#### Abstract

This paper investigates the adaptive consensus for networked mobile agents with heterogeneous nonlinear dynamics. Using tools from matrix, graph, and Lyapunov stability theories, sufficient consensus conditions are obtained under adaptive control protocols for both first-order and second-order cases. We design an adaptive strategy on the coupling strengths, which can guarantee that the consensus conditions do not require any global information except a connection assumption. The obtained results are also extended to networked mobile agents with identical nonlinear dynamics via adaptive pinning control. Finally, numerical simulations are presented to illustrate the theoretical findings.

#### 1. Introduction

In recent years, distributed cooperative control for multiagent systems has been intensively investigated by researchers from various disciplines. This is due to its broad potential applications in sensor networks, combat intelligence, surveillance, and so forth [1, 2]. Various scenarios about distributed cooperative control are studied, such as leaderless consensus [3–5], leader-following consensus [6–8], containment [9–13], and flocking [14, 15].

Consensus is one of the most fundamental problems in distributed cooperative control, which means that the states of the agents reach an agreement on a common physical quantity of interest by implementing an appropriate consensus protocol based on the information from local neighbors. Numerous interesting results about consensus algorithm were presented in the past decade. The earlier studies on consensus problem are mainly about multiagent systems with first-order dynamics [16–18]. And consensus for multiagent systems with second-order dynamics has been investigated recently [19–21], which is more challenging than the first-order case. Meanwhile, consensus for linear multiagent systems is discussed in [22, 23], which is considered more general than first-order and second-order multiagent systems.

However, in reality, mobile agents may be governed by more complicated intrinsic dynamics, so second-order consensus problems for multiagent with nonlinear dynamics have been investigated [24–33], in which the authors assume that all the agents have identical nonlinear dynamics. Recently, second-order consensus of multiagent systems with heterogeneous nonlinear dynamics and time-varying delay was investigated by introducing novel decentralized adaptive control in [28]. In [29], a distributed cooperative tracking problem was studied for a group of second-order multiagents networked system with nonidentical nonlinear dynamics and bounded external disturbances. Impulsive consensus for first-order multiagent with heterogeneous nonlinear dynamics was discussed [30], in which the information of the leader is known by every following agent. This assumption is also adopted in [31]. Motivated by [28–33], we investigate the second-order consensus for networked mobile agents with heterogeneous nonlinear dynamics without assuming that all the nonlinear functions have a common equilibrium solution. A distributed adaptive strategy and a pinning-like control is introduced in the consensus protocol. The protocol is composed of two parts. One is designed to compensate the errors of the nonlinear dynamics and the other one is designed like a pinning controller.

The rest of the paper is organized as follows. In Section 2, we state the model considered in the paper and give some basic definitions, lemmas, and assumptions. In Section 3, new results on the second-order consensus problem are addressed. And numerical example is given in Section 4. Finally, we conclude the paper in Section 5.

#### 2. Preliminaries and Model Description

In this section, some notations and preliminaries are introduced. The following notations are used throughout this paper. denotes the identity matrix. For a matrix (or a vector ), (or ) represents the transpose of (or ).

Let be a directed graph with a nonempty set of nodes , a set of edges , and a weighted adjacent matrix . An edge is denoted by in a directed graph which means that vertex can obtain information from vertex but not necessarily vice versa. represents the weight of the edge and . Node is called the parent node, node is the child node, and is a neighbor of . The neighbor set . A graph is undirected if implies . The elements of Laplace matrix is defined as , , and .

Before moving on, some assumptions and Lemmas are introduced.

*Assumption 1. *Suppose that the undirected graph is connected.

Lemma 2 (see [33]). *If is a symmetric irreducible matrix with , , then is semipositive definite and then for any matrix with , all eigenvalues of the matrix are positive.*

Lemma 3 (see [34]). *The matrix of an undirected graph is irreducible if and only if the undirected graph is connected.*

Lemma 4 (see [35]). *If a scalar function satisfies the following conditions, then , as . (a) is lower bounded; (b) is negative semidefinite; (c) is uniformly continuous in .*

#### 3. Main Results

In this section, consensus conditions for both first-order and second-order multiagent systems with heterogeneous nonlinear dynamics are obtained by designing pinning-like adaptive control protocols. Pinning-like adaptive consensus for first-order multiagent systems with heterogeneous nonlinear dynamics is investigated in the first subsection and the second-order case is investigated in the second subsection. For the first-order case, the th node can be described as where , , is the state vector of the th node. is a continuous vector-value function, which describes the local dynamics of the nodes in the th node.

For the second-order case, the th node can be described as where , are the position, velocity, and input vector of the th follower, respectively. is a continuous vector-value function, which describes the local dynamics of the nodes in the th node.

##### 3.1. Pinning-Like Adaptive Consensus for First-Order Multiagent Systems with Heterogeneous Nonlinear Dynamics

In this subsection, we investigate consensus criteria for first-order multiagent systems with heterogeneous nonlinear dynamics via distributed adaptive pinning control.

As mentioned in many literatures, consensus or synchronization for networked systems with nonidentical nodes cannot be realized without control if the nonidentical dynamic functions do not have a common solution. Thereby, a distributed adaptive pinning-like control protocol is proposed, under which leader-following consensus for networked system (1) can be achieved; that is, there exists a desired state , such that where satisfies , and can be an equilibrium, a limit cycle, or even a chaotic attractor.

The following assumption is necessary for our main results.

*Assumption 5. *For arbitrary and every , there exists a constant such that
We choose the control protocol as
In the controller (4), for some , then the th node is named as pinning-like node and for others. and are determined by the following equations:
where , are the weights of adaptive laws for parameters and . denotes the coupling strength between node and node . We define a new matrix, named adaptive weighted Laplace matrix of the network topology, as the following:
The elements is defined as , , , and is defined as , so is zero-row-sum. The initial values of the elements of should be chosen symmetrically for the symmetry of .

*Remark 6. *From Lemmas 2 and 3, if the undirected topology graph is connected, the adaptive weighted Laplace matrix is semipositive definite.

The controlled network for system (1) under protocol (5) with adaptive strategy (6) can be rewritten as Let error state be defined as From (1) and ((8)-(9)), the error systems can be described as Noting that is zero-row-sum, (10) can be rewritten as

Theorem 7. *Suppose that Assumptions 1 and 5 hold. Then consensus for multiagent system (1) can be achieved under the control protocol (5) with adaptive strategy (6) if there exists at least one pinned-like node in the network.*

*Proof. *Let , according to Lemma 2, is positive definite. Consider the following Lyapunov function candidate:
where , and denotes the minimum eigenvalue of .

Differentiating with respect to along (11), we have
Noting that is symmetric and zero-row-sum, we have
According to (14), we have
So we have
From ((14)–(16)), one can easily conclude the following:
From (13), (16), and (17), we have
Since , is negative definite, which means . This implies that , for all . Therefore, , , and are bounded. Noting that and are dependent on and is bounded, then, from Assumption 1, (13), and (18), one can know that is bounded. This means that is uniformly continuous in . Hence, according to Lemma 4, as . Again noting that is negative definite, one can conclude that , as , which means that the consensus of networked system (1) is achieved.

*Remark 8. *From Theorem 7, we can know all the nodes in network (1) will asymptotically converge to the desired node . When all the nonlinear dynamic have a common solution, that is, there exists a vector , satisfied , we can choose as the desired node. And if all the nonlinear dynamic do not have a common solution, we can choose as the desired node. And we also can obtain the distributed consensus protocol for nonlinear multiagent systems with identical nodes. Considering the network with identical nodes, the th node of which is described as
We choose the control protocol as
where the desired state satisfies and the adaptive strategy is designed as (6).

According to Theorem 7, we have the following corollary.

Corollary 9. *Suppose that the topology is undirected and connected and satisfies Lipschitz condition with a Lipschitz constant . Then, under the control protocol (20) with adaptive strategy (6), consensus for multiagent system (19) can be achieved if there exists at least one pinned node in the network.*

##### 3.2. Pinning-Like Adaptive Consensus for Second-Order Multiagent Systems with Heterogeneous Nonlinear Dynamics

In this subsection, we investigate second-order consensus of networked nonlinear multiagent system (2) via pinning-like adaptive control.

*Assumption 10. *For every in (2) and arbitrary , there exist corresponding constants such that

*Remark 11. *Choosing in [24], we can get the well-definition of Assumption 10. In fact, this condition can be satisfied by many systems. Examples include the pendulum system with a control torque, car-like robots, the Chuas circuit, the Lorenz system, and the Chen system and so forth [27].

Let
Synchronization or consensus for network with nonidentical nodes cannot be realized without control if the nonidentical dynamic functions do not have a common equilibrium. The purpose of this paper is to design distributed tracking consensus algorithm for networked system (1). Therefore, a distributed adaptive control protocol is proposed, under which leader-following second-order consensus for network (1) can be achieved; that is, there exists a desired state , satisfying , , such that
The following consensus protocol is designed for networked system (2):
In the controller (3), for some , then the th node is named as pinning-like node and for others. The adaptive weights , , are determined by the following equations:
where , are the weights of adaptive laws for parameters and .

Under consensus protocol (24), the error systems can be rewritten as
The following theorem addresses a sufficient condition for the consensus of networked system (2).

Theorem 12. *Suppose that Assumptions 1 and 5 hold. Then consensus of heterogeneous nonlinear networked system (2) can be achieved under the control protocol (24) with adaptive strategy (25) if there exists at least one pinned-like node in the network.*

*Proof. *Let , . According to [19, Theorem ], for an undirected graph , the Laplace matrix is irreducible if and only if is connected. Noting that the corresponding graph of has the same connectivity with , is irreducible under Assumption 5. Then one can conclude that is semipositive definite according to Lemma 2. And then by virtue of* Schur* Complement Lemma, we can know that is positive definite.

Consider the following Lyapunov function candidate:
where , , and denotes the minimum eigenvalue of .

Differentiating with respect to along (28), we have
Noting that is symmetric and zero-row-sum, similarly as (16), we have
From ((11)–(13)), we have
According to Lemma 2, is positive definite. Since , is negative definite, which means . The same to Theorem 7, by in virtue of Lemma 4, one can conclude that and , as , which means that the consensus of networked system (2) is achieved.

*Remark 13. *From Theorem 12, we can know that all the nodes in network (2) will asymptotically converge to the desired node . When all the nonlinear dynamic have a common solution, we can choose it as the desired node. And if all the nonlinear dynamics do not have a common solution, we can choose average state as the desired node. And we also can obtain the pinning adaptive consensus control protocol for nonlinear mobile agents networked system with identical nonlinear dynamics.

When all the nonlinear function , are identical, that is, the th node in system (1) can be described as
We choose the control protocol as
In the protocol, the adaptive weights , , , are determined by (25). Then, from Theorem 12, one can easily obtain the following corollary.

Corollary 14. *Suppose that Assumptions 1 and 5 hold. Then consensus for nonlinear mobile multiagent networked system (31) can be achieved under the control protocol (32) with adaptive strategy (25) if there exists at least one pinned node in the network.*

#### 4. Simulations

In this section, several simulation results are presented to illustrate the previous theoretical results.

*Example 1. *We choose a network with 4 nodes and the reference node is chosen as . The nonlinear dynamic functions are chosen as , , and nonlinear coupled function . The topology is described by the following adjacent matrix:

In Figure 1, the dash dotted line denotes the state of the reference node. And other lines represent the trajectories of the nodes in the network. The trajectories of the adaptive gains are drawn in Figure 2. According to the simulation, we can know that the consensus for the proposed network can be achieved rapidly, which results in that the adaptive gains are stabilized rapidly.

*Example 2. *The topology is chosen the same as Example 1 and the reference node is chosen as

The nonlinear dynamic functions are chosen as , . We choose the first node as the pinning-like node. In Figures 3 and 4, the dash dotted line denotes the position trajectory and the velocity trajectory of the reference node. And other lines represent the trajectories of the nodes in the network. The trajectories of the adaptive gains are drawn in Figure 5. According to Figures 3 and 4, the consensus can be achieved.

#### 5. Conclusions

Adaptive consensus for networked mobile agents with heterogeneous nonlinear dynamics was investigated in this paper. Sufficient consensus conditions for both first-order and second-order networked mobile agents with heterogeneous nonlinear dynamics were obtained. By designing an adaptive strategy on the coupling strengths, the consensus can be achieved without requiring any global information except a connection assumption. We also extended the results to the consensus for nonlinear mobile agents with identical nonlinear dynamics. Simulation examples were given to demonstrate the feasibility and effectiveness of the proposed consensus scheme.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work is supported by National Natural Science Foundation (NNSF) of China under Grant no. 61174075, 51375186, Natural Science Foundation of Ministry of Education in Hunan Province (12C0077), and NSF of Hunan University of Technology (2012HZX18).