Second-Order Multiagent Systems with Event-Driven Consensus Control

Abstract

Event-driven control scheduling strategies for multiagent systems play a key role in future use of embedded microprocessors of limited resources that gather information and actuate the agent control updates. In this paper, a distributed event-driven consensus problem is considered for a multi-agent system with second-order dynamics. Firstly, two kinds of event-driven control laws are, respectively, designed for both leaderless and leader-follower systems. Then, the input-to-state stability of the closed-loop multi-agent system with the proposed event-driven consensus control is analyzed and the bound of the inter-event times is ensured. Finally, some numerical examples are presented to validate the proposed event-driven consensus control.

1. Introduction

Recently, synthesis and analysis of multi-agent systems have drawn great attention in many disciplines, such as mathematics, physics, computer science, systems biology, engineering, and social science. Roughly speaking, multi-agent systems are a class of networked dynamic systems consisting of a group of autonomous agents, which interact with each other locally and achieve an emergence behavior over a communication network. The controlled multi-agent systems have a broad range of applications including flocking and swarming in animal groups, vehicle formation, satellite reconfiguration, and unmanned aerial vehicles for rescue and surveillance.

Consensus problems have a long history originated from management science and statistics in 1960s [1]. In the context of multi-agent systems, consensus generally means to reach an agreement regarding a certain quantity of interest that depends on the state of all agents [2]. In the literature of consensus control of multi-agent systems, many works have been focused on an important issue, that is, to investigate the coordination behavior of agents governed by different order dynamics. Especially, first-order multi-agent systems are extensively considered as a representative multi-agent consensus model in, for example, [3–8] and references therein. More recently, the consensus problems of multi-agent systems with second-order dynamics (e.g., [9–14]) and high order dynamics (e.g., [15–19]) have been paid much attention, which is mainly because in many real applications mass-point models are invalid for agents and more complex dynamics should be considered. Generally, the dynamics of a second-order multi-agent system is described by a second-order differential equation or difference equation, which contains both the position and the velocity information.

One potential application of multi-agent control is to equip each autonomous agent with a small embedded micro-processor to collect information from neighboring agents for actuating the controller updates. However, micro-processors are generally resource- and energy-limited [20], which requires a time- or an event-triggered scheduling strategy to update the control. A time-triggered update scheduling involves sampling at predefined time instances while an event-triggered one executes the control task whenever a certain error becomes large compared with the state norm. Time-triggered consensus problems were studied in [14, 21, 22] via data-sampled method. However, event-triggered strategies seem more favorable in applications. A distributed event-triggered control was considered for a first-order multi-agent system in [23–25]. Up to date, there are few contributions devoted to designing an event-triggered consensus control for multi-agent systems with second-order dynamics.

In this paper, we consider an event-driven consensus problem of a second-order leaderless and leader-follower multi-agent system with a fixed directed communication network. Firstly, the event-driven consensus problem is formulated. Secondly, an event-driven consensus control is designed for each agent to achieve consensus. Then the closed-loop multi-agent system is proven to be input-to-state stable with respect to the measurement error and, simultaneously, a positive lower bound is found for the event-time between two consecutive actuation updates.

Throughout this paper, the following notations are used. 𝐼_𝑛×𝑛 denotes an 𝑛×𝑛 identity matrix; 0_𝑛×𝑚 denotes an 𝑛×𝑚 zero matrix; col(⋅) denotes a column vector; 𝟏_𝑛=col(1,…,1)∈ℝ^𝑛. The norm of a vector 𝑥∈ℝ^𝑛 is defined as ‖𝑥‖=𝑥^𝑇𝑥. The spectral norm of matrix 𝐴∈ℝ^𝑚×𝑛 is defined as ‖𝐴‖=max_{1≤𝑖≤𝑛}𝜆_𝑖, where 𝜆_𝑖 are eigenvalues of 𝐴^𝑇𝐴. For a matrix 𝐴∈ℝ^𝑛×𝑛, 𝜆_min(𝐴) and 𝜆_max(𝐴) denote, respectively, its minimal and maximal eigenvalues.

2. Preliminaries and Problem Formulation

2.1. Some Preliminaries

Let 𝒢=(𝒱,ℰ,𝐴) be a weighted directed graph with a set of vertices 𝒱={1,…,𝑛}, a set of arcs ℰ⊆𝒱×𝒱, and a weighted adjacency matrix 𝐴=[𝑎_𝑖𝑗]∈ℝ^𝑛×𝑛. In the mapping of graph 𝒢 to the interconnection topology of a multi-agent system, vertex 𝑖∈𝒱 represents agent 𝑖, and arc (𝑖,𝑗), which starts from vertex 𝑖 and ends on vertex 𝑗, is in ℰ if and only if agent 𝑖 can receive information from agent 𝑗. In this case, agent 𝑗 is called a neighbor of agent 𝑖, and, accordingly, 𝒩_𝑖={𝑗|(𝑖,𝑗)∈ℰ} denotes the neighboring set of agent 𝑖. The element 𝑎_𝑖𝑗 in the adjacency matrix 𝐴 is associated with the arc (𝑖,𝑗), that is, 𝑎_𝑖𝑗>0 if and only if (𝑖,𝑗)∈ℰ. Moreover, we assume that 𝑎_𝑖𝑖=0 for all 𝑖∈𝒱. When a single leader-agent is involved in the multi-agent systems, a vertex 0 is added to represent the leader-agent and the interconnection topology is denoted by 𝒢=(𝒱,ℰ) with 𝒱={0,1,2,…,𝑛} and ℰ={(𝑖,𝑗)|𝑖,𝑗∈𝒱}.

Definition 1 (see [26]). A path from vertex 𝑖 to vertex 𝑗 is a sequence of arcs (𝑖₀,𝑖₁), (𝑖₁,𝑖₂),…,(𝑖_𝑙−1,𝑖_𝑙) in the directed graph 𝒢 with distinct vertexes 𝑖_𝑘,𝑘=0,1,…,𝑙 and 𝑖₀=𝑖, 𝑖_𝑙=𝑗. A directed graph 𝒢 is strongly connected if there exists a path from vertex 𝑖 to vertex 𝑗, for every 𝑖,𝑗∈𝒱.

Definition 2 (see [26]). Vertex 𝑗 is said to be reachable from vertex 𝑖 if there exists a path from vertex 𝑖 to vertex 𝑗 in the directed graph 𝒢. Vertex 𝑗 is globally reachable if there exists a path from every other vertex to vertex 𝑗 in 𝒢.

According to Definitions 1 and 2, a directed graph 𝒢 is strongly connected if and only if each vertex in 𝒢 is globally reachable, which shows that the global reachability of a directed graph is much weaker than the strong connectedness.

A diagonal matrix 𝐷=diag(𝑑₁,…,𝑑_𝑛)∈ℝ^𝑛×𝑛 is a degree matrix of 𝒢, whose diagonal elements 𝑑_𝑖=∑_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗 for 𝑖=1,…,𝑛. Then the Laplacian matrix of a weighted directed graph is defined as 𝐿=𝐷−𝐴. (1)

The next lemma shows an important property of Laplacian matrix 𝐿 associated with directed graph 𝒢.

Lemma 3 (see [4]). Laplacian matrix 𝐿 has least one zero eigenvalue with 𝟏_𝑛 as its eigenvector, and all the non-zero eigenvalues of 𝐿 have positive real parts. 𝐿 has a simple zero eigenvalue if and only if 𝒢 has a globally reachable vertex.

In the leader-follower consensus literature, it is always assumed that the leader-agent is self-active, that is, the leader does not need information feedback from other agents and thus, the adjacency coefficients 𝑎_0𝑖=0 for every follower-agent 𝑖=1,…,𝑛. For followers, we define a diagonal matrix 𝐵=diag{𝑎₁₀,…,𝑎_𝑛0} to represent the leader-follower adjacency relationship. Let 𝐻=𝐿+𝐵.

Lemma 4 (see [11]). If vertex 0 is globally reachable in 𝒢, then all eigenvalues of 𝐻 have positive real-parts.

A Schur-complement lemma will be used in the stability analysis of the close-loop multi-agent systems and is given to end this subsection.

Lemma 5 (see [27]). Consider a symmetric matrix ΔΔ=¹¹Δ₁₂Δ^𝑇₁₂Δ₂₂, (2) where Δ₁₁ and Δ₂₂ are square. Then Δ is positive definite if and only if both Δ₁₁−Δ₁₂Δ⁻¹₂₂Δ^𝑇₁₂ and Δ₂₂ are positive definite.

2.2. Problem Formulation

In a leaderless consensus problem, a group of 𝑛 identical agents are moving with a continuous-time dynamics described by a second-order differential equation as follows: 𝑥̇_𝑖(𝑡)=𝑣_𝑖(𝑡),𝑣̇_𝑖(𝑡)=𝑢_𝑖(𝑡),𝑖=1,…,𝑛, (3) where 𝑥_𝑖(𝑡)∈ℝ^𝑚, 𝑣_𝑖(𝑡)∈ℝ^𝑚, and 𝑢_𝑖(𝑡)∈ℝ^𝑚 are, respectively, the position, velocity, and control input of agent 𝑖.

In a leader-follower consensus problem, the dynamics of follower-agents are given as (3) while the kinematics of the self-active leader is described by the following second-order differential equation: 𝑥̇₀(𝑡)=𝑣₀(𝑡),𝑣̇₀(𝑡)=𝑎₀(𝑡), (4) where 𝑥₀(𝑡)∈ℝ^𝑚, 𝑣₀(𝑡)∈ℝ^𝑚, and 𝑎₀(𝑡)∈ℝ^𝑚 are, respectively, the position, velocity, and acceleration. Here for notation simplicity, let 𝑚=1.

When agents are equipped with resource-limited micro-processors, it is preferable to design an event-driven consensus controls for all agents such that the consensus controls need no update in continuous-time. For agent 𝑖∈𝒱, we define a state measurement error 𝑒_𝑖(𝑡) and let 𝑒(𝑡)=col(𝑒₁(𝑡),…,𝑒_𝑛(𝑡))∈ℝ^𝑛. Then an event-trigger condition is defined as 𝑓(𝑒(𝑡),𝑥(𝑡),𝑣(𝑡))=0, where 𝑥(𝑡),𝑣(𝑡)∈ℝ^𝑛 are, respectively, the concatenation vector of the state variables in system (3). All agents update their consensus controls 𝑢_𝑖(𝑡) at a series of event-times 𝜏(𝑠)(𝑠=0,1,…) which are implicitly defined by 𝑓(𝑒(𝜏(𝑠)),𝑥(𝜏(𝑠)),𝑣(𝜏(𝑠)))=0. Between control updates at two consecutive event-times, the event-driven consensus control is held constant until the next event is triggered, that is, 𝑢_𝑖(𝑡)=𝑢_𝑖(𝜏(𝑠)),∀𝑡∈[𝜏(𝑠),𝜏(𝑠+1)). (5)

We say that the event-driven consensus problem of leaderless multi-agent system (3) is solved if an event-driven consensus control can be found to ensure that lim_𝑡→∞𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)=0,lim_𝑡→∞𝑣_𝑖(𝑡)−𝑣_𝑗(𝑡)=0, (6) for 𝑖,𝑗∈𝒱. Similarly, the event-driven consensus problem of leader-follower multi-agent system (3)-(4) is solved if lim_𝑡→∞𝑥_𝑖(𝑡)−𝑥₀(𝑡)=0,lim_𝑡→∞𝑣_𝑖(𝑡)−𝑣₀(𝑡)=0, (7) for 𝑖∈𝒱.

3. Control Design and Stability Analysis

3.1. Leaderless Consensus Control

We assume that the consensus laws 𝑢_𝑖(𝑡) in system (3) are updated only at discrete event-times. Suppose that the event-times of all agents are modeled as a sequence 𝜏(𝑠) for 𝑠=0,1,…. For agent 𝑖 with dynamics (3), define two measurement errors 𝑒_𝑖,𝑥(𝑡)=𝑥_𝑖(𝜏(𝑠))−𝑥_𝑖(𝑡), 𝑒_𝑖,𝑣(𝑡)=𝑣_𝑖(𝜏(𝑠))−𝑣_𝑖(𝑡), 𝑡∈[𝜏(𝑠),𝜏(𝑠+1)). The event-times 𝜏(𝑠) are defined by the function 𝑓(𝑒(𝜏(𝑠)),𝑥(𝜏(𝑠)),𝑣(𝜏(𝑠)))=0, which will be determined in sequel.

An event-driven consensus control of agent 𝑖 is proposed as follows: 𝑢_𝑖(𝑡)=𝑘²_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑥_𝑗(𝜏(𝑠))−𝑥_𝑖(𝜏(𝑠))+𝑘³_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑣_𝑗(𝜏(𝑠))−𝑣_𝑖(𝜏(𝑠)), (8) for 𝑡∈[𝜏(𝑠),𝜏(𝑠+1)). The control gain parameter 𝑘 will be determined in sequel.

Then a compact form of (3) using control (8) is given by 𝑥̇(𝑡)(𝑡),𝑣̇(𝑡)=𝑣=−𝑘²𝐿𝑥(𝜏(𝑠))−𝑘³𝐿𝑣(𝜏(𝑠)), (9) where 𝑥,𝑣∈ℝ^𝑛, 𝐿 is the Laplacian matrix.

Denote 𝑒_𝑥(𝑡)=col(𝑒_1,𝑥(𝑡),…,𝑒_𝑛,𝑥(𝑡)), 𝑒_𝑣(𝑡)=col(𝑒_1,𝑣(𝑡),…,𝑒_𝑛,𝑣(𝑡)). One has 𝑥(𝜏(𝑠))=𝑒_𝑥(𝑡)+𝑥(𝑡), 𝑣(𝜏(𝑠))=𝑒_𝑣(𝑡)+𝑣(𝑡). Furthermore, system (9) can be rewritten as 𝑥̇(𝑡)(𝑡),𝑣̇(𝑡)=𝑣=−𝑘²𝐿𝑥(𝑡)−𝑘³𝐿𝑣(𝑡)−𝑘²𝐿𝑒_𝑥(𝑡)−𝑘³𝐿𝑒_𝑣(𝑡), (10) or 𝑥̇(𝑡)𝑣̇(𝑡)=𝐹₁𝑥(𝑡)𝑣(𝑡)+𝐽₁𝑒_𝑥(𝑡)𝑒_𝑣(𝑡), (11) where 𝐹₁=0_𝑛×𝑛𝐼_𝑛−𝑘²𝐿−𝑘³𝐿,𝐽₁=0_𝑛×𝑛0_𝑛×𝑛−𝑘²𝐿−𝑘³𝐿. (12)

Now we will analyze the convergence of the closed-loop multi-agent system (11) under the event-driven consensus control (8). To facilitate the convergence analysis, we describe the following lemma. Its proof is quite obvious and omitted here.

Lemma 6. For Laplacian matrix 𝐿 associated with the directed graph 𝒢, there exists a non-singular matrix 𝑈=1∗⋯∗1∗⋯∗⋮⋮⋮1∗⋯∗∈ℝ^𝑛×𝑛 (13) such that 𝑈⁻¹𝐿𝑈=0ℎ^𝑇0_𝑛−1𝐻=Λ∈ℝ^𝑛×𝑛, (14) where ℎ∈ℝ^𝑛−1, 𝐻∈ℝ^{(𝑛−1)×(𝑛−1)}.

From Lemma 6, the submatrix 𝐻 of Λ is a full-rank matrix if and only if 𝒢 has a global reachable vertex. Moreover, the eigenvalues of 𝐻 have positive real-parts, or equivalently, −𝐻 is Hurwitz stable. Therefore, there exists a positive definite matrix 𝑃∈ℛ^{(𝑛−1)×(𝑛−1)} such that 𝑃𝐻+𝐻^𝑇𝑃=𝐼_𝑛−1. (15)

Take a coordinate transformation 𝑥(𝑡)=𝑈⁻¹𝑥(𝑡),𝑣(𝑡)=𝑈⁻¹𝑣(𝑡),𝑒_𝑥(𝑡)=𝑈⁻¹𝑒_𝑥(𝑡),𝑒_𝑣(𝑡)=𝑈⁻¹𝑒_𝑣(𝑡), (16) and then system (10) becomes 𝑥̇‌(𝑡)=𝑣(𝑡),𝑣̇(𝑡)=−𝑘²Λ𝑥(𝑡)−𝑘³Λ𝑣(𝑡)−𝑘²Λ𝑒_𝑥(𝑡)−𝑘³Λ𝑒_𝑣(𝑡). (17)

Essentially, system (17) can be regarded as a series interconnection of two subsystems: 𝑥̇‌₁(𝑡)=𝑣₁(𝑡),𝑣̇₁(𝑡)=−𝑘²ℎ^𝑇𝑥₂(𝑡)−𝑘³ℎ^𝑇𝑣₂(𝑡)−𝑘²ℎ^𝑇𝑒²_𝑥(𝑡)−𝑘³ℎ^𝑇𝑒²_𝑣(𝑡),𝑥̇₂(𝑡)=𝑣₂(𝑡),𝑣̇₂(𝑡)=−𝑘²𝐻𝑥₂(𝑡)−𝑘³𝐻𝑣₂(𝑡)−𝑘²𝐻𝑒²_𝑥(𝑡)−𝑘³𝐻𝑒²_𝑣(𝑡), (18)(19) where 𝑥₁,𝑣₁,𝑒¹_𝑥,𝑒¹_𝑣∈ℝ and 𝑥₂,𝑣₂,𝑒²_𝑥,𝑒²_𝑣∈ℝ^𝑛−1, 𝑥=𝑥₁𝑥₂,𝑣=𝑣₁𝑣₂,𝑒_𝑥=𝑒¹_𝑥𝑒²_𝑥,𝑒_𝑣=𝑒¹_𝑣𝑒²_𝑣. (20)

Let 𝜖=col(𝑥₂,𝑣₂), 𝑒=col(𝑒²_𝑥,𝑒²_𝑣), then subsystem (19) becomes 𝜖̇=𝐹₂𝜖+𝐽₂𝑒, (21) where 𝐹₂=0_{(𝑛−1)×(𝑛−1)}𝐼_𝑛−1−𝑘²𝐻−𝑘³𝐻,𝐽₂=0_{(𝑛−1)×(𝑛−1)}0_{(𝑛−1)×(𝑛−1)}−𝑘²𝐻−𝑘³𝐻. (22)

Now a main result is obtained for system (10).

Theorem 7. Assume that the interconnection topology 𝒢 associated with multi-agent system (3) has a globally reachable vertex. If the control gain 𝑘 satisfies 𝑘>𝜆_max𝑃2+1 (23) with 𝑃 given in (15), then the event-driven consensus problem of the leaderless multi-agent system (3) is solved with the control (8), that is, lim_𝑡→∞𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)=0,lim_𝑡→∞𝑣_𝑖(𝑡)−𝑣_𝑗(𝑡)=0. (24)

Proof. For system (19), or equivalently (21), take a Lyapunov function 𝑉(𝜖)=𝜖^𝑇𝑃𝜖, (25) where 𝑃=⊗𝑘11𝑘𝑃, (26) where 𝑃 satisfies (15). Thus, 𝑃 is positive definite with 𝑘 given in (23).
Differentiating 𝑉(𝜖) leads to 𝑉̇(𝜖)∣₍₂₁₎=𝜖^𝑇𝐹^𝑇₂𝑃+𝑃𝐹₂𝜖+2𝜖^𝑇𝑃𝐽₂e=−𝜖^𝑇𝑄𝜖+2𝜖^𝑇𝑃𝐽₂e, (27) where 𝑄=−𝐹^𝑇₂𝑃+𝑃𝐹₂=𝑘²𝐼_𝑛−1𝑘³𝐼_𝑛−1−𝑘𝑃𝑘³𝐼_𝑛−1−𝑘𝑃𝑘⁴𝐼_𝑛−1−2𝑃. (28) According to Schur complement Lemma 5, 𝑄 is positive definite with 𝑘 given in (23).
Therefore, 𝑉̇(𝜖)∣₍₂₁₎≤−𝜆_min(𝑄)‖𝜖‖²+2𝜆_max(𝑃)𝐽₂𝜖𝑒‖‖𝜖𝜆‖‖‖=−‖_min(𝑄)‖𝜖‖−2𝜆_max(𝑃)𝐽₂‖𝑒‖. (29) Due to the fact that ‖𝐽₂‖²=𝑘⁴(𝑘²+1)‖𝐻‖², enforcing the measurement error 𝑒 to satisfy ‖𝑒‖𝜆≤𝜎^min(𝑄)‖𝜖‖2𝑘²𝑘²+1𝜆_max(𝑃)𝐻 (30) with 𝜎∈(0,1), one has 𝑉̇(𝜖)≤−(1−𝜎)𝜆_min(𝑄)‖𝜖‖²()𝜆≤−1−𝜎_min(𝑄)𝜆_max(𝑃)𝑉(𝜖). (31) From (31), we conclude that lim_𝑡→∞𝜖(𝑡)=0, that is, as 𝑡→∞, 𝑥₂(𝑡)→0, 𝑣₂(𝑡)→0.
On the other hand, for system (18), let 𝑥₁(0),𝑣₁(0) be the initial values of 𝑥₁(𝑡),𝑣₁(𝑡) and take a variable change 𝑥^∼₁(𝑡)=𝑥₁(𝑡)−(𝑣₁(0)𝑡+𝑥₁(0)), 𝑣^∼₁(𝑡)=𝑣₁(𝑡)−𝑣₁(0). Then the solution can be described by the following integral equation: 𝑥^∼_1∼𝑣₁=^𝑡₀×01𝑡−𝜍01−𝑘²ℎ^𝑇𝑥₂(𝜍)−𝑘³ℎ^𝑇𝑣₂(𝜍)−𝑘²ℎ^𝑇𝑒²_𝑥(𝜍)−𝑘³ℎ^𝑇𝑒²_𝑣(𝜍)𝑑𝜍. (32)
Since the system (19) is exponentially stable, based on the event-triggered condition (30), solution (32) has an exponential decay term with respect to time 𝑡. Consequently, the solution is convergent to 0 as 𝑡→∞, and thus, 𝑥₁(𝑡)𝑥₂(𝑡)−𝑣₁(0)𝑡+𝑥₁(0)0_𝑛−1⟶0_𝑛,𝑣₁(𝑡)𝑣₂(𝑡)⟶𝑣₁(0)0_𝑛−1, (33) as 𝑡→∞. Furthermore, from the variable change (16), one has 𝑥−𝑈𝑣₁(0)𝑡+𝑥₁(0)0_𝑛−1=𝑥−𝟏_𝑛⊗𝑣₁(0)𝑡+𝑥₁(0)⟶0_𝑛,𝑣−𝑈𝑣₁(0)0_𝑛−1=𝑣−𝟏_𝑛⊗𝑣₁(0)⟶0_𝑛. (34) Therefore, when the event-driven consensus control (8) is applied to each agent, one has 𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)→0, 𝑣_𝑖(𝑡)−𝑣_𝑗(𝑡)→0 as 𝑡→∞. The proof is complete.

Remark 8. The gain 𝑘 in (23) can be taken without exact knowledge of the interconnection topology associated with the multi-agent system in real application of the control (8). In fact, the spectral norm of 𝑃 in (23) can be obtained by estimating the bound of the solution of Lyapunov equation (15) (see [28]), which is closely related with the Laplacian spectrum. Fortunately, there have been many results on the bounds of the eigenvalues of a Laplacian matrix [29].

Next, we show that the inter-event times 𝜏(s+1)−𝜏(𝑠) have a lower bound for 𝑠=0,1,…. It is noticed that, from (16), 𝑥(𝑡)𝑣(𝑡)=𝐼₂⊗𝑈⁻¹𝑥(𝑡)𝑣(𝑡),𝑒_𝑥(𝑡)𝑒_𝑣(𝑡)=𝐼₂⊗𝑈⁻¹𝑒_𝑥(𝑡)𝑒_𝑣(𝑡). (35) Define a matrix 𝑊=[0_𝑛−1𝐼_𝑛−1]∈ℝ^{(𝑛−1)×𝑛}. Then, based on (35), one has 𝜖=𝐼₂⊗𝑊𝑥(𝑡)𝑣(𝑡)=𝐼₂⊗𝑊𝑈⁻¹𝑥(𝑡)𝑣(𝑡),𝑒=𝐼₂⊗𝑊𝑈⁻¹𝑒_𝑥(𝑡)𝑒_𝑣(𝑡). (36) For convenience of simplified calculation, let 𝑝(𝑡)=col(𝑥(𝑡),𝑣(𝑡)), 𝑒_𝑝(𝑡)=col(𝑒_𝑥(𝑡),𝑒_𝑣(𝑡)) and 𝑧(𝑡)=‖𝑒_𝑝(𝑡)‖/‖𝑝(𝑡)‖. Take the derivative of 𝑧(𝑡): ̇𝑑𝑧=𝑒𝑑𝑡^𝑇𝑝𝑒_𝑝^1/2𝑝^𝑇𝑝^1/2𝑒=−^𝑇𝑝𝑝̇𝑒_𝑝𝑝−𝑒_𝑝𝑝𝑝^𝑇𝑝̇𝑝²≤𝑒1+^𝑝𝑝𝑝̇𝑝=(1+𝑧)𝐹₁+𝐽₁𝑧. (37)

Assume that 𝜙(𝑡,𝜙₀) is the solution of 𝜙̇=1+𝜙𝐹₁+𝐽₁𝜙,𝜙0,𝜙₀=𝜙₀. (38) Then, one has 𝑧(𝑡)≤𝜙(𝑡,𝜙₀) for 𝑡≥0. From the variable change (36) and the event-driven condition (30), 𝑧(𝑡) satisfies 𝑧(𝑡)=𝑒_𝑝(𝑡)𝑝(𝑡=‖𝑒(𝑡))‖‖𝜖(𝑡,)‖ (39) and the inter-event times are bounded by the time 𝜏_𝐷∈ℝ⁺ it takes for 𝜙 to evolve from 0 to 𝜙(𝜏_𝐷,0)=(𝜎𝜆_min(𝑄)/2𝑘²𝑘²+1𝜆_max(𝑃)‖𝐻‖). Solving the differential equation (38) leads to 𝜏_𝐷=1𝐹₁−𝐽₁𝜏ln1+𝜙_𝐷,01+𝐽₁/𝐹₁𝜙𝜏_𝐷,0, (40) which is a positive bound due to the fact that ‖𝐹₁‖>‖𝐽₁‖.

Then a result is stated as follows.

Theorem 9. When the event-driven consensus control (8) is applied to the multi-agent system (3), the inter-event times 𝜏(𝑠+1)−𝜏(𝑠) for 𝑠=0,1,…, have a lower bound given by (40).

3.2. Leader-Follower Consensus Control

In the leader-following problem, we assume that the state information, that is, 𝑥₀(𝑡), 𝑣₀(𝑡), and 𝑎₀(𝑡) of the leader can be measured in continuous-time by the followers. Thus, we propose the following event-driven consensus control of the follower 𝑖: 𝑢_𝑖(𝑡)=𝑎₀(𝑡)+𝑘²_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑥_𝑗(𝜏(𝑠))−𝑥_𝑖(𝜏(𝑠))+𝑘²𝑎_𝑖0𝑥₀(𝑡)−𝑥_𝑖(𝑡)+𝑘³_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑣_𝑗(𝜏(𝑠))−𝑣_𝑖(𝜏(𝑠))+𝑘³𝑎_𝑖0𝑣₀(𝑡)−𝑣_𝑖(𝑡) (41) for 𝑡∈[𝜏(𝑠),𝜏(𝑠+1)).

Remark 10. For the leader-following problem (3)-(4) under investigation, the leader is assumed to be self-active, which means that the leader is moving according to its own (predesigned) policy and needs no feedback information from any other agent. In some sense, the leader plays the role of an external commander of all the followers. Thus, no event-driven strategy is applied to update the state of the leader. Though the followers are using the event-driven relative information from the neighboring followers, but they can obtain the real-time relative position and velocity measurements from the leader only if the followers are connected to the leader. Therefore, in the control (41), it would be preferable to assume that 𝑥₀(𝑡)−𝑥_𝑖(𝑡) and 𝑣₀(𝑡)−𝑣_𝑖(𝑡) are measured in continuous time.

Still denote the measurement errors 𝑒_𝑖,𝑥(𝑡)=𝑥_𝑖(𝜏(𝑠))−𝑥_𝑖(𝑡), 𝑒_𝑖,𝑣(𝑡)=𝑣_𝑖(𝜏(𝑠))−𝑣_𝑖(𝑡) for 𝑖=1,…,𝑛, 𝑡∈[𝜏(𝑠),𝜏(𝑠+1)). Applying control (41) to system (3) leads to 𝑥̇(𝑡)(𝑡),𝑣̇(𝑡)=𝑣=𝑎₀(𝑡)𝟏−𝑘²𝐻𝑥(𝑡)+𝑘²𝐵𝟏𝑥₀(𝑡)−𝑘³𝐻𝑣(𝑡)+𝑘³𝐵𝟏𝑣₀(𝑡)−𝑘²𝐿𝑒_𝑥(𝑡)−𝑘³𝐿𝑒_𝑣(𝑡), (42) where 𝐻=𝐿+𝐵. Define 𝑥^∼_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑥₀(𝑡), 𝑣^∼_𝑖(𝑡)=𝑣_𝑖(𝑡)−𝑣₀(𝑡) and 𝑥^∼(𝑡)=col(𝑥^∼₁(𝑡),…,𝑥^∼_𝑛(t)), 𝑣^∼(𝑡)=col(𝑣^∼₁(𝑡),…,𝑣^∼_𝑛(𝑡)). Based on the fact that −𝐻𝑥^∼(𝑡)=−𝐻𝑥(𝑡)+𝐵𝟏𝑥₀(𝑡), −𝐻𝑣^∼(𝑡)=−𝐻𝑣(𝑡)+𝐵𝟏𝑣₀(𝑡), one has 𝑥̇^∼(𝑡)𝑣̇^∼(𝑡)=𝐹₃^∼𝑥(𝑡)_∼𝑣(𝑡)+𝐽₃𝑒_𝑥(𝑡)𝑒_𝑣(𝑡), (43) where 𝐹₃=0_𝑛×𝑛𝐼_𝑛−𝑘²𝐻−𝑘³𝐻,𝐽₃=0_𝑛×𝑛0_𝑛×𝑛−𝑘²𝐿−𝑘³𝐿. (44)

Theorem 11. Assume that the vertex 0 is globally reachable in the leader-follower interconnection topology 𝒢. If the control gain 𝑘 satisfies 𝑘>𝜆𝑃_max^∼2+1, (45) where 𝑃^∼ is the solution of the Lyapunov equation 𝑃^∼𝐻+𝐻𝑃^𝑇∼=𝐼_𝑛, then the event-driven consensus problem of the leader-follower multi-agent system (3)-(4) is solved with the control (41), that is, lim_𝑡→∞𝑥_𝑖(𝑡)−𝑥₀(𝑡)=0,lim_𝑡→∞𝑣_𝑖(𝑡)−𝑣₀(𝑡)=0. (46)

Proof. For system (43), take a Lyapunov function 𝑉(𝜀)=𝜀^𝑇𝑃_𝐿𝜀, (47) where 𝜀(𝑡)=^∼𝑥(𝑡)_∼𝑣(𝑡),𝑃_𝐿=⊗𝑘11𝑘^∼𝑃. (48) It is not difficult to show that 𝑃_𝐿 is positive definite with 𝑘 given in (45).
Differentiating 𝑉(𝜀) leads to 𝑉̇(𝜀)∣₍₄₃₎=𝜀^𝑇𝐹^𝑇₃𝑃_𝐿+𝑃_𝐿𝐹₃𝜀+2𝜀^𝑇𝑃_𝐿𝐽₃𝑒=−𝜀^𝑇𝑄_𝐿𝜀+2𝜀^𝑇𝑃_𝐿𝐽₃𝑒, (49) where 𝑄_𝐿=−𝐹^𝑇₃𝑃_𝐿+𝑃_𝐿𝐹₃=𝑘²𝐼_𝑛𝑘³𝐼_𝑛−𝑘𝑃^∼𝑘³𝐼_𝑛−𝑘𝑃^∼𝑘⁴𝐼_𝑛−2𝑃^∼. (50) According to Schur complement Lemma 5, 𝑄_𝐿 is positive definite with 𝑘 given in (45).
Therefore, 𝑉̇(𝜀)|₍₄₃₎≤−𝜆_min𝑄_𝐿‖𝜖‖²+2𝜆_max𝑃_𝐿𝐽₃𝜀𝑒‖‖𝜀𝜆‖‖‖=−‖_min𝑄_𝐿‖𝜖‖−2𝜆_max𝑃_𝐿𝐽₃‖𝑒‖. (51) Due to the fact that ‖𝐽₃‖²=𝑘⁴(𝑘²+1)‖𝐻‖², enforcing the measurement error 𝑒(𝑡) to satisfy ‖𝑒(𝑡𝜆)‖≤𝜎^min𝑄_𝐿‖𝜀‖2𝑘²𝑘²+1𝜆_max𝑃_𝐿𝐻 (52) with 𝜎∈(0,1), one has 𝑉̇(𝜀)≤−(1−𝜎)𝜆_m𝑖𝑛𝑄_𝐿‖𝜀‖²()𝜆≤−1−𝜎_min𝑄_𝐿𝜆_max𝑃_𝐿𝑉(𝜀). (53) From (53), we conclude that lim_𝑡→∞𝜀(𝑡)=0, that is, as 𝑡→∞, 𝑥^∼₂(𝑡)→0,𝑣^∼₂(𝑡)→0. The proof is thus competed.

Similar to Theorem 9, a conclusion about the lower bound of the event-times can also be found true, which is omitted here.

4. Simulations

Example 12. Consider four agents whose dynamics is described by (3). The interconnection topology is shown in Figure 1. Obviously, 𝒢 has a globally vertex 4; however, it is not strongly connected.

Figure 1 

The interconnection topology 𝒢.

Assume that the weighted adjacency matrix 𝐴 reduces to a 0−1 matrix. Then the Laplacian matrix 𝐿 of 𝒢 is 𝐿=100−1−11000−12−10000, (54) which is an asymmetric matrix. A non-singular matrix 𝑈 in Lemma 6 can be easily found as 𝑈=1000110010101001. (55)

Then the submatrices 𝐻=^101−120₀₀₁ and ℎ=col(0,0,−1). The positive definite matrix satisfying (15) is given by 𝑃=0.58330.0833−0.30560.08330.2500−0.0278−0.3056−0.02780.8056, (56) whose minimal and maximal eigenvalues are, respectively, 𝜆_min(𝑃)=0.2248,𝜆_max(𝑃)=1.0260. According to (23), take the gain 𝑘=1.5.

The initial conditions of system (9) are 𝑥(0)=col(2,−4,−9,9), 𝑣(0)=col(2,6,3,−2) and 𝜏(0)=0. In the event triggered condition (30), we take 𝜎=0.8. It can be seen that, from Figures 2 and 3, the four agents reach consensus on the position and velocity states with the proposed event-driven consensus control (8). Additionally, since the vertex 4 has no link starting from it, the agent 4 is moved according to its own dynamics and the initial conditions, as shown in Figures 2, 3, and 5. The evolution of the measurement error vector 𝑒(𝑡) is depicted in Figure 4, which shows that the error 𝑒(𝑡) is bounded by the specified threshold (𝜎𝜆_min(𝑄)‖𝜖‖/2𝑘²𝑘²+1𝜆_max(𝑃)‖𝐻‖). In Figure 5, the consensus controllers are illustrated for the four agents, whose event-driven update frequencies are decreasing as time evolves.

Figure 2 

The evolution of position states 𝑥𝑖(𝑡) for 𝑖=1,…,4.

Figure 3 

The evolution of velocity state 𝑣𝑖(𝑡) for 𝑖=1,…,4.

Figure 4 

Evolution of the measurement error norm ‖𝑒(𝑡)‖.

Figure 5 

Evolution of the consensus controllers 𝑢𝑖(𝑡) for 𝑖=1,…,4.

Example 13. Consider four followers and one leader whose dynamics are, respectively, described by (3) and (4). The leader-follower interconnection topology 𝒢 is shown in Figure 6, which has a globally reachable vertex 0.

Figure 6 

𝒱(𝒢)={0,1,2,3,4} and 𝒱(𝒢)={1,2,3,4}.

It is not difficult to obtain the Laplacian matrix 𝐿 as 𝐿=1−100−1100001−100−11, (57) and the leader adjacency matrix 𝐵=diag{1,0,0,1}. Since −𝐻=−(𝐿+𝐵) is a stable matrix, one solves the Lyapunov equation 𝑃^∼𝐻+𝐻𝑃^𝑇∼=𝐼₄ and has the positive definite solution as 𝑃^∼=0.750.25000.250.5000001.00.50000.500.50 (58) whose maximal eigenvalue 𝜆_max(𝐻)=1.309. From (45), still take 𝑘=1.5.

The acceleration of the active leader is assumed to be 𝑎₀(𝑡)=𝑒^−𝑡sin(𝑡). The initial values of system (3) is same as those in Example 12 and the initial values of the leader is given as 𝑥₀(0)=0,𝑣₀(0)=1 and 𝑎₀(0)=0. Figures 7 and 8 show that the followers and the self-active leader reach consensus on the position and velocity under the event-driven control (41). The evolution of ‖𝑒(𝑡)‖ is also presented in Figure 9.

Figure 7 

The evolution of position states 𝑥𝑖(𝑡) for 𝑖=0,1,…,4.

Figure 8 

The evolution of velocity states 𝑣𝑖(𝑡) for 𝑖=0,1,…,4.

Figure 9 

Evolution of the measurement error norm ‖𝑒(𝑡)‖.

5. Conclusions

An event-driven consensus problem of second-order multi-agent systems with/without a self-active leader was considered in this paper. The consensus controllers have been proposed for all autonomous mobile agents based on an event-driven control update strategy. The input-to-state stability of the closed-loop multi-agent system has been analyzed by employing an ISS Lyapunov function. Some numerical examples have been presented to validate the proposed event-driven controls. However, it is noted that the event-driven condition depends on the states of the whole multi-agent group and all agents have identical event-times. The result is somewhat preliminary due to the centralized information gathering, so further work will be devoted to designing a decentralized event-driven consensus control for a second-order multi-agent system in the future.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant 61104104 and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China.