Distributed Coordination Control of First- and Second-Order Multiagent Systems with External Disturbances

This paper is devoted to the coordination control problem of heterogeneous firstand second-order multiagent systems with external disturbances. First, by applying the theory of eigenvalue and the method of model transformation, the consensus state of heterogeneous multiagent systems is obtained. Then, based on the consensus state, the control output is defined, and sufficient conditions are derived to make all agents reach consensus with H ∞ performance. Finally, simulation results are provided to demonstrate the effectiveness of the presented results.


Introduction
In the past decade, the consensus problem of multiagent systems is a hot issue in the field of systems and control, partly due to their potential applications in practical systems, such as unmanned vehicles and automated highway systems.For multiagent systems with first-order, second-order, highorder, and fractional-order integrator, many protocols have been designed based on the information of each agent and its neighbors [1].For example, Vicsek et al. proposed a simple model for phase transition of a group of self-driven particles and numerically demonstrated complex dynamics of the model [2], and then Jadbabaie et al. provided a theoretical explanation for the consensus behavior of the Vicsek model using graph theory [3].Hong et al. studied the consensus problem of multiagent systems with switching jointly connected topologies [4].The consensus problem for multiagent systems with time delays is investigated in [5][6][7][8][9][10].And then the leader-following consensus protocols are given in [11,12].Recently, the consensus problem for fractionalorder multiagent systems is also investigated, such as [13].In [14][15][16], rotating and constrained consensus problems are considered, respectively, and some new protocols are proposed.
Most of results of consensus problem are on multiagent systems with the same-order dynamics.Recently, the consensus of heterogeneous multiagent systems has received more and more attention because the dynamics of the agents coupled with each other are different in the practical systems.The consensus problem of heterogeneous multiagent systems with linear consensus protocol and saturated consensus protocol was studied in [17]; some sufficient conditions for consensus are established when the communication topologies are undirected connected graphs.Then, the consensus problem of heterogeneous multiagent systems with directed communication graphs was investigated in [18].And the consensus problem of heterogeneous multiagent systems composed of first-order, second-order, and nonlinear Euler-Lagrange agents was considered in [19].In [20], the consensus problem for heterogeneous multiagent systems with switching jointly connected interconnection is studied.In engineering, it is necessary to know the consensus state of the system.However, most of results have not given the consensus state of heterogeneous multiagent systems.In the past decade, the consensus states are deduced for some protocols, however, most of which include the negative feedback of velocity for every second-order agent.But what is the consensus state when the protocol does not include the 2 Mathematical Problems in Engineering negative feedback of velocity?This paper will investigate this problem.
On the other hand, multiagent systems are often subjected to various disturbances such as actuator bias, measurement/calculation errors, and the variation of the communication topology.For homogeneous multiagent systems, robust  ∞ consensus problems for first-order and high-order multiagent systems with external disturbances were studied in [21][22][23].And some conditions are derived to make all agents reach consensus with  ∞ performance.However, to the best of our knowledge, there is little result on the  ∞ consensus problem of heterogeneous multiagent systems.
Compared with literatures, agents considered in this paper not only have different order integrator dynamics but also are influenced by external disturbances.First, we deduced the consensus values of heterogeneous multiagent systems by applying the theory of linear algebra and the method of model transformation.Based on the consensus state, the control output is defined, and then we gave the sufficient conditions under which all agents can reach consensus with  ∞ performance by applying the robust control method.Finally, we demonstrated the effectiveness of the presented results by simulation.

Graph Theory.
To solve the coordination problems, graph theory is useful (see [24] for details).Consider a dynamical system consisting of  agents.With regard to the  agents as the vertices  = {V  ,  = 1, 2, . . ., }, the interconnection topology of  agents can be conveniently described by an undirected graph  = {, }, where  ⊂  ×  is the set of edges of the graph.(V  , V  ) defines one of the graph's edges if V  and V  can communicate with each other.  () = { | (V  , V  ) ∈ } denotes the set of labels of those agents which are neighbors of agent  ( = 1, 2, . . ., ) at time .Let  = [  ] ∈  × be the weighted adjacency matrix of the graph , and  = diag{ 1 ,  2 , . . .,   } ∈  × is its degree matrix, where   = ∑  =1   ,  = 1, . . ., .Then the Laplacian of the weighted graph is defined as  =  − , which is symmetric.A path that connects V  and V  in the graph  is a sequence of distinct vertices V  0 , . . ., V   , where If there exists a path between any two vertices V  and V  ( ̸ = ), then the graph is said to be connected.

System Model.
Consider a heterogeneous multiagent system composed of first-order and second-order integrator agents.The number of agents is .Assume that there are  second-order integrator agents ( < ).And each secondorder agent dynamics is given as follows: where   ∈ R, V  ∈ R, and   ∈ R are the position, velocity, and control input of agent , respectively, I  = {1, 2, . . ., }, and   () ∈  2 is the external disturbance.Each first-order agent dynamics is given as follows: where   ∈ R and   ∈ R are the position and control input of agent , respectively,   () ∈  2 is the external disturbance, and ]  be the initial condition of system (( 1)-( 2)).
Definition 1.The heterogeneous multiagent system (( 1)-( 2)) with   = 0 (∀) is said to reach consensus if, for any initial condition, one has lim Partition the adjacency matrix  of  as  = [ ) , and   ∈ R (−)× .Let   and   be the Laplacian matrix of the second-order and first-order agents networks, respectively.For agent , let  , be its neighbors with second-order agents and let  , be its neighbors with first-order agents.Thus, the Laplacian matrix of  can be written as where In this paper, we apply the following linear consensus protocol to the heterogeneous multiagent system ((1)-( 2)): where  = [  ] is the corresponding weighted adjacency matrix and  1 and  2 are the control parameters.
Remark 2. In [17], the convergence of system ((1)-( 2)) with protocol ( 6) is proved when  = 0, but the consensus state is not provided.In [18], the consensus states are deduced for some protocols; however, all protocols considered there include the absolute information of velocity for every agent.
In fact, for practical systems, the absolute information of velocity can not be received sometimes.Therefore, it is meaningful to study the consensus state of system ((1)-( 2)) with protocol (6). ,   ()  ]  , and () = [ 1 , . . .,   ]  .Then, the dynamics of system ((1)-( 2)) can be written as where and   is the identity matrix of order .If () = 0, then system (7) can be changed into the following form:

Main Results
In this section, firstly, the final consensus values of heterogeneous multiagent system (9) are deduced by applying matrix theory.The following lemmas are necessary in the following analysis.
Lemma 5. Suppose that the communication network  is an undirected and connected graph.Then the system matrix Γ has a simple zero eigenvalue.In addition, all the other eigenvalues are less than zero.
Proof.It is easy to see that zero is the eigenvalue of Γ.By the elementary transformation of matrix, we have where  is the Laplacian matrix of .From Lemma 3, we have rank() =  − 1; thus, rank(Γ) =  +  − 1.So zero is the simple eigenvalue of Γ.On the other hand, system (9) is stable by Lemma 4, so the eigenvalues of Γ have nonpositive real parts.Therefore, all the other eigenvalues of Γ have negative real parts.
Next, we will give the main result of this section.
Proof.It is easy to see that  = [1   , 0, 1  − ]  is the right eigenvector of Γ associated with zero eigenvalue and   = [0,  2 1   , 1  − ] is the left eigenvector of Γ associated with zero eigenvalue, where  is a parameter to be determined.
Remark 8.In [18], the consensus states of heterogeneous multiagent systems are deduced for some protocols which include the absolute information of velocity of every secondorder agent, and the analysis was complex.In the paper, we consider different protocol (6), which only includes relative information of agents, and the analysis is more comprehensive and obvious.And protocol ( 6) is not the special case of [18] because formula (10) in [18] can not be satisfied herein.
Lemma 9 (see [25]).Consider the system where () ∈ R  , () ∈ R  , and () ∈ R  are the state, input, and output of system (15).The systems matrices , ,  are constant matrices of appropriate dimensions.Given  > 0, system ( 15) is asymptotically stable with ‖  ()‖ <  for all nonzero () ∈  2 [0, ∞), if there exists a positive definite matrix  ∈ R × , such that the following linear matrix inequality holds: The objective of this paper is to find appropriate parameters of protocol (6) to attenuate disturbances of agents and make all agents reach consensus.Considering the consensus state of system ( 9), a natural way to define control output   () is as follows: Let () = [ 1 (), . . .,  + ()]; then (17) can be summarized as where  =  −   .
Remark 12.In the proof of Theorem 10, we first separate the agreement states from disagreement states by using a model transformation.Then, we studied the new states related to  ∞ consensus problem instead of all states.This technique is very important for solving the problem of  ∞ consensus.
Choose the performance index  = 0.53.By MATLAB, it is easy to verify that linear matrix inequality (19) Take   () = 0.8().Figure 4 shows the corresponding energy trajectories for the heterogeneous multiagent systems with external disturbance ().That is, consensus is achieved with the performance ‖  ()‖ ∞ < .

Conclusions
In this paper, we investigated the distributed coordination control problems of first-and second-order multiagent systems with external disturbances.A dynamic neighbor-based protocol is adopted for the agents.The consensus state is given by utilizing matrix theory and model transformation when external disturbance does not exist.Based on the consensus state, the control output is defined, and then sufficient conditions are derived to make all agents reach consensus while satisfying desired  ∞ performance.Finally, simulation results are provided to demonstrate the effectiveness of the presented results.The method used in this paper can be easily extended to the heterogeneous multiagent systems with switching topologies.The future work will focus on the timedelay systems with directed network.