Adaptive Modified Function Projective Synchronization for Uncertain Complex Dynamic Networks With Multiple Time-Varying Delay Couplings Under Input Nonlinearity

This paper is concerned with the modified function projective synchronization of the uncertain complex dynamic networks with both input nonlinearity and multiple time-varying delay couplings. Firstly, the model of the complex dynamic networks with sector nonlinear input, multiple time-varying delay couplings, model uncertainty and external disturbance is constructed. Then, the adaptive controller is formulated based on the Lyapunov stability theory and the matrix inequality theory, by which the network nodes of the driving system and the response system can realize modified function projective synchronization according to the corresponding function scaling factors. Finally, a four-dimensional hyperchaotic system is considered as the nodes of the complex dynamic networks to achieve numerical simulation. The corresponding theoretical proof and computer simulation are worked out to demonstrate the effectiveness and feasibility of the proposed scheme.


I. INTRODUCTION
Complex network is a large-scale network with complex topological structure and dynamic behavior, which exists widely in nature and society. The complex dynamic networks can represent almost any natural and man-made structure, and have been an active research topic because of their flexibility and versatility. It is well known that complex network usually consists of a series of coupled interconnected nodes, where each node is a dynamical system [1]- [4]. Chaotic system is a special kind of nonlinear system, which has the characteristics of initial value sensitivity, unpredictability, strong attractor and so on, so it is often used as the node of complex network in the research of complex network [5]- [6]. Synchronization is a typical and important dynamical The associate editor coordinating the review of this manuscript and approving it for publication was Shuping He . behavior of complex dynamical networks. Complex network synchronization research is one of the most important research directions on complex networks [7]- [8]. A variety types of synchronization have been investigated and lots of valuable theoretical results have been gained, such as antisynchronization, combination synchronization, finite-time synchronization, lag synchronization, impulsive synchronization, global synchronization, hybrid synchronization and so on [9]- [19].
Modified function projective synchronization is a more general synchronization form which means that the driving system realize synchronization with the response system by different functional scaling factors. The unpredictability of the functional scaling factor can additionally improve the reliability of secure communication [20]- [23]. Reference [24] investigated the cluster-modified function projective synchronization of a generalized linearly coupled network with 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/ asymmetric coupling and nonidentical dynamical nodes. Reference [25] investigated robust modified function projective lag synchronization between two nonlinear complex networks with different-dimensional nodes and disturbances. Reference [26] achieved modified function projective synchronization for two fractional-order complex dynamical networks with unknown parameters and unknown bounded external disturbances. Considering that time delays often occur in dynamical networks due to the finite speed of information propagation or processing, it is imperative to incorporate time delays into the models of complex dynamical networks [27]- [29]. However, the absolute constant delay may be scarce in the practical networks, it makes more sense to study complex networks with multiple time-varying delay couplings [30]- [33]. Reference [34] realized the modified function projective synchronization of complex dynamical network with multiple time-delay couplings and external disturbances. Reference [35] investigated modified function projective synchronization for complex dynamical networks with mixed time-varying and hybrid asymmetric coupling delays. Moreover, the time-varying coupling strength was also considered in [36]- [38].
On the other hand, in real control systems, actuators and sensors often have nonlinear characteristics such as deadzone, backlash, and hysteresis. Owing to physical limitation, there usually exist nonlinearities in the control input. The effect of input nonlinearities usually results in control performance degeneration or instability in the controlled system. Hence, the existence of input nonlinearities cannot be ignored in the actual control system [39]- [42]. Sector nonlinear input refers to the system input within a sector which can represents a large class of input functions with nonlinearity. Sector nonlinear input exists in many actual control systems, such as air-breathing hypersonic vehicles, flexible robotic manipulator, mechanical connections, electric servo motors, magnetic levitation, etc. [43]- [47]. Therefore, it is of great significance to study the nonlinear system control with sector nonlinear input. Reference [48] designed a controller for a class of uncertain multi-input multi-output chaotic systems with both sector nonlinearities and dead-zones. Reference [49] achieved synchronization for a class of fractional-order chaotic systems with sector nonlinearities. What is more, there always exist model uncertainty and external disturbance in most real systems, which should not be ignored in the controller design [50], [51]. Reference [52] analyzed the synchronization control for a class of master-slave chaotic systems with parameter uncertainties, external noise disturbances and sector nonlinear input. Up to now, the research of sector nonlinear input is mainly concentrated on the chaotic system, and there is relatively little research on the complex network.
Motivated by the above discussion, an adaptive modified function projective synchronization strategy for a class of uncertain complex dynamic networks with sector nonlinear input and multiple time-varying delay couplings is studied in this paper. We first proposed a new complex dynamic network model which has sector nonlinear input, multiple time-varying delay couplings, model uncertainty and external disturbance. What is more, we investigated the function projective synchronization of two complex dynamic network based on the Lyapunov stability theory and the matrix inequality theory. The theoretical proof and simulation results are given to verify the effectiveness and feasibility of the synchronization scheme.
Compared with previous work, the main contribution of this study is: (a) Sector nonlinear input is introduced into the complex network, which makes the model get close to the engineering reality. (b) The effect of multiple time-varying delay couplings is considered in the modeling, which is like the real-world systems. (c) In the theoretical analysis, the Lyapunov stability theory and the matrix inequality theory are flexibly used to realize the modified function projective synchronization. (d) The controller does not include time delay terms in our work, so the proposed method is more general and realistic.
The rest of this paper is organized as follows: Section 2 introduces the new network model and some other preliminaries. In Section 3, the adaptive controller is designed to realize the modified function projective synchronization of the complex dynamic networks. In Section 4, a practical example is provided to demonstrate the effectiveness and feasibility of the proposed control method. Some conclusions and suggestions for future work are given in Section 5.
Notations. All the symbols are assumed to be standard and the matrices are with compatible dimensions in this paper. R n represents an n dimensional Euclidean space; R n×h represents an n × h real matrix; A T is the transpose of the matrix A; · denotes the Euclidean norm of a vector; I represents the unit matrix; diag {...} represents a block-diagonal matrix; ⊗ represents the Kronecker product; λ max (Q) is the maximum eigenvalue of Matrix Q.

II. MODEL DESCRIPTION
We consider a complex dynamical network consisting of N identical nodes, and the driving system can be described as follows: For the driving system, the response system model can be written as follows: . . , f s in (y in (t) , t) represent model uncertainty of the system model; the complex network is divided into m subnetworks by τ l (t), τ l (t) ≥ 0, (l = 0, 1, 2, . . . , m − 1) denotes different time-varying delays, and especially τ 0 (t) = 0 means that the coupling delay is 0; c l (t) represents different coupling strengths, which can be constant or change with time; l is a matrix that describes the internal coupling of individual node; A l = (a l ij ) N ×N is the weight configuration matrix which represents the topological structure of the network. If nodes i and j have a connection, then a ij = a ji = 0, (i = j), otherwise, a ij = a ji = 0, (i = j), and the diagonal elements of matrix Definition 1: For complex network model (1) and (2), if there exists a continuously differentiable scaling function matrix M i (t) = diag {m 1 (t) , m 2 (t) , . . . , m n (t)}, i = 1, 2, . . . , N , such that (3) holds, it means that the driving system and the response system realize modified function projective synchronization.
where · denotes the Euclidean norm of a vector, m i (t) is a continuously differentiable scaling function.
, t) and f s i (y i (t) , t) are bounded, and there exist positive constants is a continuously differentiable scaling function, there exists a positive constant µ, such that Assumption 3: The time-varying coupling strength c l (t) is bounded, and there exists a positive constant c l , such that where 0 < ε < 1 is positive constant. This assumption is still satisfied if τ l (t) is zero or some other constants.
Lemma 1 [34]:. For any vectors X , Y ∈ R n and a positive definite matrix Q ∈ R n×n , the following matrix inequality holds:

III. CONTROLLER DESIGN
In order to achieve the modified function projective synchronization of the complex dynamic network model (1) and (2), we can design the adaptive controller as follows: VOLUME 8, 2020 where r 0 > 0, r 1 > 0, r 2 > 0, r 3 > 0 are four positive constants; θ i is the estimated parameter for θ i ;α i is the estimated parameter for α i ;β i is the estimated parameter for β i ;k i is adaptive feedback control gain. From Eq. (7), we can let u i (t) = ξ i sgn e T i (t) (ξ i < 0), where . When e i (t) = 0, the equation obviously holds, that is This completes the proof. Theorem 1: For any initial conditions, if assumptions 1-4 are satisfied, the driving system (1) and the response system (2) can realize modified function projective synchronization with the controller (7) and adaptive laws (8)- (11).
Proof: From Definition 1, we have the error term: The time derivative of e i (t) is: Choose the following Lyapunov function: where θ i = θ i − θ i , and k * is the positive constant to be designed later. The time derivative of V (t) is: By Corollary 1 and 2, and substituting (9) and (10) into (17), we havė By the Assumption 3 and Lemma 2, we havė Substituting ξ i into (19), we havė By Assumption 4, and substituting (8) and (11) into (20), we havė Let e (t) = e T 1 (t) , e T 2 (t) , . . . , e T N (t) VOLUME 8, 2020 where ⊗ represents the Kronecker product.
By Lemma 1, we have So, we havė where λ max (Q) is the maximum eigenvalue of Matrix Q. Therefore, by taking appropriate k * such that we can obtainV (t) ≤ 0. According to Lyapunov stability theory, we can obtain e i (t) → 0 as t → ∞, which means that the modified function projective synchronization between the driving system (1) and the response system (2) is achieved. This completes the proof.
where m(t) is a continuously differentiable scaling function, then the modified function projective synchronization problem will be changed into the function projective synchronization problem, and the controller (7) and adaptive laws (8)- (11) are also practicable.
Remark 2: If m 1 (t) = m 2 (t) = · · · = m n (t) = m, where m is a constant, then the modified function projective synchronization problem will be changed into the projective synchronization problem, and the controller (7) and adaptive laws (8)- (11) are also practicable.
Remark 3: Time delay always varies in many practical applications, so it is difficult to measure the delay and implement the delay term. In our work, the controller does not include τ (t), so the proposed method is more general and realistic.

IV. ILLUSTRATIVE EXAMPLE
In this section, we present an example to illustrate the correctness of the results which obtained in this paper. We take a four-dimensional hyperchaotic system as reference node. The four-dimensional hyperchaotic system is described as follows:v where v 1 , v 2 , v 3 , v 4 are state variables and ρ 1 , ρ 2 , ρ 3 , ρ 4 are real constants. The system is in a hyperchaotic state when ρ 1 = 35, ρ 2 = 35, ρ 3 = 3, ρ 4 = 8. The state phase diagram of this system is shown in figure 1. Consider the complex dynamical network consisting of four four-dimensional hyperchaotic systems and two different time-varying delays, that is, N = 4, m = 3. The driving system and the response system have described as follows: State phase diagram of the four-dimensional hyperchaotic system: where In the numerical simulation, we set r 0 = 3, r 1 = 6, r 2 = 2, 2+e t , and thenτ 2 (t) = 2e t (2+e t ) 2 ∈ (0 , 1 2 . And network inner-coupling matrix 0 = 1 = 2 = I 4×4 .

Remark 4:
In the simulation process, the hyperbolic tangent function tanh (e i (t)) was used instead of the signum function sgn (e i (t)) to reduce chattering.

V. CONCLUSIONS
In this paper, we introduced a more representative complex dynamical network model, which has sector nonlinear input, multiple time-varying delay couplings, model uncertainty and external disturbance. And the modified function projection synchronization of this model was investigated. The sufficient condition to realize the modified function projective synchronization of the driving system and the response system is strictly deduced. The robust controller does not include time delay terms, which makes the synchronization strategy more general and realistic. The simulation results demonstrated the correctness and superiority of the proposed control scheme. The work of this article provides a theoretical reference for the control and application of complex networks. In future work, we will focus on secure communication and information processing based on complex networks. He is currently an Associate Professor with the School of Electrical and Information Engineering, Zhengzhou University of Light Industry. He has been involved in a wide range of research projects in the chaotic, neural networks, and memristors. He has published over 50 SCI journal articles in the areas of chaotic, dynamics, and control.
WEI WANG received the Ph.D. degree from Concordia University, Canada, in 2002. He is currently a Professor with the School of Electrical and Information Engineering, Zhengzhou University of Light Industry. His research interests include computer vision, artificial intelligence, tactile sensors, and tactile recognition. VOLUME 8, 2020