Delay-Distribution-Dependent Consensus for Second-Order Leader-Follower Nonlinear Multiagent Systems via Pinning Control

Abstract

This paper investigates the consensus problem for second-order leader-follower nonlinear multiagent systems with general network topologies. A pinning control algorithm is proposed, where it includes time-varying delay information. By using the information of delay-partition and delay-distribution and constructing an appropriate Lyapunov-Krasovskii functional, the consensus criteria are derived to achieve leader-follower consensus for multiagent systems, which are in the form of linear inequalities that can be solved by employing the semidefinite programme method. Moreover, this paper addresses what kind of agents and how many agents should be pinned. Two numerical examples are presented to further demonstrate the effectiveness of the proposed approach.

1. Introduction

The coordinated and cooperative control of multiple autonomous vehicles is a problem which has been much researched in recent years due to its applications in the formation control of unmanned air vehicles, the cooperative control of mobile robots, the design of distributed sensor networks, and so on. Particularly, consensus in multiagent systems, which means all the agents can reach an agreement on certain concern or interest, has been extensively studied in the past few years [1–8].

A particularly interesting topic is the consensus of a group of agents with a leader, where the leader is a special agent whose motion is independent of all the other agents and thus is followed by all the other ones, and such a problem is commonly called leader-following consensus problem [9–13]. Leader-following consensus problem is investigated in [9], and the corresponding conclusion is that if all the agents are jointly connected with their leader, their states would converge to that of the leader. Reference [11] provides a rigorous proof for the consensus using an extension of LaSalle’s invariance principle. A neighbor-based observer design approach is proposed in [12] and discusses a group of mobile agents with an active leader moving with an unknown velocity. In [13], the leader-following consensus problem is considered by designing distributed controllers using local information to ensure that all the agents follow the leader. It is well known that time delay often causes undesirable dynamic behaviors such as oscillation, performance degradation, and instability of the network [14–16]; therefore, various approaches to consensus analysis for multiagent systems with time delay have been investigated in the literature [17, 18]. In [17], two different cases of coupling topologies are investigated for coordination problem of multiagent system with time-varying coupling delays. Reference [18] studies the leader-following consensus for a multiagent system with a varying-velocity leader and time-varying delays. However, less consideration has been paid to the delay-dependent consensus conditions for multiagent systems with time-varying delays. It is noted that all physical systems are nonlinear in nature [19] and that even different agents may have different dynamics. It is very challenging task to discuss consensus problem of nonlinear multiagent systems [20, 21]. Reference [21] proposes leader-following consensus algorithms based on pinning control for second-order nonlinear multiagent systems and provides a pinned-agent selection scheme to determine what kind of followers and how many followers should be pinned. Considering that it is practically impossible to apply control actions to all agents in a large-scale multiagent system, and some authors develop consensus algorithms based on pinning control [22–25]. It is well known that pinning control is an effective method to solve the problem of nonlinear system, it would be beneficial to apply these control techniques to study the consensus problem of multiagent system. In many practical systems, such as networked control systems, the probability distribution of the time delay in the interval is an important characteristic for the network conditions, and its probability distribution can be measured by the statistical method [26–28]. Therefore, the information of probability distribution of the delay should be employed in the model. To the best of the authors’ knowledge, no result has been reported for the consensus problem for second-order leader-follower nonlinear multiagent systems via pinning control, when both the information of variation range of the time delay and the information of variation probability of the time delay in an interval can be observed, which motivates the present study.

Motivated by the above discussions, in this paper, we investigate the consensus problem for second-order leader-follower nonlinear multiagent systems. Based on the probability distribution of delay, a new type of multiagent system model with stochastic parameters in the coefficient matrices is proposed. A pinning control algorithm is proposed, and the delay-partition-dependent and delay-distribution-dependent consensus criteria are derived by combining delay-partitioning method and Lyapunov-Krasovskii functional method. Moreover, this paper addresses what kind of agents and how many agents should be pinned.

The main contributions of this paper can be summarized as follows. (1) Based on the probability distribution of delay, a novel multiagent system model with stochastic parameters in the coefficient matrices is introduced. (2) The new pinning control algorithm is proposed, and this paper gives what kind of agents and how many agents should be pinned.

The rest of this paper is organized as follows. In Section 2, problem formulation and preliminaries are briefly outlined. In Section 3, the consensus criteria are derived to achieve leader-follower consensus for multiagent systems. In Section 4, two simulation examples are provided to show the advantages of the obtained results, and some conclusions are drawn in Section 5.

Notation. The notation used in the paper is fairly standard. ℝ^𝑛 denotes the 𝑛-dimensional Euclidean space, and ℝ^𝑛×𝑚 is a set of real 𝑛×𝑚 matrices. The notation 𝑋>0 (resp., 𝑋<0), for 𝑋∈ℝ^𝑛×𝑛, means that the matrix 𝑋 is real symmetric positive definite (resp., negative definite). diag{⋯} stands for a block-diagonal matrix. ‖⋅‖ denotes the Euclidean norm in ℝ^𝑛. The superscript “𝑇” stands for matrix transposition. 𝐸{⋅} stands for mathematical expectation. The Kronecker product of matrices 𝑄∈ℝ^𝑚×𝑛 and 𝑅∈ℝ^𝑝×𝑞 is a matrix in ℝ^{𝑚𝑝×𝑛𝑞} and denoted as  𝑄⊗𝑅. In this paper, if not explicitly stated, matrices are assumed to have compatible dimensions.

2. Preliminaries

Let 𝑔={𝑣,𝜀,𝐴} be a weighted diagraph of order 𝑁, with the set of nodes 𝑣={𝑣₁,𝑣₂,…,𝑣_𝑁}, an edge set 𝜀⊆𝑣×𝑣, and a weighted adjacency matrix 𝐴=(𝑎_𝑖𝑗)_𝑁×𝑁 with nonnegative elements, directed edge 𝑒_𝑖𝑗 in network 𝑔, is denoted by the ordered pair of nodes (𝑣_𝑖,𝑣_𝑗), which means that node 𝑣_𝑗 can receive information from node 𝑣_𝑖. The elements of the adjacency matrix 𝐴 are defined as 𝑎_𝑖𝑗>0 if and only if there is a directed edge (𝑣_𝑗,𝑣_𝑖) in 𝑔; otherwise 𝑎_𝑖𝑗=0. We assume that 𝑎_𝑖𝑖=0 for all 𝑖∈𝑣. The neighbor set of node 𝑖 is defined by 𝑁_𝑖={𝑗∈𝑣∣(𝑣_𝑗,𝑣_𝑖)∈𝜀}, and the in-degree and out-degree of node 𝑖 are defined as deg_in(𝑖)=^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗,deg_out(𝑖)=^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑗𝑖. (1) A diagraph is called balanced if deg_in(𝑖)=deg_out(𝑖) for all 𝑖∈𝑣.

The Laplacian matrix 𝐿=(𝑙_𝑖𝑗)_𝑁×𝑁 associated with the adjacency matrix 𝐴 is defined as 𝑙_𝑖𝑗=−𝑎_𝑖𝑗𝑙𝑖≠𝑗_𝑖𝑖=−^𝑁_{𝑗=1,𝑗≠𝑖}𝑙_𝑖𝑗𝑖,𝑗=1,2,…,𝑁 (2) which ensures the diffusion property that ∑^𝑁_𝑗=1𝑙_𝑖𝑗=0.

A directed path from node 𝑣_𝑖 to 𝑣_𝑗 in 𝑔 is a sequence of edges ((𝑣_𝑖,𝑣_𝑖1),(𝑣_𝑖1,𝑣_𝑖2),…,(𝑣_𝑖𝑙,𝑣_𝑗)) in the directed graph which is strongly connected if for any two distinct nodes 𝑣_𝑗 and 𝑣_𝑖, there exists a directed path form node 𝑣_𝑗 to 𝑣_𝑖. A directed graph has a directed spanning tree if there exists at least one node called root which has a directed path to all other nodes.

The dynamics of the virtual leader is described by the following second-order nonlinear system: 𝑥̇_𝑟(𝑡)=𝑣_𝑟(𝑡),𝑣̇_𝑟(𝑡)=𝑓𝑡,𝑥_𝑟(𝑡),𝑣_𝑟(𝑡), (3) where 𝑥_𝑟(𝑡)=(𝑥_𝑟1(𝑡),𝑥_𝑟2(𝑡),…,𝑥_𝑟𝑛(𝑡))^𝑇∈ℝ^𝑛 and 𝑣_𝑟(𝑡)=(𝑣_𝑟1(𝑡),𝑣_𝑟2(𝑡),…,𝑣_𝑟𝑛(𝑡))^𝑇∈ℝ^𝑛 are the position and velocity states of the virtual leader, respectively. 𝑓(𝑡,𝑥_𝑟(𝑡),𝑣_𝑟(𝑡)) = (𝑓₁(𝑡,𝑥_𝑟(𝑡),𝑣_𝑟(𝑡)),𝑓₂(𝑡,𝑥_𝑟(𝑡),𝑣_𝑟(𝑡)),…,𝑓_𝑛(𝑡,𝑥_𝑟(𝑡),𝑣_𝑟(𝑡)))^𝑇 is a nonlinear vector-valued continuous bounded function.

The second-order dynamics of multiagent systems consisting of 𝑁-coupled autonomous agents is given as follows: 𝑥̇_𝑖(𝑡)=𝑣_𝑖(𝑡)+𝑢_1𝑖(𝑡),𝑣̇_𝑖(𝑡)=𝑓𝑡,𝑥_𝑖(𝑡),𝑣_𝑖(𝑡)+𝑢_2𝑖(𝑡)(𝑖=1,2,…,𝑁), (4) where 𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡))^𝑇∈ℝ^𝑛 and 𝑣_𝑖(𝑡)=(𝑣_𝑖1(𝑡),𝑣_𝑖2(𝑡),…,𝑣_𝑖𝑛(𝑡))^𝑇∈ℝ^𝑛 are the position and velocity states of agent 𝑖, respectively. 𝑢_1𝑖(𝑡) and 𝑢_2𝑖(𝑡) are the control inputs for agent 𝑖 to be designed.

Definition 1. The multiagent systems (4) are said to achieve second-order leader-following consensus if, for any initial conditions, the state of each following agent 𝑖(𝑖=1,2,…,𝑁) satisfies lim_𝑡→∞𝑥_𝑖(𝑡)−𝑥_𝑟(𝑡)=0lim_𝑡→∞𝑣_𝑖(𝑡)−𝑣_𝑟(𝑡)=0. (5)

Motivated by the coupling rule used in [17], the consensus algorithm is designed based on pinning control such that all agents in nonlinear multiagent systems (4) can asymptotically follow the virtual leader (3). Without loss of generality, rearrange the order of all agents, and the sort method will be given in Section 3. Let the first 𝑙(1≤𝑙<𝑁) agents in multiagent systems (4) be controlled, and the controllers have the following form: 𝑢_1𝑖(𝑡)=−_{𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡+𝑥_𝑖)(𝑡−𝜏(𝑡))−𝑥_𝑗(𝑡−𝜏(𝑡))−𝛿_𝑖𝑥_𝑖(𝑡)−𝑥_𝑟(𝑡)+𝑥_𝑖(𝑡−𝜏(𝑡))−𝑥_𝑟((𝑡𝑢𝑡−𝜏))_2𝑖(𝑡)=−_{𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑣_𝑖(𝑡)−𝑣_𝑗(𝑡)+𝑣_𝑖(𝑡−𝜏(𝑡))−𝑣_𝑗(𝑡−𝜏(𝑡))−𝛿_𝑖𝑣_𝑖(𝑡)−𝑣_𝑟(𝑡)+𝑣_𝑖(𝑡−𝜏(𝑡))−𝑣_𝑟(𝑡−𝜏(𝑡)), (6) where the local feedback gains satisfy 𝛿_𝑖>0(𝑖=1,2,…,𝑙) and 𝛿_𝑖=0(𝑖=𝑙+1,𝑙+2,…,𝑁). Also 𝜏(𝑡)∈[0,𝜏] is the bounded time-varying delay, and 𝜏 is a positive scalar.

For notational convenience, we define 𝑥(𝑡)=𝑥^𝑇₁(𝑡),𝑥^𝑇₂(𝑡),…,𝑥^𝑇_𝑁(𝑡)^𝑇∈ℝ^𝑁𝑛,𝑣(𝑡)=𝑣^𝑇₁(𝑡),𝑣^𝑇₂(𝑡),…,𝑣^𝑇_𝑁(𝑡)^𝑇∈ℝ^𝑁𝑛,1_𝑁=(1,1,…,1)^𝑇∈ℝ^𝑁,𝐹((𝑡)(𝑡=𝑓𝑡,𝑥,𝑣))_𝑇𝑡,𝑥₁(𝑡),𝑣₁(𝑡),𝑓^𝑇𝑡,𝑥₂(𝑡),𝑣₂(𝑡)𝑓,…,_𝑇𝑡,𝑥_𝑁(𝑡),𝑣_𝑁(𝑡)^𝑇∈ℝ^𝑁𝑛𝛿=diag𝛿₁,𝛿₂,…,𝛿_𝑙,0,…,0∈ℝ^𝑁×𝑁. (7)

Let 𝜉(𝑡)=𝑥(𝑡)−1_𝑁⊗𝑥_𝑟(𝑡), 𝜂(𝑡)=𝑣(𝑡)−1_𝑁⊗𝑣_𝑟(𝑡). Combining (3) and (4), one has 𝜉̇(𝑡)=𝑥̇(𝑡)−1_𝑁⊗𝑥̇_𝑟(𝑡)=𝑣(𝑡)−(𝐿+𝛿)⊗𝐼_𝑛(𝜉(𝑡)+𝜉(𝑡−𝜏(𝑡)))−1_𝑁⊗𝑣_𝑟(𝑡)=𝜂(𝑡)−(𝐿+𝛿)⊗𝐼_𝑛𝜉(𝑡)((𝑡,𝜂̇(𝑡)=𝑣̇(𝑡)(+𝜉𝑡−𝜏)))−1_𝑁⊗𝑣̇_𝑟(𝑡)=𝐹(𝑡,𝑥(𝑡),𝑣(𝑡))−1_𝑁⊗𝑓𝑡,𝑥_𝑟(𝑡),𝑣_𝑟(𝑡)−(𝐿+𝛿)⊗𝐼_𝑛𝜂(𝑡)+𝜂(𝑡−𝜏(𝑡)). (8)

Based on (8), the augment system can be obtained as follows: 𝜉̇(𝑡)𝜂̇(𝑡)^𝑇=−(𝐿+𝛿)⊗𝐼_𝑛𝐼_𝑁𝑛0−(𝐿+𝛿)⊗𝐼_𝑛𝜉(𝑡)𝜂(𝑡)+−(𝐿+𝛿)⊗𝐼_𝑛00−(𝐿+𝛿)⊗𝐼_𝑛𝜉((𝑡𝜂((𝑡+0𝐹((𝑡)(𝑡𝑡−𝜏))𝑡−𝜏))𝑡,𝑥,𝑣))−1_𝑁⊗𝑓𝑡,𝑥_𝑟(𝑡),𝑣_𝑟(𝑡). (9)

Setting 𝑒(𝑡)=𝜉(𝑡)𝜂(𝑡)−()𝐼𝐴=𝐿+𝛿_𝑁()00−𝐿+𝛿𝐹=𝐹𝐴_𝜏=−()0()𝐿+𝛿0−𝐿+𝛿𝐹=𝐹(𝑡,𝑥(𝑡),𝑣(𝑡))−1_𝑁⊗𝑓𝑡,𝑥_𝑟(𝑡),𝑣_𝑟(𝑡), (10) then the system (9) can be rewritten in the following compact form: 𝑒̇(𝑡)=𝐴⊗𝐼_𝑛𝑒(𝑡)+𝐴_𝜏⊗𝐼_𝑛𝑒(𝑡−𝜏(𝑡))+𝐹. (11)

Remark 2. In many practical systems, such as networked control systems, the probability distribution of the delay in the interval is an important characteristic. Inspired by [26–28], in this paper, it is assumed that the probability of the delay appearing in some intervals can be observed. Then new multiagent systems with stochastic time-varying delays can be proposed.

For a given positive integer 𝑀 and a partition 0=𝜏₀<𝜏₁<𝜏₂⋯<𝜏_𝑀=𝜏, we assume that the probabilities of 𝜏(𝑡)∈[𝜏_𝑖−1,𝜏_𝑖)(𝑖=1,2,…,𝑁) are known. Define the set Ω_𝑖=𝑡∶𝜏(𝑡)∈𝜏_𝑖−1,𝜏_𝑖(𝑖=1,2,…,𝑀). (12)

It can easily be seen that ⋃Ω_𝑖=𝑅⁺, Ω_𝑖⋂Ω_𝑗=𝜙(𝑖≠𝑗). Define 𝜏_𝑖(𝑡)=𝜏(𝑡)𝑡∈Ω_𝑖0𝑡∈Ω_𝑖𝛽_𝑖(𝑡)=1𝑡∈Ω_𝑖0𝑡∈Ω_𝑖. (13)

Assumption 3. Stochastic variable 𝛽(𝑡)∶=[𝛽₁(𝑡),𝛽₂(𝑡),…,𝛽_𝑁(𝑡)]^𝑇 is a distributed sequence with Prob𝛽_𝑖(𝑡)=1=𝐸𝛽_𝑖(𝑡)=𝛽_𝑖^𝑀_𝑖=1𝛽_𝑖=1. (14)

Remark 4. By some computations, we derived from Assumption 3 that 𝐸𝛽_𝑖(𝑡)−𝛽_𝑖²=𝛽_𝑖1−𝛽_𝑖𝐸𝛽_𝑖(𝑡)−𝛽_𝑖()𝐸𝛽=0𝑖=1,2,…,𝑀_𝑖(𝑡)−𝛽_𝑖𝛽_𝑗(𝑡)−𝛽_𝑗=−𝛽_𝑖𝛽_𝑗𝑖,𝑗=1,2,…,𝑀;𝑖≠𝑗. (15)

By using the new functions 𝛽_𝑖(𝑡)(𝑖=1,2,…,𝑀), system (11) can be rewritten as 𝑒̇(𝑡)=𝒜+^𝑀−1_𝑖=1𝛽_𝑖(𝑡)−𝛽_𝑖𝒜_𝜏𝑖𝜉(𝑡)+𝐹, (16) where 𝒜=𝐴,𝛽₁𝐴_𝜏,0,𝛽₂𝐴_𝜏,0,…,0,𝛽_𝑀−1𝐴_𝜏,0,𝛽_𝑀𝐴_𝜏⊗𝐼_𝑛𝒜_𝜏𝑖=0_{𝑁,(2𝑖−1)𝑁},𝐴_𝜏,0_{𝑁,(2𝑀−2𝑖−1)𝑁},−𝐴_𝜏,𝜉^𝑇(𝑡)=𝑒^𝑇(𝑡),𝑒^𝑇𝑡−𝜏₁(𝑡),𝑒^𝑇𝑡−𝜏₁,𝑒^𝑇𝑡−𝜏₂(𝑡),…,𝑒^𝑇𝑡−𝜏_𝑀−1,𝑒^𝑇𝑡−𝜏_𝑀(𝑡). (17)

Let 𝐶([−𝜏,0];ℝ^𝑛) denote the family of continuous functions 𝜑 from [−𝜏,0] to ℝ^𝑛. It is not difficult to show that, given any 𝜑∈𝐶([−𝜏,0];ℝ^𝑛), the system (16) has a unique continuous solution 𝑒(𝑡) on 𝑡≥−𝜏. Clearly, 𝑒(𝑡) is a continuous ℝ^𝑛-valued stochastic process on 𝑡∈[−𝜏,0]. Let 𝑒_𝑡={𝑒(𝑡+𝜃)∶−𝜏≤𝜃≤0} for 𝑡≥0. Then {𝑒_𝑡} is a 𝐶([−𝜏,0];ℝ^𝑛)-valued Markov process. Its infinitesimal operator 𝐿, acting on functional 𝑉∶𝐶([−𝜏,0];ℝ^𝑛)×[0,+∞)→ℝ, is defined by 𝐿𝑉𝑒_𝑡,𝑡=lim_Δ→0+1Δ𝐸𝑉𝑒_{𝑡+Δ,𝑡+Δ}∣𝑒_𝑡−𝑉𝑒_𝑡,𝑡. (18)

The following lemmas and assumption are needed to derive our main results.

Lemma 5. For matrices 𝐴, 𝐵, 𝐶, and 𝐷 with appropriate dimensions and a positive scalar 𝛼, by the definition of Kronecker product, the following properties can be proved: ()(),()()=()⊗(),()𝛼𝐴⊗𝐵=𝐴⊗𝛼𝐵𝐴+𝐵⊗𝐶=𝐴⊗𝐶+𝐵⊗𝐶,𝐴⊗𝐵)(𝐶⊗𝐷𝐴𝐶𝐵𝐷𝐴⊗𝐵^𝑇=𝐴^𝑇⊗𝐵^𝑇. (19)

Lemma 6. Let 𝑅∈ℝ^𝑛×𝑛 satisfy 𝑅>0. If a ℝ^𝑛-valued function 𝑥(𝑡) is integral on the interval [𝑎,𝑏], then −^𝑏_𝑎𝑥^𝑇(𝑠)(𝑠)1𝑅𝑥𝑑𝑠≤−𝑏−𝑎^𝑏_𝑎𝑥^𝑇(𝑠)𝑑𝑠𝑅^𝑏_𝑎𝑥(s)𝑑𝑠. (20)

Assumption 7. For the nonlinear function 𝑓(𝑡,⋅,⋅) and for all 𝑥,𝑣,𝑦,𝑧∈ℝ^𝑛, there exist nonnegative constants 𝜌₁ and 𝜌₂, such that the following inequality holds: 𝑓(𝑡,𝑥,𝑣)−𝑓𝑡,𝑦,𝑧²≤𝜌₁𝑥−𝑦²+𝜌₂‖𝑣−𝑧‖². (21)

3. Main Results

Theorem 8. Suppose that Assumption 7 holds. Second-order leader-following consensus is achieved under the pinning controllers (6), if for some given positive scalars 𝜏_𝑖 and 𝛽_𝑖∈[0,1](𝑖=1,2,…,𝑀) there exist some positive matrices 𝐏=diag{𝑃,𝑃}, 𝐐_𝑖=diag{𝑄_𝑖,𝑄_𝑖}(𝑖=1,2,…,𝑀−1), 𝐑_𝑗=diag{𝑅_𝑗,𝑅_𝑗}(𝑗=1,2,…,𝑀+1) and a positive 𝛼>0 such that the following linear matrix inequality holds:ΦΦ=¹¹Φ∗∗∗⋯∗∗₂₁Φ₂₂∗∗⋯∗∗00Φ₃₃Φ∗⋯∗∗₄₁0Φ₄₃Φ₄₄Φ⋯∗∗⋮⋮⋮⋮⋱⋮⋮_2𝑀,1000⋯Φ_2𝑀,2𝑀∗0Φ_2𝑀+1,20Φ_2𝑀+1,4⋯Φ_{2𝑀+1,2𝑀}Φ_{2𝑀+1,2𝑀+1}<0, (22)where Φ₁₁=Ω₁+𝛼𝜌₁𝐼_𝑁𝑃−(𝐿+𝛿)^𝑇Δ𝑃−Δ(𝐿+𝛿)Ω₁+𝛼𝜌₁𝐼_𝑁,Φ₂₁=Ω₂−𝛽₁(𝐿+𝛿)^𝑇Δ0Ω₂,Φ₂₂=𝛽₁(𝐿+𝛿)^𝑇Δ()−𝑅𝐿+𝛿¹𝜏₁00𝛽₁(𝐿+𝛿)^𝑇Δ()−𝑅𝐿+𝛿¹𝜏₁,Φ₃₃=−𝑄₁−𝑅₂𝜏₂−𝜏₁00−𝑄₁−𝑅₂𝜏₂−𝜏₁,Φ₄₁=Ω₃−𝛽₂(𝐿+𝛿)^𝑇Δ0Ω₃,Φ₄₃=𝑅₂𝜏₂−𝜏₁00𝑅₂𝜏₂−𝜏₁,Φ₄₄=𝑅₂𝜏₂−𝜏₁+𝛽₂(𝐿+𝛿)^𝑇Δ()00𝑅𝐿+𝛿₂𝜏₂−𝜏₁+𝛽₂(𝐿+𝛿)^𝑇Δ(),Φ𝐿+𝛿_2𝑀,1=Ω₄−𝛽_𝑀(𝐿+𝛿)^𝑇Δ0Ω₄,Φ_2𝑀,2𝑀=Ω₅00Ω₅,Φ_2𝑀+1,2=()0(),Φ𝑃−Δ𝐿+𝛿0𝑃−Δ𝐿+𝛿_2𝑀+1,4=−𝛽₁Δ(𝐿+𝛿)00−𝛽₁Δ(),Φ𝐿+𝛿_{2𝑀+1,2𝑀}=−𝛽_𝑀Δ(𝐿+𝛿)00−𝛽_𝑀Δ(),Φ𝐿+𝛿_{2𝑀+1,2𝑀+1}=−𝛼𝐼_𝑁+Δ00−𝛼𝐼_𝑁,+ΔΔ=^𝑀_𝑖=1𝜏_𝑖−𝜏_𝑖−1𝑅_𝑖+𝜏_𝑀𝑅_𝑀+1,Ω₁=−𝑃(𝐿+𝛿)−(𝐿+𝛿)^𝑇𝑃+^𝑀−1_𝑖=1𝑄_𝑖−𝑅₁𝜏₁−𝑅_𝑀+1𝜏_𝑀+(𝐿+𝛿)^𝑇Δ(),Ω𝐿+𝛿₂=−𝛽₁(𝐿+𝛿)^𝑇𝑅𝑃+¹𝜏₁+𝛽₁(𝐿+𝛿)^𝑇Δ(),Ω𝐿+𝛿₃=−𝛽₂(𝐿+𝛿)^𝑇𝑃+𝛽₂(𝐿+𝛿)^𝑇Δ(),Ω𝐿+𝛿₄=−𝛽_𝑀(𝐿+𝛿)^𝑇𝑅𝑃+^𝑀+1𝜏_𝑀+𝛽_𝑀(𝐿+𝛿)^𝑇Δ(),Ω𝐿+𝛿₅𝑅=−^𝑀+1𝜏_𝑀−𝑅_𝑀𝜏_𝑀−𝜏_𝑀−1+𝛽_𝑀−2^𝑀−1_{𝑖,𝑗=1;𝑖≠𝑗}𝛽_𝑖𝛽_𝑗(𝐿+𝛿)^𝑇Δ(𝐿+𝛿). (23)

Proof. Consider the following Lyapunov function candidate: 𝑉𝑡,𝑒_𝑡=𝑉₁𝑡,𝑒_𝑡+𝑉₂𝑡,𝑒_𝑡+𝑉₃𝑡,𝑒_𝑡, (24) where 𝑉₁𝑡,𝑒_𝑡=𝑒^𝑇(𝑡)𝐏⊗𝐼_𝑛𝑒(𝑡),𝑉₂𝑡,𝑒_𝑡=^𝑀−1_𝑖=1^𝑡_{𝑡−𝜏𝑖}𝑒^𝑇(𝑠)𝐐_𝐢⊗𝐼_𝑛𝑒(𝑠)𝑉𝑑𝑠,₃𝑡,𝑒_𝑡=^𝑀_𝑖=1^{𝑡−𝜏𝑖−1}_{𝑡−𝜏𝑖}^𝑡_𝑠𝑒^𝑇̇(𝑣)𝐑_𝑖⊗𝐼_𝑛𝑒̇(𝑣)+𝑑𝑣𝑑𝑠_{𝑡𝑡−𝜏𝑀}𝑒^𝑇̇(𝑣)𝐑_𝑀+1⊗𝐼_𝑛𝑒̇(𝑣)𝑑𝑣𝑑𝑠. (25)
Taking time derivative of 𝑉_𝑖(𝑡,𝑒_𝑡)(𝑖=1,2,3) along the trajectory (16) yields 𝐿𝑉₁𝑡,𝑒_𝑡=2𝑒^𝑇(𝑡)𝐏⊗𝐼_𝑛𝒜𝜉(𝑡)+2𝑒^𝑇(𝑡)𝐏⊗𝐼_𝑛𝐹=2𝑒^𝑇(𝑡)𝐏⊗𝐼_𝑛𝒜𝜉(𝑡)+2𝑒^𝑇(𝑡)×𝐏0𝐼_𝑁⊗𝐼_𝑛𝐹,𝐿𝑉₂𝑡,𝑒_𝑡=^𝑀−1_𝑖=1𝑒^𝑇(𝑡)𝐐_𝐢⊗𝐼_𝑛𝑒(𝑡)−𝑒^𝑇𝑡−𝜏_𝑖𝐐_𝐢⊗𝐼_𝑛𝑒𝑡−𝜏_𝑖,𝐿𝑉₃𝑡,𝑒_𝑡=𝑒^𝑇̇(𝑡)Δ⊗𝐼_𝑛𝑒̇(𝑡)−^𝑀_𝑖=1^{𝑡−𝜏𝑖−1}_{𝑡−𝜏𝑖}𝑒^𝑇̇(𝑠)𝐑_𝑖⊗𝐼_𝑛𝑒̇(𝑠)−𝑑𝑠_{𝑡𝑡−𝜏𝑀}𝑒^𝑇̇(𝑠)𝐑_𝑀+1⊗𝐼_𝑛𝑒̇(𝑠)𝑑𝑠, (26)(27)(28) where Δ=∑^𝑀_𝑖=1(𝜏_𝑖−𝜏_𝑖−1)𝐑_𝑖+𝜏_𝑀𝐑_𝑀+1
By Lemma 6, the following inequalities hold: −^{𝑡−𝜏𝑖−1}_{𝑡−𝜏𝑖}𝑒^𝑇̇(𝑠)𝐑_𝑖⊗𝐼_𝑛𝑒̇(𝑠)1𝑑𝑠≤−𝜏_𝑖−𝜏_𝑖−1^{𝑡−𝜏𝑖−1}_{𝑡−𝜏𝑖}𝑒^𝑇̇(𝑠)𝑑𝑠𝐑_𝑖⊗𝐼_𝑛^{𝑡−𝜏𝑖−1}_{𝑡−𝜏𝑖}𝑒̇(𝑠)1𝑑𝑠≤−𝜏_𝑖−𝜏_𝑖−1^{𝑡−𝜏𝑖−1}_{𝑡−𝜏𝑖(𝑡)}𝑒^𝑇̇(𝑠)𝑑𝑠𝐑_𝑖⊗𝐼_𝑛^{𝑡−𝜏𝑖−1}_{𝑡−𝜏𝑖(𝑡)}𝑒̇(𝑠)1𝑑𝑠=−𝜏_𝑖−𝜏_𝑖−1𝜉^𝑇(𝑡)0_{4(𝑖−1)𝑁𝑛,2𝑁𝑛}𝐼_2𝑁𝑛−𝐼_2𝑁𝑛0_{4(𝑀−𝑖)𝑁𝑛,2𝑁𝑛}×𝐑_𝑖⊗𝐼_𝑛0_{4(𝑖−1)𝑁𝑛,2𝑁𝑛}𝐼_2𝑁𝑛−𝐼_2𝑁𝑛0_{4(𝑀−𝑖)𝑁𝑛,2𝑁𝑛}^𝑇𝜉(𝑡),−^𝑡_{𝑡−𝜏𝑀}𝑒^𝑇̇(𝑠)𝐑_𝑀+1⊗𝐼_𝑛𝑒̇(𝑠)1𝑑𝑠≤−𝜏_𝑀^𝑡_{𝑡−𝜏𝑀}𝑒^𝑇̇(𝑠)𝑑𝑠𝐑_𝑀+1⊗𝐼_𝑛^𝑡_{𝑡−𝜏𝑀}𝑒̇(𝑠)1𝑑𝑠≤−𝜏_𝑀^𝑡_{𝑡−𝜏𝑀(𝑡)}𝑒^𝑇̇(𝑠)𝑑𝑠𝐑_𝑀+1⊗𝐼_𝑛^𝑡_{𝑡−𝜏𝑀(𝑡)}𝑒̇(𝑠)1𝑑𝑠=−𝜏_𝑀𝜉^𝑇(𝑡)𝐼_2𝑁𝑛0_{4(𝑀−1)𝑁𝑛,2𝑁𝑛}−𝐼_2𝑁𝑛×𝐑_𝑀+1⊗𝐼_𝑛𝐼_2𝑁𝑛0_{4(𝑀−1)𝑁𝑛,2N𝑛}−𝐼_2𝑁𝑛^𝑇𝜉(𝑡). (29)
From (16) and (28), we can obtain 𝑒^𝑇̇(𝑡)Δ⊗𝐼_𝑛𝑒̇(𝑡)=𝜉^𝑇(𝑡)𝒜^𝑇+^𝑀−1_𝑖=1𝛽_𝑖(𝑡)−𝛽_𝑖𝒜^𝑇_𝜏𝑖×Δ⊗𝐼_𝑛𝒜+^𝑀−1_𝑖=1𝛽_𝑖(𝑡)−𝛽_𝑖𝒜_𝜏𝑖𝜉(𝑡)+2𝜉^𝑇(𝑡)𝒜^𝑇+^𝑀−1_𝑖=1𝛽_𝑖(𝑡)−𝛽_𝑖𝒜^𝑇_𝜏𝑖Δ⊗𝐼_𝑛𝐹+𝐹^𝑇Δ⊗𝐼_𝑛𝐹=𝜉^𝑇(𝑡)𝒜^𝑇Δ𝒜𝜉(𝑡)+2^𝑀−1_𝑖=1𝛽_𝑖(𝑡)−𝛽_𝑖𝜉^𝑇(𝑡)×𝒜^𝑇Δ⊗𝐼_𝑛𝒜_𝜏𝑖𝜉(𝑡)+2_𝑖≠𝑗𝛽_𝑖(𝑡)−𝛽_𝑖𝛽_𝑗(𝑡)−𝛽_𝑗𝜉^𝑇(𝑡)𝒜^𝑇_𝜏𝑖Δ⊗𝐼_𝑛𝒜_𝜏𝑖𝜉(𝑡)+2^𝑀−1_𝑖=1𝛽_𝑖(𝑡)−𝛽_𝑖²𝜉^𝑇(𝑡)𝒜^𝑇_𝜏𝑖Δ⊗𝐼_𝑛𝒜_𝜏𝑖𝜉(𝑡)+2𝜉^𝑇(𝑡)𝒜^𝑇+^𝑀−1_𝑖=1𝛽_𝑖(𝑡)−𝛽_𝑖𝒜^𝑇_𝜏𝑖Δ⊗𝐼_𝑛𝐹+𝐹^𝑇𝐹. (30)
Taking the mathematical expectation on both sides of (30) yields 𝐸𝑒^𝑇̇(𝑡)Δ⊗𝐼_𝑛𝑒̇(𝑡)=𝜉^𝑇(𝑡)𝒜^𝑇Δ⊗𝐼_𝑛𝒜−2^𝑀−1_{𝑖,𝑗=1;𝑖≠𝑗}𝛽_𝑖𝛽_𝑗𝒜^𝑇_𝜏𝑖Δ⊗𝐼_𝑛𝒜_𝜏𝑖+^𝑀−1_𝑖=1𝛽_𝑖1−𝛽_𝑖𝒜^𝑇_𝜏𝑖Δ⊗𝐼_𝑛𝒜_𝜏𝑖𝜉(𝑡)+2𝜉^𝑇(𝑡)𝒜^𝑇Δ⊗𝐼_𝑛0𝐼_𝑁𝐹+𝐹^𝑇𝐹. (31)
By Assumption 3, we can obtain the following inequality: 𝐹^𝑇𝐹=𝐹(𝑡,𝑥(𝑡),𝑣(𝑡))−1_𝑁⊗𝑓𝑡,𝑥_𝑟(𝑡),𝑣_𝑟(𝑡)²=^𝑁_𝑖=1𝑓𝑡,𝑥_𝑖(𝑡),𝑣_𝑖(𝑡)−𝑓𝑡,𝑥_𝑟(𝑡),𝑣_𝑟(𝑡)²≤𝜌₁^𝑁_𝑖=1𝑥_𝑖(𝑡)−𝑥_𝑟(𝑡)²+𝜌₂^𝑁_𝑖=1𝑣_𝑖(𝑡)−𝑣_𝑟(𝑡)²=𝜌₁𝑥(𝑡)−1_𝑁⊗𝑥_𝑟(𝑡)²+𝜌₂𝑣(𝑡)−1_𝑁⊗𝑣_𝑟(𝑡)²=𝜌₁𝜉^𝑇(𝑡)𝜉(𝑡)+𝜌₂𝜂^𝑇(𝑡)𝜂(𝑡)=𝑒^𝑇(𝑡)𝜌₁𝐼_𝑁𝑛00𝜌₂𝐼_𝑁𝑛𝑒(𝑡). (32)
For a given 𝛼>0, the following inequality holds based on (32): 𝛼𝑒^𝑇(𝑡)𝜌₁𝐼_𝑁𝑛00𝜌₂𝐼_𝑁𝑛𝑒(𝑡)−𝛼𝐹^𝑇𝐹≥0. (33)
It follows from (26)–(33) that 𝐿𝑉𝑡,𝑒_𝑡≤2𝑒^𝑇(𝑡)𝐏⊗𝐼_𝑛𝒜𝜉(𝑡)+2𝑒^T(𝑡)𝐏0𝐼_𝑁⊗𝐼_𝑛𝐹+^𝑀−1_𝑖=1𝑒^𝑇(𝑡)𝐐_𝐢⊗𝐼_𝑛𝑒(𝑡)−𝑒^𝑇𝑡−𝜏_𝑖𝐐_𝐢⊗𝐼_𝑛𝑒𝑡−𝜏_𝑖+𝑒^𝑇̇(𝑡)Δ⊗𝐼_𝑛𝑒̇(𝑡)−^𝑀_𝑖=11𝜏_𝑖−𝜏_𝑖−1𝜉^𝑇(𝑡)0_{4(𝑖−1)𝑁𝑛,2𝑁𝑛}𝐼_2𝑁𝑛−𝐼_2𝑁𝑛0_{4(𝑀−𝑖)𝑁𝑛,2𝑁𝑛}×𝐑_𝑖⊗𝐼_𝑛0_{4(𝑖−1)𝑁𝑛,2𝑁𝑛}𝐼_2𝑁𝑛−𝐼_2𝑁𝑛0_{4(𝑀−𝑖)𝑁𝑛,2𝑁𝑛}^𝑇𝜉(𝑡)−1𝜏_𝑀𝜉^𝑇(𝑡)𝐼_2𝑁𝑛0_{4(𝑀−1)𝑁𝑛,2𝑁𝑛}−𝐼_2𝑁𝑛×𝐑_𝑀+1⊗𝐼_𝑛𝐼_2𝑁𝑛0_{4(𝑀−1)𝑁𝑛,2𝑁𝑛}−𝐼_2𝑁𝑛^𝑇𝜉(𝑡)+𝛼𝑒^𝑇(𝑡)𝜌₁𝐼_𝑁𝑛00𝜌₂𝐼_𝑁𝑛𝑒(𝑡)−𝛼𝐹^𝑇𝐹=𝜉^𝑇(𝑡)Ψ⊗𝐼_𝑛𝜉(𝑡)+2𝑒^𝑇(𝑡)𝐏0𝐼_𝑁⊗𝐼_𝑛𝐹+2𝜉^𝑇(𝑡)𝒜^𝑇Δ⊗𝐼_𝑛0𝐼_𝑁𝐹+𝛼𝑒^𝑇(𝑡)𝜌₁𝐼_𝑁𝑛00𝜌₂𝐼_𝑁𝑛𝑒(𝑡)+(1−𝛼)𝐹^𝑇𝐹, (34) where Ψ=Φ₁₁Φ∗∗∗⋯∗₂₁Φ₂₂∗∗⋯∗00Φ₃₃Φ∗⋯∗₄₁0Φ₄₃Φ₄₄Φ⋯∗⋮⋮⋮⋮⋱⋮_2𝑀,1000⋯Φ_2𝑀,2𝑀Φ₁₁=Φ₁₁𝜌−𝛼¹𝐼_𝑁𝑛00𝜌₂𝐼_𝑁𝑛. (35)
Let 𝜁(𝑡)=[𝜉^𝑇(𝑡),𝐹^𝑇]^𝑇, and the inequality (34) can further be written as 𝐿𝑉𝑡,𝑒_𝑡≤𝜁^𝑇(𝑡)Ψ∗ΓΦ_{2𝑀+1,2𝑀+1}⊗𝐼_𝑛𝜁(𝑡)=𝜁^𝑇(𝑡)Φ⊗𝐼_𝑛𝜁(𝑡)<0, (36) where Γ=0,Φ_2𝑀+1,2,Φ_2𝑀+1,4,…,0,Φ_{2𝑀+1,2𝑀}. (37)
From (22), we have 𝐿𝑉(𝑡,𝑒_𝑡)<0. That is, ‖𝜉(𝑡)‖→0, ‖𝜂(𝑡)‖→0 as 𝑡→∞. Therefore, the states of pinning-controlled multiagent systems (4) globally asymptotically approach state of virtual leader (3). That is, ‖𝑥_𝑖(𝑡)−𝑥_𝑟(𝑡)‖→0, ‖𝑣_𝑖(𝑡)−𝑣_𝑟(𝑡)‖→0 as 𝑡→∞. So the second-order leader-following consensus in pinning-controlled multiagent systems (4) is achieved. The proof is completed.

Remark 9. It can be seen from Theorem 8 that the feasibility of (22) dependents on not only 𝜏_𝑖 and 𝛿 but also the probability distribution of the delay 𝛽_𝑖, and more information of time delays is involved (22).

Let us consider one special case. When 𝑓(𝑡,⋅,⋅)≡0, then the virtual leader (3) reduces to 𝑥̇_𝑟(𝑡)=𝑣_𝑟(𝑡)𝑣̇_𝑟(𝑡)=0, (38) which indicates that the reference velocity is constant.

Each agent in multiagent systems is described by 𝑥̇_𝑖(𝑡)=𝑣_𝑖(𝑡)+𝑢_1𝑖(𝑡),𝑣̇_𝑖(𝑡)=𝑢_2𝑖(𝑡)(𝑖=1,2,…,𝑁). (39)

The pinning controllers are given as follows: 𝑢_1𝑖(𝑡)=−_{𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)+𝑥_𝑖(𝑡−𝜏(𝑡))−𝑥_𝑗(𝑡−𝜏(𝑡))−𝛿_𝑖𝑥_𝑖(𝑡)−𝑥_𝑟(𝑡)−𝑥_𝑖(𝑡−𝜏(𝑡))−𝑥_𝑟((𝑡𝑢𝑡−𝜏))_2𝑖(𝑡)=−_{𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑣_𝑖(𝑡)−𝑣_𝑗(𝑡)−𝑣_𝑖(𝑡−𝜏(𝑡))−𝑣_𝑗(𝑡−𝜏(𝑡))−𝛿_𝑖𝑣_𝑖(𝑡)−𝑣_𝑟(𝑡)−𝑣_𝑖(𝑡−𝜏(𝑡))−𝑣_𝑟(𝑡−𝜏(𝑡)), (40) where the parameters 𝛿_𝑖 are defined in Section 2.

Let 𝜉(𝑡)=𝑥(𝑡)−1_𝑁⊗𝑥_𝑟(𝑡), 𝜂(𝑡)=𝑣(𝑡)−1_𝑁⊗𝑣_𝑟(𝑡). Then we can obtain from (38)–(40) that 𝑒̇(𝑡)=𝐴⊗𝐼_𝑛𝑒(𝑡)+𝐴_𝜏⊗𝐼_𝑛𝑒(𝑡−𝜏(𝑡)). (41)

From Theorem 8 and its proof, we have the following corollary.

Corollary 10. For some given positive scalars 𝛽_𝑖∈[0,1](𝑖=1,2,…,𝑀), there exist some positive matrices 𝐏=diag{𝑃,𝑃}, 𝐐_𝑖=diag{𝑄_𝑖,𝑄_𝑖}(𝑖=1,2,…,𝑀−1), 𝐑_𝑗=diag{𝑅_𝑗,𝑅_𝑗}(𝑗=1,2,…,𝑀+1) such that the following linear matrix inequality holds: Φ₁₁Φ∗∗∗⋯∗₂₁Φ₂₂∗∗⋯∗00Φ₃₃Φ∗⋯∗₄₁0Φ₄₃Φ₄₄Φ⋯∗⋮⋮⋮⋮⋱⋮_2𝑀,1000⋯Φ_2𝑀,2𝑀<0. (42) Then the second-order leader-following consensus is achieved under the adaptive pinning controllers (40). That is, 𝑥_𝑖(𝑡)→𝑥_𝑟(0)+𝑣_𝑟(0)𝑡 and 𝑣_𝑖(𝑡)→𝑣_𝑟(0), as 𝑡→∞.

Up to this point, a question arises naturally: how to choose a set of pinned agents such that pinning conditions (22) and (42) are satisfied?

Proposition 11 (see [21]). For a diagraph 𝑔, let 𝑣 and 𝐷 denote the node set of 𝑔 and the pinned-node set, respectively. All nodes in 𝑣/𝐷 should have access to the pinned-node set 𝐷. That is, for any node 𝑖∈𝑣/𝐷, one can always find node 𝑗∈𝐷 such that there exists a directed path from node 𝑗 to node 𝑖.

According to Proposition 11, we give the following procedure to select a set of pinning-agents and to design the parameters 𝛿_𝑖  (𝑖=1,2,…,𝑙).(1)Define a degree-difference vector deg_dif(𝑖)=deg_out−deg_in(𝑖)  (𝑖=1,2,…,𝑁).(2)Pick all agents with zero in-degrees as pinned agents, and rearrange the remaining agents in descending order according to their degree-differences.(3)Find the minimum number of agents 𝑙₀ which satisfies (22) or (42), and let 𝑙=𝑙₀.(4)From Theorem 8 and Corollary 10, by using LMI Toolbox in MATLAB, we can obtain 𝛿_𝑖(𝑖=1,2,…,𝑙₀).(5)If 𝛿_𝑖(𝑖=1,2,…,𝑙₀) is not good enough for practical use, add more agents to the pinned-agent set, and repeat STEP (4) until we find proper 𝛿_𝑖.

4. Numerical Results

In this section, two numerical examples are given to verify the effectiveness of the proposed pinning control techniques.

Example 12. We consider multiagent systems consisting of six agents described by 𝑥̇_𝑖(𝑡)=𝑣_𝑖(𝑡)+𝑢_1𝑖(𝑡),𝑣̇_𝑖(𝑡)=𝑓𝑡,𝑥_𝑖(𝑡),𝑣_𝑖(𝑡)+𝑢_2𝑖(𝑡)(𝑖=1,2,…,6), (43) where 𝑓𝑡,𝑥_𝑖(𝑡),𝑣_𝑖(𝑡)()−√=2cos2𝑡55sin𝑥_𝑖(𝑡)−√105𝑣_𝑖(𝑡)𝑢_1𝑖(𝑡)=−_{𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡−𝜏(𝑡))−𝑥_𝑗(𝑡−𝜏(𝑡))−𝛿_𝑖𝑥_𝑖(𝑡)−𝑥_𝑟(𝑡)−𝑥_𝑖(𝑡−𝜏(𝑡))−𝑥_𝑟((𝑡𝑢𝑡−𝜏))_2𝑖(𝑡)=−_{𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑣_𝑖(𝑡)−𝑣_𝑗(𝑡)−𝑣_𝑖(𝑡−𝜏(𝑡))−𝑣_𝑗(𝑡−𝜏(𝑡))−𝛿_𝑖𝑣_𝑖(𝑡)−𝑣_𝑟(𝑡)−𝑣_𝑖(𝑡−𝜏(𝑡))−𝑣_𝑟(𝑡−𝜏(𝑡)), (44) where 𝑥_𝑖(𝑡)∈ℝ and 𝑣_𝑖(𝑡)∈ℝ are the position and velocity states of agent 𝑖, respectively.
The interaction diagraph of multiagent systems (43) is shown in Figure 1. We examine what agents should be pinned, and note that agent 6 has zero in-degree and that the out-degree of agents 2 is bigger than their in-degree. According to Proposition 11, agent 6 should be pinned first. Because its state is not affected by others, agent 2 can be chosen as pinned candidates, furthermore. Based on the pinned-agent selection scheme, we rearrange six agents and the new order is 6, 2, 4, 5, 3, 1. By some calculation, we can obtain 𝜌₁=0.1, 𝜌₂=0.2, for given 𝑀,  𝜏_𝑖(𝑖=1,2,…,𝑀−1) and 𝛽_𝑗(𝑗=1,2,…,𝑀) (when 𝑀=1, let 𝜏₀=0; when 𝑀=2, let 𝜏₀=0, 𝜏₁=0.1, 𝛽₁=0.1, 𝛽₂=0.9; when 𝑀=3, let 𝜏₀=0, 𝜏₁=0.1, 𝜏₂=0.2, 𝛽₁=0.1, 𝛽₂=0.1, 𝛽₃=0.8 ). By employing the LMI Toolbox in Theorem 8, we can obtain Table 1, it can be seen that (1) the minimum number of pinned agents is three, and we chose agents 6, 2, and 4 as pinned agents; (2) the allowable upper bound 𝜏 increases with increasing 𝑀. If we can obtain more information of time delay, it will lead to a larger allowable upper bound of time delay. When we chose agents 6, 2, and 4 as pinned agents, the evolutions of positions and velocities of six agents are shown in Figures 2, 3, 4, and 5.

Table 1 

The allowable upper bound of time delay for different 𝑀.

Figure 1 

The topology structure of the six agents.

Figure 2 

The position curves 𝑥𝑖(𝑡)(𝑖=1,2,…,6) for nonlinear multiagent systems (43) with 𝜏0=0, 𝜏1=0.3.

Figure 3 

The velocity curves 𝑣𝑖(𝑡)(𝑖=1,2,…,6) for nonlinear multiagent systems (43) with 𝜏0=0, 𝜏1=0.3.

Figure 4 

The position curves 𝑥𝑖(𝑡)(𝑖=1,2,…,6) for nonlinear multiagent systems (43) with 𝜏0=0, 𝜏1=0.2, 𝜏2=0.3,  𝛽1=0.1, and 𝛽2=0.9.

Figure 5 

The velocity curves 𝑣𝑖(𝑡)(𝑖=1,2,…,6) for nonlinear multiagent systems (43) with 𝜏0=0, 𝜏1=0.2, 𝜏2=0.3, 𝛽1=0.1, and 𝛽2=0.9.

Example 13. We consider the following multiagent systems consisting of six agents described by 𝑥̇_𝑖(𝑡)=𝑣_𝑖(𝑡)+𝑢_1𝑖(𝑡)𝑣̇_𝑖(𝑡)=𝑢_2𝑖(𝑡)(𝑖=1,2,…,6), (45) where 𝑢_1𝑖(𝑡)=−_{𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡−𝜏(𝑡))−𝑥_𝑗(𝑡−𝜏(𝑡))−𝛿_𝑖𝑥_𝑖(𝑡)−𝑥_𝑟(𝑡)−𝑥_𝑖(𝑡−𝜏(𝑡))−𝑥_𝑟((𝑡𝑢𝑡−𝜏))_2𝑖(𝑡)=−_{𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑣_𝑖(𝑡)−𝑣_𝑗(𝑡)−𝑣_𝑖(𝑡−𝜏(𝑡))−𝑣_𝑗(𝑡−𝜏(𝑡))−𝛿_𝑖𝑣_𝑖(𝑡)−𝑣_𝑟(𝑡)−𝑣_𝑖(𝑡−𝜏(𝑡))−𝑣_𝑟(𝑡−𝜏(𝑡)), (46) where 𝑥_𝑖(𝑡)∈ℝ and 𝑣_𝑖(𝑡)∈ℝ are the position and velocity states of agent 𝑖, respectively.
Similar to Example 12, by using the LMI Toolbox in Corollary 10, we can obtain Table 1. When we chose agents 6, 2, and 4 as pinned agents, the evolutions of positions and velocities of six agents are shown in Figures 6, 7, 8, and 9.

Figure 6 

The velocity curves 𝑣𝑖(𝑡)(𝑖=1,2,…,6) for nonlinear multiagent systems (45) with 𝜏0=0, 𝜏1=0.3.

Figure 7 

The velocity curves 𝑣𝑖(𝑡)(𝑖=1,2,…,6) for nonlinear multiagent systems (45) with 𝜏0=0, 𝜏1=0.3.

Figure 8 

The position curves 𝑥𝑖(𝑡)(𝑖=1,2,…,6) for nonlinear multiagent systems (45) with 𝜏0=0, 𝜏1=0.2, 𝜏2=0.3, 𝛽1=0.1, and 𝛽2=0.9.

Figure 9 

The velocity curves 𝑣𝑖(𝑡)(𝑖=1,2,…,6) for nonlinear multiagent systems (45) with 𝜏0=0, 𝜏1=0.2, 𝜏2=0.3, 𝛽1=0.1, and 𝛽2=0.9.

5. Conclusions

By employing the information of the probability distribution of the time delay, this paper investigates the consensus problem for second-order leader-follower nonlinear multiagent systems with general network topologies. Different from the common assumptions on the time delay in the existing literatures, it is assumed in this paper that the delay is random and its probability distribution is known a prior. Based on graph theory, a pinning control algorithm is proposed, and the consensus criteria are derived to achieve leader-follower consensus for multiagent systems with nonlinear second-order dynamics. Moreover, this paper addresses what kind of agents and how many agents should be pinned.

Acknowledgments

This work was jointly supported by National Natural Science Foundation of China under Grant nos. 61272034 and 11301226 and Zhejiang Provincial Natural Science Foundation of China under Grant no. LY13F030012 and LQ13A010017.