement previous results effectively. Moreover, the existing FMFCNNs model with no fuzzy logic, no time-varying delay, or no memristor can all be regarded as the special case of our model.
• The settling time in this article is easy to estimate. In addition, compared with the classical results, the estimation bound of the settling time given in our paper is more accurate and effective. Numerical examples are given to demonstrate the effectiveness of the proposed approaches.
The remainder of the paper is as follows. The response-drive system introduced in Section 2. Furthermore, few definitions and assumptions are presented, and some useful lemmas needed are also introduced. In Section 3, some fixed-time synchronization criteria for delayed FMFCNNs are proposed on the basis of the fixed-time stability theory. Section 4 presents two simulation examples. Finally, some conclusions of this article are drawn, and some future research works are proposed in the last section.

II. PROBLEM FORMULATION AND PRELIMINARIES
The fractional-order integral-differentiation can be considered as an extension of integer integral-differentiation. The most common definitions of fractional calculus are Caputo, Riemann-Liouville, and Grünwald-Letnikov. In practical engineering applications, we generally use the Caputo derivative instead of the Riemann-Liouville derivative. The reason is that Laplace transform of Caputo's derivative only requires integer-order derivatives, while the Laplace transform of Riemann-Liouville involves fractional-order derivatives which are difficult to be physically interpreted. In this article, the Caputo's derivative is considered. Let the lower limit of the fractional calculus be 0. Then the fractional-order integral with fractional order α can be defined as where (·) is the Euler's Gamma function. The corresponding Caputo fractional derivative can be expressed by where α is the fractional order, and n is an integer satisfying n − 1 ≤ α < n.

A. THE DRIVE-RESPONSE FMFCNNs MODELS
The fractional-order memristor-based fuzzy cellular neural networks with time-varying delay in this article is as follows: where x i (t) is the state of the ith neuron at time t, 0 < α ≤ 1 is the fractional order, c i denotes the passive decay rates of the ith neuron, I i and ν j represent the bias value and input of the ith neuron, respectively; T ij and S ij denote the element of fuzzy feed-forward MIN and MAX template, respectively; α ij and β ij are the element of fuzzy feedback MIN and MAX template, respectively; ∨ and ∧ represent the fuzzy AND operation and fuzzy OR operation, respectively; f i and g i are activation functions; represents a continuous function set from the interval [−τ, 0] to R, a ij (x j (t)) and b ij (x j (t − τ j (t))) denote the non-delayed and delayed memristor-based synaptic connection weights, respectively, and given by Remark 1: The weight of FMFCNNs includes a memristor, and the resistance of the memristor can be changed according to the state of the system. Therefore, FMFCNNs can be regarded as a special switching neural networks.
From the definitions of a ij (x j (t)) and b ij (x j (t − τ j (t))), the model (1) is a switching differential equation with discontinuous right-hand sides. For this kind of equation, Filippov gave an effective treatment [36]. In order to simplify the processing of such equations, we define the following set-valued mappings Based on the differential inclusions theory, the model (1) can be rewritten According to the measurable selection theorem [37], there exist measurable functions Let the FMFCNNs (1) be the drive system. The corresponding response system is selected as follows.
where u i (t) is the fixed-time synchronization controller to be designed, a ij and b ij are as follows.
Similarly, we have ) are also closed, convex and compact about y j (t) and y j (t − τ j (t)), respectively. Similarly, There exist measurable For the drive-response systems (1) and (4), we have the following assumptions.
Assumption 1: The activation functions f i (x) and g i (x) are Lipschitz continuous on R, i.e., there exist positive constants l i , m i that make the following inequality hold Assumption 2: For any switching jump T i , i = 1, 2, . . . , n, the activations f i (x) and g i (x) satisfy the following conditions Remark 2: Generally speaking, in order to study the stability or synchronization control of the memristor-based neural networks, we must first put forward relevant assumptions about the activation function of the memristor-based neural networks. Assumption 1 and 2 are rational, and one can see the references [33], [38], [41].

B. THE DEFINITION OF FIXED-TIME SYNCHRONIZATION
In this subsection, the definition of fixed-time synchronization and some useful lemmas are given. According to the drive system (3) and response system (5), we give the definition of the error system as follows Definition 1: (See [39].) The origin of error system (6) is said to be globally fixed-time stable if it is globally uniformly finite-time stable and the settling time T is globally bounded, i.e. ∃T max ∈ R + such that T (e 0 ) ≤ T max , ∀e 0 ∈ R n . Remark 3: From the above definition, we can see that fixed-time stability is actually a special finite-time stability, but the corresponding the settling time T has a certain upper bound, which does not depend on the initial value of the system, and it is only related to system parameters and controller parameters.
According to the definition of the fixed-time stability, the following definition is given.
Definition 2: The drive-response systems (1) and (4) are said to achieve fixed-time synchronization if there exists T (e 0 (t)) in some finite time such that where T max is the settling time, · represents the Euclidean norm.

Remark 4:
In order to study the fixed-time synchronization problem between the driving system (1) and the response system (4), we convert the fixed-time synchronization problem to the fixed-time stability problem of the error system (6).
Lemma 1 (See [31]): If there exists a regular, positive definite and radially unbounded function V (x) : R n → R and constants a, b, δ, k > 0 and δk > 1 meeṫ then the origin is fixed-time stable, and the settling time T max is estimated by Lemma 2 (See [6]): Suppose x j (t) and y j (t) are two states of system(6), then we have Lemma 3 (See [40]): Let b 1 , b 2 , . . . , b N ≥ 0, q ≥ 1, then the following inequality hold where α 1 , α 2 > 0 and α 1 + α 2 ≤ 1. Lemma 6 (See [42]): Suppose that x(t) ∈ C 1 [0, T ], where T is a positive constant and the fractional order α lies in (0, 1]. then the following equations hold:  [35], the fixed-time synchronization of FCNNs are studied. Our results differ from others in two aspects at least. First, the synchronization controller used in the our paper is more complex, and the derivation process is more sophisticated. Second, our research object is the fractional-order neural network, which is superior in describing the complicated input-output signal relations and the dynamics of neural networks.

III. THE FIXED-TIME SYNCHRONIZATION OF THE DRIVE-RESPONSE FMFCNNs
In this section, the sufficient conditions of fixed-time synchronization of the drive-response FMFCNNs will be given. In order to obtain these synchronization criteria, the fixedtime synchronization controller u i (t) is designed as follow where k i ≥ 0, ν i > 0, γ i > 0, ρ i > 0, and l > 1. sign(·) is the symbolic function.
In order to facilitate the main results of our paper, we first simplified the error system (6). Combined with Eqs. (3) and (5), we have β ij g j (y j (t − τ j (t))) Substituting the controller (10) into (11), we obtain According to Assumption 1, Lemmas 2 and 4, we obtain |α ij | · |g j (y j (t − τ j (t))) − g j (x j (t − τ j (t)))| VOLUME 8, 2020 Theorem 1: Suppose that Assumptions 1 and 2 hold, then the error system (6) Furthermore, the settling time can be calculated by the approximate formula Obviously, V (e(t)) ≥ 0 and V (e(t)) = 0 if and only if e(t) = 0. V (e(t)) is regular, positive definite and radially unbounded. According to Lemmas 5 and 6, we obtaiṅ Calculating the upper right-hand derivative of Eq.(16) along the error system (6) and replacing the 0 D α t e i (t) with the inequality (13), we havė When choosing the appropriate controller parameters to meet the condition (14), we have the following inequalitẏ where r is an arbitrary positive number. To give a less conservative and more accurate estimation of T s , let which shows that ϕ(r) reaches to its minimum valueφ given byφ On the other hand, we can optimize the estimation of the settling time by choosing the suitable parameters a, b, k, δ in applications. In fact, if a > be and δ = , a less conservative estimation of the settling time function T s can be obtained by Corollary 1: It should be noted that, when the conditions (14) hold, the fixed-time synchronization between system (1) and (4) can be achieved, and the settling time can be calculated by Eq. (15).
If there are not fuzzy feedback MAX template, fuzzy feedback MIN template and delayed feedback template in the system (1), i.e. β ij = α ij = b ij = 0, then the corresponding drive-response system can be rewritten as follow Here, the fixed-time synchronization controller (12) can also be reduced to the following form Moreover, the settling time T max can be obtained from Eq. (15)

IV. NUMERICAL SIMULATIONS
In this section, two numerical examples are given in order to verify the effectiveness of the theoretical results of Theorem 1 and Corollary 2. Example 1: Consider the following three-dimensional drive-response memristor-based fractional-order fuzzy cellular neural network: where the non-delayed and delayed memristor-based synaptic connection weights are listed as follows.
The elements of fuzzy feedback MIN template, fuzzy feedback MAX template, feed-forward template, fuzzy feed-forward MIN template and fuzzy Feed-forward MAX template are listed as follows. The activation functions f i (x), g i (x), the initial values φ i (s), ψ i (s), and the time-varying delay τ i (t) of system (20) are given by The following controller u i (t) is designed as follow  Fig. 1 presents the phase diagrams of the free-controller driveresponse system in (20), and Figs. 2(a), 3(a) and 4(a) show the trajectories of the same system as above. Fig. 1 indicates that the example drive-response systems have limit cycles with current parameters. Figs. 2(a), 3(a) and 4(a) indicate that the drive-response system can not achieve synchronization.
It is easy to verify that the condition (14) is satisfied. Thus, the synchronization of the example system can be achieved within a predetermined time. The phase diagrams and the trajectories of the example system with controller are shown in Fig. 5 and Figs. 2(b), 3(b) and 4(b), respectively.
The errors of the drive-response system system (20) are presented in Fig. 6. The same limit cycle will eventually be reached for various system initial points under controller (21). Compared with Figs. 2(a), 3(a) and 4(a), Figs. 2(b), 3(b) and 4(b) show that the state trajectory curves of (20) achieve synchronization within a fixed time  T max = 0.7211, which can be also confirmed by error system curves in Fig. 6.  Example 2: Consider the following three-dimensional drive-response memristor-based fractional-order fuzzy cellular neural network without delay items.
(22) VOLUME 8, 2020    Fig. 7 shows the phase diagram of the drive system and response system without controller, respectively.
To guarantee the fixed-time synchronization of driveresponse system, we apply the controller scheme in    The phase diagrams and the error trajectories of the drive-response system with controller (18) are shown in Fig. 11 and Fig. 12, respectively.
The numerical simulations show that the fixed-time synchronization of the drive-response system are obtained in the settling time T max = 0.5000.

V. CONCLUSION
In this article, the problem of fixed-time synchronization control for fractional-order memristor-based fuzzy cellular neural networks (FMFCNNs) with time-varying delay has been studied extensively. By designing an appropriate state feedback controller, based on the fixed time stability theory and the Lyapunov method, we have derived some new and useful fixed-time synchronization criteria for FMFCNNs with time-varying delays. Finally, the simulation results show the effectiveness of the proposed fixed-time synchronization control scheme. It should be pointed out that most of the existing fixed time synchronization results are based on the assumption that the neuron activation functions are Lipschitz continuous. However, this assumption is not necessarily true in the actual environment. Therefore, in future work, based on the research results of this article, we will further study fixed-time synchronization problem of FMFCNNs with discontinuous activation functions.