Synchronization of Complex Dynamical Networks with Nonidentical Nodes and Derivative Coupling via Distributed Adaptive Control

Abstract

Adaptive synchronization control is proposed for a new complex dynamical network model with nonidentical nodes and nonderivative and derivative couplings. The distributed adaptive learning laws of periodically time-varying and constant parameters and distributed adaptive control are designed. The new method which can obtain the synchronization error of closed-loop complex network system is asymptotic convergence in the sense of square error norm. What is more, the coupling matrix is not assumed to be symmetric or irreducible. Finally, a simulation example shows the feasibility and effectiveness of the approach.

1. Introduction

Complex dynamical networks have become a focus issue and have attracted increasing attention in many fields. The structure of many real systems in nature can be modeled by complex networks, such as biological neural networks, ecosystems, food webs, and World Wide Web [1, 2]. It has been demonstrated that many cooperative behavior mechanisms and complex phenomena in nature and society have close relationships with network synchronization [3]. Thus, the synchronization of complex systems has been widely studied, and many control schemes, such as adaptive control [4, 5], sliding mode control [6], and state observer-based control [7], have been focused on this topic. However, many existing works on synchronization are based on the networks with same nodes. Since almost all complex dynamical networks in engineering have different nodes, it is unrealistic to assume that all network nodes are the same. For example, in a multirobot system, the robots can have distinct structures or have different parameters such as masses and inertias. With much more complicated behaviors, the synchronization schemes for complex networks with identical nodes cannot be directly applied to the dynamical networks with nonidentical nodes. Therefore, it is strongly necessary to conduct further investigation of new synchronization schemes for a complex dynamical network with nonidentical nodes.

Recently, some papers have studied the synchronization problem with nonidentical nodes. A pinning control scheme was developed to realize the cluster synchronization of community networks with nonidentical nodes [8]. Moreover, other weaker forms of synchronization, namely, output synchronization and bounded synchronization, were proposed in [9, 10]. It is so difficult for a complex network with nonidentical nodes to achieve asymptotic synchronization that little work has been reported to deal with the problem. By constructing a common Lyapunov function for all the nodes, a synchronization criterion for a nonidentical nodes system was given in [11]. Some local robust controllers were designed for the network with strictly different nodes to achieve generalized synchronization in [12]. In [13], the asymptotic synchronization for complex dynamical networks with nonidentical nodes was studied. Based on solving a number of lower dimensional matrix inequalities and scalar inequalities, global synchronization criteria were given. Except the propriety of nonidentical nodes, the time-varying parameter and unknown system parameters should be considered meanwhile. Reference [14] introduced the adaptive synchronization of complex network systems with unknown time-varying nonlinear coupling. Although it considered nonlinearly parameterized complex dynamical networks with unknown time-varying parameters, the nonidentical nodes were not considered yet.

On the other hand, the aforementioned papers only considered nonderivative coupling. Taking into account the complexity of the network, such as population ecology [15] and robots in contact with rigid environments [16], there always exist delayed coupling [17], derivative coupling, and so on. In [18], the authors proposed a general complex dynamical network with non-derivative and derivative coupling. Under the designed adaptive controllers, the network system can asymptotically synchronize to a given trajectory. The paper was limited to the inner coupling matrix of derivative being identity matrix. A complex dynamic network model with derivative coupling and nonidentical nodes was studied in [19]. By designing the adaptive strategy with time derivative terms, the authors have dealt with the known derivative coupling successfully. However, they did not consider nonlinear parameterized complex network with time-varying coupling strength and the unknown derivative coupling strength.

In recent years, the adaptive learning control has been proposed to deal with the unknown constant or time-varying parameters of the system in order to obtain control input [20, 21]. Under the designed adaptive learning control, the convergence of tracking error can be proved by using the composite energy function. Meanwhile, the sliding mode control (SMC) strategy has been successfully applied to a class of systems with nonlinearities and uncertainties [22–24]. The advantage of using SMC method is the concepts of sliding mode surface design and equivalent control. In this paper, under the adaptive control strategy and inequality technique, we solve the proposed problem successfully. For the nonlinearities function, further work can focus on SMC method.

Motivated by the above discussion, the synchronization problem via new distributed adaptive control is proposed for nonlinear parametric complex dynamical networks with derivative coupling and nonidentical nodes. The main difficulty is how to deal with the unknown derivative coupling. Using the adaptive control and skill of inequality zoom, we find a way to overcome this challenge. By designing the distributed adaptive laws, the effect of derivative coupling can be eliminated. Based on a Lyapunov-Krasovskii-like composite energy function (CEF) [25, 26], the adaptive asymptotical synchronization is obtained for nonlinearly parameterized complex dynamical networks with nonidentical nodes and derivative coupling. The theoretical analysis shows that the designed effective distributed adaptive strategy guarantees that the system asymptotically stable in the 𝐿²_𝑇 norm sense and all closed-loop signals are bounded. And it is worth mentioning that the coupling configuration matrix may be arbitrary matrix with appropriate dimensions [27].

The rest of the paper is organized as follows. The problem formulation and preliminaries are given in Section 2. In Section 3, adaptive synchronization scheme of the complex dynamical networks is presented. In Section 4, a simulation example is provided to illustrate the effectiveness of the proposed controller. Finally, conclusion is given in Section 5.

2. Problem Statement and Preliminaries

A complex dynamical network consisting of 𝑁 nonidentical nodes is described as 𝑥̇_𝑖(𝑡)=𝑓_𝑖𝑡,𝑥_𝑖(𝑡)+^𝑁_𝑗=1𝑎_𝑖𝑗Γ𝑔𝑥_𝑗(𝑡),𝜑_𝑖(𝑡)+^𝑁_𝑗=1𝛼_𝑖𝑏_𝑖𝑗Γ𝑥̇_𝑗(𝑡),𝑖=1,2,…,𝑁, (1) where 𝑥_𝑖(𝑡)=[𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡)]^𝑇∈𝑅^𝑛 is the state vector of the 𝑖th node, 𝑓_𝑖∶𝑅⁺×𝑅^𝑛→𝑅^𝑛 is a vector-valued continuous function, 𝑔∶𝑅^𝑛×𝑅→𝑅^𝑛 is an unknown continuous nonlinear vector-valued function, 𝜑_𝑖(𝑡) represents the unknown time-varying function, and 𝛼_𝑖 is an unknown constant parameter. The inner coupling matrix Γ=(𝛾_𝑖𝑗)∈𝑅^𝑛×𝑛 is nonnegative definite. 𝐴=(𝑎_𝑖𝑗)_𝑁×𝑁 and 𝐵=(𝑏_𝑖𝑗)_𝑁×𝑁 are the coupling configuration constant matrices, which are defined as follows: if there is a connection from node 𝑖 to node 𝑗  (𝑖≠𝑗), then 𝑎_𝑖𝑗>0, 𝑏_𝑖𝑗>0; otherwise, 𝑎_𝑖𝑗=𝑏_𝑖𝑗=0. And the coupling matrices that satisfy ∑^𝑁_𝑗=1𝑎_𝑖𝑗=∑^𝑁_𝑗=1𝑏_𝑖𝑗=0.

Remark 1. In this paper, we assume that the strength of matrix Γ is unknown, which is more general than the inner coupling matrices assumed to be diagonal matrix in [18]. As the authors in papers [19] and [21] only consider the known coupling matrix strength, it should be pointed out that the unknown derivative coupling consists of more information than the dynamical model in [19, 21]. The coupling configuration matrices 𝐴 and 𝐵 need not be symmetric or irreducible. Thus, the results can be suitable to more complex dynamical networks.
Let 𝑠(𝑡)∈𝑅^𝑛 be a solution of the dynamics of the isolated node to which all 𝑥_𝑖(𝑡) are expected to synchronize.
Define synchronization error vectors 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑠(𝑡),𝑖=1,2,…,𝑁. (2)
In order to obtain the synchronized dynamics of network (1), we add an adaptive control strategy to nodes in network (1). Then, the controlled network is given by 𝑥̇_𝑖(𝑡)=𝑓_𝑖𝑡,𝑥_𝑖(𝑡)+^𝑁_𝑗=1𝑎_𝑖𝑗Γ𝑔𝑥_𝑗(𝑡),𝜑_𝑖(𝑡)+^𝑁_𝑗=1𝛼_𝑖𝑏_𝑖𝑗Γ𝑥̇_𝑗(𝑡)+𝑢_𝑖(𝑡),𝑖=1,2,…,𝑁, (3) where 𝑢_𝑖(𝑡)=[𝑢_𝑖1(𝑡),𝑢_𝑖2(𝑡),…,𝑢_𝑖𝑛(𝑡)]^𝑇∈𝑅^𝑛, 𝑖=1,2,…,𝑁 are the adaptive controllers to be designed.
Since the row sum of coupling matrix is zero, we have the following dynamical error equation: 𝑒_𝑖̇(𝑡)=𝑥̇_𝑖(𝑡)−𝑠(̇𝑡)=𝑓_𝑖𝑡,𝑥_𝑖(𝑡)+^𝑁_𝑗=1𝑎_𝑖𝑗Γ𝑔𝑥_𝑗(𝑡),𝜑_𝑖(𝑡)−𝑔𝑠(𝑡),𝜑_𝑖(𝑡)+^𝑁_𝑗=1𝛼_𝑖𝑏_𝑖𝑗Γ𝑥̇_𝑖(𝑡)−𝑠(̇𝑡)−𝑠(̇𝑡)+𝑢_𝑖(𝑡)=𝑓_𝑖𝑡,𝑥_𝑖(𝑡+_𝑁𝑗=1)𝑎_𝑖𝑗Γ𝑔𝑥_𝑗(𝑡),𝜑_𝑖(𝑡)−𝑔𝑠(𝑡),𝜑_𝑖(𝑡+_𝑁𝑗=1)𝛼_𝑖𝑏_𝑖𝑗Γ𝑒_𝑗̇(𝑡)−𝑠(̇𝑡)+𝑢_𝑖(𝑡). (4)
Our objective is to design adaptive control laws 𝑢_𝑖(𝑡) such that the synchronization errors converge to zero in the norm sense.
In order to derive the main results, some assumptions and lemmas are introduced as follows.

Assumption 2. For the unknown continuous function 𝑔(⋅,⋅), the following inequality holds: 𝑔𝑥_𝑗(𝑡),𝜑_𝑖(𝑡)−𝑔𝑠(𝑡),𝜑_𝑖(𝑡)²≤𝑥_𝑗(𝑡)−𝑠(𝑡)²ℎ𝑥_𝑗(𝑡),𝑠(𝑡)𝜆𝜑_𝑖(𝑡). (5) Here ℎ(⋅,⋅) is a known nonnegative continuous function with constant upper bounded 𝐻, and 𝜆(⋅) is an unknown nonnegative continuous function.

Remark 3. According to separation principle [28], Assumption 2 can be easily satisfied. Under the assumption, we can “separate” the parameters from the nonlinear function. For the unknown parameters, we can solve the problem using adaptive method.

Assumption 4 (Lipschitz condition). For system (1), there exist positive constants 𝑙_𝑖>0, 𝑖=1,…,𝑁 such that𝑓_𝑖(𝑡,𝑥)−𝑓_𝑖𝑡,𝑦≤𝑙_𝑖𝑥−𝑦,∀𝑥,𝑦∈𝑅^𝑛,𝑖=1,2,…,𝑁. (6)

Assumption 5. In network (1), 𝜑_𝑖(𝑡) is an unknown time-varying function with a known period 𝑇; thus, 𝜆(𝜑_𝑖(𝑡)) is also a periodic function with the same period 𝑇. Suppose 𝜆𝜑_𝑖(𝑡)=𝜙_𝑖(𝑡)+𝜃_𝑖, (7) where 𝜙_𝑖(𝑡) is an unknown continuous function with a known period 𝑇 and 𝜃_𝑖 is an unknown constant parameter.

Assumption 6. In the network (1), the inner coupling matrix Γ satisfies max‖Γ‖,max𝛾_𝑖𝑗≤𝛾,∀𝑖,𝑗=1,2,…,𝑁, (8) where 𝛾 is positive constant.

Assumption 7. Assume that the state and the state derivative of system (1) are measurable.

Remark 8. This assumption is necessary to design controller and adaptive laws. Assumption 7 seems to be restrictive. The observer for state derivative will be considered in the near future.

Lemma 9 (Young’s inequality). For any vectors 𝑥,𝑦∈𝑅^𝑛, and any 𝑐>0, the following matrix inequality holds: 𝑥^𝑇𝑦≤𝑐𝑥^𝑇1𝑥+𝑦4𝑐^𝑇𝑦. (9)

Lemma 10. For a positive constant 𝑇, if for all 𝑡≥0, 𝑓(𝑡) is a continuous function,∫^𝑡_𝑡−𝑇𝑓(𝜏)𝑑𝜏 exists and is bounded with 𝑀. Then, 𝑓(𝑡) is bounded.

Proof. In fact, if 𝑓(𝑡) is unbounded, we can get, for all 𝑁>0, ∃𝜏∈[𝑡−𝑇,𝑡], 𝑓(𝜏)>𝑁. As 𝑓(𝑡) is a continuous function, according to the properties of the continuous functions, we have ∃𝜀>0 for all 𝑘∈[𝜏−𝜀,𝜏+𝜀]⊂[𝑡−𝑇,𝑡], 𝑓(𝑘)>𝑁; then ∫^𝑡_𝑡−𝑇𝑓(𝜏)𝑑𝜏≥∫^𝜏+𝜀_𝜏−𝜀𝑓(𝑡)𝑑𝑡≥2𝜀𝑁. We can choose 𝑁=𝑀/2𝜀+1; then ∫^𝑡_𝑡−𝑇𝑓(𝜏)𝑑𝜏>𝑀. This is a contradiction with the assumption. The proof is completed.

3. Adaptive Synchronization of the Complex Dynamical Networks

In this section, adaptive controller is designed to make the states of (1) synchronize to the trajectory 𝑠(𝑡), and some sufficient conditions are given to ensure the asymptotic stability of the synchronization process.

For network (1), we design the controller 𝑢_𝑖(𝑡)∈𝑅^𝑛 as follows: 𝑢_𝑖(𝑡)=𝑠(̇𝑡)−𝑓_𝑖(𝑡,𝑠(𝑡))−𝛾𝐻𝑒_𝑖(𝑡)^𝑁_𝑗=1𝑎²_𝑗𝑖𝜙_𝑗(𝑡)+𝜃_𝑗(𝑡)−^𝑁_𝑗=1𝑏_𝑖𝑗𝛾𝛼_𝑖(𝑡)sign𝑒_𝑖(𝑡)𝑒̇_𝑗⋅(𝑡), (10) where 𝛾, 𝐻 satisfy Assumption 6, 𝑁 is the node number of network (1), and 𝜙_𝑖(𝑡), 𝜃_𝑖(𝑡), 𝛼_𝑖(𝑡) are the estimations of 𝜙_𝑖(𝑡), 𝜃_𝑖, 𝛼_𝑖, respectively. We denote sign𝑒_𝑖(𝑡)=sign𝑒_𝑖1(𝑡),…,sign𝑒_𝑖𝑛(𝑡)^𝑇,𝑒̇_𝑗⋅(𝑡)=^𝑛_𝑘=1𝑒̇_𝑗𝑘(𝑡). (11)

Remark 11. The main design difficulty is how to deal with the unknown derivative coupling and Γ. In this paper, we use the known upper bound to magnify the inner coupling matrix. Thus, the controller is related to every element of the synchronization error vector. Under the new controller, we can deal with the derivative terms easily.

The constant parameter distributed update laws are designed as follows: ̇𝜃_𝑖(𝑡)=𝑟_𝑖^𝑁_𝑗=1𝑎²_𝑖𝑗𝑒^𝑇_𝑗(𝑡)𝑒_𝑗(𝑡),𝛼̇_𝑖(𝑡)=𝑔_𝑖^𝑛_𝑘=1𝑒_𝑖𝑘(𝑡)^𝑁_𝑗=1𝑏_𝑖𝑗𝑒̇_𝑗⋅(𝑡). (12)(13) The time-varying distributed periodic adaptive learning law is designed as 𝜙_𝑖(𝑡)=[),𝑞0,𝑡∈−𝑇,0_𝑖0(𝑡)^𝑁_𝑗=1𝑎²_𝑖𝑗𝑒^𝑇_𝑗(𝑡)𝑒_𝑗(𝑡)[),𝜙,𝑡∈0,𝑇_𝑖(𝑡−𝑇)+𝑞_𝑖^𝑁_𝑗=1𝑎²_𝑖𝑗𝑒^𝑇_𝑗(𝑡)𝑒_𝑗(𝑡),𝑡∈[𝑇,+∞), (14) where 𝑟_𝑖, 𝑔_𝑖, 𝑞_𝑖 are positive constants and 𝑞_𝑖0(𝑡) is a continuous and strictly increasing nonnegative function which satisfies 𝑞_𝑖0(0)=0, 𝑞_𝑖0(𝑇)=𝑞_𝑖, which ensure 𝜙_𝑖(𝑡) is continuous when 𝑡=𝑖𝑇, 𝑖=1,2,…,𝑁.

The following theorem will give a sufficient condition of asymptotical synchronization for the controlled network (3).

Theorem 12. Under Assumptions 2–6, the control law (10) with the distributed adaptive learning laws (12)–(14) guarantees that the controlled network (4) achieves asymptotic synchronization in the 𝐿²_𝑇 norm sense; that is to say, lim_𝑡→∞^𝑡_𝑡−𝑇𝑒_𝑖(𝜏)²𝑑𝜏=0,𝑖=1,2,…,𝑁. (15) Moreover, all closed-loop signals are bounded.

Proof. Define a Lyapunov-Krasovskii function as follows: for 𝑡∈[0,∞)𝑉(𝑡)=12^𝑁_𝑖=1𝑒^𝑇_𝑖𝑒_𝑖+𝛾𝐻2^𝑁_𝑖=1^𝑡_𝑡−𝑇𝑞_𝑖⁻¹𝜙²_𝑖(𝜏)+𝑑𝜏𝛾𝐻2^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡)+𝐿²+𝛾2^𝑁_𝑖=1𝑔_𝑖⁻¹𝛼²_𝑖(𝑡), (16) where 𝜙_𝑖(𝑡)=𝜙_𝑖(𝑡)−𝜙_𝑖(𝑡),𝜃_𝑖(𝑡)=𝜃_𝑖−𝜃_𝑖(𝑡),𝛼_𝑖(𝑡)=𝛼_𝑖−𝛼_𝑖(𝑡), (17) and 𝐿 is a sufficiently large positive constant which will be determined later.
It should be noted that we define 𝜙_𝑖(𝑡)=𝜙_𝑖(0),𝑡∈[−𝑇,0), (18) from the distributed adaptive law (14), and we get 𝜙²_𝑖(𝑡−𝑇)=𝜙²_𝑖(𝑡−𝑇)=𝜙²_𝑖(0),𝑡∈[0,𝑇]. (19)
Firstly, the finiteness property of 𝑉(𝑡) for the period [0,𝑇) is given. Consider (4) and the proposed distributed control laws (12)–(14), it can be seen that the right-hand side of (4) is continuous with respect to all arguments. According to the existence theorem of differential equation, (4) has unique solution in the interval [0,𝑇₁)⊂[0,𝑇), with 0<𝑇₁≤𝑇. This can guarantee the boundedness of 𝑉(𝑡) over [0,𝑇₁). Therefore, we need only focus on the interval [𝑇₁,𝑇).
For any ∈[𝑇₁,𝑇), the derivative of 𝑉(𝑡) with respect to time is given by 𝑉̇(𝑡)=^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖̇(𝑡)+𝛾𝐻2^𝑁_𝑖=1𝑞_𝑖⁻¹𝜙²_𝑖(𝑡)−^𝑁_𝑖=1𝑞_𝑖⁻¹𝜙²_𝑖(𝑡−𝑇)+𝛾𝐻^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡)̇+𝐿𝜃_𝑖(𝑡)+𝛾^𝑁_𝑖=1𝑔_𝑖⁻¹𝛼_𝑖(𝑡)̇𝛼_𝑖(𝑡). (20)
Let us introduce some notations as Φ=𝑓_𝑖𝑡,𝑥_𝑖(𝑡)−𝑓_𝑖((𝑡,𝑥𝑡,𝑠))Λ=𝑔_𝑗(𝑡),𝜑_𝑖(𝑡)−𝑔𝑠(𝑡),𝜑_𝑖(𝑡). (21)
Along the trajectory of (4), we get ^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖̇(𝑡)=^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑓_𝑖𝑡,𝑥_𝑖(𝑡)+^𝑁_𝑗=1𝑎_𝑖𝑗+ΓΛ_𝑁𝑗=1𝛼_𝑖𝑏_𝑖𝑗Γ𝑒_𝑗̇(𝑡)−𝑠(̇𝑡)+𝑢_𝑖(𝑡). (22)
Substituting (10) into the above equation, from Assumption 6 and Lemma 9, we have^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖̇(𝑡)≤^𝑁_𝑖=1𝑒^𝑇_𝑖Φ+^𝑁_𝑗=1𝑎_𝑖𝑗𝑒^𝑇_𝑖ΓΛ+^𝑁_𝑗=1𝛼_𝑖𝑏_𝑖𝑗𝑒^𝑇_𝑖Γ𝑒_𝑗̇−𝛾𝐻𝑒^𝑇_𝑖𝑒_𝑖^𝑁_𝑗=1𝑎²_𝑗𝑖𝜙_𝑗(𝑡)+𝜃_𝑗(𝑡)−𝑒^𝑇_𝑖^𝑁_𝑗=1𝑏_𝑖𝑗𝛾𝛼_𝑖(𝑡)sign𝑒_𝑖(𝑡)𝑒̇_𝑗⋅(𝑡)^𝑁_𝑗=1≤^𝑁_𝑖=1𝑐𝑒^𝑇_𝑖𝑒_𝑖+1Φ4𝑐^𝑇Φ+^𝑁_𝑗=1𝑑𝑒^𝑇_𝑖𝑒_𝑖+1𝑎4𝑑^2𝑖𝑗Λ^𝑇Γ^𝑇+ΓΛ_𝑁𝑗=1𝛼_𝑖𝑏_𝑖𝑗𝑒^𝑇_𝑖(𝑡)Γ𝑒_𝑗̇(𝑡)−𝛾𝐻𝑒^𝑇_𝑖𝑒_𝑖^𝑁_𝑗=1𝑎²_𝑗𝑖𝜙_𝑗(𝑡)+𝜃_𝑗(𝑡)−^𝑁_𝑗=1𝑏_𝑖𝑗𝛾𝛼_𝑖(𝑡)^𝑛_𝑘=1𝑒_𝑖𝑘(𝑡)𝑒̇_𝑗⋅(𝑡). (23)
Using inequality properties, we obtain ^𝑁_𝑗=1𝛼_𝑖𝑏_𝑖𝑗𝑒^𝑇_𝑖(𝑡)Γ𝑒_𝑗̇(𝑡)≤^𝑁_𝑗=1𝑏_𝑖𝑗𝛾𝛼_𝑖^𝑛_𝑘=1𝑒_𝑖𝑘(𝑡)𝑒̇_𝑗⋅(𝑡). (24)
Then, according to Assumptions 2–5 and the previous inequality, we have ^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖̇(𝑡)≤^𝑁_𝑖=1𝑐𝑒^𝑇_𝑖𝑒_𝑖+1𝑙4𝑐^2𝑖𝑒^𝑇_𝑖𝑒_𝑖+^𝑁_𝑗=1𝑑𝑒^𝑇_𝑖𝑒_𝑖+^𝑁_𝑗=1𝑎²_𝑖𝑗𝜙_𝑖(𝑡)+𝜃_𝑖𝐻𝛾²𝑒4𝑑^𝑇𝑗𝑒_𝑗+^𝑁_𝑗=1𝑏_𝑖𝑗𝛾𝛼_𝑖^𝑛_𝑘=1𝑒_𝑖𝑘(𝑡)𝑒̇_𝑗⋅(𝑡−_𝑁𝑗=1)𝑎²_𝑗𝑖𝛾𝐻𝜙_𝑗(𝑡)+𝜃_𝑗(𝑡)𝑒^𝑇_𝑖𝑒_𝑖−^𝑁_𝑗=1𝑏_𝑖𝑗𝛾𝛼_𝑖(𝑡)^𝑛_𝑘=1𝑒_𝑖𝑘(𝑡)𝑒̇_𝑗⋅(𝑡), (25) where 𝑐 and 𝑑 are positive constants. If choosing 1𝑐=2𝛾,𝑑=4, (26) we have ^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖̇(𝑡)≤^𝑁_𝑖=112+12𝑙²_𝑖+𝑁𝛾4𝑒^𝑇_𝑖𝑒_𝑖+^𝑁𝑁_𝑖=1𝑗=1𝑎²_𝑖𝑗𝛾𝐻𝜙_𝑖(𝑡)+𝜃_𝑖(𝑡)𝑒^𝑇_𝑗𝑒_𝑗+^𝑁_𝑗=1𝑏_𝑖𝑗𝛾𝛼_𝑖^𝑛_𝑘=1𝑒_𝑖𝑘(𝑡)𝑒̇_𝑗⋅(𝑡). (27)
From (12), the third term on the right-hand side of (20) satisfies 𝛾𝐻^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡)̇+𝐿𝜃_𝑖(𝑡)=−𝛾𝐻^𝑁_𝑖=1𝜃_𝑖(𝑡)+𝐿^𝑁_𝑗=1𝑎²_𝑖𝑗𝑒^𝑇_𝑗𝑒_𝑗, (28) while the fourth term on the right-hand side of (20) satisfies 𝛾^𝑁_𝑖=1𝑔_𝑖⁻¹𝛼_𝑖(𝑡)̇𝛼_𝑖(𝑡)=−^𝑁_𝑖=1𝛾𝛼_𝑖(𝑡)^𝑛_𝑘=1𝑒_𝑖𝑘(𝑡)^𝑁_𝑗=1𝑏_𝑖𝑗𝑒̇_𝑗⋅(𝑡). (29)
Since 𝑞_𝑖0(𝑡) is a continuous function and strictly increasing in [𝑇₁,𝑇)⊂[0,𝑇), 𝑞_𝑖⁻¹≤𝑞⁻¹_𝑖0(𝑡)<∞ is ensured. Focusing on the second term on the right-hand side of (20), from (19), we can get, when 𝑡∈[0,𝑇), 𝛾𝐻2^𝑁_𝑖=1𝑞_𝑖⁻¹𝜙²_𝑖(𝑡)−^𝑁_𝑖=1𝑞_𝑖⁻¹𝜙²_𝑖()≤𝑡−𝑇𝛾𝐻2^𝑁_𝑖=1𝑞⁻¹_𝑖0𝜙²_𝑖(𝑡)=𝛾𝐻2^𝑁_𝑖=1𝑞⁻¹_𝑖0𝜙²_𝑖(𝑡)+2𝜙²_𝑖(𝑡)−2𝜙_𝑖(𝑡)𝜙_𝑖(𝑡)−𝜙²_𝑖(𝑡≤)𝛾𝐻2^𝑁_𝑖=1𝑞⁻¹_𝑖0𝜙²_𝑖(𝑡)+2𝜙²_𝑖(𝑡)−2𝜙_𝑖(𝑡)𝜙_𝑖(𝑡=)𝛾𝐻2^𝑁_𝑖=1𝑞⁻¹_𝑖0𝜙²_𝑖(𝑡)−𝛾𝐻^𝑁𝑁_𝑖=1𝑗=1𝑎²_𝑖𝑗𝜙_𝑖(𝑡)𝑒^𝑇_𝑗𝑒_𝑗. (30)
Substituting (27)–(30) into (20), we have 𝑉̇(𝑡)≤^𝑁_𝑖=112+12𝑙²_𝑖+𝑁𝛾4−𝐿^𝑁_𝑗=1𝑎²_𝑗𝑖𝑒^𝑇_𝑖𝑒_𝑖+𝛾𝐻2^𝑁_𝑖=1𝑞⁻¹_𝑖0(𝑡)𝜙²_𝑖(𝑡). (31)
We can choose the appropriate 𝐿 such that ^𝑁_𝑖=112+12𝑙²_𝑖+𝑁𝛾4−𝐿^𝑁_𝑗=1𝑎²_𝑗𝑖<0. (32)
According to (31), we have 𝑉̇(𝑡)≤𝛾𝐻2^𝑁_𝑖=1𝑞⁻¹_𝑖0(𝑡)𝜙²_𝑖(𝑡),𝑡∈[0,𝑇). (33)
Since 𝜙_𝑖(𝑡) is continuous and periodic, the boundedness can be obtained. The boundedness of 𝜙_𝑖(𝑡) leads to the boundedness of 𝑉̇(𝑡). For 𝑉(𝑇₁) is bounded, the finiteness of 𝑉(𝑡) is obvious, for all 𝑡∈[0,𝑇).
Next, we will prove the asymptotical convergence of 𝑒(𝑡).
According to (16), for all 𝑡≥𝑇, we get (𝑡)(𝑡)()=1Δ𝑉=𝑉−𝑉𝑡−𝑇2^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)−12^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡−𝑇)𝑒_𝑖()+𝑡−𝑇𝛾𝐻2^𝑁_𝑖=1^𝑡_𝑡−𝑇𝑞_𝑖⁻¹𝜙²_𝑖(𝜏)−𝑞_𝑖⁻¹𝜙²_𝑖(+𝜏−𝑇)𝑑𝜏𝛾𝐻2^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡)+𝐿²−𝛾𝐻2^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡−𝑇)+𝐿²+𝛾2^𝑁_𝑖=1𝑔_𝑖⁻¹𝛼²_𝑖(𝑡)−𝛾2^𝑁_𝑖=1𝑔_𝑖⁻¹𝛼²_𝑖(𝑡−𝑇). (34)
Considering the first two terms on the right-hand side of (34), with Newton-Leibniz formula, we obtain 12^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)−12^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡−𝑇)𝑒_𝑖()=𝑡−𝑇_{𝑁𝑖=1𝑡𝑡−𝑇}𝑒^𝑇_𝑖(𝜏)𝑒_𝑖̇(𝜏)≤𝑑𝜏_{𝑁𝑖=1𝑡𝑡−𝑇}12+12𝑙²_𝑖+𝑁𝛾4𝑒^𝑇_𝑖(𝜏)𝑒_𝑖(𝜏)+^𝑁_𝑗=1𝑎²_𝑖𝑗𝛾𝐻𝑒^𝑇_𝑗(𝜏)𝑒_𝑗(𝜏)𝜙_𝑖(𝜏+)𝑑𝜏_{𝑁𝑁𝑖=1𝑗=1𝑡𝑡−𝑇}𝑎²_𝑖𝑗𝛾𝐻𝑒^𝑇_𝑗(𝜏)𝑒_𝑗(𝜏)𝜃_𝑖(𝜏)+𝑑𝜏_{𝑁𝑁𝑖=1𝑗=1𝑡𝑡−𝑇}𝑏_𝑖𝑗𝛾𝛼_𝑖^𝑛_𝑘=1𝑒_𝑖𝑘(𝜏)𝑒̇_𝑗⋅(𝜏)𝑑𝜏. (35)
Use the algebraic relation (𝑎−𝑏)^𝑇𝐻(𝑎−𝑏)−(𝑎−𝑐)^𝑇𝐻()=()𝑎−𝑐𝑐−𝑏^𝑇𝐻[2(𝑎−𝑏)+(𝑏−𝑐)], (36) where 𝑎,𝑏,𝑐∈𝑅^𝑝, 𝐻∈𝑅^𝑝×𝑝.
Choosing 𝐻=1, 𝑎=𝜙_𝑖(𝜏), 𝑏=𝜙_𝑖(𝜏), 𝑐=𝜙_𝑖(𝜏−𝑇), 𝜙_𝑖(𝜏)=𝜙_𝑖(𝜏−𝑇), then 𝑎−𝑐=𝜙_𝑖(𝜏−𝑇), 𝑎−𝑏=𝜙_𝑖(𝜏). The following equality can be obtained: 𝛾𝐻2^𝑁_𝑖=1𝑞_𝑖⁻¹𝜙²_𝑖(𝜏)−𝛾𝐻2^𝑁_𝑖=1𝑞_𝑖⁻¹𝜙²_𝑖()=𝜏−𝑇𝛾𝐻2^𝑁_𝑖=1𝑞_𝑖⁻¹2𝜙_𝑖(𝜏)+𝜙_𝑖(𝜏)−𝜙_𝑖(×𝜙𝜏−𝑇)_𝑖(𝜏−𝑇)−𝜙_𝑖(𝜏)=−𝛾𝐻^𝑁_𝑖=1𝜙_𝑖(𝜏)+12𝑞_𝑖^𝑁_𝑗=1𝑎²_𝑖𝑗𝑒^𝑇_𝑗𝑒_𝑗^𝑁_𝑘=1𝑎²_𝑖𝑘𝑒^𝑇_𝑘𝑒_𝑘. (37) Taking the third term on the right-hand side of (34), from (37), one obtains 𝛾𝐻2^𝑁_𝑖=1^𝑡_𝑡−𝑇𝑞_𝑖⁻¹𝜙²_𝑖(𝜏)−𝑞_𝑖⁻¹𝜙²_𝑖(𝜏−𝑇)𝑑𝜏=−𝛾𝐻^𝑁_𝑖=1^𝑡_𝑡−𝑇𝜙_𝑖(𝜏)+12𝑞_𝑖^𝑁_𝑗=1𝑎²_𝑖𝑗𝑒^𝑇_𝑗𝑒_𝑗×^𝑁_𝑘=1𝑎²_𝑖𝑘𝑒^𝑇_𝑘𝑒_𝑘𝑑𝜏. (38)
The other terms on right-hand side of (34) can be simplified as follows: 𝛾𝐻2^𝑁_𝑖=1𝑟_𝑖⁻¹(𝜃_𝑖(𝑡)+𝐿)²−𝛾𝐻2^𝑁_𝑖=1𝑟_𝑖⁻¹(𝜃_𝑖(𝑡−𝑇)+𝐿)²=𝛾𝐻^𝑁_𝑖=1^𝑡_𝑡−𝑇𝑟_𝑖⁻¹𝜃_𝑖(𝜏)̇+𝐿𝜃_𝑖(𝜏)𝑑𝜏=−𝛾𝐻^𝑁_𝑖=1^𝑡_𝑡−𝑇𝜃_𝑖(𝜏)+𝐿^𝑁_𝑗=1𝑎²_𝑖𝑗𝑒^𝑇_𝑗(𝜏)𝑒_𝑗(𝜏)𝛾𝑑𝜏,2^𝑁_𝑖=1𝑔_𝑖⁻¹𝛼²_𝑖(𝑡)−𝛾2^𝑁_𝑖=1𝑔_𝑖⁻¹𝛼²_𝑖(𝑡−𝑇)=𝛾^𝑁_𝑖=1^𝑡_𝑡−𝑇𝑔_𝑖⁻¹𝛼_𝑖(𝜏)̇𝛼_𝑖(𝜏)𝑑𝜏=−𝛾^𝑁𝑁_𝑖=1𝑗=1^𝑡_𝑡−𝑇𝑏_𝑖𝑗𝛼_𝑖^𝑛_𝑘=1𝑒_𝑖𝑘(𝑡)𝑒̇_𝑗⋅(𝑡)𝑑𝜏. (39)
Substituting (35), (38)-(39) into (34), we can attain Δ𝑉(𝑡)≤−^𝑁_𝑖=1^𝑡_𝑡−𝑇𝐿^𝑁_𝑗=1𝑎²_𝑗𝑖−12−12𝑙²_𝑖−𝑁𝛾4×𝑒^𝑇_𝑖(𝜏)𝑒_𝑖(𝜏)𝑑𝜏. (40)
Obviously, there exists 𝐿 such that ()𝑙𝐿>1/2+1/2²_𝑖+𝑁𝛾/4∑^𝑁_𝑗=1𝑎²_𝑗𝑖,𝑖=1,…,𝑁. (41)
We can obtain Δ𝑉(𝑡)<0. (42)
Applying (40) repeatedly for any 𝑡∈[𝑙𝑇,(𝑙+1)𝑇], 𝑙=1,2,… and denoting 𝑡₀=𝑡−𝑙𝑇, we have 𝑉(𝑡)=𝑉𝑡₀+^𝑙−1_𝑗=0Δ𝑉𝑡−𝑗𝑇. (43)
Considering 𝑡₀∈[0,𝑇) and the positive of 𝑉(𝑡), according to (43), we obtain 𝑉(𝑡)<max_{𝑡0∈[)0,𝑇}𝑉𝑡₀−^𝑙−1𝑁_{𝑗=0𝑖=1𝑇}^{𝑡−𝑗𝑇}_{𝑡−𝑗+1}𝐿^𝑁_𝑗=1𝑎²_𝑗𝑖−12−12𝑙²_𝑖−𝑁𝛾4×𝑒^𝑇_𝑖(𝜏)𝑒_𝑖(𝜏)𝑑𝜏. (44)
Since 𝑉(𝑡₀) is bounded in the interval [0,𝑇), according to the convergence theorem of the sum of series and (44), the error 𝑒_𝑖(𝑡) converges to zero asymptotically in the 𝐿²_𝑇 norm sense. That is to say, we have lim_𝑡→∞^𝑡_𝑡−𝑇𝑒^𝑇_𝑖(𝜏)𝑒_𝑖(𝜏)𝑑𝜏=0. (45)
Finally, for all 𝑡∈[𝑇,∞), we prove that all the closed-loop signals are bounded, and the derivative of 𝑉(𝑡) is 𝑉̇(𝑡)=^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖̇(𝑡)+𝛾𝐻2^𝑁_𝑖=1𝑞_𝑖⁻¹𝜙²_𝑖(𝑡)−𝛾𝐻2^𝑁_𝑖=1𝑞_𝑖⁻¹𝜙²_𝑖(𝑡−𝑇)+𝛾𝐻^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡)̇+𝐿𝜃_𝑖(𝑡)+𝛾^𝑁_𝑖=1𝑔_𝑖⁻¹𝛼_𝑖(𝑡)̇𝛼_𝑖(𝑡)≤−𝛾𝐻2^𝑁_𝑖=1𝑞_𝑖^𝑁_𝑗=1𝑎²_𝑖𝑗𝑒^𝑇_𝑗𝑒_𝑗²−^𝑁_𝑖=1𝐿^𝑁_𝑗=1𝑎²_𝑗𝑖−12−12𝑙²_𝑖−𝑁𝛾4𝑒^𝑇_𝑖𝑒_𝑖. (46)
By (46), one can obtain 𝑉(𝑡)≤𝑉(𝑇)−𝛾𝐻2^𝑁_𝑖=1^𝑡_𝑇𝑞_𝑖^𝑁_𝑗=1𝑎²_𝑖𝑗𝑒^𝑇_𝑗𝑒_𝑗²−𝑑𝜏_{𝑁𝑖=1𝑡𝑇}𝐿^𝑁_𝑗=1𝑎²_𝑗𝑖−12−12𝑙²_𝑖−𝑁𝛾4𝑒^𝑇_𝑖𝑒_𝑖𝑑𝜏. (47)
Choosing ()𝑙𝐿>1/2+1/2²_𝑖+𝑁𝛾/4∑^𝑁_𝑗=1𝑎²_𝑗𝑖,𝑖=1,…,𝑁, (48) we have 𝑉(𝑡)<𝑉(𝑇). (49)
From the boundedness of 𝑉(𝑡) and (16), we conclude that 𝑒_𝑖, ∫^𝑡_𝑡−𝑇𝜙²_𝑖(𝜏)𝑑𝜏, 𝜃_𝑖(𝑡)𝛼_𝑖(𝑡) are all bounded. Since 𝜙_𝑖(𝑡) is continuous function and 𝜃_𝑖, 𝛼_𝑖 are constants, it implies the boundedness of 𝜃_𝑖(𝑡) and 𝛼_𝑖(𝑡). According to Lemma 10, the boundedness of 𝜙_𝑖(𝑡) is obviously obtained. As 𝜙_𝑖(𝑡) is a continuous periodic function, we can get that 𝜙_𝑖(𝑡) is bounded. According to (10), the boundedness of the control input 𝑢_𝑖(𝑡) is obtained. Since 𝑒_𝑖(𝑡) is bounded, the boundedness of 𝑥_𝑖(𝑡) is received. So the proof is completed.

4. Simulation Example

To demonstrate the theoretical result obtained in Section 3, we consider the following dynamical network with six nonidentical nodes 𝑥̇_𝑖(𝑡)=𝑓_𝑖𝑡,𝑥_𝑖(𝑡)+^𝑁_𝑗=1𝑎_𝑖𝑗Γexp−𝜑_𝑖(𝑡)𝑥²_𝑗1(𝑡)𝑥²_𝑗2(𝑡)𝑥²_𝑗3(𝑡)+^𝑁_𝑗=1𝛼_𝑖𝑏_𝑖𝑗Γ𝑥̇_𝑗(𝑡)+𝑢_𝑖(𝑡),𝑖=1,…,6, (50) where 𝑓₁𝑡,𝑥₁=−2.5𝑥₁₁+0.3𝑥₁₂+0.9𝑥₁₃+3𝑥²₁₃0.6𝑥₁₁−2.6𝑥₁₂+3𝑥²₁₂+𝑥₁₃−2.8𝑥₁₁+sin𝑥₁₁cos𝑥₁₂−2.2𝑥₁₃,𝑓₂𝑡,𝑥₂=−2.5𝑥₂₁+𝑥₂₂+𝑥₂₁𝑥₂₂+𝑥₂₃0.5𝑥₂₁−0.8𝑥₂₂−sin𝑥₂₂+𝑥₂₃−𝑥²₂₃−2𝑥₂₁−0.8𝑥₂₃−0.5sin2𝑥₂₃,𝑓₃𝑡,𝑥₃=−2.5𝑥₃₁+𝑥₃₁sin𝑥₃₁+𝑥₃₂+1.4𝑥₃₃0.5𝑥₃₁−2.5𝑥₃₂+𝑥₃₃−2𝑥₃₁+𝑥₃₂𝑥₃₃−0.6𝑥₃₃−𝑥₃₃cos𝑥₃₃,𝑓₄𝑡,𝑥₄=−2.5𝑥₄₁+1.5𝑥₄₂+1.5𝑥₄₃𝑥₄₁−2.5𝑥₄₂+𝑥²₄₂+1.5𝑥₄₃+𝑥²₄₃−2𝑥₄₁+𝑥₄₁𝑥₄₃−2.5𝑥₄₃,𝑓₅𝑡,𝑥₅=−2.1𝑥₅₁+0.5𝑥²₅₁+1.4𝑥₅₂+𝑥₅₃0.9𝑥₅₁−2.1𝑥₅₂+𝑥₅₂𝑥₅₃+𝑥₅₃−2𝑥₅₁+0.4𝑥₅₂−2.1𝑥₅₃,𝑓₆𝑡,𝑥₆=−3.2𝑥₆₁+10𝑥₆₂+2.95𝑥₆₁+1−𝑥₆₁𝑥−1₆₁−𝑥₆₂+𝑥₆₃−14.7𝑥₆₂,,,.Γ=100010001𝐴=−3021002−5002103−4001110−5210013−4001110−3𝐵=−3201000−3001201−3200102−4010011−2020001−3 (51)

Nonlinearly parameterized function satisfies exp−𝜑_𝑖(𝑡)𝑥²_𝑗1(𝑡)𝑥²_𝑗2(𝑡)𝑥²_𝑗3(𝑡)−exp−𝜑_𝑖(𝑡)𝑠²(𝑡)𝑠²(𝑡)𝑠²(𝑡)²≤𝑒_𝑗(𝑡)²2𝜑_𝑖(𝑡)exp(−1). (52)

We choose 𝑠(̇𝑡)=𝑓₆(𝑡,𝑠(𝑡)), and the parameters are selected as follows: 𝜙𝑁=6,𝑇=0.5,𝛾=1,𝐻=5,₁(𝑡)=0.2sin8𝜋𝑡+2,𝜙₂(𝑡)𝜙=2cos4𝜋𝑡+2,₃(𝑡)=−sin8𝜋𝑡+2,𝜙₄(𝑡)𝜙=cos8𝜋𝑡+2,₅(𝑡)=−2sin8𝜋𝑡+2,𝜙₆(𝑡)()=2sin4𝜋𝑡+2,𝜃=1,2,3,1.1,1.5,1.3^𝑇,𝛼=(0.01,0.01,0.01,0.01,0.01,0.01)^𝑇. (53)

In the following simulation, we choose 𝑞₁=1,𝑞₂=3,𝑞₃𝑞=2,₄=5,𝑞₅=1,𝑞₆𝑞=2,₁₀(𝑡)=2𝑡𝑞₁,𝑞₂₀(𝑡)=2𝑡𝑞₂,𝑞₃₀(𝑡)=2𝑡𝑞₃,𝑞₄₀(𝑡)=2𝑡𝑞₄,𝑞₅₀(𝑡)=2𝑡𝑞₅,𝑞₆₀(𝑡)=2𝑡𝑞₆. (54)

The initially estimated values of the unknown parameters are 𝜙(0)=111111^𝑇,𝜃(0)=(1,2,3,1.1,1.5,0)^𝑇,𝛼(0)=(0,0,0,0,0,0)^𝑇, (55) and the initial states are chosen as 𝑥₁=0.010.030.02^𝑇,𝑥₂=0.030.020.03^𝑇,𝑥₃=0.040.020^𝑇,𝑥₄=00.050.03^𝑇,𝑥₅=0.100.03^𝑇,𝑥₆=0.10.010^𝑇,𝑠=0.10.50^𝑇. (56)

According to Theorem 12, the synchronization of the complex dynamical network can be guaranteed by the distributed adaptive controllers (10) and the distributed adaptive learning laws (12)–(14). Figure 1 shows the error evolutions under the designed controllers. Figures 2-3 depict the time evolution of the controllers, and Figures 4, 5, and 6 show the evolution of the estimated time-varying parameters. Figures 2–6 show that all signals in the network are bounded.

Figure 1 

The change of synchronization error 𝑒𝑖(𝑡) along period number 𝑖=1,…,6.

(a)

(b)

(c)

Figure 2 

The change of control 𝑢𝑖(𝑡) along time: (a) 𝑢1𝑖, 𝑖=1,2,3, (b) 𝑢2𝑖, 𝑖=1,2,3, and (c) 𝑢3𝑖, 𝑖=1,2,3.

(a)

(b)

(c)

Figure 3 

The change of curve of control 𝑢𝑖(𝑡) along time: (a) 𝑢4𝑖, 𝑖=1,2,3, (b) 𝑢5𝑖, 𝑖=1,2,3, and (c) 𝑢6𝑖, 𝑖=1,2,3.

Figure 4 

The evaluations of parameters 𝛼𝑖 along time with 𝑖=1,…,6.

Figure 5 

The evaluations of parameters 𝜃𝑖 along time with 𝑖=1,…,6.

Figure 6 

The evaluations of parameters 𝜙𝑖 along time with 𝑖=1,…,6.

5. Conclusion

In this paper, a new distributed adaptive learning control method is applied to the synchronization of complex dynamical networks with nonidentical nodes, nonlinear nonderivative coupling, and derivative coupling. The coupling matrix is not assumed to be symmetric or irreducible. By combining inequality techniques and the parameter separation, introducing the composite energy function, the convergence of the tracking error and the boundedness of the system signals are derived. Simulation results demonstrate the effectiveness of the proposed control method. Future effort is needed to design the observer for state derivative.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant no. 60974139.