Consensus of Continuous-Time Multiagent Systems with General Linear Dynamics and Nonuniform Sampling

Abstract

This paper studies the consensus problem of multiple agents with general linear continuous-time dynamics. It is assumed that the information transmission among agents is intermittent; namely, each agent can only obtain the information of other agents at some discrete times, where the discrete time intervals may not be equal. Some sufficient conditions for consensus in the cases of state feedback and static output feedback are established, and it is shown that if the controller gain and the upper bound of discrete time intervals satisfy certain linear matrix inequality, then consensus can be reached. Simulations are performed to validate the theoretical results.

1. Introduction

Consensus is one of the fundamental issues in the study of distributed control of multiagent systems, and it has wide applications in formation control of multiple robots, communication among sensor networks, cooperative control of unmanned aerial vehicles, and so forth. Much research work on consensus has been emerged, and most of the existing work focuses on the consensus problems of multiple agents with special dynamics, such as first-order dynamics (or single integrator) [1–10], second-order dynamics (or double integrators) [11–19], and high-order dynamics [20, 21]. Consensus problems of second-order continuous-time multiagent systems are studied in [12–14], where the information transmission among agents is intermittent. In [15–17], only the partial state of second-order agent can be measured, and thus, some static or dynamic output feedback controllers are designed. In [20–22], consensus is studied for high-order and nonlinear multiagent systems, respectively.

In recent years, the consensus of multiple agents with general linear dynamics has been paid more and more attention, such as [23–28], and the analysis of such multiagent systems is more challenging than the case of special dynamics. In [24], consensus is considered in the case of static output feedback, and it is proved that the consensus is equivalent to the Hurwitz stable or Schur stable of a constant matrix, which is determined by the topology and the system dynamics. By studying the stability of the constant matrix, it is shown that consensus can be reached for continuous-time multiagent systems if and only if the system is stabilizable and detectable and the topology has a spanning tree under some rank constraints, and a necessary condition is also provided for consensus of discrete-time multiagent systems. In [25], consensus of continuous-time and discrete-time multiagent systems under a dynamic output feedback controller, which is actually a state estimator, is investigated, respectively. By applying the result in [29], some sufficient and necessary conditions are presented in [25]. In [26], the joint effect of network topology, agent dynamics, and communication data rate on consensus of discrete-time multiagent systems is analyzed, and it is shown that under perfect state feedback, consensus is reached if and only if the dynamics of each agent is stabilizable and the unstable eigenvalues of each agent satisfy some constraints. In [27], the 𝐻_∞ consensus problem of continuous-time multiagent systems is studied under dynamic output feedback by applying the robust control theory and linear matrix inequality technique. In [28], consensus of discrete-time multiagent systems under a dynamic output feedback controller, which is actually an observer-type controller, is discussed, and the discrete-time consensus region is analyzed for neurally stable agents and unstable agents, respectively.

It should be mentioned that [23–25, 27] all study the consensus of continuous-time multiagent systems, and the information transmission among agents is continuous. However, due to the limitation of bandwidth, the cost of communication, the technique constraints, and so forth, it is possible to transmit information in the intermittent manner. In addition, sampled-data control has many favorable properties, such as flexibility, robustness, and low cost see [30] for further details. Hence, it is also necessary to study the consensus of general continuous-time multiagent systems with intermittent information transmission. To the authors’ best knowledge, there is little research work reported on this problem. Based on the previous consideration, we analyze the consensus of continuous-time multiagent systems with general dynamics, where each agent can only obtain the information of other agents at discrete times. Moreover, the discrete time intervals may not be equal, which often occurs in the event-driven systems or networked control systems [31]. The sufficient condition for consensus and the method to design controller gain are presented.

This paper is organized as follows. In Section 2, we present some concepts in graph theory and formulate the model to be studied. In Section 3, main results are stated. In Section 4, simulations are provided to illustrate the effectiveness of the theoretical results. Conclusion remarks are made in Section 5.

Notations. Let 𝐼 or 𝐼_𝑛∈ℝ^𝑛×𝑛 be an identity matrix, and 𝟏_𝑛=[1,…,1]^𝑇∈ℝ^𝑛; for symmetric matrices 𝐴 and 𝐵, 𝐴<𝐵 (resp., 𝐴≤𝐵) means that 𝐴−𝐵 is a negative definite (resp., negative semidefinite) matrix; ⊗ denotes the Kronecker product operator, and 𝐶⊗𝐷=[𝑐_𝑖𝑗𝐷], where 𝐶=[𝑐_𝑖𝑗] and 𝐷=[𝑑_𝑖𝑗] are two matrices.

2. Preliminaries

2.1. Graph Theory

Some basic definitions in graph theory [32] are first introduced.

A directed graph 𝒢 consists of a vertex set 𝒱(𝒢) and an edge set ℰ(𝒢), where 𝒱(𝒢)={𝑣₁,…,𝑣_𝑛} and ℰ(𝒢)⊂{(𝑣_𝑗,𝑣_𝑖)∶𝑣_𝑗,𝑣_𝑖∈𝒱(𝒢)}. For edge (𝑣_𝑗,𝑣_𝑖), 𝑣_𝑗 is called the parent vertex of 𝑣_𝑖 and 𝑣_𝑖 is called the child vertex of 𝑣_𝑗. The set of neighbors of vertex 𝑣_𝑖 is defined by 𝑁(𝒢,𝑣_𝑖)={𝑣_𝑗∶(𝑣_𝑗,𝑣_𝑖)∈ℰ(𝒢)and𝑗≠𝑖}, and the associated index set is denoted by 𝑁(𝒢,𝑖)={𝑗∶𝑣_𝑗∈𝑁(𝒢,𝑣_𝑖)}. A (directed) path from 𝑣_𝑖1 to 𝑣_𝑖𝑘 is a sequence, 𝑣_𝑖1,…,𝑣_𝑖𝑘, of distinct vertices such that (𝑣_𝑖𝑗,𝑣_𝑖𝑗+1)∈ℰ(𝒢),𝑗=1,…,𝑘−1. A directed graph 𝒢 is strongly connected if there is a path from every vertex to every other vertex. A directed tree is a directed graph, where every vertex except one special vertex has exactly one parent vertex, and the special vertex, called root vertex, has no parent vertices and can be connected to any other vertices via paths. A subgraph 𝒢_𝑠 of 𝒢 is a graph such that 𝒱(𝒢_𝑠)⊂𝒱(𝒢) and ℰ(𝒢_𝑠)⊂ℰ(𝒢). 𝒢_𝑠 is said to be a spanning subgraph if 𝒱(𝒢_𝑠)=𝒱(𝒢). For any 𝑣_𝑖,𝑣_𝑗∈𝒱(𝒢_𝑠), if (𝑣_𝑖,𝑣_𝑗)∈ℰ(𝒢_𝑠)⇔(𝑣_𝑖,𝑣_𝑗)∈ℰ(𝒢), then 𝒢_𝑠 is said to be an induced subgraph of 𝒢 and 𝒢_𝑠 is also said to be induced by 𝒱(𝒢_𝑠). A spanning tree of 𝒢 is a directed tree which is a spanning subgraph of 𝒢. 𝒢 is said to have a spanning tree if some edges form a spanning tree of 𝒢.

A matrix is called nonnegative if each of its elements is nonnegative. A weighted directed graph 𝒢(𝐹) is a directed graph 𝒢 plus a nonnegative matrix 𝐹=[𝑓_𝑖𝑗]∈ℝ^𝑛×𝑛, where 𝑓_𝑖𝑗>0⇔(𝑣_𝑗,𝑣_𝑖)∈ℰ(𝒢), and 𝑓_𝑖𝑗 is called the weight of edge (𝑣_𝑗,𝑣_𝑖). The Laplacian matrix 𝐿=[𝑙_𝑖𝑗]∈ℝ^𝑛×𝑛 of 𝒢(𝑓) is defined as 𝑙_𝑖𝑗=−𝑓_𝑖𝑗,𝑖≠𝑗,_{𝑛𝑠=1,𝑠≠𝑖}𝑓_𝑖𝑠,𝑖=𝑗. (1)

The Laplacian matrix of 𝒢(𝐹) has the following properties.

Lemma 1 (see [2]). (i)Zero is an eigenvalue of 𝐿, and 𝟏_𝑛 is the associated right eigenvector.(ii)Zero is an algebraically simple eigenvalue of 𝐿, and all the other eigenvalues are with positive real parts if and only if 𝒢(𝐹) has a spanning tree.

2.2. Model

Consider a group of agents with the following general continuous-time dynamics: 𝑥̇_𝑖(𝑡)=𝐴𝑥_𝑖(𝑡)+𝐵𝑢_𝑖(𝑡),𝑦_𝑖(𝑡)=𝐶𝑥_𝑖(𝑡),𝑖=1,2,…,𝑛, (2) where 𝑥_𝑖∈ℝ^𝑚, 𝑢_𝑖∈ℝ^𝑝 and 𝑦_𝑖∈ℝ^𝑞, are the state, control input, and output of agent 𝑖, respectively, and 𝐴, 𝐵, 𝐶 are constant matrices.

Given 𝑢_𝑖, 𝑖=1,…,𝑛, we say that 𝑢_𝑖 or multiagent system (2) solves a consensus problem asymptotically if lim_𝑡→∞(𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡))=0, 𝑖,𝑗=1,2,…,𝑛, for any initial states.

In this paper, we consider the case of intermittent information transmission; namely, each agent can only obtain the information of its neighbors at some discrete times 𝑡₀,𝑡₁,…, where 0≤𝑡₀<𝑡₁<𝑡₂<⋯. Let 𝑇_𝑘=𝑡_𝑘+1−𝑡_𝑘, and assume that 𝑇_𝑘≤𝑑, 𝑘=0,1,…; namely, all discrete time intervals have a common upper bound. Note that all 𝑇_𝑘 may not be equal.

In the case that the (relative) state of each agent can be measured directly, we consider the following control input: 𝑢_𝑖(𝑡)=𝐾_{()𝑗∈𝑁𝒢,𝑖}𝑎_𝑖𝑗𝑥_𝑗𝑡_𝑘−𝑥_𝑖𝑡_𝑘,𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑖=1,2,…,𝑛, (3) where 𝐾 is the controller gain to be designed.

In some times, the state of each agent cannot be measured directly, and thus, we also consider the following static output feedback controller: 𝑢_𝑖(𝑡)=𝐾_{()𝑗∈𝑁𝒢,𝑖}𝑎_𝑖𝑗𝑦_𝑗𝑡_𝑘−𝑦_𝑖𝑡_𝑘,𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑖=1,2,…,𝑛. (4)

Remark 2. Obviously, if 𝐴 is Hurwitz stable, then consensus can be reached for 𝐾=0. Hence, we assume that 𝐴 is not Hurwitz stable in this paper.

3. Main Results

In this section, we will present a sufficient condition for consensus under controllers (3) and (4), respectively, and the methods to design controller gains are also provided.

Let 𝑋(𝑡)=(𝑥^𝑇₁(𝑡),𝑥^𝑇₂(𝑡),…,𝑥^𝑇_𝑛(𝑡))^𝑇, then multiagent system (2) under controller (3) can be written as 𝑋̇(𝑡)=𝐼_𝑛⊗𝐴𝑋(𝑡)−(𝐿⊗(𝐵𝐾))𝑋𝑡_𝑘,𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑘=0,1,…. (5)

By Lemma 1, there exists an invertible matrix 𝑈, the first column of which is 𝟏_𝑛, such that 𝑈⁻¹𝐿𝑈=0𝛼0𝐻, (6) where 𝐻∈ℝ^{(𝑛−1)×(𝑛−1)}. Let 𝛿(𝑡)=(𝑈⁻¹⊗𝐼_𝑚)𝑋(𝑡); then 𝛿̇₁(𝑡)=𝐴𝛿₁(𝑡)−(𝛼⊗(𝐵𝐾))𝛿𝑡_𝑘,̇𝛿(𝑡)=𝐼_𝑛−1⊗𝐴𝛿(𝑡)−(𝐻⊗(𝐵𝐾))𝛿𝑡_𝑘, (7)(8) where 𝛿(𝑡)=(𝛿₁(𝑡)^𝑇,𝛿(𝑡)^𝑇)^𝑇, 𝛿₁(𝑡)∈ℛ^𝑚.

By the previous state transformation, it is easy to obtain the following lemma.

Lemma 3. Controller (3) solves a consensus problem asymptotically if and only if system (8) is asymptotically stable.

Proof. Sufficiency. Let 𝑈=[𝟏_𝑛,𝑈₁], where 𝑈₁∈ℛ^{𝑛×(𝑛−1)}. By 𝑋(𝑡)=(𝑈⊗𝐼_𝑚)𝛿(𝑡), we have 𝑋(𝑡)=𝟏_𝑛⊗𝛿₁(𝑡)+(𝑈₁⊗𝐼_𝑚)𝛿(𝑡). Clearly, if system (8) is asymptotically stable, namely, lim_𝑡→∞𝛿(𝑡)=0, then lim_𝑡→∞(𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡))=0, 𝑖,𝑗=1,2,…,𝑛.
Necessity. Let 𝑈⁻¹=(𝑢₂,𝑈^𝑇₂)^𝑇, where 𝑈₂∈ℛ^{(𝑛−1)×𝑛}. From 𝑈⁻¹𝑈=𝐼_𝑛, we have 𝑢^𝑇₂𝟏_𝑛=1, 𝑈₂𝟏_𝑛=0. Since consensus is reached, there exists 𝜙(𝑡)∈ℛ^𝑚 such that lim_𝑡→∞(𝑥_𝑖(𝑡)−𝜙(𝑡))=0, 𝑖=1,2,…,𝑛. By 𝛿(𝑡)=(𝑈⁻¹⊗𝐼_𝑚)𝑋(𝑡), 𝛿(𝑡)−𝑈⁻¹⊗𝐼_𝑚𝟏_𝑛(𝑡=𝑈⊗𝜙)₋₁⊗𝐼_𝑚𝑋(𝑡)−𝟏_𝑛(𝑡)(𝑡)−𝑢⊗𝜙⇒𝛿^𝑇2𝟏_𝑛𝑈₂𝟏_𝑛(𝑡)=𝑈⊗𝜙₋₁⊗𝐼_𝑚𝑋(𝑡)−𝟏_𝑛(𝑡)(𝑡)−10(𝑡)=𝑈⊗𝜙⇒𝛿⊗𝜙₋₁⊗𝐼_𝑚𝑋(𝑡)−𝟏_𝑛(𝑡)⇒𝛿⊗𝜙₁(𝑡)(𝑡)𝛿(𝑡)=𝑈−𝜙₋₁⊗𝐼_𝑚𝑋(𝑡)−𝟏_𝑛⊗𝜙(𝑡)⇒lim_𝑡→∞𝛿₁(𝑡)(𝑡)𝛿(𝑡)=𝑢−𝜙_𝑇2⊗𝐼_𝑚𝑈₂⊗𝐼_𝑚lim_𝑡→∞𝑋(𝑡)−𝟏_𝑛⊗𝜙(𝑡). (9) By lim_𝑡→∞(𝑋(𝑡)−𝟏_𝑛⊗𝜙(𝑡))=0, it is easy to obtain lim_𝑡→∞𝛿(𝑡)=0, which means that system (8) is asymptotically stable.

By Lemma 3, we will analyze the stability of system (8) by applying the input delay approach [33], which is an effective method to deal with the stability of continuous-time systems with intermittent input.

Let 𝜏(𝑡)=𝑡−𝑡_𝑘, and let 𝑡∈[𝑡_𝑘,𝑡_𝑘+1); then 𝑡_𝑘=𝑡−𝜏(𝑡), and system (8) can be rewritten as ̇𝛿(𝑡)=𝐼_𝑛−1𝛿(𝑡)−((𝛿((𝑡=𝐼⊗𝐴𝐻⊗𝐵𝐾))𝑡−𝜏))_𝑛−1()𝛿(𝑡)+((𝛿(𝑡)−𝛿((𝑡=𝐴𝛿(𝑡)+((𝜉(𝑡),⊗𝐴−𝐻⊗𝐵𝐾𝐻⊗𝐵𝐾))𝑡−𝜏))𝐻⊗𝐵𝐾)) (10) where 𝐴=𝐼_𝑛−1⊗𝐴−𝐻⊗(𝐵𝐾), 𝜉(𝑡)=𝛿(𝑡)−𝛿(𝑡−𝜏(𝑡)).

By 𝑇_𝑘≤𝑑, we have 0≤𝜏(𝑡)≤𝑑, 𝑘=0,1,…. Obviously, the stability of system (8) is equivalent to that of system (10). By analyzing the stability of system (10), we obtain the following main result.

Theorem 4. Assume that (𝐴,𝐵) is stabilizable and the topology graph has a spanning tree. Controller (3) solves a consensus problem asymptotically if there exist positive definite matrices 𝑃, 𝑅 such that 𝐾 and 𝑑 satisfy the following linear matrix inequality: 𝑃𝐴+𝐴^𝑇𝑃𝑃(𝐻⊗(𝐵𝐾))𝐴^𝑇𝑅1∗−𝑑𝑅(𝐻⊗(𝐵𝐾))^𝑇𝑅1∗∗−𝑑𝑅. (11)

Proof. Consider the following Lyapunov-Krasovskii functional for system (10): 𝑉(𝑡)=𝛿^𝑇(𝑡)𝑃𝛿(𝑡)+^𝑡_𝑡−𝑑()̇𝑠−𝑡+𝑑𝛿^𝑇(𝑠)𝑅̇𝛿(𝑠)𝑑𝑠, (12) where 𝑃>0, 𝑅>0, and then 𝑉̇(𝑡)=2𝛿^𝑇(𝑡)𝑃𝛿̇(𝑡)+−^𝑡_𝑡−𝑑̇𝛿^𝑇(𝑠)𝑅̇𝛿(𝑠)̇𝑑𝑠+𝑑𝛿^𝑇(𝑡)𝑅̇𝛿(𝑡). (13) By Lemma  4 in [34], ^𝑡_𝑡−𝑑̇𝛿^𝑇(𝑠)𝑅̇𝛿(𝑠)≥1𝑑𝑠𝑑𝛿(𝑡)−𝛿(𝑡−𝜏(𝑡))^𝑇𝑅𝛿(𝑡)−𝛿(𝑡−𝜏(𝑡)), (14) and thus, 𝑉̇(𝑡)≤2𝛿^𝑇(𝑡)𝑃𝛿̇(𝑡)−1𝑑𝛿(𝑡)−𝛿(𝑡−𝜏(𝑡))^𝑇𝑅𝛿(𝑡)−𝛿((𝑡̇𝛿𝑡−𝜏))+𝑑^𝑇(𝑡)𝑅̇𝛿(𝑡)=𝛿^𝑇(𝑡),𝜉^𝑇(𝑡)𝑄𝛿^𝑇(𝑡),𝜉^𝑇(𝑡)^𝑇, (15) where 𝑃𝐴𝑄=𝐴+^𝑇((1𝑃𝑃𝐻⊗𝐵𝐾))∗−𝑑𝑅𝐴+𝑑^𝑇(𝐻⊗(𝐵𝐾))^𝑇𝑅𝐴𝐻⊗(𝐵𝐾). (16) By Schur complement, 𝑄<0 if and only if (11) is satisfied. Hence, 𝑉̇(𝑡)<0; namely, system (10) is asymptotically stable. By Lemma 3, consensus is reached.

Remark 5. By [24], if (𝐴,𝐵) is stabilizable and the topology graph has a spanning tree, then there exists 𝐾 such that 𝐴 is Hurwitz stable; namely, there exists 𝑃>0 such that 𝑃𝐴+𝐴^𝑇𝑃<0. Obviously, if 𝑑>0 is small enough, then (11) must be satisfied. Hence, by (11), we can find 𝐾 and 𝑑 which ensure consensus.

Theorem 4 shows that if (𝐴,𝐵) is stabilizable and the topology graph has a spanning tree, then there exists controller gain 𝐾 and discrete times 𝑡₀,𝑡₁,…, where 𝑡_𝑘+1−𝑡_𝑘≤𝑑, 𝑘=0,1,…, such that consensus is reached. Moreover, 𝐾 and 𝑑 can be obtained by (11), which can be solved easily by the feasp solver in Matlab LMI Toolbox.

Similar to the analysis in the case of state feedback, the consensus under controller (4) is equivalent to the asymptotic stability of the following system: ̇𝛿(𝑡)=𝐴𝛿(𝑡)+(𝐻⊗(𝐵𝐾𝐶))𝜁(𝑡), (17) where 𝐴=𝐼_𝑛−1⊗𝐴−𝐻⊗(𝐵𝐾𝐶), 𝜁(𝑡)=𝛿(𝑡)−𝛿(𝑡−𝜏(𝑡)). By analyzing the stability of system (17), we obtain the following result.

Theorem 6. Assume that (𝐴,𝐵,𝐶) is stabilizable and detectable and the topology graph has a spanning tree. Controller (4) solves a consensus problem asymptotically if there exist positive definite matrices 𝑃,𝑅 such that 𝐾 and 𝑑 satisfy the following LMI: 𝑃𝐴+𝐴^𝑇𝑃𝑃(𝐻⊗(𝐵𝐾𝐶))𝐴^𝑇𝑅1∗−𝑑𝑅(𝐻⊗(𝐵𝐾𝐶))^𝑇𝑅1∗∗−𝑑𝑅. (18)

Remark 7. Although only the synchronous case is considered, the method in our paper can be applied to study the asynchronous case; namely, the discrete times of each agent are independent of the others.

4. Simulations

Consider the system of four agents, where the topology among four agents is shown in Figure 1. The dynamics of each agent are 𝑥̇_𝑖=𝐴𝑥_𝑖+𝐵𝑢_𝑖,𝑖=1,2,3,4, (19) where 𝑥_𝑖=(𝑥_𝑖1,𝑥_𝑖2,𝑥_𝑖3)^𝑇, and 121𝐴=10221110−2,𝐵=. (20) Since rank(𝐵,𝐴𝐵,𝐴²𝐵)=3, (𝐴,𝐵) is stabilizable. It is easy to verify that 𝐴 is Hurwitz stable for 𝐾=(−4,5,2). By using the feasp solver in Matlab LMI Toolbox, (11) is feasible for 𝑑=0.0625 and it is infeasible for 𝑑=0.0626, which means that for 𝐾=(−4,5,2), the maximum 𝑑 satisfying (11) is 𝑑=0.0625. By Theorem 4, for 𝐾=(−4,5,2), consensus can be reached if the maximum discrete time interval is not larger than 0.0625.

Figure 1 

Topology.

Without loss of generality, the discrete time intervals are chosen from (0,0,0625] randomly. Then the state strategies of four agents during time interval [0,3] are shown in Figures 2, 3, and 4, and the state strategies of four agents during time interval [7.5,9.5] are shown in Figures 5, 6, and 7, which validate our theoretical results.

Figure 2 

State trajectories 𝑥11, 𝑥21, 𝑥31, 𝑥41 during time interval [0,3].

Figure 3 

State trajectories 𝑥12, 𝑥22, 𝑥32, 𝑥42 during time interval [0,3].

Figure 4 

State trajectories 𝑥13, 𝑥23, 𝑥33, 𝑥43 during time interval [0,3].

Figure 5 

State trajectories 𝑥11, 𝑥21, 𝑥31, 𝑥41 during time interval [7.5,9.5].

Figure 6 

State trajectories 𝑥12, 𝑥22, 𝑥32, 𝑥42 during time interval [7.5,9.5].

Figure 7 

State trajectories 𝑥13, 𝑥23, 𝑥33, 𝑥43 during time interval [7.5,9.5].

5. Conclusion

This paper has studied the consensus problem of continuous-time multiagent systems with general linear dynamics and nonuniform sampling. By applying a state transformation and the input-delay approach, the consensus under consideration is equivalent to the asymptotic stability of a continuous-time system with time-varying delay. By analyzing the asymptotic stability of the continuous-time system, it is shown that there exist a controller gain and discrete times such that consensus can be reached. Furthermore, the controller gain and the upper bound of discrete time intervals can be obtained easily by solving a linear matrix inequality. Simulations have been provided to illustrate the effectiveness of the theoretical results.

Acknowledgments

This work was supported by the National Natural Science Foundation (61170113, 61203150, and 61104141), Beijing Natural Science Foundation (4122019), the Science and Technology Development Plan Project of Beijing Education Commission (no. KM201310009011), and the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHR201108055).