New Trends in Networked Control of Complex Dynamic Systems: Theories and Applications
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Zhaoxia Wang, Minrui Fei, Dajun Du, Lili Shang, "EventTriggered Average Consensus for Multiagent Systems with TimeVarying Delay", Mathematical Problems in Engineering, vol. 2014, Article ID 131586, 14 pages, 2014. https://doi.org/10.1155/2014/131586
EventTriggered Average Consensus for Multiagent Systems with TimeVarying Delay
Abstract
The paper investigates average consensus for multiagent systems with timevarying delay. A reducing dimension multiagent systems model is presented firstly. Using eventtriggered mechanism to reduce network load, a comprehensive model is then proposed, which considers communication delay and triggered issue. Furthermore, the eventtriggered average consensus stability of multiagent systems with fixed directed/undirected graph is analyzed, and sufficient conditions are provided. Moreover, the upper bound of timevarying delay can be obtained conveniently. Finally, simulation results confirm the feasibility and effectiveness of the proposed method.
1. Introduction
Recently distributed coordination of multiagent systems has attracted significant interest of many researchers. It is becoming increasingly important in multivehicles, sensor network, and formation flying [1–5]. Consensus (or synchronization) problem [1, 6–8] is one of the most important issues of multiagent systems. In practice, it is very important for multiagent systems to achieve average consensus [9].
With the development of industrial largescale dynamical multiagent systems, the agent is responsible for collecting a lot of information from its neighboring nodes and transmission signals in the shared wireless communication network are massive. However, computing resources of each agent and the bandwidth of wireless network are limited. Therefore, it is necessary to study communication mechanism of multiagent systems [10], which is one of the most important issues in the implementation of average consensus algorithms. Moreover, communication networks are not always reliable, and communication delay is inevitably introduced [11–14]. It is well known that time delay may degrade the system performance or even cause the system instability [15–19]. Therefore, it is necessary to investigate the average consensus problem with time delay. Some researchers have investigated the average consensus problem for multiagent systems with time delay and communication scheme, respectively.
Considering the average consensus problem with time delay, there are some research results that have been reported [9, 20–23]. For an undirected network with fixed/switching graphs, an upper bound of communication time delays was obtained [20, 21]. Average consensus problem for directed networked multiagent systems with fixed/switching graphs and constant/varying time delays was investigated based on the LMI method [22, 23]. A necessary and sufficient condition was derived for multiagent systems with heterogeneous time delays to achieve average consensus [9]. The average consensus problems have been investigated with time delays and other issues such as noise [24–26]. Moreover, some researches have investigated consensus or average consensus for highorder multiagent systems with timevarying delays [27–30]. However, how to get the quantitative relationship between maximum time delay and system stability for directed networks with fixed/switching graphs is still an open issue.
For communication schemes, the eventtriggered mechanism has been proposed [31–34]. Compared with traditional timetriggered mechanism (i.e., the fixed sampling period), the eventtriggered mechanism can reduce the communication burden. This is because the signal can be transmitted only when the triggered condition is satisfied. Eventtriggered mechanism has been employed in multiagent systems [10, 35–37] without time delay, which was an effective methodology for multiagent systems with limited computational and communicational resources. Furthermore, based on eventtriggered mechanism, a tracking control problem of leaderfollower multiagent continuoustime systems with communication delays was investigated [38], and the consensus problem of discretetime multiagent systems with random communication delays was studied [39].
In this paper, the eventtriggered average consensus problem for continuoustime multiagent systems in a directed network with timevarying delay is studied. The main contributions include the following. Firstly, a reducing dimension multiagent systems model with eventtriggered average consensus protocol and timevarying delay is provided. Secondly, a LyapunovKrasovskii functional is constructed, and sufficient conditions are obtained, and all the agents can achieve the average consensus asymptotically. Different from [38, 39], an upper bound of timevarying communication delay is derived in this paper.
The paper is organized as follows. Section 2 presents the background and system model. The proposed approach is provided in Section 3. Section 4 gives the main results of the paper. Simulation results are described in Section 5. Section 6 concludes the paper.
2. Background and System Model
2.1. Preliminaries of Graph Theory
Let be a weighted digraph of order , with the set of nodes , and set of edges , where is ordered, is the edge’s tail, and is the head. The set of neighbors of node is denoted as . The node indexes of belong to a finite index set . is a weighted communication adjacency matrix with nonnegative adjacency elements , where and if and only if . The indegree and outdegree of are defined as
The degree matrix is a diagonal matrix with
The graph Laplacian of is defined as . It can be easily obtained that the element of satisfies
By definition, every row sum of is zero and thus is an eigenvector of associated with the eigenvalue . This therefore means that .
Definition 1 (balanced graph [20]). We say the node of a graph is balanced if and only if . A graph is called balanced if and only if all of its nodes are balanced. Obviously any connected undirected graph is balanced.
Definition 2 (balanced matrix [22]). A square matrix is said to be a balanced matrix if and only if and .
It is easy to know that the Laplacian of a balanced graph satisfies and .
Lemma 3 (see [22]). Consider the matrix
The following statements hold.(1)The eigenvalues of are with multiplicity and 0 with multiplicity 1. The vectors and are left and the right eigenvectors of associated with the zero eigenvalue, respectively.(2)There exists an orthogonal matrix such that and is the matrix of eigenvectors of . For any balanced matrix
Remark 4. For the Laplacian of a connected graph , it is easy to have the following equation by Lemma 3: where is an orthogonal matrix of eigenvectors of , and we define it as , in which is the first columns of .
2.2. System Model
Suppose that the network system consists of agents. Each agent is regarded as a node in the graph . Let denote the state (or value) of node . The value of a node might represent physical quantities including position, temperature, and voltage. Moreover, assume that each node is an agent with dynamics: where is the control input (or protocol). The consensus control protocol without communication time delay in [20] was given by Here with the consensus protocol, each agent consists of a controller and dynamics as shown in Figure 1.
We say the nodes of a network have reached a consensus if and only if for all , . Whenever the nodes of a network are all in agreement, the value of all nodes is called the group decision value. Particularly, for all , if there exists protocol asymptotically solves the average consensus problem.
Further, the consensus control law with communication time delay was given by [9] where is the communication time delay with which the state of node passes through a channel before getting to node . is the selfdelay, which occurs when node compares its selfinformation and information of node over the network.
In this paper, we consider the case that the selfdelay is equal to the communication delay , that is, , which is discussed in [9, 20, 22, 23]. Then the consensus control law (11) becomes
Given protocol (9) and (11), the network dynamics of agent is summarized as where denotes the value of all nodes and is the Laplacian of graph .
The system (13) without communication time delay under eventtriggered strategy has been investigated in [36, 37].
In the following section, we will redefine protocol (12) and model (14) to take into account eventtriggered mechanism. The consensus control formulation and problem statement for centralized eventtriggered cooperative control are described in the following section.
3. The Proposed Approach
3.1. EventTriggered Mechanism and Modeling of Multiagents Systems with Communication Time Delay
Each agent of multiagent systems is equipped with a small embedded microprocessor and regularly sampled with period by microprocessor in practice, where the monotone increasing sampling sequence is described by the set , . Using eventtriggered mechanism, only the successful transmitted signals are available, where the successful transmitted instants sequences satisfy , . Considering the bounded delay , which denotes the time delay at successful transmitted instant , the transmitted finishing instant is described by the set . Moreover, the control input is generated by a zeroorder holder (ZOH) with the holding time . The relationship between , , and is shown in Figure 2.
Remark 5. Specially, if , it means that all sampled data are transmitted, that is, a timetriggered transmission scheme. If , it means that not all sampled data are transmitted; that is, the numbers of sampled data are reduced. Therefore, the eventtriggered scheme can reduce communication burden.
3.1.1. Centralized EventTriggered Mechanism
The state measurement error is defined by where is the error between the current sampled value and the latest transmitted sampled value .
The eventtriggered transmission mechanism in [40, 41] is designed as where is a positive definite matrix and is a given and bounded positive scalar parameter.
When the local state measurement error signal exceeds the given threshold, that is, condition (16) is satisfied, the current sampled information is transmitted. Obviously, the current successful transmitted state value is the subsequences of the latest transmitted sampled value, which is denoted by . Then each agent with eventtriggered consensus protocol is shown in Figure 3.
Remark 6. The eventtriggered mechanism (16) is characterized by the parameters , , and . Specially, if in (16), this leads to , and it becomes a timetriggered transmission mechanism.
3.1.2. Modeling of Hybrid EventTriggered Multiagent Systems
Here we considered the system (12) with fixed topology and timevarying delay. We know that the input is held constant in a control period; that is,
The proposed eventtriggered control law of (12) in the centralized case is defined as so that
The network dynamics of system (14) is then given by
Obviously, and .
Next, two cases are discussed.
(1) If , there exists , satisfying
For a detailed timing analysis, consider the following interval: where some sets are defined as follows: and , .
Define
From (23) and (24): From (25), it can be obtained that where and .
Define the state measurement error
(2) If , define , .
It is clear that ; that is, , . Then there exists .
According to the definition and analysis of the aforementioned two cases, it is seen that (19) and (20) can be equivalently written as where and . The initial conditions of system (28) are assumed to satisfy , , . Moreover, between and , no control signal is triggered; that is,
Therefore, in the eventtriggered formulation, the event time is defined by with , , .
3.2. Problem Setting
The following task is to find the conditions for system (28) to reach average consensus with the eventtriggered communication mechanism (29). Considering a directed network system (28) with fixed graph that is strongly connected and balanced, denote by the average of the agents’ states. Due to the balanced graph of , the Laplacian satisfies from Definition 2. For system (28), it implies ; that is, . Then the time derivative of is given by . So that ; that is, is an invariant quantity. Then the state vector can be decomposed as [20] where and satisfies . For a strongly connected digraph, the vector belongs to an ()dimensional subspace called the disagreement eigenspace of .
Submitting (31) to (28), system (28) is equivalent to
Moreover, between and (), no control signal is triggered; that is,
Because is a positive definite matrix, there exists invertible matrix satisfying . Substituting into (33) yields that is, So when the condition satisfies one can get Between and (), no control signal is triggered, where
For system (32), it follows that, from Remark 4, where is an orthogonal matrix of eigenvectors of , and we define it as , in which is the first columns of . Denote .
Noting that , , we have
Then, system (32) is equivalent to where and .
Condition (37) is equivalent to that is, where and .
Lemma 7. If , then .
Proof. From , we have , and then . Therefore, Lemma 7 is obtained. This completes the proof.
Lemma 8 (see [42]). Let be a balanced digraph; then is strongly connected if and only if is weakly connected.
Remark 9. The requirement of graph that we discuss is strongly connected, but it can be obtained that it should be weakly connected based on Lemma 8.
Definition 10 (see [43]). Let be a given finite number of functions such that they have positive values in an open subset of . Then, a reciprocally convex combination of these functions over is a function of the form where the real numbers satisfy and .
For a reciprocally convex combination of scalar positive functions , Lemma 11 is obtained by Definition 10.
Lemma 11 (see [43]). Let have positive values in an open subset of . Then, the reciprocally convex combination of over satisfies subject to Lemma 11 can be applied to handle the double integral terms of the LyapunovKrasovskii functional for time delay systems.
4. Main Results
Theorem 12. Considering a network system of agents, the network has a balanced and weakly connected weighted digraph with uncertain timevarying communication delay. For some given positive constants , , and , under the eventtriggered mechanism (43), the system (41) is asymptotically stable, if there exist positive definite matrixes , , , () , matrix (), () with () dimensions and , (), such that
where
and , is the first columns of , and is an orthogonal matrix of eigenvectors of .
The proof of Theorem 12 is presented in the appendix.
Corollary 13. Consider system (28) with the conditions (29), and assume that the network has a balanced and weakly weighted digraph with uncertain timevarying communication delay. Then, the system asymptotically achieves average consensus; that is,
Proof. Using Theorem 12, we have that . According to Lemma 7, , so we have . This completes the proof.
From Remark 6, when in (29), the eventtriggered mechanism becomes a timetriggered sampling mechanism. The following Corollary 14 provides the results for this case.
Corollary 14. Considering a network system of agents, the network has a balanced and weakly connected weighted digraph with timevarying communication delay. For some given positive constants , , the system (28) asymptotically achieves average consensus, if there exist positive definite matrixes , (), , (), and matrix (), () with dimensions, such that where and , is the first columns of , and is an orthogonal matrix of eigenvectors of .
Remark 15. It is known that any connected undirected graph is balanced by Definition 1. Therefore, the proposed Theorem 12 and Corollaries 13 and 14 can be applied to the connected undirected network.
5. Numerical Example
To verify the effectiveness of proposed method, numerical example was operated on.
Example. Considering a directed network of six agents as shown in Figure 4, it has balanced and weakly connected digraph with 01 weights. Set the initial condition ; then .
Set , using Theorem 12, and maximum allowable delay can be easily gained. Simulation results are listed in Table 1. It is found that the maximum allowable delay decreases with the increasing of .

For the first case , , , , using Theorem 12, the corresponding feasible solution isThe state trajectories of the network are shown in Figure 5. It is seen obviously that average consensus is asymptotically achieved. Figure 6 shows the eventtriggered time instant and time intervals. It is seen that the sampled data that need be transmitted reduce importantly. Evolution of the error norm is seen in Figure 7. It is seen that the solid line represents the evolution of , which stays below the specified statedependent threshold which is represented by the dotted line.
For the another case , , , , using Theorem 12, the corresponding feasible solution isSimilarly, Figure 5 shows the state trajectories of the network with , , . The network also asymptotically achieved average consensus. Eventtriggered time instant and time intervals are shown in Figure 8.
It is found from Figures 6 and 9 that the numbers of eventtriggered time instant reduce and the average value of eventtriggered time intervals increases with the increasing of . Simulation results are listed in Table 2. Therefore, the proposed eventtriggered mechanism can reduce much signal transmission and thus reduce the multiagent network load.

6. Conclusions
This paper has mainly investigated the eventtriggered average consensus problem in a directed/undirected network for multiagent systems with fixed topology and timevarying delay. Sufficient conditions for average consensus are presented, and an upper bound of timevarying communication delay is derived. Furthermore, due to unreliable information channels and limited bandwidth, communication between agents may produce data packet dropout and outoforder. Considering these issues, how to study the average consensus problem is the further work. Moreover, how to solve decentralized eventtriggered mechanism with networkedrelated nondeterministic issues is another important direction in the future.
Appendix
Proof of Theorem 12. The system (28) is equivalent to (41). Based on the system (41), we construct a LyapunovKrasovskii functional candidate as
where , (), are positive definite matrices:
Define
Then .
The derivative of (A.1) with respect to is
where
Define
Then the system (41) is rewritten as .
Using Jensens inequality [44] to handle the integral items in (A.6). Next, two cases are discussed.
(1) When , , , we can get
Applying Lemma 11 to (A.10) yields
where