Distributed event-triggered scheme for discrete-time second-order multi-agent systems

Abstract

In order to achieve the consistency of multi-agent systems, each agent needs to communicate with its adjacent agents, which will consume energy of sensors embedded on the agents and occupy network bandwidth of multi-agent systems. Both resources are limited. To solve the above problem, a novel distributed event-triggered scheme of discrete-time second-order multi-agent systems are proposed in this article. The characteristics of the scheme have two aspects. Firstly, the event-triggered conditions are considered for the state and the velocity separately. Secondly, when the event is triggered on an agent, the agent only communicates with its local neighbors. Then, the agent and its local neighbors update their controls while the other agents' controllers remain unchanged. So the scheme can maximize reduction of the sensor energy consuming and communication burden in the multi-agent network. Based on the Lyapunov functional method, a sufficient condition is obtained to achieve the stability of the second-order multi-agent systems in terms of linear matrix inequality. Finally, numerical examples are presented to validate the proposed event-triggered consensus control.

Keywords

Event-triggered control multi-agent systems distributed control consensus

Introduction

In recent years, various research groups have paid attention to multi-agent systems, because multi-agent systems have wide applications in different fields, such as aircraft cooperative control,¹ formation control,² and distributed sensor networks.³ Consensus is one of the typical characteristic of multi-agent systems. Its main function is to design control protocol, control the updates of agents' status, and finally let all agents achieve the same status to complete a common task.⁴ Many scholars have focused on the study of consensus problems of multi-agent systems about different orders. Among them, coordination behavior of the first-order multi-agent systems is most widely investigated.^5–8 But in many practical applications, the agents may be described in the forms of double integrals, such as torque motors and gas jets. They adjust their behavior by acceleration speed rather than speed. As a result, second-order^9–14 even high-order^15–19 dynamics of multi-agent systems are worth investigating. In this article, we discussed the second-order dynamics of multi-agent systems. Generally, second-order multi-agent systems are denoted by differential equations, including the state and velocity information.

Generally, embedded microprocessor is installed on each agent to gather information from its neighbors and update the protocol. Communications between the various agents need to take up network bandwidth. However, the energy resources of embedded microprocessor and the bandwidth of network are limited. In order to solve the above problems, an event-triggered strategy is proposed. In event-triggered strategies, error measurement functions are designed firstly. Then, the agent keeps the same control protocol until the error accumulated to a certain extent. When this event happened, the agent updates its own control protocol and sends its latest state and velocity information to its neighbors. This strategy saves the energy consumption and network bandwidth by reducing communications between agents. In previous studies, the event triggered scheme for first-order multi-agent systems have extensively studied in the literature.^20–30 However, there are a few research of event-triggered scheme for second-order multi-agent systems. Existing research achievement includes literatures.^31–35 Event-triggered strategies can be divided into the following three cases: (1) Centralized event-triggered strategy.³¹ With this strategy, a virtual center is used to collect the error information of each agent and a global event-triggered condition is designed to determine the trigger time. When the event is triggered, all agents update their control protocol at the same trigger time. (2) Decentralized event-triggered strategy.^32,33 With this strategy, local trigger function is designed to determine the trigger time and each agent can update its control at a sequence of separate event-times. But lower bounds on the time between two execution updates need to be given to ensure the overall switched system does not exhibit zeno behavior.³⁴ (3) Distributed event-triggered strategy.^35,36 With this strategy, each agent has its own event-triggered function and respective event time which depend on its own state and the information gathering from its neighboring agents. The above literatures have focused on the event-triggered scheme for second-order multi-agent systems on premise of continuous time condition. But in practical applications, the communication channels may be not always keeping stable and the energy of the microprocessors embedded on the agents is limited. These will cause that information transmission is not continuous. The discrete-time sampled data control is more feasible.

In this article, the distributed event-triggered control of discrete-time second-order multi-agent systems with a fixed directed communicative network will be discussed. In the proposed scheme, each agent has two event-triggered functions for the state and the velocity separately and the event time of each agent is different. The event-triggered condition depends on the information of the agent itself and the information which is gathered from its neighbors during the latest event of its neighbors. When the event (state or velocity of one agent) is triggered, the agent updates its own controller and sends the latest information (state or velocity) to its neighbors. So the strategy we proposed is helpful to save network communication bandwidth and energy consumption of the agents by reducing communication with agents each other.

The reminder of the article is organized as follows. In Preliminaries and problem formulation section, we give some preliminaries on algebraic graph theory and the formulation of the consensus problem. The control design and stability analysis is given in Control design and stability analysis section. A numerical simulation is given in Numerical examples and simulations section to illustrate the consensus algorithm. Finally, concluding remarks are stated in Conclusions section.

Preliminaries and problem formulation

Algebraic graph theory

In this article, second-order multi-agent systems with direct topology network will be discussed. The structure of the network is represented by a directed graph G.³⁷ Assumed that there are a directed graph G which contains n agents. A vertex set $V = {1, 2, \dots, n}$ represents the set of n agents; an edge set $E \subseteq V \times V$ represents the set of edge. $(j, i) \in E$ represents an edge which starts from agent i and ends on agent j. It means that there is a communication link between agent i and agent j, so agent j can receive information from agent i. In this case, agent i is called a neighbor of agent j. Therefore, the neighboring set of agent i can be defined as $N i = {j | (j, i) \in E}$ . If there exists vertexes $i k$ , $k = 1, 2, \dots, l$ and an ordered set of edges $(i, i 1)$ , $(i 1, i 2)$ , …, $(i l, j)$ between agent i and agent j in a directed graph, we called it a directed path from agent i to agent j. The directed network has a directed spanning tree if at least one node has a directed path to all the other nodes.

Next, definitions of some matrix associated with a directed graph will be reviewed subsequently, such as the adjacency matrix, the degree matrix and the Laplacian matrix.

The adjacency matrix $A = [a ij] \in R n \times n$ can be defined as where $a ij$ is the weight of the edge from agent j to agent i. It is noted that we assume $a ii = 0$ for all $i \in V$ .

The degree matrix D is a diagonal matrix, it can be defined as

[b 1 00 b n]

where

b i = j \in N i a ij

for i = 1, 2, … , n. Notation diag{b₁, b₂, … , b_n} denotes the diagonal matrix.

The Laplacian matrix $L = [l ij] \in R n \times n$ is defined as

L = D - A

where D is the degree matrix and A is the adjacency matrix.

l ij

satisfies

l ij = {- a ij i \neq j \sum_{j \in N i} a ij i = j

Problem formation

Discrete-time second-order multi-agent systems can be expressed as

{x i (k + 1) = x i (k) + T v i (k) + \frac{T 2}{2} u i (k) v i (k + 1) = v i (k) + T u i (k)

(1)

where

i = 1, 2, \dots, n

x i (k) \in R m

indicates the state vector of the ith agent at time k;

v i (k) \in R m

indicates the velocity vector of the ith agent at time k;

u i (k) \in R m

indicates control vector of the system;

R m

represents the m-dimensional Euclidean space.

In order to save the communication bandwidth of the multi-agent network and reduce energy consumption of embedded microprocessor, an event-triggered scheme is proposed in this article. Unlike some existing algorithms, we consider the distributed event-triggered communication scheme for the state and the velocity separately. For agent $i \in V$ , we define a state measurement error $e i, x (k)$ and a velocity measurement error $e i, v (k)$ , respectively. Then a state event-triggered condition is defined as $f i (x i (k), e i, x (k)) < 0$ and a velocity event-triggered condition is defined as $h i (v i (k), e i, v (k)) < 0$ . When the state events are triggered for agent i, the communication control of the agent i and its neighbors are updated and the same thing happens when the velocity events are triggered. Between control updates, the event-triggered consensus control is held constant until the next event is triggered.

The event-triggered consensus problem is solved if an event-triggered consensus control can be found to ensure that

lim k \to \infty ‖ x i (k) - x j (k) ‖ = 0 lim k \to \infty ‖ v i (k) - v j (k) ‖ = 0

(2)

for

i, j = 1, 2, \dots, n

Control design and stability analysis

Consensus control

In this section, consensus control algorithms of system (1) under event-triggered conditions will be discussed. Using periodic sampling techniques and zero-order hold circuit, discrete system (1) can be expressed as

u i (k) = - K 1 {\sum_{j \in N i} [a (i, j) (x i (k) - x j (k))] + x i (k)} - K 1 {\sum_{j \in N i} [a (i, j) (v i (k) - v j (k))] + v i (k)}

(3)

where K₁ is feedback gain matrix and N_i is neighboring agent of agent i.

Remark 1

The network topology is assumed that the fixed weighted directed graph including a directed spanning tree. This is a basic condition to ensure the consistency of the system (1).^38,39 If the directed graph does not contain a spanning tree, at least one agent cannot obtain information from other agents, which may lead to the consistency of the system cannot be achieved.

The goal of event-triggered scheme is to reduce useless communication through the network. According to the event-triggered scheme mentioned in Algebraic graph theory section, the error measurement functions of agent i on state and velocity are defined. Firstly, we define the error measurement formula as equation (4)

e i, x (k) = \tilde{x} i (k) - x i (k) e i, v (k) = \tilde{v} i (k) - v i (k)

(4)

In equation (4), the $\tilde{x} i (k)$ , $\tilde{v} i (k)$ are defined by the following functions

\tilde{x} i (k) = x i (k_{m}^{i}), for k_{m}^{i} \leq k < k_{m + 1}^{i} \tilde{v} i (k) = v i (k_{n}^{i}), for k_{n}^{i} \leq k < k_{n + 1}^{i}

(5)

where

k_{m}^{i}

k_{n}^{i}

are denoted the mth state event instant and the nth velocity event instant for agent i respectively, k is denoted the latest sampling instant. Secondly, the error measurement functions are defined as equation (6)

h i (x i (k), \tilde{x} i (k)) = e_{i, x}^{T} (k) Ω i 1 e i, x (k) - σ 1 x_{i}^{T} (k) Ω i 1 x i (k) g i (v i (k), \tilde{v} i (k)) = e_{i, v}^{T} (k) Ω i 1 e i, v (k) - σ 2 v_{i}^{T} (k) Ω i 2 v i (k)

(6)

where

σ 1

σ 2

are positive scalars which satisfy

0 < σ 1, σ 2 < 1

Ω i 1

Ω i 2

are positive definite trigger-matrices to be determined later.

Each agent computes the values of the triggered functions by its current information on state and velocity ( $x i (k)$ and $v i (k)$ ) and the state and the velocity information when the last event triggered ( $\tilde{x} i (k)$ and $\tilde{v} i (k)$ ). When the values of the triggered functions $f i (x i (k), e i, x (k)) < 0$ or $h i (v i (k), e i, v (k)) < 0$ , the event on agent i is triggered. The agent i broadcast its own information to its neighbors as soon as the event occurs. Then both the triggered agent and its neighbors update their controls. In summary, each agent (i.e., agent i) only knows three types of information: its current information on state and velocity ( $x i (k)$ and $v i (k)$ ), the state and the velocity information when the last event triggered ( $\tilde{x} i (k)$ and $\tilde{v} i (k)$ ), and the broadcast values of its neighbors j, $j \in N i$ ( $\tilde{x} j (k)$ and $\tilde{v} j (k)$ ). Then the consensus algorithm with the controller (3) can be replaced under the event-triggered communication scheme by using the constant function (5). So the event-triggered control can be described as

u i (k) = - K 1 {\sum_{j \in N i} [a (i, j) (x i (k_{m}^{i}) - x j (k_{m'}^{j}))] + x i (k_{m}^{i})} - K 1 {\sum_{j \in N i} [a (i, j) (v i (k_{n}^{i}) - v j (k_{n'}^{j}))] + v i (k_{n}^{i})} = - K 1 {\sum_{j \in N i} [a (i, j) (\tilde{x} i (k) - \tilde{x} j (k))] + \tilde{x} i (k)} - K 1 {\sum_{j \in N i} [a (i, j) (\tilde{v} i (k) - \tilde{v} j (k))] + \tilde{v} i (k)}

(7)

The event trigged instants for agent i and j are defined by

k_{m'}^{j} = max {k | k \in {k_{m}^{j}, m = 0, 1, 2, \dots}, k \leq k_{m + 1}^{i}} k_{n}^{i} = max {k | k \in {k_{n}^{i}, n = 0, 1, 2, \dots}, k \leq k_{m + 1}^{i}} k_{n'}^{j} = max {k | k \in {k_{n}^{j}, n = 0, 1, 2, \dots}, k \leq k_{m + 1}^{i}}

(8)

where

k_{0}^{i} = 0

is the initial value. Obviously, the time interval of events can be larger than the sample period T. The controller of agent i may be updated when the neighbors' events occurred, that means the values of

x j (k_{m'}^{j})

v i (k_{n}^{i})

, and

v j (k_{n'}^{j})

may change for

k \leq k_{m + 1}^{i}

Remark 2

The advantages of the event-triggered scheme proposed in the article are discussed herein. Firstly, different from event-triggered conditions on the premise of continuous time, which require continuous event detection and continuous communication between neighboring agents, the event-triggered scheme proposed in this article only needs discrete time instants. Secondly, different from centralized event detectors, where every agent must be aware of global information, the method proposed in this article only needs the information from its local neighbors; therefore the method proposed in the article is distributed. Thirdly, compared with the existing distributed event-triggered strategies, the proposed scheme has the following two characteristics: (1) The triggered conditions of the state and the velocity are considered separately, when an event occurs, only the state or the velocity information deliveries in the network, which can further save network bandwidth; (2) When an event is triggered on a agent, only the agent and its adjacent agent update their protocols, without having to update all agent agreements together, the energy consumption of the sensor can be further reduced. It can conclude that the method proposed can greatly minimize the sensor energy consuming and reduce the usage of network bandwidth. Furthermore, it is worth noting that the feedback gain matrix K₁ of the proposed event-triggered consensus algorithm can be solved by using the linear matrix inequality (LMI) method.

Stability analysis

This section presents the analysis of the convergence of the closed-loop multi-agent system (1) under the event-triggered consensus control (7).

According to equation (4), the event-triggered control protocol (7) can be written as

u i (k) = - K 1 {\sum_{j \in N i} [a (i, j) ((x i (k) - x j (k)) + (e i, x (k) - e j, x (k)))] + x i (k) + e i, x (k)} - K 1 {\sum_{j \in N i} [a (i, j) ((v i (k) - v j (k)) + (e i, v (k) - e j, v (k)))] + v i (k) + e i, v (k)}

(9)

where

k_{m}^{i} \leq k < k_{m + 1}^{i}

Now, we define the augmented variables used in equation (9) as

y i (k) = [x_{i}^{T} (k), v_{i}^{T} (k)] T \in R 2 p w i (k) = [e_{i, x}^{T} (k), e_{i, v}^{T} (k)] T \in R 2 p Y (k) = [y_{1}^{T} (k), y_{2}^{T} (k), \dots, y_{l}^{T} (k)] T \in R 2 lp W (k) = [w_{1}^{T} (k), w_{2}^{T} (k), \dots, w_{l}^{T} (k)] T \in R 2 lp

(10)

Equation (1) can be expressed in matrix form as

y i (k + 1) = [x i (k + 1) v i (k + 1)] = [I p T I p 0 I p] [x i (k) v i (k)] + [\frac{T 2}{2} u i (k) T u i (k)]

(11)

where

I p

denotes an

p \times p

identity matrix.

Combining equations (9) to (11), we can obtain that

Y (k + 1) = I l \otimes [I p T I p 0 I p] Y (k) + [- \frac{T 2}{2} K 1 {\sum_{j \in N 1} [a (1, j) (x 1 (k) - x j (k) + e 1, x (k) - e j, x (k))] + (x 1 (k) + e 1, x (k))} - \frac{T 2}{2} K 1 {\sum_{j \in N 1} [a (1, j) (v 1 (k) - v j (k) + e 1, v (k) - e j, v (k))] + (v 1 (k) + e 1, v (k))} - T K 1 {\sum_{j \in N 1} [a (1, j) (x 1 (k) - x j (k) + e 1, x (k) - e j, x (k))] + (x 1 (k) + e 1, x (k))} - T K 1 {\sum_{j \in N 1} [a (1, j) (v 1 (k) - v j (k) + e 1, v (k) - e j, v (k))] + (v 1 (k) + e 1, v (k))} : - \frac{T 2}{2} K 1 {\sum_{j \in N l} [a (l, j) (x l (k) - x j (k) + e l, x (k) - e j, x (k))] + (x l (k) + e l, x (k))} - \frac{T 2}{2} K 1 {\sum_{j \in N l} [a (l, j) (v l (k) - v j (k) + e l, v (k) - e j, v (k))] + (v l (k) + e l, v (k))} - T K 1 {\sum_{j \in N l} [a (l, j) (x l (k) - x j (k) + e l, x (k) - e j, x (k))] + (x l (k) + e l, x (k))} - T K 1 {\sum_{j \in N l} [a (l, j) (v l (k) - v j (k) + e l, v (k) - e j, v (k))] + (v l (k) + e l, v (k))}] = I l \otimes [I p T I p 0 I p] Y (k) + I l \otimes [- \frac{T 2}{2} K 1 - \frac{T 2}{2} K 1 - T K 1 - T K 1] Y (k) - L \otimes [\frac{T 2}{2} K 1 \frac{T 2}{2} K 1 T K 1 T K 1] Y (k) + I l \otimes [- \frac{T 2}{2} K 1 - \frac{T 2}{2} K 1 - T K 1 - T K 1] W (k) - L \otimes [\frac{T 2}{2} K 1 \frac{T 2}{2} K 1 T K 1 T K 1] W (k)

(12)

where the notation ⊗ stands for the Kronecker product. At last, we can obtain matrix form from equation (12) as

Y (k + 1) = MY (k) + KNY (k) + KNW (k)

(13)

where

M = I l \otimes [I p T I p 0 I p], N = [I l \otimes [- \frac{T 2}{2} - \frac{T 2}{2} - T - T] - L \otimes [\frac{T 2}{2} \frac{T 2}{2} T T]], K = I l \otimes [K 1 00 K 1]

Based on the preceding preparations, next we will derive sufficient conditions to ensure consensus for discrete-time second-order multi-agent systems by using the Lyapunov functional method and the Kronecker product techniques.

Theorem 1

Assume that the fixed weighted directed graph associated with multi-agent system (1) contains a directed spanning tree. The event-triggered control protocol (7) with given trigger parameters $σ i$ (i = 1,2) can solve the consensus problem for discrete-time second-order multi-agent systems if there exist symmetrical positive definite matrix P, where $P = diag {I l \otimes P 1, I l \otimes P 1} \in R 2 lp \times 2 lp$ , matrices $Ω ij > 0$ , i = 1,2,…,l, j = 1,2 and $Y = diag {I l \otimes Y 1, I l \otimes Y 1}$ with suitable dimensions such that the following LMI is satisfied

[- P + \tilde{Ω} * * 0 - Ω * PM + YN YN - P] < 0

(14)

where * denotes the entries determined by symmetry,

\tilde{Ω} = diag {σ 1 Ω 11, σ 2 Ω 12, \dots, σ 1 Ω l 1, σ 2 Ω l 2}

(15)

Ω = diag {Ω 11, Ω 12, \dots, Ω l 1, Ω l 2}

(16)

Furthermore, the matrix $K = diag {I l \otimes K 1, I l \otimes K 1}$ which composed by the feedback gain matrix K₁ are given as

K = P - 1 Y

Proof

Construct the following Lyapunov function

V (k) = Y T (k) PY (k)

where P is a positive definite matrix and satisfied

P = diag {I l \otimes P 1, I l \otimes P 1} \in R 2 lp \times 2 lp

Δ V (k) = V (k + 1) - V (k) \leq Y T (k + 1) PY (k + 1) - Y T (k) PY (k) + max i = 1, 2, \dots, l {σ 1 x_{i}^{T} (k) Ω i 1 x i (k) - e_{i, x}^{T} (k) Ω i 1 e i, x (k), σ 2 v_{i}^{T} (k) Ω i 2 v i (k) - e_{i, v}^{T} (k) Ω i 1 e i, v (k)} \leq Y T (k + 1) PY (k + 1) - Y T (k) PY (k) + \sum_{i = 1, 2, \dots, l} {σ 1 x_{i}^{T} (k) Ω i 1 x i (k) - e_{i, x}^{T} (k) Ω i 1 e i, x (k), σ 2 v_{i}^{T} (k) Ω i 2 v i (k) - e_{i, v}^{T} (k) Ω i 2 e i, v (k)} = [MY (k) + KNY (k) + KNW (k)] T \times P [MY (k) + KNY (k) + KNW (k)] - Y T (k) PY (k) + Y T (k) \tilde{Ω} Y (k) - W T (k) Ω W (k) = [Y T (k) W T (k)] \times [(M + KN) T P (M + KN) - P + \tilde{Ω} * (NK) T P (M + KN) (KN) T P (KN) - Ω] \times [Y T (k) W T (k)]

It is easy to obtain that

[(M + KN) T P (M + KN) - P + \tilde{Ω} * (KN) T P (M + KN) (KN) T P (KN) - Ω] = [- P + \tilde{Ω} * 0 - Ω] + [(M + KN) T (KN) T] \cdot P \cdot P - 1 \cdot P \cdot [M + KN KN]

Finally, by using the Schur complement, the sufficient condition to ensure $Δ V (k) < 0$ is that the following inequality holds

[- P + \tilde{Ω} * * 0 - Ω * PM + PKN PKN - P] < 0

(17)

Let $Y = PK$ , we observed that equation (17) is equivalent to equation (14). Accordingly, we can derive the form of $Y = diag {I l \otimes Y 1, I l \otimes Y 1} \in R 2 lp \times 2 lp$ . We can conclude that equation (14) is the sufficient condition to ensure asymptotically stable of the multi-agent system (13). That is to say equation (2) is satisfied simultaneously.

Numerical examples and simulations

Next, computer simulation is illustrated on the event-triggered program proposed. The multi-agent system which contains four agents is considered. The network topology of the multi-agent system is shown in Figure 1. It is a directed weighed graph which contains a spanning tree. It is not difficult to get the graph's adjacency matrix

A = [0 0.8 00 1.2 0 1.5 0300002 2.5 0]

so the Laplacian matrix is thus given by

L = [0.8 - 0.8 00 - 1.2 2.7 - 1.5 0 - 3 0300 - 2 - 2.5 4.5]

Figure 1.

Directed communication graph G.

The initial states and the initial velocities are chosen as $x 1 (0) = [- 12, 1] T$ , $x 2 (0) = [- 2, - 1] T$ , $x 3 (0) = [3, 4] T$ , and $x 4 (0) = [12, - 1.5] T$ and $v 1 (0) = [- 0.3, 0.9] T$ , $v 2 (0) = [- 0.6, 0.4] T$ , $v 3 (0) = [0.6, 0.3] T$ , and $v 4 (0) = [0.2, 0.8] T$ , respectively. Event-triggered controller case and traditional periodic-triggered controller case are compared in the following cases.

Case I: event-triggered control

The event-triggered controller (7) is applied to the four agents. Event-triggered conditions (6) are used in the experiment, which is characterized by separately considering the state and the velocity. We set the sampling period T = 0.2, feedback gain matrix

K 1 = [0.4 0.1 0.1 0.5]

and parameters

σ 1 = 0.04

σ 2 = 0.05

Ω ij = [1001]

, where

i = 1, 2, 3, 4; j = 1, 2

. The state and the velocity responses of the four agents are depicted in Figures 2 and 3, respectively. The sampled and event numbers for states and velocities are given in Tables 1 and 2, respectively.

Figure 2.

The state responses of the four agents under Case I.

Figure 3.

The velocity responses of the four agents under Case I.

Table 1.

Sample and event times for states.

Time/agents	Agent 1	Agent 2	Agent 3	Agent 4
Sample times	250	250	250	250
Event times	129	128	133	134

Table 2.

Sample and event times for velocities.

Time/agents	Agent 1	Agent 2	Agent 3	Agent 4
Sample times	250	250	250	250
Event times	130	135	133	141

Case II: traditional periodic-triggered

The controllers (3) are applied to the four agents in the Case II. Under this case, we use the same feedback gain matrix K and the same simulation time as in Case I (i.e., $t \in [0, 50]$ ) for comparison. Because it is not event-triggered scheme, all agents must communicate with its neighbors at every sampling time and then update their protocols. Figures 4 and 5 show the state and the velocity responses of the four agents, respectively.

Figure 4.

The state responses of the four agents under Case II.

Figure 5.

The velocity responses of the four agents under Case II.

Next, we compare the performance in the two cases. When the total sampling number is 250, all of the state and velocity signals need to be transmitted to the adjacent agents in Case I, while only 53.15% of the state and velocity signals need to be transmitted in Case II. That is, the communication bandwidth used by the proposed event-triggered strategy is reduced about 46.85% in comparison with the total number of state and velocity signals transmission by the periodic-triggered scheme. Then, we make a comparison between Figures 2 and 3 and Figures 4 and 5, respectively. It can be easily seen that the state and velocity responses of the four agents have no clear differences under event-triggering scheme and periodic-triggering scheme. Therefore, it can be concluded that when the event-triggered communication scheme is applied, the communication bandwidth is reduced while the performance of the algorithm is not compromised. This indicates that the event-triggered strategy proposed in the article is effective.

Conclusions

In order to reduce the communication bandwidth of network and save energy consumption of the sensors, this article proposes a distributed event-triggered control of discrete-time second-order multi-agent systems under directed topology with a spanning tree. By constructing Lyapunov function and utilization of Kronecker product method, the sufficient condition of the system convergence is given in the form of LMI. Finally, the effectiveness of the proposed event-triggered control is validated by several numerical examples.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported partly by Top-notch Academic Programs Project of Jiangsu Higher Education Institutions and partly by Joint Innovation Project Foundation of Jiangsu Province under Grant BY2014024.

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