𝐻_∞ Consensus for Multiagent Systems with Heterogeneous Time-Varying Delays

Abstract

We apply the linear matrix inequality method to consensus and 𝐻_∞ consensus problems of the single integrator multiagent system with heterogeneous delays in directed networks. To overcome the difficulty caused by heterogeneous time-varying delays, we rewrite the multiagent system into a partially reduced-order system and an integral system. As a result, a particular Lyapunov function is constructed to derive sufficient conditions for consensus of multiagent systems with fixed (switched) topologies. We also apply this method to the 𝐻_∞ consensus of multiagent systems with disturbances and heterogeneous delays. Numerical examples are given to illustrate the theoretical results.

1. Introduction

In recent years, decentralized coordination of multiagent systems has received many researchers' attention in the areas of system control theory, biology, communication, applied mathematics, computer science, and so forth. In cooperative control of multiagent systems, a critical problem is to design appropriate protocols such that multiple agents in a group can reach consensus. So far, by using the matrix theory, the graph theory, the frequency-domain analysis method, the Lyapunov direct method, and so forth, consensus problems for various kinds of multiagent systems have been studied extensively [1–3].

In the field of systems and control theory, the pioneering work was done by Borkar and Varaiya [4] and Tsitsiklis and Athans [5], where the asynchronous consensus problem with an application in distributed decision-making systems was considered. Later, Vicsek et al. [6] proposed a simple but interesting discrete-time model of multiple agents which can be viewed as a special case of a computer model mimicking animal aggregation. Jadbabaie et al. [7] provided a theoretical explanation of the consensus property of the Vicsek model.

For the case of single integrator multiagent systems, Olfati-Saber and Murray [8] discussed the consensus problem for networks of dynamic agents by defining a disagreement function. Ren and Beard [9] established some more relaxable consensus conditions under dynamically changing interaction topologies. Hui and Haddad [10], Liu et al. [11], and Bauso et al. [12] investigated the consensus problem for nonlinear multiagent systems. Tan and Liu [13] studied consensus of networked multiagent systems via the networked predictive control. Li and Zhang [14] gave the necessary and sufficient condition of mean square average-consensus for multiagent systems with noises. Zheng and Wang [15] studied finite-time consensus of heterogenous multiagent systems with and without velocity measurements. The constrained consensus problem for multiagent systems in unbalanced networks was investigated in Lin and Ren [16, 17]. Liu et al. [18] considered the consensus problem for multiagent systems with inherent nonlinear dynamics under directed topologies. A distributed shortest-distance consensus problem under dynamically changing network topologies was studied by Lin and Ren [19]. Li et al. [20] studied the distributed consensus problem of multiagent systems with general continuous-time dynamics for both the case without and with a leader.

When considering communication delays in the feedback, three types of consensus protocols have been analyzed: (i) both the state of the agent and its neighbors are affected by identical delays [21–27]; (ii) communication delays only affect the state received from neighbors of the agent [28–32]; (iii) the state of the agent and its neighbors are affected by heterogeneous delays. Note that (i) and (ii) are special cases of (iii). Compared to the cases of (i) and (ii), consensus for the case of (iii) receives less attention. Under the restricted assumptions that the graph is undirected and the communication delays are constants, Münz et al. [32, 33] investigated the robustness of the third kind of consensus protocols in both identical and nonidentical agent dynamics.

We are here concerned with the third type of consensus protocols, where the heterogeneous delays are time varying, and the involved graph is directed. It seems to us that the frequency-domain analysis method in [32, 33] becomes invalid in this case, and it is difficult to employ the Lyapunov method directly. In this paper, we will apply the linear matrix inequality method to this problem. In [21–25], the linear matrix inequality method has been used successfully in the case of (i). When applying this method to the case of (iii), we have to overcome some difficulties caused by heterogeneous time-varying delays. For this sake, we first skillfully rewrite the multiagent system by virtue of a partially-reduced-order system and an integral system. Then, by defining a particular Lyapunov function based on the above partially-reduced-order system and the integral system, sufficient conditions to consensus are derived in both cases of fixed topology and switching topologies by using the linear matrix inequality method.

We also consider the 𝐻_∞ consensus problem for the single integrator multiagent system with heterogeneous time-varying delays. So far, there are few 𝐻_∞ consensus results for continuous time multiagent systems. Lin et al. [34] studied distributed robust 𝐻_∞ consensus problem in directed networks of agents with identical delays. Ugrinovskii [35] considered a problem of design of distributed robust filters with 𝐻_∞ consensus by using the recent vector dissipativity theory. Sun and Wang [36] investigated the 𝐻_∞ consensus for the double integrator multiagent system with asymmetric delays.

To the best of our knowledge, little has been known about the 𝐻_∞ consensus problem for the single integrator multiagent systems with heterogeneous time-varying delays and directed network topologies. Therefore, another purpose of this paper is to establish 𝐻_∞ consensus criteria in the cases of heterogeneous delays and directed fixed topology (or switching topologies) by using the linear matrix inequality technique.

This paper is structured as follows. The problem statement and the transformation of the multiagent system are summarized in Section 2. In Section 3, sufficient conditions in terms of linear matrix inequalities are established for consensus and 𝐻_∞ consensus of the single integrator multiagent system with heterogeneous delays. These results are also extended to the case of switching topologies. Numerical examples and simulation results are given in Section 4. The paper is concluded in Section 5.

The notation used throughout this paper is fairly standard. 𝐴^𝑇 means the transpose of the matrix 𝐴. 𝐼_𝑚 is an 𝑚×𝑚-dimensional identity matrix. 𝐚_𝑚=(𝑎,𝑎,…,𝑎)^𝑇 is an 𝑚-dimensional column vector with 𝑎∈ℝ. We say 𝑋>𝑌 if 𝑋−𝑌 is positive definite, where 𝑋 and 𝑌 are symmetric matrices of the same dimensions. We use an asterisk ∗ to represent a term that is induced by symmetry and diag{⋯} stands for a block-diagonal matrix. 𝐿₂[0,∞) denotes the space of square-integrable vector functions over [0,∞).

2. Preliminaries

Throughout this paper, we denote a weighted digraph by 𝒢=(𝑉,𝐸,𝐴), where 𝑉={𝑣₁,𝑣₂,…,𝑣_𝑛} is the set of nodes with 𝑛≥2 and node 𝑣_𝑖 represents the 𝑖th agent; 𝐸⊆𝑉×𝑉 is the set of edges, an edge of 𝒢 is denoted by an order pair (𝑖,𝑗), and (𝑖,𝑗)∈𝐸 if and only if 𝑎_𝑗𝑖>0; 𝐴=[𝑎_𝑖𝑗] is an 𝑛×𝑛-dimensional weighted adjacency matrix with 𝑎_𝑖𝑖=0. If (𝑖,𝑗) is an edge of 𝒢, node 𝑖 is called the parent of node 𝑗. A directed tree is a directed graph, where every node, except one special node without any parent, which is called the root, has exactly one parent, and the root can be connected to any other nodes through paths. The Laplacian matrix 𝐿=[𝑙_𝑖𝑗] of digraph 𝒢 is defined by 𝑙_𝑖𝑖=−∑^𝑛_𝑗=1𝑎_𝑖𝑗 and 𝑙_𝑖𝑗=𝑎_𝑖𝑗 for 𝑖≠𝑗, 𝑖,𝑗∈𝒩={1,2,…,𝑛}.

Consider the following multiagent system with heterogeneous delays: 𝑥̇_𝑖(𝑡)=^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗𝑡−𝑑₁(𝑡)−𝑥_𝑖𝑡−𝑑₂(𝑡),𝑖∈𝒩, (1) where 𝑡≥0, 𝑎_𝑖𝑗≥0, 𝑖,𝑗∈𝒩, are entries of the weighted adjacency matrix 𝐴, 𝑑₁(𝑡) and 𝑑₂(𝑡) are different piecewise continuous communication delays satisfying 0≤𝑑₁(𝑡)≤ℎ₁ and 0≤𝑑₂(𝑡)≤ℎ₂ for 𝑡≥0, and ℎ₁ and ℎ₂ are positive constants.

Denote 𝑥=(𝑥₁,𝑥₂,…sd,𝑥_𝑛)^𝑇 and 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛−1)^𝑇. If we set 𝑦_𝑖=𝑥_𝑖+1−𝑥₁ for 𝑖=1,2,…,𝑛−1, by the straightforward computation, we get 𝑦=𝐸𝑥,𝑥=𝑥₁𝟏_𝑛+𝐹𝑦, (2) where the (𝑛−1)×𝑛-dimensional matrix 𝐸 and the 𝑛×(𝑛−1)-dimensional matrix 𝐹 are defined as follows: 𝐸=−𝟏_𝑛−1,𝐼_𝑛−1𝟎,𝐹=^𝑇𝑛−1𝐼_𝑛−1. (3)

Next, we make use of transformation (2) to derive two descriptor systems of the system (1). Rewrite (1) as follows: 𝑥̇_𝑖(𝑡)=^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗𝑡−𝑑₁(𝑡)−𝑥_𝑗(𝑡+_𝑛𝑗=1)𝑎_𝑖𝑗𝑥_𝑖(𝑡)−𝑥_𝑖𝑡−𝑑₂(𝑡),𝑖∈𝒩. (4) Therefore, by the Newton-Leibniz formula, we get a matrix form of (4): 𝑥̇(𝑡)=𝐿𝑥(𝑡)+𝐴^𝑡−𝑑1_𝑡^(𝑡)𝑥̇(𝑠)𝑑𝑠+Δ^𝑡_{𝑡−𝑑2(𝑡)}𝑥̇(𝑠)𝑑𝑠, (5) where Δ=𝐴−𝐿.

Denote 𝑧(𝑡)=𝑥̇(𝑡). By transformation (2) and the property of the Laplacian matrix 𝐿 that 𝐿𝟏_𝑛=𝟎_𝑛, we get from (5) the following two descriptor systems: 𝑦̇(𝑡)=𝐸𝐿𝐹𝑦(𝑡)+𝐸𝐴^𝑡−𝑑1_𝑡^(𝑡)𝑧(𝑠)𝑑𝑠+𝐸Δ^𝑡_{𝑡−𝑑2(𝑡)}𝑧(𝑠)𝑧(𝑡)(𝑡)𝑑𝑠,=𝐿𝐹𝑦+𝐴^𝑡−𝑑1_𝑡^(𝑡)𝑧(𝑠)𝑑𝑠+Δ^𝑡_{𝑡−𝑑2(𝑡)}𝑧(𝑠)𝑑𝑠. (6)

When the system involves disturbance input, we consider the following multiagent system of the form 𝑥̇_𝑖(𝑡)=^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗𝑡−𝑑₁(𝑡)−𝑥_𝑖𝑡−𝑑₂(𝑡)+𝑤_𝑖𝑗(𝑡),𝑖∈𝒩, (7) where 𝑎_𝑖𝑗, 𝑑₁(𝑡), and 𝑑₂(𝑡) are defined as above and 𝑤_𝑖𝑗(𝑡) is the disturbance input satisfying 𝑤_𝑖𝑗(𝑡)∈𝐿₂[0,∞).

Denote the 𝑖th row of the matrix 𝐴 by 𝛼_𝑖, Σ=diag{𝛼₁,𝛼₂,…,𝛼_𝑛}, 𝑤_𝑖(𝑡)=(𝑤_𝑖1(𝑡),𝑤_𝑖2(𝑡),…,𝑤_𝑖𝑛(𝑡)]^𝑇, and 𝑤(𝑡)=(𝑤^𝑇₁(𝑡),𝑤^𝑇₂(𝑡),…,𝑤^𝑇_𝑛(𝑡)]^𝑇. Similar to the above procedure, we can get the following matrix form of (7): 𝑦̇(𝑡)=𝐸𝐿𝐹𝑦(𝑡)+𝐸𝐴^𝑡−𝑑1_𝑡^(𝑡)𝑧(𝑠)𝑑𝑠+𝐸Δ^𝑡_{𝑡−𝑑2(𝑡)}𝑧(𝑠)(𝑡),𝑧(𝑡)(𝑡)𝑑𝑠+𝐸Σ𝑤=𝐿𝐹𝑦+𝐴^𝑡−𝑑1_𝑡^(𝑡)𝑧(𝑠)𝑑𝑠+Δ^𝑡_{𝑡−𝑑2(𝑡)}𝑧(𝑠)𝑑𝑠+Σ𝑤(𝑡). (8)

In this paper, say that system (1) achieves consensus asymptotically if lim_𝑡→∞𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)=0,∀𝑖≠𝑗,𝑖,𝑗∈𝒩. (9) Say that system (7) solves 𝐻_∞ consensus if system (1) achieves consensus asymptotically, and there exists a constant 𝛾>0 such that ^∞₀𝑦^𝑇(𝑡)𝑦(𝑡)𝑑𝑡≤𝛾^∞₀𝑤^𝑇(𝑡)𝑤(𝑡)𝑑𝑡 (10) for all nonzero 𝑤∈𝐿₂[0,∞) under zero initial condition.

3. Main Results

The following two lemmas can be concluded from [21, 22].

Lemma 1. The digraph 𝒢={𝑉,𝐸,𝐴} has a spanning tree if and only if the matrix 𝐸𝐿𝐹 is Hurwitz.

Lemma 2. For any continuous vector 𝑢(𝑡)∈ℝ^𝑚 on [𝑡−ℎ_𝑘,𝑡] for 𝑘=1,2 and 𝑡≥0 and any 𝑚×𝑚-dimensional matrix 𝑊>0, the following inequality holds: ^𝑡−𝑑1_𝑡^(𝑡)𝑢(𝛼)𝑑𝛼^𝑇𝑊^𝑡−𝑑1_𝑡^(𝑡)𝑢(𝛼)𝑑𝛼≤ℎ₁^𝑡_𝑡−ℎ1𝑢^𝑇(𝛼)𝑊𝑢(𝛼)𝑑𝛼,_{𝑡𝑡−𝑑2(𝑡)}𝑢(𝛼)𝑑𝛼^𝑇𝑊^𝑡_{𝑡−𝑑2(𝑡)}𝑢(𝛼)𝑑𝛼≤ℎ₂^𝑡_𝑡−ℎ2𝑢^𝑇(𝛼)𝑊𝑢(𝛼)𝑑𝛼. (11)

Based on Lemmas 1 and 2 and the preliminaries given in Section 2, we have the following consensus result for system (1).

Theorem 3. If the digraph 𝒢 has a spanning tree, system (1) achieves consensus asymptotically for appropriate constants ℎ₁>0 and ℎ₂>0 which satisfy the following matrix inequality: 𝑃𝐿+𝐿^𝑇𝑃𝑃𝐴𝑃Δℎ₁𝐿^𝑇𝑄₁ℎ₂𝐿^𝑇𝑄₂∗−𝑄₁0ℎ₁𝐴^𝑇𝑄₁ℎ₂𝐴^𝑇𝑄₂∗∗−𝑄₂ℎ₁Δ^𝑇𝑄₁ℎ₂Δ^𝑇𝑄₂∗∗∗−𝑄₁0∗∗∗∗−𝑄₂<0, (12) where 𝐿=𝐸𝐿𝐹, 𝐴=𝐸𝐴, Δ=𝐸Δ, 𝐿=𝐿𝐹, and 𝑃, 𝑄₁, and 𝑄₂ are positive-definite matrices of appropriate dimensions.

Proof. First, we see that (12) is equivalent to Ω+ℎ²₁Φ^𝑇𝑄₁Φ+ℎ²₂Φ^𝑇𝑄₂Φ<0, (13) where Φ=𝑃𝐿𝐿,𝐴,Δ,Ω=𝐿+^𝑇𝑃𝑃𝐴𝑃Δ∗−𝑄₁0∗∗−𝑄₂. (14)
On the other hand, we show that (13) is feasible if the digraph 𝒢 has a spanning tree. By Lemma 1, we see that there exists a positive-definite matrix 𝑃 such that 𝑃𝐿+𝐿^𝑇𝑃<0. Once the positive-definite matrix 𝑃 is determined, it is not difficult to see that Ω<0 for certain positive-definite matrices 𝑄₁ and 𝑄₂. Once the matrices 𝑃>0, 𝑄₁>0, and 𝑄₂>0 are determined, (13) is valid for sufficiently small ℎ₁>0 and ℎ₂>0. Therefore, (13) always holds for appropriate matrices 𝑃>0, 𝑄₁>0, and 𝑄₂>0 and constants ℎ₁>0, ℎ₂>0 if 𝒢 has a spanning tree.
Next, by the definition of vector 𝑦, it is sufficient to show that lim_𝑡→∞𝑦(𝑡)=0 if (13) holds. We construct the Lyapunov function as follows: 𝑉(𝑡)=𝑦^𝑇(𝑡)𝑃𝑦(𝑡)+ℎ₁^𝑡_𝑡−ℎ1𝑠−𝑡+ℎ₁𝑧^𝑇(𝑠)𝑄₁𝑧(𝑠)𝑑𝑠+ℎ₂^𝑡_𝑡−ℎ2𝑠−𝑡+ℎ₂𝑧^𝑇(𝑠)𝑄₂𝑧(𝑠)𝑑𝑠, (15) where positive constants ℎ₁ and ℎ₂ and positive-definite matrices 𝑃, 𝑄₁, and 𝑄₂ satisfy (13). We now consider the derivative of 𝑉(𝑡) along the trajectory of system (6). For the sake of convenience, set 𝜂₁(𝑡)=^𝑡−𝑑1_𝑡^(𝑡)𝑧(𝑠)𝑑𝑠,𝜂₂(𝑡)=^𝑡_{𝑡−𝑑2(𝑡)}𝑧(𝑠)𝑑𝑠. (16) A straightforward computation yields that 𝑉̇(𝑡)=2𝑦^𝑇(𝑡)𝑃𝐿𝑦(𝑡)+𝐴𝜂₁(𝑡)+Δ𝜂₂(𝑡)+ℎ²₁𝑧^𝑇(𝑡)𝑄₁𝑧(𝑡)−ℎ₁^𝑡_𝑡−ℎ1𝑧^𝑇(𝑠)𝑄₁𝑧(𝑠)𝑑𝑠+ℎ²₂𝑧^𝑇(𝑡)𝑄₂𝑧(𝑡)−ℎ₂^𝑡_𝑡−ℎ2𝑧^𝑇(𝑠)𝑄₂𝑧(𝑠)𝑑𝑠. (17) By Lemma 1, (6), and (17), we have that 𝑉̇(𝑡)≤2𝑦^𝑇(𝑡)𝑃𝐿𝑦(𝑡)+𝐴𝜂₁(𝑡)+Δ𝜂₂(𝑡)+ℎ²₁𝑧^𝑇(𝑡)𝑄₁𝑧(𝑡)−𝜂^𝑇₁(𝑡)𝑄₁𝜂₁(𝑡)+ℎ²₂𝑧^𝑇(𝑡)𝑄₂𝑧(𝑡)−𝜂^𝑇₂(𝑡)𝑄₂𝜂₂(𝑡)=𝜉^𝑇(𝑡)Ω+ℎ²₁Φ^𝑇𝑄₁Φ+ℎ²₂Φ^𝑇𝑄₂Φ𝜉(𝑡), (18) where 𝜉(𝑡)=(𝑦^𝑇(𝑡),𝜂^𝑇₁(𝑡),𝜂^𝑇₂(𝑡))^𝑇 and Φ and Ω are defined by (14). From (13) and (18), we have that 𝑉̇(𝑡)<0 for 𝑡≥0. Therefore, we eventually conclude that lim_𝑡→∞𝑦(𝑡)=0. This completes the proof of Theorem 3.

The method used in Theorem 3 can also be applied to the 𝐻_∞ consensus problem of system (7).

Theorem 4. For given constants ℎ₁>0 and ℎ₂>0, if there exist positive-definite matrices 𝑃, 𝑄₁, and 𝑄₂ of appropriate dimensions and positive constants 𝛼>0 and 𝛽>0 such that the following linear matrix inequality holds: 𝑃𝐿+𝐿^𝑇𝑃+𝛼𝐼_𝑛−1𝑃𝐴𝑃Δ𝑃Σℎ₁𝐿^𝑇𝑄₁ℎ₂𝐿^𝑇𝑄₂∗−𝑄₁00ℎ₁𝐴^𝑇𝑄₁ℎ₂𝐴^𝑇𝑄₂∗∗−𝑄₂0ℎ₁Δ^𝑇𝑄₁ℎ₂Δ^𝑇𝑄₂∗∗∗−𝛽𝐼_𝑛−1ℎ₁Σ^𝑇𝑄₁ℎ₂Σ^𝑇𝑄₂∗∗∗∗−𝑄₁0∗∗∗∗∗−𝑄₂<0, (19)where Σ=𝐸Σ, 𝐴, Δ, 𝐿, and 𝐿 are defined as above, then system (7) solves 𝐻_∞ consensus with 𝛾=𝛽/𝛼.

Proof. Choose the Lyapunov function defined by (15). First, (19) implies that (12) holds. By Theorem 3, system (1) achieves consensus asymptotically. Next, we show that (10) holds with 𝛾=𝛽/𝛼 for all nonzero 𝑤∈𝐿₂[0,∞) under zero initial condition. In fact, similar to the computation in Theorem 3, we have 𝑉̇(𝑡)≤2𝑦^𝑇(𝑡)𝑃𝐿𝑦(𝑡)+𝐴𝜂₁(𝑡)+Δ𝜂₂(𝑡)+Σ𝑤(𝑡)+ℎ²₁𝑧^𝑇(𝑡)𝑄₁𝑧(𝑡)−𝜂^𝑇₁(𝑡)𝑄₁𝜂₁(𝑡)+ℎ²₂𝑧^𝑇(𝑡)𝑄₂𝑧(𝑡)−𝜂^𝑇₂(𝑡)𝑄₂𝜂₂(𝑡), (20) which implies that 𝑉̇(𝑡)+𝛼𝑦^𝑇(𝑡)𝑦(𝑡)−𝛽𝑤^𝑇(𝑡)𝑤(𝑡)≤𝜉̃^𝑇Ω+ℎ²₁Φ^𝑇𝑄₁Φ+ℎ²₂Φ^𝑇𝑄₂Φ𝜉̃, (21) where 𝜉̃=(𝑦^𝑇(𝑡),𝜂^𝑇₁(𝑡),𝜂^𝑇₂(𝑡),𝑤^𝑇(𝑡))^𝑇, Φ=,𝑃𝐿𝐿,𝐴,Δ,ΣΩ=𝐿+^𝑇𝑃+𝛼𝐼_𝑛−1𝑃𝐴𝑃Δ𝑃Σ∗−𝑄₁00∗∗−𝑄₂0∗∗∗−𝛽𝐼_𝑛−1. (22) By (19), (21), and (22), we have that 𝑉̇(𝑡)+𝛼𝑦^𝑇(𝑡)𝑦(𝑡)−𝛽𝑤^𝑇(𝑡)𝑤(𝑡)<0,𝑡≥0. (23) Integrating (23) from 0 to ∞ under zero initial condition, we get ^∞₀𝑦^𝑇(𝑡)𝑦(𝑡)𝛽𝑑𝑡≤𝛼^∞₀𝑤^𝑇(𝑡)𝑤(𝑡)𝑑𝑡. (24) This completes the proof of Theorem 4.

Remark 5. Similar to the analysis in the proof of Theorem 3, we see that a necessary condition to Theorem 3 is that the digraph 𝒢 has a spanning tree. That is, the linear matrix inequality (19) implies that 𝒢 has a spanning tree.

Remark 6. Unlike most of consensus analysis for multiagent systems, it does not require that 𝑎_𝑖𝑗≥0 for all 𝑖≠𝑗 in the proofs of Theorems 3 and 4. Therefore, even when 𝑎_𝑖𝑗<0 for some 𝑖≠𝑗, system (7) can also solve 𝐻_∞ consensus under appropriate conditions.

The method used in this paper can also be extended to the case of switching topology. Consider the following multiagent system with switched topologies and heterogeneous delays: 𝑥̇_𝑖(𝑡)=^𝑛_𝑗=1𝑎^𝜎_𝑖𝑗𝑥_𝑗𝑡−𝑑₁(𝑡)−𝑥_𝑖𝑡−𝑑₂(𝑡),𝑖∈𝒩, (25) where 𝑡≥0, 𝜎(𝑡)∶[0,∞)→Γ={1,2,…,𝑝} is a switching signal that determines which subsystem is active at time 𝑡, and 𝑎^𝑘_𝑖𝑗≥0, 𝑖,𝑗∈𝒩, 𝑘∈Γ, are entries of the weighted adjacency matrix 𝐴_𝑘. When 𝜎(𝑡)=𝑘∈Γ, we denote the involved digraph by 𝒢_𝑘=(𝑉,𝐴_𝑘,𝐸_𝑘).

Under the reduced-order transformation (2), system (25) can be described by the following two systems: 𝑦̇(𝑡)=𝐸𝐿_𝜎𝐹𝑦(𝑡)+𝐸𝐴_𝜎^𝑡−𝑑1_𝑡^(𝑡)𝑧(𝑠)𝑑𝑠+𝐸Δ_𝜎^𝑡_{𝑡−𝑑2(𝑡)}𝑧(𝑠)𝑧(𝑡)𝑑𝑠,=𝐿_𝜎𝐹𝑦(𝑡)+𝐴_𝜎^𝑡−𝑑1_𝑡^(𝑡)𝑧(𝑠)𝑑𝑠+Δ_𝜎^𝑡_{𝑡−𝑑2(𝑡)}𝑧(𝑠)𝑑𝑠, (26) where 𝑦 and 𝑧 are defined as above and 𝐿_𝑘, 𝐴_𝑘, and Δ_𝑘 (𝑘∈Γ) relative to the digraph 𝒢_𝑘 are defined as 𝐿, 𝐴, and Δ, respectively. Let the Lyapunov function defined by (15) be the common Lyapunov function for the switched system (26). Then, similar to the proof of Theorem 3, we have the following consensus result in the case of switching topology.

Theorem 7. For given constants ℎ₁>0 and ℎ₂>0, system (25) solves consensus under arbitrary switching, if there exist positive-definite matrices 𝑃, 𝑄₁, and 𝑄₂ of appropriate dimensions such that the following linear matrix inequality 𝑃𝐿_𝑘+𝐿^𝑇_𝑘𝑃𝑃𝐴_𝑘𝑃Δ_𝑘ℎ₁𝐿^𝑇_𝑘𝑄₁ℎ₂𝐿^𝑇_𝑘𝑄₂∗−𝑄₁0ℎ₁𝐴^𝑇_𝑘𝑄₁ℎ₂𝐴^𝑇_𝑘𝑄₂∗∗−𝑄₂ℎ₁Δ^𝑇_𝑘𝑄₁ℎ₂Δ^𝑇_𝑘𝑄₂∗∗∗−𝑄₁0∗∗∗∗−𝑄₂<0 (27) holds for 𝑘∈Γ, where 𝐿_𝑘=𝐸𝐿_𝑘𝐹, 𝐴_𝑘=𝐸𝐴_𝑘, Δ_𝑘=𝐸Δ_𝑘, and 𝐿_𝑘=𝐿_𝑘𝐹.

Similarly, for the following switched multiagent system with disturbance input and heterogeneous delays 𝑥̇_𝑖(𝑡)=^𝑛_𝑗=1𝑎^𝜎_𝑖𝑗𝑥_𝑗𝑡−𝑑₁(𝑡)−𝑥_𝑖𝑡−𝑑₂(𝑡)+𝑤_𝑖𝑗(𝑡),𝑖∈𝒩, (28) we have the following 𝐻_∞ consensus result.

Theorem 8. For given constants ℎ₁>0 and ℎ₂>0, if there exist positive-definite matrices 𝑃, 𝑄₁, and 𝑄₂ of appropriate dimensions and positive constants 𝛼>0 and 𝛽>0 such that the following linear matrix inequality𝑃𝐿_𝑘+𝐿^𝑇_𝑘𝑃+𝛼𝐼_𝑛−1𝑃𝐴_𝑘𝑃Δ_𝑘𝑃Σ_𝑘ℎ₁𝐿^𝑇_𝑘𝑄₁ℎ₂𝐿^𝑇_𝑘𝑄₂∗−𝑄₁00ℎ₁𝐴^𝑇_𝑘𝑄₁ℎ₂𝐴^𝑇_𝑘𝑄₂∗∗−𝑄₂0ℎ₁Δ^𝑇_𝑘𝑄₁ℎ₂Δ^𝑇_𝑘𝑄₂∗∗∗−𝛽𝐼_𝑛−1ℎ₁Σ^𝑇_𝑘𝑄₁ℎ₂Σ^𝑇_𝑘𝑄₂∗∗∗∗−𝑄₁0∗∗∗∗∗−𝑄₂<0 (29)holds for 𝑘∈Γ, where 𝐿_𝑘, 𝐴_𝑘, Δ_𝑘, Σ_𝑘, Σ_𝑘, and 𝐿_𝑘 relative to the digraph 𝒢_𝑘 are defined as 𝐿, 𝐴, Δ, Σ, Σ, and 𝐿 in Theorem 4, then system (28) solves 𝐻_∞ consensus with 𝛾=𝛽/𝛼 under arbitrary switching.

4. Numerical Examples

Consider the following four digraphs with six nodes shown in Figure 1, where the weights relative to the edges shown by solid lines and dashed lines are 1 and −0.5, respectively.

(a)

(b)

(c)

(d)

Figure 1 

Four digraphs: (a) 𝒢𝑎, (b) 𝒢𝑏, (c) 𝒢𝑐, and (d) 𝒢𝑑.

When 𝒢=𝒢_𝑎, we have that (12) is feasible for given ℎ₁≤0.38 and ℎ₂≤0.24. By Theorem 3, system (1) achieves consensus asymptotically. The state trajectory under the stochastic initial condition is shown in Figure 2. In Remark 6, we show that Theorem 3 can also be applied to the extreme case when part of weights 𝑎_𝑖𝑗 is negative. For example, if 𝒢=𝒢_𝑏, we have that (12) is still feasible for given ℎ₁≤0.26 and ℎ₂≤0.22. Therefore, system (1) also solves consensus asymptotically even when there exist negative weights 𝑎₅₁=𝑎₅₃=−0.5. The state trajectory under the stochastic initial condition is shown in Figure 3.

Figure 2 

State trajectory of system (1) with 𝒢=𝒢𝑎.

Figure 3 

State trajectory of system (1) with 𝒢=𝒢𝑏.

For the case of disturbance input, assume that 𝒢=𝒢_𝑐 and 𝑤_𝑖𝑗(𝑡)=𝑤_𝑗(𝑡) for 𝑖,𝑗∈𝒩. Therefore, Σ=𝐴. For given ℎ₁=0.2 and ℎ₂=0.1, we get from (19) that 𝛾=43.0731. By Theorem 4, we have that system (7) solves 𝐻_∞ consensus with 𝛾=43.0731. If we let 𝑤(𝑡)=(0.1sin𝑡)𝟏,0≤𝑡≤30,0.0,otherwise, (30) the trajectory of system (7) under a stochastic initial condition is shown in Figure 4.

Figure 4 

State trajectory of system (7) with 𝒢=𝒢𝑐 and given disturbance.

For the case of switching topologies {𝒢_𝑐,𝒢_𝑑}, we have that (27) is feasible for given ℎ₁≤0.24 and ℎ₂≤0.18. By Theorem 7, system (25) solves consensus under arbitrary switching.

5. Conclusions

In this paper, we first apply the linear matrix inequality method to consensus of the single integrator multiagent system with heterogeneous time-varying delays in directed networks. Unlike the case of identical delays, the multiagent system with heterogeneous delays usually cannot be transformed to a reduced-order system. To overcome such difficulty, we introduce a partially-reduced-order system and an integral system. As a result, the linear matrix inequality method becomes useful in the analysis of consensus and 𝐻_∞ consensus by constructing a particular Lyapunov function. The main results are also extended to the case of switching topologies. Finally, numerical examples and simulation results are given to illustrate the theoretical results.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant nos. 61174217 and 61374074) and Natural Science Foundation of Shandong Province (Grant nos. ZR2010AL002 and JQ201119).