Locally and Globally Exponential Synchronization of Moving Agent Networks by Adaptive Control

Abstract

The exponential synchronization problem is investigated for a class of moving agent networks in a two-dimensional space and exhibits time-varying topology structure. Based on the Lyapunov stability theory, adaptive feedback controllers are developed to guarantee the exponential synchronization between each agent node. New criteria are proposed for verifying the locally and globally exponential synchronization of moving agent networks under the constraint of fast switching. In addition, a numerical example, including typical moving agent network with the Rössler system at each agent node, is provided to demonstrate the effectiveness and applicability of the proposed design approach.

1. Introduction

Over the past decade years, the analysis of complex systems from the viewpoint of networks has become an important interdisciplinary issue [1]. Complex networks have been intensively studied in many fields, such as social, biological, mathematical, and engineering sciences. Generally, a complex network is made up of interconnected nodes in which a node is a basic unit with detailed contents. These interactions between nodes determine many basic properties of a network. To better understand the complex dynamical behaviors of many natural systems, we need to study their operating mechanism, dynamic behavior, synchronization, antijamming ability, and so on. Recently, synchronization of complex dynamical networks has received a great deal of attentions from various fields of science and engineering [1–3]. Particularly, synchronization of large-scale complex networks of coupled chaotic oscillators has been extensively investigated in the fields of science and engineering.

The synchronization properties of a complex network are mainly determined by its topological structures connections between nodes. In the current study of complex networks, most of the existing works on synchronization consider static networks, whose topological structures do not change as time evolves [4–10]. The Master-stability function (MSF) approach [11] allows us to determine the stability of a linearly coupled dynamical network with a constant coupling (or Laplacian) matrix. However, numerous real-world networks such as biological, communication, social, and epidemiological networks generally evolve with time-varying topological structures. Henceforth, researchers have devoted more and more efforts to complex networks with time-varying topologies. Stilwell et al. [12] prove that if the network of oscillators synchronizes for the static time-average of the topology, then the network will synchronize with the time-varying topology if the time-average is achieved sufficiently fast. At the same conditions, Lu et al. [13, 14] found that the directed network with switching topology can reach global synchronization for sufficiently large coupling strength if there exists a spanning-directed tree in the network.

Inspired by the above discussions, in this paper we investigate the adaptive exponential synchronization problem for a specific time-varying network model. The model arises from the interaction of mobile agents proposed by Frasca et al. [15] and can be widely used to explore various practical problems, for example, clock synchronization in mobile robots [16], synchronized bulk oscillations [17], and task coordination of swarming animals [18]. How one controls the appearance of synchronized states of the dynamical network is of great significance in theory and potential applications. For the mobile agent network model, we introduce adaptive control to regulate its synchronization, as an attempt to explain the control of complex time-varying systems.

Although synchronization control of moving agent network has great application potential in a variety of areas, there has been very little existing literature on the exponential synchronization problem. Therefore, we adopt the constraint of fast switching to derive exponent synchronization conditions. By using Lyapunov stability theory, adaptive controllers are designed for synchronization of moving agent network with time-varying topological structures. The adaptive controllers can ensure the states of moving agent network fast synchronization.

The current paper is organized as follows. A general moving agent network model and several mathematical preliminaries are introduced in Section 2. In Section 3, several locally and globally adaptive synchronization criteria for the moving agent networks are deduced. A representative example is given to show the effectiveness of the proposed network synchronization criteria in Section 4. Conclusions are finally drawn in Section 5.

2. Moving Agent Network Model and Preliminaries

We consider 𝑁 as moving agents distributed in a two-dimensional planar space of size 𝐿Γ={(𝑦₁,𝑦₂)∈𝑅²∶0≤𝑦₁≤𝐿,0≤𝑦₂≤𝐿}, with periodic boundary conditions. Each agent moves with velocity 𝑣_𝑖(𝑡) and direction of motion 𝜃_𝑖(𝑡). The velocity 𝑣_𝑖(𝑡) is the same for all individuals (denoted by 𝑣) and is updated in direction through the angle 𝜃_𝑖(𝑡) for each time unit. The agents are considered as random walkers. Hence, the motion law of the 𝑖th agent is given as follows: 𝑦_𝑖1𝑡+Δ𝑡_𝑀=𝑦_𝑖1(𝑡)+𝑣cos𝜃_𝑖(𝑡)Δ𝑡_𝑀,𝑦_𝑖2𝑡+Δ𝑡_𝑀=𝑦_𝑖2(𝑡)+𝑣sin𝜃_𝑖(𝑡)Δ𝑡_𝑀,𝜃_𝑖𝑡+Δ𝑡_𝑀=𝜂_𝑖𝑡+Δ𝑡_𝑀, (1) where 𝑖=1,2,…,𝑁, 𝑦_𝑖(𝑡)=(𝑦₁,𝑦₂)∈Γ is the position of agent 𝑖 in the plane at time 𝑡, 𝜂_𝑖(𝑡) are 𝑁 independent random variables chosen at each time unit with uniform probability in the interval [0,2𝜋], and Δ𝑡_𝑀 is the motion integration step size.

Each agent interacts at a given time with only those agents located within a neighborhood of an interaction radius, defined as 𝑅 [15, 19, 20]. When two agents interact, the state equations of each agent are changed to include diffusive coupling with the neighboring agent. Under these hypotheses, the state dynamics of agent node 𝑖 can be formulated as 𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡),𝑡−𝜎^𝑁_𝑗=1𝑔_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)+𝑢_𝑖(𝑡), (2) where 𝑖=1,…,𝑁, 𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡))^𝑇∈𝑅^𝑛 are the state variables of the node 𝑖; 𝑓∶𝐷×𝑅⁺→𝑅^𝑛 is a smooth nonlinear vector-valued function which governs the local dynamics of oscillator; 𝜎 is the coupling strength; 𝑔_𝑖𝑗(𝑡) are the elements of a time-varying Laplacian matrix: 𝐺(𝑡)=[𝑔_𝑖𝑗(𝑡)]∈𝑅^𝑁×𝑁 which defines the neighborhood of agents at a given time 𝑡 and depends on the trajectory of each agent. In detail, for arbitrary two agents 𝑖, 𝑗 with distance 𝑑 at time 𝑡, 𝑔_𝑖𝑗(𝑡)=−1, 𝑔_𝑗𝑖(𝑡)=−1 if 𝑑<𝑅; 𝑔_𝑖𝑗(𝑡)=0, 𝑔_𝑗𝑖(𝑡)=0 if 𝑑>𝑅; 𝑔_𝑖𝑖(𝑡)=ℎ, where ℎ is the number of neighbors of the 𝑖th agent at time 𝑡; 𝑢_𝑖(𝑡)∈𝑅^𝑛 are the control inputs.

In this paper, the control objective is to make the states of network (2) exponentially synchronize with a manifold defined in (3) by introducing a simple adaptive controller into each individual node 𝑥₁(𝑡)=𝑥₂(𝑡)=⋯=𝑥_𝑁(𝑡)=𝑠(𝑡), (3) where 𝑠(𝑡) is a solution of an isolated node, 𝑠(̇𝑡)=𝑓(𝑠(𝑡),𝑡). (4)

We assume that 𝑠(𝑡) is an arbitrary desired state which can be an equilibrium point, a periodic orbit, an aperiodic orbit, or even a chaotic orbit in the phase space.

Next, the rigorous mathematical definition of exponential synchronization for dynamical network (2) is introduced.

Definition 1. Let 𝑋(𝑡,𝑡₀,𝜙)=(𝑥₁(𝑡,𝑡₀,𝜙)^𝑇,𝑥₂(𝑡,𝑡₀,𝜙)^𝑇,…,𝑥_𝑁(𝑡,𝑡₀,𝜙)^𝑇)^𝑇∈𝑅^𝑛𝑁 be a solution of dynamical network (2), where 𝜙=(𝜙^𝑇₁,𝜙^𝑇₂,…,𝜙^𝑇_𝑁)^𝑇 are initial conditions of node 𝑖 and 𝑓∶𝐷×𝑅⁺→𝑅^𝑛 is continuously differentiable on 𝐷⊆𝑅^𝑛. If there are control inputs 𝑢_𝑖,𝑖=1,2,…,𝑁, and further there exist constants 𝛼>0, 𝜆>0 and a nonempty subset Λ⊆𝐷 with 𝜙_𝑖∈Λ, 𝑖=1,2,…,𝑁, such that 𝑋(𝑡,𝑡₀,𝜙)∈𝐷×𝐷×⋯×𝐷 for all 𝑡≥𝑡₀, and 𝑋𝑡,𝑡₀,𝜙−𝑠𝑡,𝑡₀,𝑠₀≤𝛼𝑒^−𝜆𝑡𝜙−𝑆₀, (5) (Hereafter, denote ‖⋅‖ as the Euclidean norm.) where 𝑆(𝑡,𝑡₀,𝑠₀)=(𝑠(𝑡,𝑡₀,𝑠₀)^𝑇,…,𝑠(𝑡,𝑡₀,𝑠₀)^𝑇)^𝑇∈𝑅^𝑛𝑁, 𝑆₀=(𝑠^𝑇₀,…,𝑠^𝑇₀)^𝑇, 𝑠(𝑡,𝑡₀,𝑠₀) is a solution of the system (4) with the initial condition 𝑠₀∈𝐷, then the dynamical network (2) is said to realize exponential synchronization such that 𝜆 is the exponential rate. And Λ×⋯×Λ is called the region of synchrony for the dynamical network (2) (see [21, 22]).

3. Synchronization of Moving Agent Networks

In this section, we discuss the exponential synchronization of moving agent network (2) by designing adaptive controllers for each agent node. Several network synchronization criteria are given.

3.1. Local Synchronization

In order to achieve the objective of synchronization on the manifold (3), let us define the error vector 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑠(𝑡),𝑖=1,2,…,𝑁. (6) Subtracting (4) from (2) yields the error dynamical system 𝑒_𝑖̇(𝑡)=𝑓𝑥_𝑖(𝑡),𝑡−𝑓(𝑠(𝑡),𝑡)−𝜎^𝑁_𝑗=1𝑔_𝑖𝑗(𝑡)𝑒_𝑗(𝑡)+𝑢_𝑖(𝑡). (7)

Then, exponential synchronization problem of the dynamical network (2) is equivalent to the problem of exponential stabilization of the error dynamical system (7).

Linearizing error system (7) about the synchronized states 𝑠(𝑡), we can get 𝑒_𝑖̇(𝑡)=𝐴(𝑡)𝑒_𝑖(𝑡)−𝜎^𝑁_𝑗=1𝑔_𝑖𝑗(𝑡)𝑒_𝑗(𝑡)+𝑢_𝑖(𝑡), (8) where 𝑖=1,2,…,𝑁 and 𝐴(𝑡)=𝐽_𝑓(𝑠,𝑡) is the Jacobian matrix of 𝑓 evaluated at 𝑥=𝑠(𝑡).

In the following, we give several useful hypotheses.

Assumption 2. Suppose that there exists a nonnegative constant 𝛼 satisfying ‖𝐴(𝑡)‖≤𝛽. (9)

Generally, Assumption 2 is likely to be satisfied. For example, in many chaotic systems such as Lorenz system, Rössler system, and Chen system, there exists a constant 𝛽 satisfying ‖𝐴(𝑡)‖≤𝛽.

Assumption 3. Suppose there exists a constant 𝑇 such that coupling matrix 𝐺(𝑡) satisfying 1𝑇_𝑡^𝑡+𝑇𝐺(𝜏)𝑑𝜏=𝐺, (10) where 𝐺 is the time-average of the coupling matrix 𝐺(𝑡).

Assumption 3 implies that the switching between all the possible network configurations is sufficiently fast as defined in [12]. According to [12] the following lemma can be given.

Lemma 4. Suppose that a coupled network with fixed topology defined by 𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡),𝑡−𝜎^𝑁_𝑗=1𝑔_𝑖𝑗𝑥_𝑗(𝑡)+𝑢_𝑖(𝑡)𝐺=𝑔_𝑖𝑗 (11) admits a stable synchronization manifold and Assumption 3 holds. Then the network with a time-variant topological structure defined by (2) admits a stable synchronization manifold.

The proof of Lemma 4 can be obtained by main results of [12]. Here we omitted proof process.

According to analysis of [15], under the constraint of fast switching, 𝐺=𝑝𝐺_𝐴, where 𝑝 is the probability that a link is activated and thus 𝑝=𝜋𝑅²/𝐿², and 𝐺_𝐴 is the all-to-all coupling matrix with zero-row sum. That is 𝐺=𝜋𝑅²𝐿²𝑁−1−1⋯−1−1𝑁−1⋯−1⋮⋮⋱⋮−1−1⋯𝑁−1. (12)

Based on Assumptions 2 and 3, a network synchronization criterion is deduced as follows.

Theorem 5. Suppose that Assumptions 2 and 3 hold. Then the dynamical moving agent network (2) is locally exponential synchronization under the following sets of adaptive controllers: 𝑢_𝑖=−𝑑_𝑖𝑒_𝑖,𝑖=1,2,…,𝑁 (13) and updating laws 𝑑̇_𝑖=𝑒^2𝜆𝑡𝑒_𝑖², (14) where 𝜆 is the exponential rate available to be designed, and 𝑑_𝑖(𝑡₀)=𝛼−𝜆, where 𝛼 is defined in the Assumption 2.

Proof. Select a Lyapunov function as follows: 𝑉(𝑡)=12^𝑁_𝑖=1𝑒^𝑇_𝑖𝑒_𝑖+12^𝑁_𝑖=1𝑒^−2𝜆𝑡𝑑_𝑖−𝑑^∗_𝑖², (15) where constant 𝑑^∗_𝑖 is to be given below. Then the time derivative of 𝑉(𝑡) along the solution of the error system (8) is given as follows: 𝑉̇=12^𝑁_𝑖=1𝑒^𝑇_𝑖̇𝑒_𝑖+𝑒^𝑇_𝑖𝑒_𝑖̇+12^𝑁_𝑖=1𝑒^−2𝜆𝑡(−2𝜆)𝑑_𝑖−𝑑^∗_𝑖²+^𝑁_𝑖=1𝑒^−2𝜆𝑡𝑑_𝑖−𝑑^∗_𝑖𝑑̇_𝑖=12^𝑁_𝑖=1𝐴(𝑡)𝑒_𝑖−𝜎^𝑁_𝑗=1𝑔_𝑖𝑗(𝑡)𝑒_𝑗+𝑢_𝑖^𝑇𝑒_𝑖+𝑒^𝑇_𝑖𝐴(𝑡)𝑒_𝑖−𝜎^𝑁_𝑗=1𝑔_𝑖𝑗(𝑡)𝑒_𝑗+𝑢_𝑖−𝜆^𝑁_𝑖=1𝑒^−2𝜆𝑡𝑑_𝑖−𝑑^∗_𝑖²+^𝑁_𝑖=1𝑒^−2𝜆𝑡𝑑_𝑖−𝑑^∗_𝑖𝑒^2𝜆𝑡𝑒^𝑇_𝑖𝑒_𝑖=^𝑁_𝑖=1𝑒^𝑇_𝑖𝐴^𝑇(𝑡)+𝐴(𝑡)2𝑒_𝑖−𝜎^𝑁_𝑗=1𝑔_𝑖𝑗(𝑡)𝑒_𝑗(𝑡)+𝑢_𝑖−𝜆^𝑁_𝑖=1𝑒^−2𝜆𝑡𝑑_𝑖−𝑑^∗_𝑖²+^𝑁_𝑖=1𝑑_𝑖−𝑑^∗_𝑖𝑒^𝑇_𝑖𝑒_𝑖. (16) According to Assumption 2,  ‖𝐴(𝑡)‖≤𝛽, so ‖(𝐴^𝑇(𝑡)+𝐴(𝑡))/2‖≤𝛽𝑉̇≤𝛽^𝑁_𝑖=1𝑒^𝑇_𝑖𝑒_𝑖−𝜎^𝑁𝑁_𝑖=1𝑗=1𝑔_𝑖𝑗(𝑡)𝑒^𝑇_𝑖𝑒_𝑗(𝑡)−^𝑁_𝑖=1𝑑_𝑖𝑒^𝑇_𝑖𝑒_𝑖+^𝑁_𝑖=1𝑑_𝑖−𝑑^∗_𝑖𝑒^𝑇_𝑖𝑒_𝑖−𝜆^𝑁_𝑖=1𝑒^−2𝜆𝑡𝑑_𝑖−𝑑^∗_𝑖²=𝛽^𝑁_𝑖=1𝑒^𝑇_𝑖𝑒_𝑖−𝜎^𝑁𝑁_𝑖=1𝑗=1𝑔_𝑖𝑗(𝑡)𝑒^𝑇_𝑖𝑒_𝑗(𝑡)−^𝑁_𝑖=1𝑑^∗_𝑖𝑒^𝑇_𝑖𝑒_𝑖−𝜆^𝑁_𝑖=1𝑒^−2𝜆𝑡𝑑_𝑖−𝑑^∗_𝑖². (17)
Here, according to Lemma 4, under the constraint of fast switching, one substitute 𝑔_𝑖𝑗(𝑡) for 𝑔_𝑖𝑗𝑉̇(𝑡)≤𝛽^𝑁_𝑖=1𝑒^𝑇_𝑖𝑒_𝑖−𝜎^𝑁𝑁_𝑖=1𝑗=1𝑔_𝑖𝑗𝑒^𝑇_𝑖𝑒_𝑗−^𝑁_𝑖=1𝑑^∗_𝑖𝑒^𝑇_𝑖𝑒_𝑖−𝜆^𝑁_𝑖=1𝑒^−2𝜆𝑡𝑑_𝑖−𝑑^∗_𝑖². (18) Since 𝑒^𝑇_𝑖𝑒_𝑗≤(1/2)(𝑒^𝑇_𝑖𝑒_𝑖+𝑒^𝑇_𝑗𝑒_𝑗), −𝜎^𝑁𝑁_𝑖=1𝑗=1𝑔_𝑖𝑗𝑒^𝑇_𝑖𝑒_𝑗=𝜎𝜋𝑅²𝐿²−(𝑁−1)𝑒^𝑇₁𝑒₁+𝑒^𝑇₁𝑒₂+𝑒^𝑇₁𝑒₃+⋯+𝑒^𝑇₁𝑒_𝑁+𝑒^𝑇₂𝑒₁−(𝑁−1)𝑒^𝑇₂𝑒₂+𝑒^𝑇₂𝑒₃+⋯−𝑒^𝑇₂𝑒_𝑁⋯+𝑒^𝑇_𝑁𝑒₁+𝑒^𝑇_𝑁𝑒₂−⋯+𝑒^𝑇_𝑁𝑒_𝑁−1−(𝑁−1)𝑒^𝑇_𝑁𝑒_𝑁≤𝜎𝜋𝑅²𝐿²−(𝑁−1)𝑒^𝑇₁𝑒₁+12𝑒^𝑇₁𝑒₁+𝑒^𝑇₂𝑒₂+12𝑒^𝑇₁𝑒₁+𝑒^𝑇₃𝑒₃1+⋯+2𝑒^𝑇₁𝑒₁+𝑒^𝑇_𝑁𝑒_𝑁+12𝑒^𝑇₂𝑒₂+𝑒^𝑇₁𝑒₁−(𝑁−1)𝑒^𝑇₂𝑒₂+12𝑒^𝑇₂𝑒₂+𝑒^𝑇₃𝑒₃1+⋯+2𝑒^𝑇₂𝑒₂+𝑒^𝑇_𝑁𝑒_𝑁⋯+12𝑒^𝑇_𝑁𝑒_𝑁+𝑒^𝑇₁𝑒₁+12𝑒^𝑇_𝑁𝑒_𝑁+𝑒^𝑇₂𝑒₂1+⋯+2𝑒^𝑇_𝑁𝑒_𝑁+𝑒^𝑇_𝑁−1𝑒_𝑁−1−(𝑁−1)𝑒^𝑇_𝑁𝑒_𝑁=0. (19) Therefore, 𝑉̇(𝑡)≤𝛽^𝑁_𝑖=1𝑒^𝑇_𝑖𝑒_𝑖−^𝑁_𝑖=1𝑑^∗_𝑖𝑒^𝑇_𝑖𝑒_𝑖−𝜆^𝑁_𝑖=1𝑒^−2𝜆𝑡𝑑_𝑖−𝑑^∗_𝑖². (20) Since 𝛽 is a nonnegative constant, one can select suitable constants 𝑑^∗_𝑖 to make 𝛽−𝑑^∗_𝑖=𝜆. Therefore, 𝑉̇≤−𝜆^𝑁_𝑖=1𝑒^𝑇_𝑖𝑒_𝑖−𝜆^𝑁_𝑖=1𝑒^−2𝜆𝑡𝑑_𝑖−𝑑^∗_𝑖²=−2𝜆𝑉(𝑡). (21) By calculating integration on both sides of the above inequality, we get 𝑉(𝑡)≤𝑒^{−2𝜆(𝑡−𝑡0)}𝑉(𝑡₀).
According to (15), one can get (1/2)∑^𝑁_𝑖=1𝑒^𝑇_𝑖𝑒_𝑖≤𝑉(𝑡), so (1/2)∑^𝑁_𝑖=1𝑒^𝑇_𝑖𝑒_𝑖≤𝑒^{−2𝜆(𝑡−𝑡0)}𝑉(𝑡₀)𝑉𝑡₀=12^𝑁_𝑖=1𝑒^𝑇_𝑖𝑡₀𝑒_𝑖𝑡₀+12^𝑁_𝑖=1𝑒^{−2𝜆𝑡0}𝑑_𝑖𝑡₀−𝑑^∗_𝑖². (22) Let 𝑑_𝑖(𝑡₀)=𝑑^∗_𝑖, so (1/2)∑^𝑁_𝑖=1𝑒^𝑇_𝑖𝑒_𝑖≤𝑒^{−2𝜆(𝑡−𝑡0)}(1/2)∑^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡₀)𝑒_𝑖(𝑡₀). That is, ‖𝑒(𝑡)‖²≤𝑒^{−2𝜆(𝑡−𝑡0)}𝑒𝑡₀². (23) So, one gets ‖𝑒(𝑡)‖≤𝛼𝑒^−𝜆𝑡, (24) where 𝛼=𝑒^𝜆𝑡0‖𝑒(𝑡₀)‖.
Therefore, in closed-loop under the controllers (13) and updating laws (14), it follows that the error system (7) is locally exponentially stable at the equilibrium set 𝑒_𝑖=0, 𝑖=1,2,…,𝑁, with the exponential rate 𝜆. Consequently, the synchronous solution 𝑆(𝑡) of the dynamical network (2) is locally exponentially stable. Then the dynamical network (2) is said to realize locally exponential synchronization under the controllers (13) and updating laws (14). The proof is thus completed.

Remark 6. Since error dynamical system (8) is linearized, the moving agent network (2) is locally exponential synchronization rather than globally exponential synchronization.

3.2. Global Synchronization

Rewrite node dynamics 𝑥̇_𝑖(𝑡)=𝑓(𝑥_𝑖(𝑡),𝑡) as 𝑥̇_𝑖(𝑡)=𝐵𝑥_𝑖(𝑡)+ℎ(𝑥_𝑖(𝑡),𝑡), where 𝐵 is a constant matrix and ℎ∶𝐷×𝑅⁺→𝑅^𝑛 is a smooth nonlinear function. Thus, network (2) is described by 𝑥̇_𝑖(𝑡)=𝐵𝑥_𝑖(𝑡)+ℎ𝑥_𝑖(𝑡),𝑡−𝜎^𝑁_𝑗=1𝑔_𝑖𝑗(𝑡)𝑥_𝑗+𝑢_𝑖(𝑡), (25) where 𝑖=1,2,…,𝑁. Similarly, one can get the error system 𝑒_𝑖̇(𝑡)=𝐵𝑒_𝑖(𝑡)+ℎ𝑥_𝑖,𝑠,𝑡−𝜎^𝑁_𝑗=1𝑔_𝑖𝑗(𝑡)𝑒_𝑗(𝑡)+𝑢_𝑖(𝑡), (26) where 𝑖=1,2,…,𝑁 and ℎ(𝑥_𝑖,𝑠,𝑡)=ℎ(𝑥_𝑖,𝑡)−ℎ(𝑠,𝑡).

Assumption 7. Suppose that there exists a nonnegative constant 𝜇, satisfying ℎ𝑥_𝑖,𝑠,𝑡≤𝜇𝑒_𝑖. (27)

Theorem 8. Suppose that Assumptions 3 and 7 hold. Then the dynamical moving agent network (2) is globally synchronized under the following adaptive controllers: 𝑢_𝑖=−𝑑_𝑖𝑒_𝑖,𝑖=1,2,…,𝑁 (28) and updating laws 𝑑̇_𝑖=𝑒^2𝜆𝑡𝑒_𝑖², (29) where 𝑑_𝑖(𝑡₀)=𝛾+𝜇−𝜆, 𝛾 is a nonnegative constant satisfying ‖(𝐵+𝐵^𝑇)/2‖≤𝛾, 𝜇 is defined in the Assumption 7, and 𝜆 is the exponential rate available to be designed.

Proof. since 𝐵 is a given constant matrix, there exists a nonnegative constant 𝛾 such that ‖(𝐵+𝐵^𝑇)/2‖≤𝛾.
Similarly, construct Lyapunov function (15); then one has 𝑉̇=^𝑁_𝑖=1𝑒^𝑇_𝑖𝐵^𝑇+𝐵2𝑒_𝑖+^𝑁_𝑖=1𝑒^𝑇_𝑖ℎ𝑥_𝑖,𝑠,𝑡−𝜎^𝑁𝑁_𝑖=1𝑗=1𝑒^𝑇_𝑖𝑔_𝑖𝑗(𝑡)𝑒_𝑗+^𝑁_𝑖=1𝑒^𝑇_𝑖𝑢_𝑖(𝑡)−𝜆^𝑁_𝑖=1𝑒^−2𝜆𝑡𝑑_𝑖−𝑑^∗_𝑖²+^𝑁_𝑖=1𝑑_𝑖−𝑑^∗_𝑖𝑒^𝑇_𝑖𝑒_𝑖. (30) And according to Assumption 7, there exists a nonnegative constant 𝜇, satisfying ‖ℎ(𝑥_𝑖,𝑠,𝑡)‖≤𝜇‖𝑒_𝑖‖; therefore, 𝑉̇≤𝛾+𝜇+𝑑_𝑖−𝑑^∗_𝑖^𝑁_𝑖=1𝑒^𝑇_𝑖𝑒_𝑖−𝜎^𝑁𝑁_𝑖=1𝑗=1𝑒^𝑇_𝑖𝑔_𝑖𝑗(𝑡)𝑒_𝑗+^𝑁_𝑖=1𝑒^𝑇_𝑖𝑢_𝑖(𝑡)−𝜆^𝑁_𝑖=1𝑒^−2𝜆𝑡𝑑_𝑖−𝑑^∗_𝑖². (31) According to Lemma 4, one substitute 𝑔_𝑖𝑗(𝑡) for 𝑔_𝑖𝑗𝑉̇≤𝛾+𝜇−𝑑^∗_𝑖^𝑁_𝑖=1𝑒^𝑇_𝑖𝑒_𝑖−𝜆^𝑁_𝑖=1𝑒^−2𝜆𝑡𝑑_𝑖−𝑑^∗_𝑖². (32) Since 𝛾, 𝜇 are nonnegative constants, one can select suitable constants 𝑑^∗_𝑖 to make 𝛾+𝜇−𝑑^∗_𝑖=𝜆. Therefore, 𝑉̇≤−𝜆^𝑁_𝑖=1𝑒^𝑇_𝑖𝑒_𝑖−𝜆^𝑁_𝑖=1𝑒^−2𝜆𝑡𝑑_𝑖−𝑑^∗_𝑖²=−2𝜆𝑉(𝑡). (33) By similar calculation, one gets 𝑉(𝑡)≤𝑒^{−2𝜆(𝑡−𝑡0)}𝑉(𝑡₀). Then, one has ‖𝑒(𝑡)‖²≤𝑒^{−2𝜆(𝑡−𝑡0)}𝑒(𝑡₀)². (34) Similar, one can get ‖𝑒(𝑡)‖≤𝛼𝑒^−𝜆𝑡, (35) where 𝛼=𝑒^𝜆𝑡0‖𝑒(𝑡₀)‖.
Therefore, in closed-loop under the controllers (28) and updating laws (29), it follows that the error system (7) is globally exponentially stable at the equilibrium set 𝑒_𝑖=0, 𝑖=1,2,…,𝑁, with the globally exponential rate 𝜆. Consequently, the synchronous solution 𝑆(𝑡) of the dynamical network (2) is globally exponential stable. Then the dynamical network (2) is said to realize globally exponential synchronization under the controllers (28) and updating laws (29). The proof is thus completed.

Remark 9. In this paper, the coupling scheme of dynamical network (2) is a linear relationship. If the network coupling scheme is general nonlinear relationship, the network (2) is rewritten as follows: 𝑥̇_𝑖=𝑓𝑥_𝑖−𝜎^𝑁_𝑗=1𝑔_𝑖𝑗(𝑡)ℎ𝑥_𝑗+𝑢_𝑖. (36) Suppose that ‖ℎ(𝑥_𝑗)‖<𝜂 (𝜂 is a nonnegative constant) hold. Then, the synchronous solution 𝑆(𝑡) of moving agent network (36) is globally exponentially stable under the adaptive controllers (28) and updated laws (29) by the similar proof.

4. Simulation

In this section, one example is given for illustrating the proposed synchronization criteria. Consider a dynamical network consisting of 5 identical Rössler oscillators, where state dynamics of each agent is described by 𝑥̇_𝑖1𝑥̇_𝑖2𝑥̇_𝑖3𝑥=𝐵𝑥^𝑖1_𝑖2𝑥_𝑖3+00𝑥_𝑖1𝑥_𝑖3+𝑏, (37) where 𝐵=^{0−1−11𝑎0}_00−𝑐, 𝑥_𝑖=(𝑥_𝑖1𝑥_𝑖2𝑥_𝑖3)^𝑇. The following parameters have been used: 𝑎=0.2, 𝑏=0.2, 𝑐=7.

Each agent node interacts at a given time with only those agents located within a neighborhood of an interaction radius. Here, we let periodic boundary conditions size 𝐿=60. The initial position 𝑦_𝑖(0) of agent 𝑖 in the plane is chosen at random. The initial orientation 𝜃_𝑖(0)=0, other time units being chosen at each time unit with uniform probability in the interval [0,2𝜋]. Each agent is moving 40 time unit (𝑇=40). Then the position of each agent during the movement is shown in Figure 1.

Figure 1 

The position each agent during the movement.

When two agents interact (let interaction radius 𝑅=30), the state equations of each agent are changed to include diffusive coupling with the neighboring agent, acting on the state variable 𝑥_𝑖1. Based on these assumptions, the state dynamics of each agent can be described in terms of the following equations: 𝑥̇_𝑖=𝑓𝑥_𝑖−𝜎^𝑁_𝑗=1𝑔_𝑖𝑗(𝑡)𝐸𝑥_𝑗+𝑢_𝑖, (38) where 𝑖=1,…,5, 𝑓∶𝑅³→𝑅³ is given by the Rössler dynamics, 𝐸=¹⁰⁰⁰⁰⁰₀₀₀, 𝑔_𝑖𝑗(𝑡) are the elements of a time-varying matrix 𝐺(𝑡), and 𝑢_𝑖=−𝑑_𝑖𝑒_𝑖, 𝑑̇_𝑖=𝑒^2𝜆𝑡‖𝑒_𝑖‖².

Obviously, one gets ℎ𝑥_𝑖=00𝑥,𝑠,𝑡_𝑖1𝑥_𝑖3−𝑠₁𝑠₃. (39) Similar to [23–25], since Rössler chaotic system has a chaotic attractor which is confined to abounded region 𝜙⊂𝑅^𝑛, there exists a constant 𝑀 satisfying |𝑥_𝑖𝑗|,|𝑠_𝑗|≤𝑀 for 𝑖=1,2,…,𝑁 and 𝑗=1,2,3. Therefore, ℎ𝑥_𝑖,𝑠,𝑡=𝑥_𝑖1𝑥_𝑖3−𝑠₁𝑠₃²=𝑒_𝑖1𝑥_𝑖3−𝑠₁𝑒_𝑖3²≤√2𝑀𝑒_𝑖. (40)𝑀 can be got from the method similar to [24]. Thus, Assumption 7 holds.

Assume that 𝑣_𝑖(𝑡)=100, Δ𝑡_𝑀=0.1 to guarantee the fast-switching condition Assumption 3. Thus, Assumptions 3 and 7 hold. According to Theorem 8, the synchronous solution 𝑠(𝑡) of dynamical moving agent network (38) is globally exponentially stable. The other parameters are assigned as follows: 𝜆=1, 𝑑_𝑖(0)=1, 𝑥_𝑖(0)=(2+0.5𝑖,3+0.5𝑖,4+0.5𝑖), and 𝜎=1. The synchronous error 𝑒_𝑖 is shown in Figures 2, 3, and 4. We learn from these figures that the synchronization errors can be globally exponentially stable for dynamical network (38). Therefore, we conclude that the states of the networks (38) can be globally exponentially synchronized with the state of each isolated Rössler system.

Figure 2 

Synchronization errors of 𝑒𝑖1 for the network.

Figure 3 

Synchronization errors of 𝑒𝑖2 for the network.

Figure 4 

Synchronization errors of 𝑒𝑖3 for the network.

5. Conclusions

Locally and globally adaptive exponential synchronization of moving agent network has been investigated in this paper. The network with decentralized controllers is considered as a large-scale nonlinear system with time-varying topological structure. An adequate Lyapunov function is constructed to deal with the problem of controlled synchronization so as to ensure the closed-loop system stability. Several network synchronization criteria for such network with time-varying topological have been obtained. And a numerical simulation of coupled Rössler system network is given, which demonstrates the effectiveness of the proposed synchronization scheme.

Acknowledgments

This research was supported in part by the Natural Science Foundation of Hebei under Grant no. F2012501030, Fundamental Research Funds for the Central Universities under Grant no. N100323012 from the Ministry of Education, and the National Natural Science Foundation of China under Grant no. 51105068.