Consensus Analysis for a Class of Heterogeneous Multiagent Systems with Time Delay Based on Frequency Domain Method

The consensus problem of heterogeneous multiagent systems composed of first-order and second-order agent is investigated. A linear consensus protocol is proposed. Based on frequency domain method, the sufficient conditions of achieving consensus are obtained. If communication topology contains spanning tree and some conditions can be satisfied on control gains, consensus can be achieved. Then, a linear consensus protocol with time delay is proposed. In this case, consensus is dependent only on system coupling strength, each agent input time delay, but independent of communication delay. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical result.


Introduction
Recently, multiagent systems have received significant attention due to their potential impact on numerous civilians, homeland securities, and military applications.Consensus plays an important role in achieving distributed coordination.The basic idea of consensus is that a team of agents reach an agreement on a common value by negotiating with their neighbors.Many literatures have investigated consensus problem on many cases, such as time variant topology, time delay, and nonlinearity [1][2][3][4][5][6][7][8][9][10].
Unfortunately, all the aforementioned multiagent systems are homogeneous; that is, all the agents share the same dynamics behavior.However, the dynamics of the agents are quite different because of various restrictions in the practical systems.Zheng et al. [11] studied the consensus problem of heterogeneous multiagent systems composed of first-order and second-order integrator agents.Zhu et al. [12] studied consensus problem of multiagent systems with two types of agents, namely, active agents and passive agents.Yin et al. [13] investigated the consensus problem for a set of discrete-time heterogeneous multiagent systems composed of two kinds of agents differed by their dynamics.Zhu et al. [14] investigated the finite-time consensus problem for heterogeneous multiagent systems composed of first-order and second-order agents.Zheng and Wang [15] studied finitetime consensus of heterogeneous multiagent systems with and without velocity measurements.C. Liu and F. Liu [16] considered stationary consensus of heterogeneous multiagent systems with bounded communication delays.Yan et al. [17] are concerned with the cooperative target pursuit problem by multiple agents based on directed acyclic graph.Kim et al. [18] studied the output consensus problem for a class of heterogeneous uncertain linear multiagent systems.Yin et al. [19] studied the consensus protocols design for a set of fractional-order heterogeneous agents, which is composed of two kinds of agents.Kim et al. [20] investigated the heterogeneous consensus problem for multiagent systems with random link failures between agents.
Consensus of heterogeneous multiagent systems is investigated in undirected graph generally.An important challenge is to study this problem in directed graph.However, this seems to be less studied in the literature.Meanwhile, frequency domain approach is an important method for analyzing stability of control system.In the existing literatures, there are so few literatures that use frequency domain method.Time delay appears in almost all practical systems, and times delay can degrade the systems performance or even destroy the system stability.
In this paper, we study consensus problem of heterogeneous multiagent systems with and without time delay.The model and algorithms in this paper are similar to [11], but in [11] they only consider the case in undirected graph mainly based on Lyapunov theory.Unlike [11], in this paper consider the cases with and without time delay in directed graph.Based on the frequency domain method, the sufficient conditions are given for the existence of consensus solution to heterogeneous multiagent systems.By these conditions, it is shown that consensus can be achieved when control gains satisfy some conditions in the case without time delay.In the case with time delay, the results show that the consensus is dependent only on system coupling strength, each agent input time delay, but independent of communication delay.
The rest of the paper is organized as follows.In Section 2, some preliminaries are introduced for the graph theory, and heterogeneous multiagent systems are formulated.Two consensus protocols are proposed in Section 3, and consensus analysis is shown for without delay time and with delay time, respectively.In Section 4, two numerical examples are studied.Finally, concluding remarks are given in Section 5.

Preliminaries
2.1.Graph Theory.Let G(V, E, A) be a weighted directed graph composed of a set of nodes V = {V 1 , V 2 , . . ., V  }, set of edges E ⊂ V × V, and a weighted adjacency matrix  = [  ] × , with nonnegative adjacency elements   .An edge of G is denoted by   = (V  , V  ) ∈ E, which means that node V  receives information from node V  and V  is called the parent of V  .The adjacency elements associated with the edges of the graph are positive; that is,   ∈ E ⇔   > 0.Moreover, we assume   = 0 for all  ∈ 1, . . ., .The set of neighbors of node V  is denoted by   = {V  ∈ V : (V  , V  ) ∈ E}.The corresponding graph Laplacian  = [  ] × can be defined as If   =   , then G becomes the undirected graph.If there is not isolated node in an undirected graph, the graph is said to be connected.
Graph with a Spanning Tree.For a directed graph, if every node, except a node called root, has exactly one parent, then it is called a directed tree.A spanning tree of a directed graph is a directed tree formed by graph edges that connect all the nodes of the graph.In heterogeneous multiagent system, the neighbors of each second-order agent  include first-order and secondorder agents, denoted by   =    ∪    , and the neighbors of each first-order agent are denoted by   =    ∪    .The Laplacian matrix can be denoted as follows: where   =   +   ,   is the Laplacian matrix of secondorder agents,   = diag(∑ ∈     ,  = 1, . . ., ), and   denotes the adjacency relations of second-order agent to firstorder agent.Meanwhile,   is the Laplacian matrix of firstorder agent,   = diag(∑ ∈     ,  =  + 1, . . ., ), and   denotes the adjacency relations of first-order agents to second-order agents.

The Heterogeneous Multiagent
System.Suppose that the heterogeneous multiagent system consists of first-order and second-order integrator agents.The number of agents is , labeled from 1 through .Firstly, the heterogeneous multiagent system and concept of consensus are established.Suppose that the number of second-order integrator agents is  ( < ).The remainder is the first-order integrator agents; the number is  − .Then, the system is given as follows: where   , V  ,   are the position, velocity, and control input, of second-order agent , respectively.  ,   are the position and control input of first-order agent , respectively.The initial conditions are Definition 1.The heterogeneous multiagent system (3) is said to reach consensus asymptotically if, for any initial conditions x(0) and v(0), one has lim  → ∞ ‖  () −   ()‖ = 0, ,  = 1, . . ., , lim  → ∞ ‖V  () − V  ()‖ = 0, ,  = 1, . . ., .
Lemma 2 (see [5]).Suppose that z = [ 1 ,  2 , . . .,   ]  with   ∈  and  ∈  × is the Laplacian matrix.Then, the following four conditions are equivalent: (i)  has a simple zero eigenvalue with an associated eigenvector 1  and all of the other eigenvalues have positive real parts; (ii)  = 0 implies that  1 =  2 ⋅ ⋅ ⋅ =   ; (iii) consensus is reached asymptotically for a system  = −; (iv) the directed graph of  has a directed spanning tree.Remark 3. Throughout the paper, we just take the positions and velocities of agents as scalars.However, all the following developments can be directly extended to the case of vectors by introducing the Kronecker product.

Consensus Protocol without Time Delay.
In this section, the protocol without time delay is proposed for system (3), as follows: where  1 ,  2 > 0 are control gains.
Theorem 4. If the following conditions can hold, then consensus of system (3) with protocol (4) can be achieved.(i) The fixed topology contains spanning tree.(ii) The control gains satisfy conditions Proof.The system (3) under protocol (4) can be written in a vector form as where where The Laplace transformation is imposed on system (6) and we have From ( 8), the following form is obtained: Let   = [ 1 , . . .,   ] T ,   = [ +1 , . . .,   ] T , and  = [ T  ,  T  ] T ; we have Let () = det( − Γ()).It is the fact that the zero point of () is the eigenvalue of Λ.Consider the following.

Consensus Protocol with Time Delay.
In this section, the case with time delay is considered.A consensus protocol with time delay is proposed for system (3).The different Mathematical Problems in Engineering communication time delay and identical input time delay among agents are considered in following protocol: where  1 ,  2 > 0 are control gains,   is communication time delay, and  is input time delay.Theorem 5.If the following conditions can be hold, then consensus of system (3) with protocol (15) can be achieved.
If the point (−, 0),  ≥ 1, is not in G , then We have If the following inequalities can hold, then (22) will be established: Note that sin / ≤ 1 can be established for all  ∈  and −1 ≤ cos  ≤ 1.Therefore, if the following conditions are satisfied, then the point (−1, 0) is not encircled by Nyquist curve of G (),  = 1, . . ., : Next, G ,  =  + 1, . . ., , are considered.If the point We have It is obvious that sin / ≤ 1 can be established for all  ∈ .So, (29) can be satisfied, and (28) will hold: Combining the above analysis, Theorem 5 is demonstrated.Remark 6.From Theorem 5, it can be seen that consensus condition of system (3) with time delay only is dependent on input time delay of each agent, but is independent of communication time delay.
In Theorem 5, each agent with identical input time delay is assumed.From the above analysis, the conclusion with different input time delay among agents can also be deduced from the conclusion of Theorem 5. Consider

Simulation
In this section, several simulation results are presented to illustrate the proposed consensus algorithm introduced in Section 3.
Example 8. Consider a heterogeneous multiagent system with 4 agents, shown in Figure 1, where nodes 1, 2 are secondorder agents and nodes 3, 4 are first-order agents.The graph has a directed spanning tree apparently.If   ∈ E, then   = 1; else   = 0.The Laplacian matrix is given as follows: According to the conditions in Theorem 4, we select  1 = 2.1,  2 = 1.2.The initial conditions are x(0) = [2, 1, 1.5, 3] T , k(0) = [2, 1] T .The simulation results (Figure 2) show that the positions of all agents and the velocities of second-order agents reach consensus asymptotically.
Example 9.These are the same communication topology and initial conditions as Example 8.The consensus protocol with time delay is considered.Firstly, control gains are selected as  1 = 2.1,  2 = 1.2.Based on the conditions in Theorem 5, the upper bound of input time delay can be obtained and  < 0.2381 can be derived.We selected  = 0.23.The communication time delay among agents is set as  12 = 0.2,  23 = 0.6,  34 = 0.4,  41 = 0.5, and  42 = 0.3.The simulation results (Figure 3) show that the positions of all agents and the velocities of second-order agents reach consensus asymptotically.It can be seen that the speed of convergence of system with time delay is slower than the system without time delay.
Based on Example 9, input time delay is set as  = 0.5, and the simulation result (Figure 4) shows that the system is divergent.So, consensus condition is dependent on input time delay.
Meanwhile, we selected  = 0.23, and the communication time delays are enlarged as  12 = 1.5,  23 = 1.2,  34 = 1.6,  41 = 1.4,and  42 = 1.2.The simulation result (Figure 5) shows that the system is convergent.Consensus condition is dependent on input time delay.But, communication time delays can affect the speed of convergence of system obviously.

Conclusion
This paper focuses on the consensus problem of heterogeneous multiagent composed of first-order agent and second-order agent.The protocols with time delay and without time delay are investigated, respectively.The sufficient conditions of consensus are obtained in two cases.Consensus of the protocol without time delay is dependent on control gains and communication topology.If the control gains satisfy some conditions and communication topology contains spanning tree, then the consensus can be achieved.Consensus of the protocol with time delay is dependent only on system coupling strength, each agent input time delay, but independent of communication delay.Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical result.From simulation, it can be seen that all communication time delays and input time delay can affect the speed of convergence of system obviously.In the future work, discrete-time heterogeneous with time delay will be considered.