Leader-Following Consensus in Networks of Agents with Nonuniform Time-Varying Delays

This paper is concerned with a leader-following consensus problem for networks of agents with fixed and switching topologies as well as nonuniform time-varying communication delays. By employing Lyapunov-Razumikhin function, a necessary and sufficient condition is derived in the case of fixed topology, and a sufficient condition is obtained in the case when the interconnection topology is switched and satisfies certain condition. Simulation results are provided to illustrate the theoretical results.


Introduction
In recent years, consensus problems of multiagent systems have received compelling attention from various research communities.This is mainly due to their wide applications in many areas such as synchronization of coupled oscillators, flocking, rendezvous, distributed sensor fusion in sensor networks, and distributed formation control see 1 .The study of consensus problems is of great importance to understand many complex phenomena related to animal behaviors, such as flocking of birds, schooling of fish, and swarming of bees.
Consensus problems have a long history in the field of computer science, particularly in automata theory and distributed computation 2 .Vicsek et al. 3 proposed a simple model of a system of several autonomous agents, and demonstrated by simulation that all agents eventually reach an agreement.Jadbabaie et al. 4 provided a theoretical explanation for the observed behavior of the Vicsek model.Up to now, a variety of topics related to edges of the graph.An edge of G is denoted by i, j , representing that agent i can directly receive information from agent j.A path in a digraph is a sequence of ordered edges of the form i k , i k 1 , k 1, . . ., m − 1.We say that node j is reachable from node i if there is a path from node i to node j.The set of neighbors of node i is denoted by N i {j ∈ V | i, j ∈ E}.
The weighted adjacency matrix of the digraph G is denoted by A a ij ∈ R n×n , where a ij > 0 if i, j ∈ E and a ij 0, otherwise.Moreover, we assume that a ii 0 for all i ∈ V.The indegree and outdegree of node i are defined as deg in i n j 1 a ji and deg out i n j 1 a ij , respectively.A digraph is said to be balanced if deg in i deg out i .The Laplacian matrix L l ij associated with digraph G is defined as The definition of L clearly implies that L must have a zero eigenvalue corresponding to a eigenvector 1, where 1 1, . . ., 1 T ∈ R n .Moreover, 0 is a simple eigenvalue of L if and only if G has a spanning tree 14 .
In order to study a leader-following problem, we also concern another digraph G, which consists of digraph G, node 0, and edges from some nodes to node 0. We say that node 0 is globally reachable in G if node 0 is reachable from any node in G.The leader adjacency matrix associated with G is defined as a diagonal matrix B with diagonal elements b i , where b i > 0 if i, 0 is an edge of G and b i 0, otherwise.
In this paper, we consider the following double integrator system of n agents: where x i , v i , u i ∈ R denote the position, velocity, and control input of agent i, respectively.The dynamics of the leader is expressed as follows: where v 0 is the desired constant velocity.Let τ ij t denote the communication time-delay from agent j to agent i. Similarly to 2 , we assume that communication delays between agents are symmetrical, that is, τ ij t τ ji t .Our control goal is to let the n agents follow the considered leader in the sense of both position and velocity, namely, x i → x 0 , v i → v 0 i 1, . . ., n as t → ∞.For this end, we study the following neighbor-based protocol: where k is a control parameter.The communication topology among the group of agents may change dynamically due to link failure or creation, for instance, because of the limited detection range of agents, existence of the obstacles.In order to describe the switching topologies, we define a piecewise constant switching signal σ t σ in short : 0, ∞ → P {1, 2, . . ., N}, where N denotes the total number of all possible interaction topologies.The collection of all possible interaction topologies {G 1 , . . ., G N } is a finite set.For convenience, let To illustrate these relationships, an example is given as follows.
Example 2.1.Consider a multiagent system consisting of four agents and a leader with the interconnection topology G 1 shown in Figure 1.We assume that G 1 has 0 − 1 weights, and  2, and one can obtain the following: and B 1 B 11 diag{1, 0, 0, 1}, B 21 B 31 0. Obviously, 2.6 is true.Write 2 can be written in the following matrix form: We can obtain an error dynamics of system 2.8 as follows: where

2.10
Before ending this section, we introduce Lyapunov-Razumikhin Theorem, which plays a key role in the convergence analysis of system 2.9 .
Consider the following system: where x t θ x t θ , ∀θ ∈ −r, 0 and f 0 0. Let C −r, 0 , R n be a Banach space of continuous functions defined on an interval −r, 0 , taking values in R n with the topology of uniform convergence, and with a norm ϕ c max θ∈ −r,0 ϕ θ .Lemma 2.2 Lyapunov-Razumikhin Theorem 15 .Let φ 1 , φ 2 , and φ 3 be continuous, nonnegative, nondecreasing functions with φ 1 s > 0, φ 2 s > 0, φ 3 s > 0 for s > 0 and φ 1 0 φ 2 0 0. For system 2.11 , suppose that the function f : 2.12 In addition, there exists a continuous nondecreasing function φ s with φ s > s, s > 0 such that the derivative of V along the solution x t of 2.11 satisfies then the solution x 0 is uniformly asymptotically stable.
Usually, V t, x is called a Lyapunov-Razumikhin function if it satisfies 2.12 and 2.13 in Lemma 2.2.

Fixed Interconnection Topology
Consider system 2.9 with fixed interconnection topology.In this case, the subscript σ can be dropped.Rewritte 2.9 as 3.1 To derive a delay-dependent stability criteria, we make the following model transformation.With the observation that The process of transforming a system represented by 3.1 to one represented by 3.5 is known as a model transformation.The stability of the system represented by 3.5 implies the stability of the original system 16 .
To get the main result of this subsection, we need the following lemmas.

3.15
Along the solution of system 2.9 , from 3.5 , we have ε t s ds.

3.16
Note that 2a T b ≤ a T Ψa b T Ψ −1 b holds for any appropriate positive definite matrix Ψ.Then, we have ε T t s Pε t s ds .

3.17
Take φ s qs for some constant q > 1.In the case of by recalling that τ p t ≤ τ.As a result, if Therefore, the conclusion follows by Lemma 2.2.Necessity .System 3.1 is asymptotically stable for any time delays τ p t < τ * , p ∈ I.In particular, let τ p t ≡ 0, p ∈ I.By 3.1 , the system ε Fε t is asymptotically stable, and hence all eigenvalues of F have negative real parts.Therefore, it follows from Lemma 3.2 that H is positive stable, and the conclusion follows by Lemma 3.3.Remark 3.5.From the proof of Theorem 3.4, we can see that many zoom techniques have to be applied during the derivation of τ * , and hence our estimate τ * may be very conservative.

Time-Varying Topology
In this subsection, we consider the case of switching topologies.Similar to the case of fixed topology, we can obtain that A i,σ ε t s − τ i t ds.

3.21
Then, from 2.9 , we have ε by noting that A p,σ A i,σ 0 for p, i 1, . . ., m. Denote the following:

3.23
where H p,σ L p,σ B p,σ .Then 3.22 can be rewritten as ε t s ds.

3.24
To obtain the main result of this subsection, we introduce the following assumption.
Assumption 3.6.The weights of digraph G satisfy the following conditions: Remark 3.8.In the study of leader-following consensus for second-order multiagent systems with switching topologies 11, 21-24 , it was assumed that G is balanced and vertex 0 is globally reachable in G so that a common Lyapunov function can be established.The theoretical base is the conclusion that H H T is positive definite if G is balanced, and vertex 0 is globally reachable in G. Noticing that n 1 j 1 a kj and n 1 j 1 a jk denote the out-degree and in-degree of node k in G, respectively, Assumption 3.6 contains that the out-degree of node k is greater or equal to its in-degree for each k ∈ V as a special case.Then Assumption 3.6 is much weaker than the balanced constraint on G, and the corresponding results in the above literatures can be improved accordingly.
For convenience, denote that μ max t≥0 {λ max H σ t H T σ t }, λ min t≥0 {λ min H σ t H T σ t }, which are well defined by noting that the set P is finite.The main result of this subsection is as follows.Theorem 3.9.Suppose that the weights of G σ satisfy (Assumption 3.6) and node 0 is globally reachable in G σ for any t ≥ 0. Consider system 2.9 and take k > k * μ 2λ 1.

3.25
If τ < τ * 1 (which will be defined in the following 3.32 ), then Proof.Take a Lyapunov-Razumikhin function V ε ε T Φε with positive definite matrix

3.27
Similar to the analysis in the proof of Theorem 3.4, we have ε T t s Φε t s ds .

3.28
Take φ s qs for some constant q > 1.In the case of where According to Schur complement 19 , Q σ is positive definite for any t ≥ 0 if k satisfies 3.25 .Hence, if which is well defined by noting that the set P is finite, then V ε ≤ −η 1 ε T ε for some η 1 .Therefore, the conclusion follows by Lemma 2.2.
Remark 3.10.For the first-order multiagent systems, it was shown that the consensus can be achieved provided that the network topology jointly contains a spanning tree 14, 25 .However, if the group of agents is governed by second-order dynamics, the consensus depends not only on the topology condition but also on the coupling strength between neighboring agents, and it was shown that consensus may fail to be achieved even if the network topology contains a spanning tree 26 .It should be pointed out that Assumption 3.6 is not necessary to ensure the consensus, and it is of great interest to consider the more general condition on the network topology.

Simulations
In this section, two examples are provided to illustrate the theoretical results.For simplicity, we assume that each interconnection topology has 0-1 weights in the following two examples.Example 4.1.Consider a multiagent system consisting of a leader and four agents with fixed topology G 1 given in Figure 1.It is clear that node 0 is globally reachable in G 1 .By simple calculation, we have τ * 0.0426 and k > 0. Let τ 1 t 0.02| sin t|, τ 2 t 0.03| cos t|, τ 3 t 0.03 0.01 sin t.The simulation results are obtained with k 2. Figure 3 shows that the four agents can follow the considered leader.
Example 4.2.Consider a multiagent system consisting of a leader and four agents.The interconnection topology of the multiagent system switches every 1s in the sequence G 2 , G 3 described as Figure 4 5 that the four agents can follow the considered leader.

Conclusion
In this paper, we study a leader-following consensus problem of second-order multiagent systems with fixed and switching topologies as well as nonuniform time-varying communication delays.With the help of Lyapunov-Razumikhin function, an explicit formula for the upper bound of admissible delays is obtained for both fixed and switching topologies.Future research issues will include the cases when the communication delays are asymmetric, and the velocity of the considered leader is time-varying.

Figure 3 :
Figure 3: Position errors and velocity errors of Example 4.1.

Figure 5 :
Figure 5: Position errors and velocity errors of Example 4.2.
Noting that A p A i 0 for p, i 1, . . ., m, we have ks μ i .It follows from Lemma 3.1 that f s, μ i is Hurwitz stable if and only if Re μ i > 0 and k > | Im μ i |/ Re μ i .Therefore, all eigenvalues of F have negative real parts if and only if Re μ i > 0 and k > | Im μ i |/ Re μ i for any μ i ∈ Λ H , which implies the conclusion. is the control parameter in protocol 2.4 .Then, there exists a constant τ * > 0 (which will be defined in the following 3.20 ) such that when τ < τ * , Lemma 3.1 see 17 .Given a complex-coefficient polynomial, f s s 2 a ib s c id, 3.7 where a, b, c, d ∈ R, f s is Hurwitz stable if and only if a > 0 and abd a 2 c − d 2 > 0. namely, the n agents can follow the leader (in the sense of both position and velocity), if and only if node 0 is globally reachable in G. Proof.Sufficiency .Since node 0 is globally reachable in G, it follows from Lemma 3.3 that H is positive stable.Thus, it follows from 3.12 and Lemma 3.2 that F is Hurwitz stable.Hence, by Lyapunov theorem 19 , there exists a positive definite matrix P ∈ R 2n×2n such that PF F T P −I 2n .
namely, I denotes the index set of neighbors of vertex 0.
Lemma 3.7 see 20 .Assume that the weights of G satisfy (Assumption 3.6), and node 0 is globally reachable in G. Then H H T is positive definite, where H L B.
. It is clear that the weights of G 2 and G 3 satisfy Assumption 3.6 and node 0 is globally reachable in G 2 and G 3 .Let τ 10 t τ 40 t 0.1| sin t|, τ 21 t τ 34 t 0.2| cos t|, τ 23 t 0.2 0.1 sin t.The simulation results are obtained with k 10.It can be seen from Figure