Second-Order Leader-Following Consensus of Multiagent Systems with Time Delays

This paper is concernedwith a leader-following consensus problemof second-ordermultiagent systemswith a constant acceleration leader and time-varying delays. At first, a distributed control protocol for every agent to track the leader is proposed; then by utilizing the Lyapunov-Razumikhin function, the convergence analysis under both fixed and switching interconnection topologies is investigated. For the case of fixed topology, a sufficient and necessary condition is obtained, and for the case of switching topology, a sufficient condition is derived under some assumptions. Finally, simulation examples are provided to demonstrate the effectiveness of the theoretical results.


Introduction
Recent years have witnessed dramatic advances concerned with distributed coordination of multiple agents due to rapid developments of computer science and communication technologies.As an important issue of distributed coordination, the consensus problem has attracted more and more attention from researchers in various fields [1][2][3][4][5].A special case of the issue is known as the leader-following consensus problems and has been investigated from different perspectives.Hong et al. in [6], for instance, designed distributed observers to track an active leader for second-order continuous multiagents systems.Also, Tang et al. studied the leader-following consensus problem via sampled-data control in [7].Besides, the authors in [8] explored first-order leader-following consensus algorithms under a directed fixed topology with a time-varying leader.Moreover, the results in [8] were furthered extended to the case of actuator saturation and switching topology in [9].
Due to the limited communication capacity, time delays are sometimes unavoidable in multiagent systems.Olfatisaber and Murray first analyzed the consensus problem with fixed undirected topology and time delays by a frequency domain approach in [1].Based on the reduced-order Lyapunov-Krasovskii function and linear matrix inequalities (LMIs), Lin and Jia considered the averaged consensus problem with a switching topology and time-varying delays in [10].In addition, Hu and Lin analyzed the second-order consensus problem for multiple agents with time-varying delays in terms of the Lyapunov-Razumikhin method in [11].Note that [1,10,11] studied leaderless consensus problem with time delays.For leader-follower networks, Hu and Hong [12] investigated the second-order consensus problems with fixed and switching topologies with time delays, and Tang et al. generalized the results of [12] to the case of nonuniform time delays in [13], while the scenario in the presence of detectable time delays was addressed in [14].It is worth pointing out that the velocity of the leader in [12][13][14] is a constant, and the time delays exist only in the transmission of position between neighbors.
In this paper, we consider the second-order leaderfollowing consensus problem for multi-agent systems, with time delays existing in the transmission of both velocity and position.Additionally, the velocity of the leader is not restricted to be a constant but a linear function of time; that is, the acceleration of the leader can be a nonzero constant.Furthermore, we investigate the consensus problem under both fixed and switching directed interconnection topologies by means of the Lyapunov-Razumikhin function and provide the bounds of the allowable time delays.
The rest of this paper is organized as follows.Section 2 provides some preliminaries and formulates the leaderfollowing consensus problem.With the proposed consensus protocol, Section 3 presents the convergence analysis of the case of fixed topology, and Section 4 deals with the case of switching topology.Then simulation results are presented in Section 5. Finally, the conclusion is drawn in Section 6.

Preliminaries and Formulation
Let G = (V, E, ) be a directed graph with a node set V = {1, 2, . . ., }, an edge set E ⊆ V × V, and a weighted adjacency matrix  = [  ] with nonnegative elements   .Node  represents agent , and an edge in E is denoted by an ordered pair (, ), where (, ) ∈ E if and only if that agent  can access the state information of agent .The index set of neighbors of node  is denoted by N  = { ∈ V : (, ) ∈ E}.For ,  ∈ V,  ∈ N  ⇔   > 0.Moreover, we assume that   = 0 for all  ∈ V.The indegree and outdegree of node  are, respectively, defined as for all of its nodes.The Laplacian matrix  = [  ] ∈ R × associated with  is defined as   = ∑  =1, ̸ =    and   = −  .Clearly,  has at least one zero eigenvalue with a corresponding eigenvector 1  .
A directed path is a sequence of edges in a directed graph of the form ( 1 ,  2 ), ( 2 ,  3 ), . .., where   ∈ V.If there exists a path from node  to node , we say that  is reachable from .Moreover,  is to said be globally reachable if there is a path from any other node to it.A directed tree is a directed graph, where there exists a node, called the root, such that any other node of the digraph can be reached by one and only one path starting at the root.A spanning tree of a digraph is a directed tree formed by graph edges that connect all the nodes of the graph.
Consider a multiagent system depicted by a graph G, which consists of  followers (related to graph G) and one leader (labeled as node 0) with directed edges from node 0 to some nodes.The leader adjacency matrix associated with graph G is defined as a matrix  = diag{ 1 , . . .,   }, where   > 0 if node 0 is a neighbor of node  and   = 0 otherwise.
The matrix  =  +  plays an important role in the analysis of leader-following consensus problem, the following lemma shows a relationship between the positive stability of  and the connectedness of graph G.
Lemma 1 (see [12]).The matrix  =  +  is positive stable if and only if node 0 is globally reachable in G.
Remark 2. It is straightforward to verify that  =  +  is positive stable if and only if the opposite graph, which is formed by changing the orientation of each arc of G, has a spanning tree rooted at node 0.
All the considered  followers in this paper move in a dimensional space, with the kinematic of each follower being described by a double integrator of the form: where   , V  ,   ∈ R  denote the position, velocity, and control input of follower , respectively.The dynamics of the leader is expressed as follows: where  0 ∈ R  is the constant acceleration.The leaderfollowing consensus of system ( 1)-( 2) is said to be achieved if the states of followers satisfy lim  → ∞ (  () −  0 ()) = 0 and lim  → ∞ (V  () − V 0 ()) = 0,  = 1, . . ., .
In this paper, we are interested in discussing the leaderfollowing consensus problem in multi-agent systems for both fixed and switching topologies, with the transmitted information time delay ().To solve such a problem, the following neighbored-based protocol is used: where  1 ,  2 > 0 are control parameters,  0 is the constant acceleration of the leader, and the time-varying delay () > 0 is a continuously differentiable and bounded function.Note that   in (3) depends on the position information and velocity information, all with time delay (), of its neighbors and itself.When each follower can get the velocity feedback without time delay, only from the leader and itself, and when the leader moves at a constant velocity (i.e.,  0 ≡ 0), protocol (3) becomes which is equivalent to protocol (6) in [12].
To describe the variable topologies, we define a piecewise constant switching function () : [0, ∞] → I Γ , where I Γ ⊂ N is the index set associated with the elements of Γ.The set Γ is finite because at most a digraph with  nodes is complete and has ( − 1) edges.
The following lemmas are needed in the subsequent sections.
Lemma 4 (see [16]).For any two real vectors  and  with the same dimension, one has where Ψ is any positive definite matrix with an appropriate dimension.

Consensus with Fixed Topology
This section focuses on the convergence analysis of (6) with fixed topology.In this case, the subscript  can be dropped for simplicity.Denote  = (  , V  )  ; we can obtain an error dynamics of (3) as follows: where By the Newton-Leibnitz formula, we have Thus, system (12) can be rewritten as where  =  + .
The matrix  plays a key role in the convergence analysis of (15); the following lemma shows a sufficient and necessary condition that can guarantee the Hurwitz stability of .

Lemma 7. The matrix 𝐹 is Hurwitz stable if and only if 𝐻 is positive stable and the control parameters satisfy 𝑘
Proof.Let  be an eigenvalue of .Then, one has where   is the th eigenvalue of .Therefore, the Hurwitz stability of matrix  is equivalent to that of the polynomial: (i) () = 0 has two distinct real roots  1 <  2 ; (ii) the interlaced condition holds; that is, where  1 is the unique root of () = 0; (iii) (0)  (0) −   (0)(0) > 0.
Considering condition (i), () = 0 has two distinct real roots if and only if Δ  =  2  2 Im 2 (  ) + 4 1 Re(  ) > 0. Noting that Re(  ) > 0, since  is positive stable, condition (i) holds obviously.In addition, the roots in condition (ii) can be expressed as Theorem 8.For system (12), suppose that the control parameters satisfy Then, there exists a constant  0 > 0 (which will be estimated in the following (22)) such that when  <  0 , namely, the leader-following consensus is reached asymptotically if and only if the graph G has a spanning tree.
Proof.Sufficiency.Since the graph G has a spanning tree, by Remark 2,  is positive stable.Then, it follows from Lemma 7 that  is Hurwitz stable.Therefore, there exists a positive definite matrix  ∈ R 2×2 such that Take the Lyapunov-Razumikhin function () =   .
Necessity.As a special case of (15), the system ε =  is asymptotical stable, which implies that  is Hurwitz stable.Hence, by Lemma 7,  is positive stable, and then it follows from Remark 2 that the graph G has a spanning tree.
Remark 9.In the special case where  ≡ 0 and   () > 0 for  = 1, . . ., , that is, the system is free of time delay and all the followers have full access to the leader, system (3) becomes which is equivalent to the protocol (9) in [19].Take () =  for some constant  > 1.In the case of (( + )) < (()), −2 ≤  ≤ 0, we have where From Lemma 5,   is positive definite if the control parameters satisfy (28).Furthermore, V() is negative definite if Therefore, the conclusion follows from Lemma 3.

Simulations
In this section, to illustrate our theoretical results derived in the above section, we provide two numerical simulations.For  Example 12.For the fixed topology case, we consider a multiagent system consisting of one leader and five followers with the interaction graph G 1 given in Figure 1.It can be noted that G 1 has a spanning tree.With simple calculations, we can obtain that  0 = 0.1295 and  0 = 0.0717.Take  2 = 0.5 and  1 = 1; then  2 2 / 1 = 0.25 > 0.1295.Figures 2(a) and 2(b) show, respectively, the position errors and velocity errors when () = 0.07| sin |.We can see that the five followers can track the leader; Figures 3(a) and 3(b) show, respectively, the position errors and velocity errors when  2 = 0.3592 and  1 = 1, and in this case  2  2 / 1 = 0.129 < 0.1295.From Figure 3, it can be seen that the tracking errors become unbounded.
Example 13.For the switching topology case with  = 2 (i.e., the agents move in a plane), we consider a multiagent system consisting of one leader and four followers.The interaction topology switches from a set {G 2 , G 3 } as shown in Figure 4. Clearly, the weights of both G 2 and G 3 satisfy the assumptions in Lemma 10.With simple calculations, we can obtain that  * = 0.6063.Take  2 = 0.65,  1 = 0.61; Figures 5(a) and 5(b) illustrate, respectively, the position evolutions and velocity evolutions with () = 0.015| cos |.we can see that the four followers can track the leader.

Conclusion
In this paper, we consider a second-order consensus problem for a multi-agent system with a constant acceleration leader and time-varying delays.Based on the Lyapunov-Razumikhin function and algebraic theory, we give a sufficient and necessary condition when the interaction topology is fixed and present a sufficient condition when the interaction topology is switching and the weights of graph satisfy some assumptions.Furthermore, two numerical simulations are provided to illustrate the results.