Adaptive Synchronization of Nonlinearly Parameterized Complex Dynamical Networks with Unknown Time-Varying Parameters

Abstract

A new adaptive learning control approach is proposed for a class of nonlinearly parameterized complex dynamical networks with unknown time-varying parameters. By using the parameter separation and reparameterization technique, the adaptive learning laws of periodically time-varying and constant parameters and an adaptive control strategy are designed to ensure the asymptotic convergence of the synchronization error in the sense of square error norm. Then, a sufficient condition of the synchronization is given by constructing a composite energy function. Finally, an example of the complex network is used to verify the effectiveness of proposed approach.

1. Introduction

Since small world network and scale-free network were discovered in the past decade, complex dynamical networks have become a focus issue and have been studied intensively in various disciplines, such as sociology, biology, mathematics, and engineering [1, 2]. As nonlinear dynamics and topological structure of the complex dynamical networks influence its dynamical behaviors, and synchronization is one of the basic forms of cooperative behavior, many of the basic mechanisms have a direct relationship with the synchronization. Research on synchronization of networks has an important significance on understanding of cooperative behavior mechanism and complex phenomena in nature and society. What is more is that it also has a very broad application prospects on nuclear magnetic resonance, wireless sensor network, and multirobot coordination. On the other hand, it can also provide the necessary theoretical basis simultaneously in multiple mobile robot systems, unmanned aircraft systems and distributed sensor array in the field of intelligent autonomous coordination control of complex systems. Up to now, synchronization can be roughly divided into complete synchronization and generalized synchronization. Generalized synchronization includes cluster synchronization, partial synchronization, and functional project synchronization.

However, there are many of real-world network systems such as biological networks, mobile communications networks, and social networks, whose structure will change over time, coupling parameter, and other system parameters. For example, the feed-forward loops (FFLs) are typical network motifs in many real world biological networks. The structures, functions, as well as noise characteristics of FFLs are dependent on the network parameters. In [3], a global relative parameter sensitivities (GRPSs) were proposed for FFLs in the genetic networks. The proposed GRPS approach sheds some light on the potential real world applications, such as the synthetic genetic circuits, predicting the effect of interventions in medicine and biotechnology, and so on. The existence of time-varying coupling parameter increases the difficulty of researching the problem of synchronization for complex networks. In the existing results [4–19], the method of synchronization is divided into two categories which are adaptive coupling and adaptive control approach. [20] studied a time-varying complex dynamical network model and its controlled synchronization criteria. It showed that synchronization of such a time-varying dynamical network was completely determined by the inner-coupling matrix and by the eigenvalues and the corresponding eigenvectors of the coupling configuration matrix of the network. [21] investigated pinning adaptive synchronization of a general complex dynamical network and indeed gave the positive answers to two questions which were how many nodes and how much coupling strength should a network with fixed structure and coupling be chosen. However, in [20, 21], they considered the controlled complex dynamical network with linearly diffusive couplings; [22] researched adaptive synchronization of an uncertain complex dynamical network. Although it considered complex dynamical network with uncertain nonlinear diffusive couplings, the uncertainty of the system with unknown time-varying parameters was not considered. [23] discussed a global payoff-based strategy updating model for studying cooperative behavior of a networked population. It characterized the interactions among individuals by time varying parameter. It is obvious that the coupling of actual complex networks is mostly complex, and the coupling parameters of the network may be not sure or change with some varying factors. Among the above results, none of them is concerning with the adaptive synchronization of complex network systems with unknown time-varying nonlinear coupling.

On the other hand, in recent years, adaptive learning control [24], a new adaptive control scheme, has been proposed to deal with the unknown periodic time-varying parameters. It used various methods to estimate the unknown constant or time-varying parameters of the system in order to obtain control input. The convergence of tracking error was proved by using the composite energy function. It is worth mentioning that learning control based on the composite energy function plays an important role in dealing with the estimates of periodically time-varying parameters.

Motivated by the above observations, in this paper, the problem of adaptive synchronization is investigated for nonlinearly parameterized complex dynamical networks with unknown time-varying parameters via adaptive learning control method. The main contribution of the paper is that, for the first time, the problem of adaptive synchronization for nonlinearly parameterized complex dynamical networks with unknown time-varying parameters is solved. By combining the parameter separation [25, 26] and reparameterization technique, the adaptive learning laws of periodically time-varying and constant parameters and the proposed adaptive controllers guarantee the asymptotic convergence of the synchronization error in the 𝐿²_𝑇 norm and that all closed-loop signals are bounded in the 𝐿²_𝑇 norm. Simulation results are provided to show the effectiveness of the proposed approach.

2. Problem Formulation and Preliminaries

In this section, we introduce the network model considered in this paper and give some useful mathematical preliminaries.

Consider a dynamical network consisting of 𝑁 identical nodes with nonlinear couplings, in which each node is an 𝑛-dimensional dynamical system. The state equations of the network are𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡)+^𝑁_𝑗=1𝑎_𝑖𝑗Γ𝑔𝑥_𝑗(𝑡),𝜑_𝑖(𝑡)𝑖=1,2…,𝑁, (2.1) where 𝑥_𝑖(𝑡)=[𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡)]^T∈𝑅^𝑛 represents the state vector of the 𝑖th node. 𝑓∶𝑅^𝑛×𝑅→𝑅^𝑛 is a smooth nonlinear vector-valued function. 𝜑_𝑖(𝑡) represents the unknown time-varying function. 𝑔∶𝑅^𝑛×𝑅×𝑅→𝑅^𝑛 is an unknown continuous nonlinear vector-valued function. 𝑎_𝑖𝑗=𝑎_𝑗𝑖 is the coupling element. If there exists a connection between node 𝑖 and node 𝑗(𝑖≠𝑗), then 𝑎_𝑖𝑗=𝑎_𝑗𝑖>0, else 𝑎_𝑖𝑗=𝑎_𝑗𝑖=0. The inner coupling matrix Γ=diag(𝛾₁,…,𝛾_𝑛) is nonnegative definite.

Let𝑎𝐴=¹¹𝑎₁₂⋯𝑎_1𝑁𝑎₂₁𝑎₂₂⋯𝑎_2𝑁𝑎⋮⋮⋱⋮_𝑁1𝑎_𝑁2⋯𝑎_𝑁𝑁, (2.2) where 𝐴^T=𝐴 is an unknown irreducible real symmetric matrix. In this model, it is only required that the elements of the coupling matrix 𝐴 satisfy𝑎_𝑖𝑖=−^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗<0,𝑖,𝑗=1,2,…,𝑁. (2.3) The system (2.1) is said to be asymptotic synchronization in 𝐿²_𝑇 norm iflim_𝑡→∞^𝑡_𝑡−𝑇𝑥_𝑖(𝜏)−𝑠(𝜏)²𝑑𝜏=0,𝑖=1,2,…,𝑁, (2.4) where ‖⋅‖ stands for the Euclidean vector norm. Synchronous evolution 𝑠(𝑡) is an arbitrary desired state, which is also an isolated node of the network (2.1) with 𝑠(̇𝑡)=𝑓(𝑠(𝑡)). We assume that for the system 𝑠(̇𝑡)=𝑓(𝑠(𝑡)) there exists stable equilibrium point, stable periodic orbit, or even chaotic attractor.

Remark 2.1. It is well known that multiagent system is a special kind of complex networks. The consensus has similar meaning with synchronization, and consensus is a fundamental natural phenomenon in nature. [27] researched the cluster consensus of discrete-time multiagent systems. Based on Markov chains and nonnegative matrix analysis, two novel cluster consensus criteria were obtained for MAS with fixed and switching topology, respectively. However, in this paper, we consider a new synchronization problem for continuous-time complex dynamic network with nonlinearly time-varying coupling parameter. This problem is quite different from [27], and the challenge problem is due to the time-varying nonlinearly coupling parameter in (2.1).

Now, the following assumptions are introduced.

Assumption 2.2. In the network (2.1), suppose that there exists 𝑙_𝑖>0, satisfying 𝑥_𝑖(𝑡)−𝑠(𝑡)^𝑇𝑓𝑥_𝑖(𝑡)−𝑓(𝑠(𝑡))≤𝑙_𝑖𝑥_𝑖(𝑡)−𝑠(𝑡)^𝑇𝑥_𝑖(𝑡)−𝑠(𝑡), (2.5) where 𝑥_𝑖(𝑡) and 𝑠(𝑡) are time varying vectors.

Lemma 2.3 (separation principle). 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), (2.6) where ℎ(⋅,⋅) is known nonnegative continuous function and 𝜆(⋅) is unknown nonnegative continuous function.

Assumption 2.4. In the network (2.1), 𝜑_𝑖(𝑡) is unknown time-varying function with a known period 𝑇, in that way 𝜆(𝜑_𝑖(𝑡)) is also a periodic function. Suppose 𝜆(𝜑_𝑖(𝑡))=𝜙_𝑖(𝑡)+𝜃_𝑖, where 𝜙_𝑖(𝑡) is unknown continuous function and 𝜃_𝑖 is unknown constant parameter.

Assumption 2.5. In the network (2.1), the inner coupling matrix Γ, coupling matrix 𝐴, and ℎ(⋅,⋅) satisfy ‖Γ‖=𝛾,ℎ𝑥_𝑗(𝑡),𝑠(𝑡)<𝐻,𝑎_𝑖𝑗≤𝑎,∀𝑖,𝑗=1,2,…,𝑁, (2.7) where 𝛾,𝐻,𝑎 are positive constants.

Lemma 2.6 (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. (2.8)

3. Adaptive Controller Design

To achieve the control objective (2.4), we need an adaptive control strategy to nodes in the network (2.1). Then the controlled network is given by𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡)+^𝑁_𝑗=1𝑎_𝑖𝑗Γ𝑔𝑥_𝑗(𝑡),𝜑_𝑖(𝑡)+𝑢_𝑖(𝑡),𝑖=1,2,…,𝑁, (3.1) 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. Let synchronization error be 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑠(𝑡), 𝑖=1,2,…,𝑁 and we have the following dynamical error equations:𝑒_𝑖̇(𝑡)=𝑓𝑥_𝑖(𝑡)−𝑓(𝑠(𝑡))+^𝑁_𝑗=1𝑎_𝑖𝑗Γ𝑔𝑥_𝑗(𝑡),𝜑_𝑖(𝑡)−𝑔𝑠(𝑡),𝜑_𝑖(𝑡)+𝑢_𝑖(𝑡). (3.2) Then we design controllers by𝑢_𝑖(𝑡)=−𝑁𝑎𝛾𝐻𝜙_𝑖(𝑡)+𝜃_𝑖(𝑡)𝑥_𝑖(𝑡)−𝑠(𝑡), (3.3) where 𝛾,𝑎,𝐻 satisfy condition (2.7), 𝑁 is the node number of the network (2.1), and 𝜙_𝑖(𝑡),𝜃_𝑖(𝑡) are estimations to 𝜙_𝑖(𝑡),𝜃_𝑖, respectively. For convenience, we denote 𝜙_𝑖(𝑡)=𝜙_𝑖(𝑡)−𝜙_𝑖(𝑡),𝜃_𝑖(𝑡)=𝜃_𝑖−𝜃_𝑖(𝑡). (3.4)

The constant parameter update law and the time-varying periodic adaptive learning law are designed as follows:𝜃̇_𝑖(𝑡)=𝑟_𝑖𝑒^T_𝑖(𝑡)𝑒_𝑖(𝑡), (3.5)𝜙_𝑖(𝑡)=𝜙_𝑖(𝑡−𝑇)+𝑞_𝑖𝑒^T_𝑖(𝑡)𝑒_𝑖(𝑡)[),𝑞,𝑡∈𝑇,+∞_𝑖0(𝑡)𝑒^T_𝑖(𝑡)𝑒_𝑖(𝑡)[),[),,𝑡∈0,𝑇0,𝑡∈−𝑇,0 (3.6) where 𝑟_𝑖,𝑞_𝑖>0 are constants and 𝑞_𝑖0(𝑡) is a continuous and strictly increasing nonnegative function satisfying 𝑞_𝑖0(0)=0, 𝑞_𝑖0(𝑇)=𝑞_𝑖, which ensure 𝜙_𝑖(𝑡) is continuous in 𝑡=𝑖𝑇, 𝑖=1,2,…,𝑁.

4. Convergence Analysis

In the section, we will give the convergence of the proposed algorithm in the following theorem.

Theorem 4.1. Under Assumptions 2.2–2.5, the control law (3.3) with update law (3.5) and the periodic adaptive law (3.6), guarantees the 𝐿²_𝑇 asymptotic synchronization of the controlled network (3.1), while keeping all closed-loop signals are 𝐿²_𝑇 bounded.

Proof. To facilitate the convergence analysis, we choose a Lyapunov-Krasovskii function as follows: 𝑉(𝑡)=12^𝑁_𝑖=1𝑒^T_𝑖𝑒_𝑖+𝑁𝑎𝛾𝐻2^𝑁_𝑖=1^𝑡₀𝑞_𝑖⁻¹𝜙²_𝑖(𝜏)𝑑𝜏+𝑁𝑎𝛾𝐻2^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡)+𝐿_𝑖²[),1,𝑡∈0,𝑇2^𝑁_𝑖=1𝑒^T_𝑖𝑒_𝑖+𝑁𝑎𝛾𝐻2^𝑁_𝑖=1^𝑡_𝑡−𝑇𝑞_𝑖⁻¹𝜙²_𝑖(𝜏)𝑑𝜏+𝑁𝑎𝛾𝐻2^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡)+𝐿_𝑖²,𝑡∈[𝑇,+∞), (4.1) where 𝐿_𝑖 is sufficiently large positive constant which will be determined.
Firstly, we will prove the finiteness of 𝑉(𝑡) in [0,𝑇). Secondly, we will prove the asymptotical convergence of 𝑒_𝑖(𝑡). Finally, we will prove all closed-loop signals are 𝐿²_𝑇 bounded.
First step, let us confirm the finiteness property of 𝑉(𝑡) for the first period [0,𝑇). From the system dynamics (3.1) and the proposed control laws (3.3), (3.5), and (3.6), it can be seen that the right-hand side of (3.1) is continuous with respect to all arguments. In light of the existence theorem of differential equation, (3.2) has unique solution in an interval [0,𝑇₁)⊂[0,𝑇), with 0<𝑇₁≤𝑇. Therefore, the boundedness of 𝑉(𝑡) over [0,𝑇₁) can be guaranteed and we need only focus on the interval [𝑇₁,𝑇).
For any 𝑡∈[𝑇₁,𝑇), the time derivative of 𝑉(𝑡) for 𝑡∈[𝑇₁,𝑇) is given by 𝑉̇(𝑡)=^𝑁_𝑖=1𝑒^T_𝑖(𝑡)𝑒_𝑖̇(𝑡)+𝑁𝑎𝛾𝐻2^𝑁_𝑖=1𝑞_𝑖⁻¹𝜙²_𝑖(𝑡)+𝑁𝑎𝛾𝐻^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡)+𝐿_𝑖𝜃̇_𝑖(𝑡),𝑡∈𝑇₁,𝑇. (4.2)
Taking the first term on the right-hand side of (4.2), one obtains ^𝑁_𝑖=1𝑒^T_𝑖(𝑡)𝑒_𝑖̇(𝑡)=^𝑁_𝑖=1𝑒^T_𝑖(𝑡)𝑓𝑥_𝑖(𝑡)−𝑓(𝑠(𝑡))+^𝑁_𝑗=1𝑎_𝑖𝑗Γ𝑔𝑥_𝑗(𝑡),𝜑_𝑖(𝑡)−𝑔𝑠(𝑡),𝜑_𝑖(𝑡)+𝑢_𝑖(𝑡). (4.3) Let 𝑔(𝑥_𝑗(𝑡),𝜑_𝑖(𝑡))−𝑔(𝑠(𝑡),𝜑_𝑖(𝑡))=Λ, and ‖Λ‖²≤𝑒_𝑗(𝑡)²ℎ𝑥_𝑗(𝑡),𝑠(𝑡)𝜆𝜑_𝑖(𝑡). (4.4)
Then according to Assumption 2.2 and Lemma 2.6, we get ^𝑁_𝑖=1𝑒^T_𝑖(𝑡)𝑒_𝑖̇(𝑡)≤^𝑁_𝑖=1𝑙_𝑖𝑒^T_𝑖𝑒_𝑖+^𝑁_𝑗=1𝑎_𝑖𝑗𝑒^T_𝑖ΓΛ−𝑁𝑎𝛾𝐻𝜙_𝑖(𝑡)+𝜃_𝑖(𝑡)𝑒^T_𝑖𝑒_𝑖≤^𝑁_𝑖=1𝑙_𝑖𝑒^T_𝑖𝑒_𝑖+^𝑁_𝑗=1𝑎²_𝑖𝑗𝑐𝑒^T_𝑖𝑒_𝑖+1Λ4𝑐^TΓ^TΓΛ−𝑁𝑎𝛾𝐻𝜙_𝑖(𝑡)+𝜃_𝑖(𝑡)𝑒^T_𝑖𝑒_𝑖≤^𝑁_𝑖=1𝑙_𝑖𝑒^T_𝑖𝑒_𝑖+𝑁𝑎²𝑐𝑒^T_𝑖𝑒_𝑖+𝑁𝐻𝛾²𝑒4𝑐^T𝑖𝑒_𝑖𝜙_𝑖(𝑡)+𝜃_𝑖−𝑁𝑎𝛾𝐻𝜙_𝑖(𝑡)+𝜃_𝑖(𝑡)𝑒^T_𝑖𝑒_𝑖, (4.5) where 𝑐 is a positive constant. If choosing 𝑐=𝛾/4𝑎, we have ^𝑁_𝑖=1𝑒^T_𝑖(𝑡)𝑒_𝑖̇(𝑡)≤^𝑁_𝑖=1𝑙_𝑖+𝑁𝑎𝛾4+𝑁𝑎𝛾𝐻𝜙_𝑖(𝑡)+𝜃_𝑖(𝑡)𝑒^T_𝑖𝑒_𝑖. (4.6)
Taking the third term on the right-hand side of (4.2), we have 𝑁𝑎𝛾𝐻^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡)+𝐿_𝑖𝜃̇_𝑖(𝑡)=−𝑁𝑎𝛾𝐻^𝑁_𝑖=1𝜃_𝑖(𝑡)+𝐿_𝑖𝑒^T_𝑖𝑒_𝑖. (4.7)
Let us focus on the second term on the right-hand side of (4.2). Since 𝑞_𝑖0(𝑡) is a continuous function and strictly increasing in [0,𝑇), 𝑞_𝑖⁻¹≤𝑞⁻¹_𝑖0(𝑡)<∞ is ensured in the interval [𝑇₁,𝑇), and 𝑁𝑎𝛾𝐻2^𝑁_𝑖=1𝑞_𝑖⁻¹𝜙²_𝑖(𝑡)≤𝑁𝑎𝛾𝐻2^𝑁_𝑖=1𝑞⁻¹_𝑖0𝜙²_𝑖(𝑡)=𝑁𝑎𝛾𝐻2^𝑁_𝑖=1𝑞⁻¹_𝑖0𝜙²_𝑖(𝑡)+2𝜙²_𝑖(𝑡)−2𝜙_𝑖(𝑡)𝜙_𝑖(𝑡)−𝜙²_𝑖(𝑡≤)𝑁𝑎𝛾𝐻2^𝑁_𝑖=1𝑞⁻¹_𝑖0𝜙²_𝑖(𝑡)+2𝜙²_𝑖(𝑡)−2𝜙_𝑖(𝑡)𝜙_𝑖(𝑡=)𝑁𝑎𝛾𝐻2^𝑁_𝑖=1𝑞⁻¹_𝑖0𝜙²_𝑖(𝑡)−𝑁𝑎𝛾𝐻^𝑁_𝑖=1𝜙_𝑖(𝑡)𝑒^T_𝑖𝑒_𝑖. (4.8) Substituting (4.6)–(4.8) into (4.2) yields 𝑉̇(𝑡)≤^𝑁_𝑖=1𝑙_𝑖+𝑁𝑎𝛾4−𝑁𝑎𝛾𝐻𝐿_𝑖𝑒^T_𝑖𝑒_𝑖+𝑁𝑎𝛾𝐻2^𝑁_𝑖=1𝑞⁻¹_𝑖0(𝑡)𝜙²_𝑖(𝑡). (4.9)
It is obviously that there exists sufficiently large positive constants 𝐿_𝑖 such that 𝑙_𝑖+(𝑁𝑎𝛾/4)−𝑁𝑎𝛾𝐻𝐿_𝑖<0. According to (4.9), we have 𝑉̇(𝑡)≤𝑁𝑎𝛾𝐻2^𝑁_𝑖=1𝑞⁻¹_𝑖0(𝑡)𝜙²_𝑖(𝑡). (4.10)
Since 𝜙_𝑖(𝑡) is continuous and periodic, its 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∈[𝑇₁,𝑇).
Next step, we prove the asymptotical convergence of 𝑒(𝑡).
According to (4.1), we can 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−𝑇)+𝑁𝑎𝛾𝐻2^𝑁_𝑖=1^𝑡_𝑡−𝑇𝑞_𝑖⁻¹𝜙²_𝑖(𝜏)−𝑞_𝑖⁻¹𝜙²_𝑖(+𝜏−𝑇)𝑑𝜏𝑁𝑎𝛾𝐻2^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡)+𝐿_𝑖²−𝑁𝑎𝛾𝐻2^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡−𝑇)+𝐿_𝑖². (4.11)
Looking into the first two terms on the right-hand side of (4.11), 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)=^𝑁_𝑖=1^𝑡_𝑡−𝑇𝑒^T_𝑖(𝜏)𝑒_𝑖̇(𝜏)≤𝑑𝜏_{𝑁𝑖=1𝑡𝑡−𝑇}𝑙_𝑖+𝑁𝑎𝛾4+𝑁𝑎𝛾𝐻𝜙_𝑖+𝜃_𝑖𝑒^T_𝑖(𝜏)𝑒_𝑖(𝜏)𝑑𝜏. (4.12)
Using the algebraic relation (𝑎−𝑏)^T𝐻(𝑎−𝑏)−(𝑎−𝑐)^T𝐻(𝑎−𝑐)=(𝑐−𝑏)^T𝐻[2(𝑎−𝑏)+(𝑏−𝑐)], (4.13) where 𝑎,𝑏,𝑐∈𝑅^𝑝, 𝐻∈𝑅^𝑝×𝑝, and taking the third term on the right-hand side of (4.11), one obtains 𝑁𝑎𝛾𝐻2^𝑁_𝑖=1^𝑡_𝑡−𝑇𝑞_𝑖⁻¹𝜙²_𝑖(𝜏)−𝑞_𝑖⁻¹𝜙²_𝑖(𝜏−𝑇)𝑑𝜏=−𝑁𝑎𝛾𝐻^𝑁_𝑖=1^𝑡_𝑡−𝑇𝜙_𝑖(𝜏)+12𝑞_𝑖𝑒^T_𝑖𝑒_𝑖𝑒^T_𝑖𝑒_𝑖𝑑𝜏. (4.14)
The last two terms of right-hand side of (4.11) can be simplified as follows: 𝑁𝑎𝛾𝐻2^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡)+𝐿_𝑖²−𝑁𝑎𝛾𝐻2^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡−𝑇)+𝐿_𝑖²=𝑁𝑎𝛾𝐻^𝑁_𝑖=1^𝑡_𝑡−𝑇𝑟_𝑖⁻¹𝜃_𝑖(𝜏)+𝐿_𝑖𝜃̇_𝑖(𝜏)𝑑𝜏=−𝑁𝑎𝛾𝐻^𝑁_𝑖=1^𝑡_𝑡−𝑇𝜃_𝑖(𝜏)+𝐿_𝑖𝑒^T_𝑖(𝜏)𝑒_𝑖(𝜏)𝑑𝜏. (4.15) Substituting (4.12)–(4.15) into (4.11), we can attain Δ𝑉(𝑡)≤−^𝑁_𝑖=1^𝑡_𝑡−𝑇𝑁𝑎𝛾𝐻𝐿_𝑖−𝑙_𝑖−𝑁𝑎𝛾4𝑒^T_𝑖(𝜏)𝑒_𝑖(𝜏)𝑑𝜏. (4.16) Choosing 𝐿_𝑖>(𝑙_𝑖/𝑁𝑎𝛾𝐻)+(1/4𝐻), we can obtain Δ𝑉(𝑡)<0. (4.17) Applying (4.16) 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. (4.18)
Considering 𝑡₀∈[0,𝑇) and the positive of 𝑉(𝑡), according to (4.18), we obtain lim_𝑡→∞𝑉(𝑡)<max_{𝑡0∈[)0,𝑇}𝑉𝑡₀−lim_𝑙→∞^𝑙−1𝑁_{𝑗=0𝑖=1𝑇}^{𝑡−𝑗𝑇}_{𝑡−𝑗+1}𝑁𝑎𝛾𝐻𝐿_𝑖−𝑙_𝑖−𝑁𝑎𝛾4𝑒^T_𝑖(𝜏)𝑒_𝑖(𝜏)𝑑𝜏. (4.19)
Since 𝑉(𝑡₀) is bounded in the interval [0,𝑇), according to the convergence theorem of the sum of series and (4.19), the error 𝑒(𝑡) converges to zero asymptotically in 𝐿²_𝑇 norm. That is to say, we have lim_𝑡→∞^𝑡_𝑡−𝑇𝑒^T_𝑖(𝜏)𝑒_𝑖(𝜏)𝑑𝜏=0. (4.20)
Finally, we prove that all the closed-loop signals are bounded in 𝐿²_𝑇 norm sense. For all 𝑡∈[𝑇,∞), the derivative of 𝑉(𝑡) is 𝑉̇(𝑡)=^𝑁_𝑖=1𝑒^T_𝑖(𝑡)𝑒_𝑖̇(𝑡)+𝑁𝑎𝛾𝐻2^𝑁_𝑖=1𝑞_𝑖⁻¹𝜙²_𝑖(𝑡)−𝑁𝑎𝛾𝐻2^𝑁_𝑖=1𝑞_𝑖⁻¹𝜙²_𝑖(𝑡−𝑇)+𝑁𝑎𝛾𝐻^𝑁_𝑖=1𝑟_𝑖⁻¹𝜃_𝑖(𝑡)+𝐿_𝑖𝜃̇_𝑖(𝑡)≤−𝑁𝑎𝛾𝐻2^𝑁_𝑖=1𝑞_𝑖𝑒^T_𝑖𝑒_𝑖²−^𝑁_𝑖=1𝑁𝑎𝛾𝐻𝐿_𝑖−𝑙_𝑖−𝑁𝑎𝛾4𝑒^T_𝑖𝑒_𝑖. (4.21)
By (4.21), one can obtain 𝑉(𝑡)≤𝑉(𝑇)−𝑁𝑎𝛾𝐻2^𝑁_𝑖=1^𝑡_𝑇𝑞_𝑖𝑒^T_𝑖(𝜏)𝑒_𝑖(𝜏)²𝑑𝜏−^𝑁_𝑖=1^𝑡_𝑇𝑁𝑎𝛾𝐻𝐿_𝑖−𝑙_𝑖−𝑁𝑎𝛾4𝑒^T_𝑖𝑒_𝑖𝑑𝜏. (4.22) Choosing 𝐿_𝑖>(𝑙_𝑖/𝑁𝑎𝛾𝐻)+(1/4𝐻), we have 𝑉(𝑡)<𝑉(𝑇). (4.23)
From the boundedness of 𝑉(𝑡) and (4.1), we can conclude that 𝑒_𝑖, ∫^𝑡_𝑡−𝑇𝜙²_𝑖(𝜏)𝑑𝜏, 𝜃_𝑖(𝑡) are all bounded. Since 𝜙_𝑖(𝑡) is continuous periodic function, it implies the 𝐿²_𝑇 boundedness of 𝜙_𝑖(𝑡) and 𝜃_𝑖(𝑡). According to (3.3), the 𝐿²_𝑇 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.

5. Simulation Examples

To demonstrate the theoretical result obtained in Section 3, the Chuas chaotic circuit is used as a dynamical node of the network.

Consider the Chuas chaos circuit𝑥̇₁=𝑝−𝑥₁+𝑥₂−𝑔𝑥₁,𝑥̇₂=𝑥₁−𝑥₂+𝑥₃,𝑥̇₃=−𝑞𝑥₂, (5.1) where 𝑔(𝑥₁)=𝑚₀𝑥₁+(1/2)(𝑚₁−𝑚₀)(|𝑥₁+1|−|𝑥₁−1|) with 𝑝=10, 𝑞=14.7, 𝑚₀=−0.68 and 𝑚₁=−1.27.

We take the system (5.1) as identical nodes of the network, which is given by 𝑥̇_𝑖1𝑥̇_𝑖2𝑥̇_𝑖3=𝑝−𝑥_𝑖1+𝑥_𝑖2−𝑔𝑥_𝑖1𝑥_𝑖1−𝑥_𝑖2+𝑥_𝑖3−𝑞𝑥_𝑖2+^𝑁_𝑗=1𝑎_𝑖𝑗Γexp−𝜑_𝑖(𝑡)𝑥²_𝑗1(𝑡)𝑥²_𝑗2(𝑡)𝑥²_𝑗3(𝑡)+𝑢_𝑖(𝑡),𝑖=1,2,…,𝑁, (5.2) where𝐴=−1100001−2100001−3200002−4110001−5400014−5Γ=100010001. (5.3) Nonlinearly parameterized function satisfiesexp−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). (5.4) The parameters are selected as follows:𝜙𝑁=6,𝑇=6,𝑎=5,𝛾=1,𝐻=5,₁(𝑡)=0.2sin2𝜋𝑡3,𝜙₂(𝑡)=2cos𝜋𝑡,𝜙₃(𝑡)=−sin2𝜋𝑡3,𝜙₄(𝑡)𝜙=cos𝜋𝑡,₅(𝑡)=−2sin2𝜋𝑡3,𝜙₆(𝑡)=2sin2𝜋𝑡3,𝜃=(2,10,8,3,2,5)^T. (5.5) In the following simulations, we choose𝑞₁=0.0001,𝑞₂=0.0003,𝑞₃=0.0002,𝑞₄=0.0005,𝑞₅=0.0001,𝑞₆𝑞=0.0002,₁₀(𝑡)=𝑡6𝑞₁,𝑞₂₀(𝑡)=𝑡6𝑞₂,𝑞₃₀(𝑡)=𝑡6𝑞₃,𝑞₄₀(𝑡)=𝑡6𝑞₄,𝑞₅₀(𝑡)=𝑡6𝑞₅,𝑞₆₀(𝑡)=𝑡6𝑞₆, (5.6)𝑟₁=𝑟₂=𝑟₃=𝑟₄=𝑟₅=𝑟₆=0.0005. The initially estimated value of the unknown parameter 𝜙(𝑡) is 𝜙(0)=(0,0,0,0,0,0)^T, and the initial states are chosen as𝑥₁(0)=(2,0,−1)^T,𝑥₂(0)=(3,1,−2)^T,𝑥₃(0)=(−3,−2,1)^T,𝑥₄(0)=(−1,0,−2)^T,𝑥₅(0)=(−1,1,0)^T,𝑥₆(0)=(3,−2,0.8)^T,𝑠(0)=(−0.2,0.2,0.5)^T. (5.7)

According to Theorem 4.1, the complex dynamical network (5.2) can be synchronized by applying the adaptive controllers (3.3) and the adaptive learning laws of parameters (3.5) and (3.6). Figure 1 shows the error evolutions under the designed controllers. We clearly show that the states of the network (5.2) asymptotically synchronize in 𝐿²_𝑇 norm with the states of the desired orbit (5.1) showed in Figure 2. Figure 3 depicts the time evolution of the controllers, and Figure 4 shows the evolution of the estimated time-varying parameters. Figure 5 displays the evolution of the estimated constant parameters. Obviously, Figures 3–5 show that all the control signals and estimated parameters are also bounded.

Figure 1 

The synchronization errors in 𝐿2𝑇 norm.

Figure 2 

The desired orbit.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3 

The curve of control inputs 𝑢𝑖.

Figure 4 

The evaluations of parameters 𝜙𝑖.

Figure 5 

The evaluations of parameters 𝜃𝑖.

6. Conclusion

In this paper, a new adaptive learning control method applied in nonlinearly parameterized complex dynamical networks with unknown time-varying parameters is presented. By using the parameter separation technique and constructing a composite energy function, the adaptive learning laws of parameters and the adaptive control strategy are designed to ensure the asymptotic convergence of the synchronization error in the sense of square error norm. An example of the complex network is finally used to verify the proposed theoretical result.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. 60974139 and the Fundamental Research Funds for the Central Universities under Grant no. 72103676.