Recently, the distributed control problem of multi-agent systems has drawn great attention for its broad potential applications in many areas such as formation control, ^{[1– 4]} flocking control, ^[5] and consensus control.^{[6, 7]} Compared with traditional control systems, agents in the multi-agent systems are coupled through networks which are usually modeled by directed or undirected graphs. The system behavior depends not only on the individual agent dynamic, but also on the structure of the networks.^[7] Due to some physical constrains such as limited resources and energy or short communication ranges, individual agent cannot obtain the global information of the system. Thus, only distributed controllers can be used with local information of neighbor agents. The distributed controllers have many advantages such as flexible scalability, high robustness, and low costs. It has been studied from different perspectives and numerous results have been obtained.

In Ref.  [1], Jadbabaie et al.  considered the coordination problem of groups of mobile autonomous agents and provided a theoretical explanation for the observed behavior of the Vicsek^[6] model. Based on algebraic graph theory, Fax and Murry analyzed the formation stability under directed communication topologies.^[2] In Ref.  [5], a theoretical framework for the design and analysis of distributed flocking algorithms was presented by Olfati-Saber. In Ref.  [7], Ren and Beard proved that the consensus of a single integrator dynamic could be achieved if the union of the directed interaction graphs had a spanning tree. A consensus of general linear invariant dynamics was studied with static feedback controllers in Refs.  [8]– [10]. An observer-type dynamic consensus protocol was proposed by Li et al. in Ref.  [11] and the concept of a consensus region was introduced. In Ref.  [12], a decomposition approach was used to study the H_∞ control problem of identical dynamically coupled systems subject to external disturbances. Under undirected graphs, based on local relative output information, Li et al. proposed dynamic H_∞ controllers.^[13] As an extension of consensus regions, Li et al. introduced the notions of the H_∞ and H₂ performance regions under undirected graphs and proved that the unbounded H_∞ performance region was independent of the communication topology as long as it was connected.^[14] In Ref.  [15], Wang et al. proposed a control protocol to solve the H_∞ consensus problem synthesized with transient performance. In Refs.  [16] and [17], the H_∞ consensus problems of multi-agent systems subjected to external disturbances were solved under directed strongly connected topologies. In Refs.  [18]– [22], the leader– following tracking problems were also considered. However, until now, the H_∞ control problems of general linear multi-agent systems with directed communication topology which just contains a directed spanning tree without the strongly connected condition have not been investigated.

In this paper, we investigate the distributed H_∞ control problem of general identical linear time invariant multi-agent systems subject to external disturbances. A directed graph is used to model the communication topology. It is assumed that the directed graph has a spanning tree and there is at least one root node with a loop. Based on the local relative states information of the neighbor agents and a subset of absolute information of the agents, distributed H_∞ controllers are constructed. Sufficient conditions are proposed based on the bounded real lemma and algebraic graph theory. Then the concept of H_∞ performance regions proposed in Ref.  [14] is extended to the directed graph situations. It is proved that the H_∞ performance regions of multi-agent systems under directed communication topologies depend on both the communication topologies and system dynamics. Compared with the existing results in Refs.  [13] and [14] where the communication topologies are assumed to be undirected and the results in Ref.  [16] and [17] where the communication topologies are assumed to be strongly connected, the main contribution of this article is that the H_∞ control problem is solved under directed communication topology which just contains a spanning tree. The conclusions obtained here can consider the results in Ref.  [14] as a special case. Then the results are applied to solve the leader– follower H_∞ consensus problem.

The remainder of the paper is organized as follows. In Section 2, some necessary concepts and the notation are introduced. In Section 3, the H_∞ control problem of multi-agent systems subjected to external disturbances is addressed. In Section 4, the results are applied to solve the leader– follower H_∞ consensus problem. In Section 5, a simulation example is presented. Section 6 is the conclusion.

In this paper, the following notations will be used. R^{n× n} and C^{n× n} denote the set of n× n real and complex matrices, respectively. For μ ∈ C, the real part is Re(μ ). ⊗ denotes the Kronecker product. I_n is the identity matrix of order n. For a square matrix A, λ (A) denotes the eigenvalues of matrix A, and denotes the maximal singular value. A > 0 (A ≥ 0) means that A is positive definite (respectively, positive semidefinite). (A, B) is said to be stabilizable if there exists a real matrix K such that A + BK is Hurwitz. (A, C) is said to be detectable if and only if (A^T, C^T) is stabilizable. Denote by ω _i(t) ∈ L₂[0, ∞ ) the space of square integrable functions over [0, ∞ ).

A directed graph contains the vertex set the directed edges set the adjacency matrix with nonnegative elements a_ij. a_ij = 1 if there is a directed edge between vertex i and j, a_ij = 0, otherwise. The vertex i has a loop, which means that vertex i can get its own information. The set of neighbors of i is defined as . The Laplacian matrix of the topology is defined as , where and . Then zero is an eigenvalue of with the eigenvector 1_N. A directed graph is said to have a spanning tree if there is a node such that there is a directed path from this node to every other node. A directed graph is strongly connected if there is a directed path from every node to every other node.

Lemma 1 (bounded real lemma^[23]) For a positive scalar γ > 0 and the transfer function G(s) = C(sI − A)^{− 1}B + D, then the following are equivalent:

(i) Re(λ (A)) < 0 and ∥ G (s)∥ _∞ < γ .

(ii) and there exists a positive definite matrix P such that

Lemma 2 (Schur complement lemma^[24]) Given a matrix , where S ∈ R^{n× n}, S₁₁ ∈ R^{r× r}, then the following are equivalent:

(i) S < 0.

(ii)

(iii)

Lemma 3^[25] Suppose that the eigenvalues of A ∈ R^{N× N} have positive real parts, then there exists a positive definite matrix Q > 0 such that A^TQ + QA > 0.

Consider a multi-agent system composed of N agents with identical general linear time invariant dynamics

where x_i(t) ∈ Rⁿ, u_i(t) ∈ R^p, and ω _i(t) ∈ L₂[0, ∞ ) are the state, the control input, and the external disturbance, respectively. z_i ∈ R^q is the performance variable. A, B, C, D₁, and D₂ are constant system matrices with compatible dimensions. It is assumed that (A, B) is stabilizable, and (A, C) is detectable. Here, a directed graph is used to model the communication topologies. The following assumption is introduced.

Assumption 1 The directed graph has a directed spanning tree and at least one root node has a loop.

Based on this assumption, the following distributed static consensus controller is proposed:

where K ∈ R^{p× n} is the feedback matrix to be designed, c is the coupling strength to be selected, and a_ij are the elements of the adjacency matrix of the communication topology. g_i = 1 means that the agent i knows its own absolute state information; g_i = 0, otherwise. Then, according to Assumption 1,

Then, the closed-loop system dynamics (1) using the controller (2) is

where is the Laplacian matrix of the graph. G = diag{g₁, g₂, … , g_N}. Denote by T_{ω z} the transfer function matrix from ω to z of system  (3).

Lemma 4^[7] Zero is a simple eigenvalue of and all the other nonzero eigenvalues have positive real parts if and only if the graph has a directed spanning tree. Furthermore, if there is a root agent i such that g_i ≠ 0, G = diag {g₁, g₂, … , g_N} then Re .

Remark 1 If the graph is assumed to be directed and strongly connected or undirected and connected, every agent can be seen as a root node. Furthermore, if then .

Lemma 5 Under Assumption 1, there exists a positive definite matrix Q and a positive scalar α such that

where .

Proof According to Lemma 4, one can obtain that , then

Thus, by Lemma  3, there exists a positive definite matrix Q such that This completes the proof.

Definition 1 The suboptimal H_∞ control problem of system  (1) is said to be solved under the controller (2) with a given allowable , if the following requirements are satisfied:

(i) The system is asymptotically stable.

(ii) The H_∞ norm of T_{ω z} is less than γ , i.e., ∥ T_{ω z}∥ _∞ < γ .

Theorem 1 The suboptimal H_∞ control problem of multi-agent system (1) with the controller (2) is solved with disturbance attenuation if a positive definite matrix P and a positive scalar τ exist such that

where , β = max {λ (Q)}, and φ = min {λ (Q)}. The Q is a positive definite solution of Eq.  (4) and the coupling strength is chosen such that c ∈ S_γ , where S_γ = [τ /α , ∞ ), and the feedback matrix is designed as K = B^TP.

Proof According to bounded real lemma, if there exists a positive definite matrix such that

where , system (3) is asymptotically stable and satisfies∥ T_{ω z} ∥ _∞ < γ . Here, we chose P̄ = Q ⊗ P and K = B^TP, where Q > 0 is a solution of Eq.  (4) and P > 0 is a solution of inequality (5). Let , then

From Lemma  5, one has

where Since Q < β I_N and Q > φ I_N, where β = max {λ (Q)}and φ = min {λ (Q)}, one can obtain that

From Eqs.  (7)– (10), one has

Since cα ≥ τ , then inequality  (5) implies that inequality  (6) holds. Thus, the H_∞ control problem for system  (1) is solved.

Remark 2 Note that the Laplacian matrix of a directed graph is not symmetric. Thus, the similarity transformation employed in Ref.  [14] is not applicable here. We chose to solve the H_∞ control problem directly using the bounded real lemma. From the proof of Theorem 1, we can see that the key tools leading to the above results rely on Lemma 5 and the properly designed matrix P̄ .

Remark 3 In Ref.  [14], the notion of H_∞ performance regions is introduced under undirected communication topologies to evaluate the performance of a multi-agent network subject to external disturbance. It is proved that the unbounded H_∞ performance region is independent of the communication topology as long as it is connected. However, under directed communication topologies, we can see that the unbounded H_∞ performance region defined as S_γ = [τ /α , ∞ ) depends on both the communication topology and system dynamics.

Remark 4 Under Assumption 1, if the communication topology is modeled by an undirected graph, matrix is positive definite. Thus, in Lemma  5, Q = I_N and β = φ = 1. Then inequality  (5) can be rewritten as

This conclusion is consistent with the existing results in Ref.  [14]. Thus, the conclusion obtained here can consider the results in Ref.  [14] as a special case. Furthermore, according to Remark 1, if the directed graph is assumed to be strongly connected, the topologies assumption in Assumption 1 can be relaxed so that there is at least one node with a loop.

Similar to Ref.  [14], for a given graph, the H_∞ performance limit can be obtained by the following corollary.

Corollary 1 Given a directed graph , solving the following optimization problem, we can obtain the H_∞ performance limit γ _min of the multi-agent system (1) under the controller (2): minimize γ subject to inequality (5), with P > 0, τ > 0, and .

In this section, the leader– follower H_∞ consensus problem is discussed. The followers’ dynamics are described in (1). The leader’ s dynamics is described as

where x₀ ∈ Rⁿ, and A is the same as the followers’ . Here, we define the consensus performance variable as

where z_i ∈ R^p, and C is a constant matrix such that (A, C) is detectable.

The following distributed consensus controllers are proposed as:

where K is the feedback gain matrix to be designed. h_i = 1 if agent i can obtain the leader’ state information; h_i = 0, otherwise.

Then the closed-loop dynamics can be written as

where and H = diag {h₁, h₂, … , h_N}. Denote by T̂ _{ω z} the transfer function matrix from ω to z of the system  (12).

Definition 2 Given a positive scalar γ > 0, the leader– follower H_∞ consensus problem of system (12) is said to be solved, if the following requirements are satisfied.

Under the leader– follower structure, we need the following assumption.

Assumption 2 The graph has a spanning tree and at least one root agent has access to the leader.

Let x̂ _i = x_i − x₀, then the closed-loop dynamics with the controller (12) can be rewritten as

where . From Assumption 2 and Lemma 4, all the eigenvalues of matrix have positive real parts. Then from Lemma 5, there exist a positive definite matrix W and a positive scalar such that

where

Until now, the leader– follower H_∞ consensus problem has been converted to an H_∞ control problem. Thus, a similar conclusion can be obtained.

Theorem 2 The leader– follower H_∞ consensus problem is solved with disturbance attenuation if there exist a positive definite matrix P̂ and a positive scalar τ such that

where , and . The W is a positive definite solution of (14) and the coupling strength is chosen such that c ∈ Ŝ _γ , where , and the feedback matrix is designed as K = B^TP̂ .

In this section, we provide some examples to illustrate the effectiveness of the above theoretical results. A multi-agent system consisting five agents is considered. The system matrices are defined as

The external disturbance is defined as , where

Fig.  1.

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	Fig.  1. Communication topologies.

Example 1 In this example, the distributed H_∞ control problem is considered. The directed communication topology is given in Fig.  1(a). Clearly, the communication topology has a directed spanning tree. Agents 1 and 5 are the roots, and agent 1 can obtain its own absolute state information. The Laplacian matrix of communication topology is

and G = diag{1, 0, 0, 0, 0}. The real part of the smallest eigenvalue of is 0.3820. According to Lemma 5, we set α = 0.6639 and obtain a feasible solution of (4),

Thus, β = max {λ (Q)} = 21.5845, and φ = min {λ (Q)} = 0.0921. Set γ = 2 and τ = 1. Solving inequality (5), we obtain a feasible solution

According to Theorem  1, the feedback matrix and coupling stength can be chosen as K = [1.7839 0.6025] and c = 1.6062, respectivly. The H_∞ performance region is [1.5062, ∞ ).

Figures  2 and 3 show the state trajectories of all the agents. Figure  4 shows the trajectories of the performance variables Z_i with external disturbances.  It is shown that states of all the agents converge to zero.

Fig.  2.

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	Fig.  2. Trajectories of the first states of all agents.

Fig.  3.

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	Fig.  3. Trajectories of the second states of all agents.

Fig.  4.

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	Fig.  4. Trajectories of performance variables.

Example 2 In this example, the leader– follower H_∞ consensus problem is considered. The directed communication topology is given in Fig.  1(b). We can see that agents 1 and 2 can obtain the leader’ s information and agent 1 is the root. The Laplacian matrix is the same as example 1, and H = diag{1, 1, 0, 0, 0}. We can obtain a feasible solution of (14)

Thus, and . Set γ = 2 and τ = 1. Solving inequality (5), we obtain a feasible solution

According to Theorem 2, the feedback matrix and coupling stength can be chosen as K = [1.4171 0.3962] and c = 1.6062, respectivly.

Figures  5 and 6 show the state trajectories of the leader and follower agents. Figures  7 and 8 show the trajectories of the tracking error. Note that the tacking error converges to zero, and all follower agents can track the leader’ s path.

Fig.  5.

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	Fig.  5. Trajectories of the first states of all agents.

Fig.  6.

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	Fig.  6. Trajectories of the second states all of agents.

Fig.  7.

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	Fig.  7. Tracking error of the first states.

Fig.  8.

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	Fig.  8. Tracking error of the second states.

In this paper, the distributed H_∞ control problems of multi-agent systems subject to external disturbances have been investigated under directed communication topologies which have a spanning tree. Based on relative states of the neighbor agents and a subset of absolute information of the agents, local H_∞ controllers have been designed. The concept of an H_∞ performance region has been analyzed. It has been proved that, under directed networks, the H_∞ performance region depends on both the communication topology and system dynamics. Finally, the results have been applied to deal with the leader– follower H_∞ consensus problem.