On Leaderless and Leader-Following Consensus for Heterogeneous Nonlinear Multiagent Systems via Discontinuous Distributed Control Protocol

This paper is concerned with consensus of heterogeneous nonlinear multiagent systems via distributed control. Both the cases of leaderless and leader-following are systematically investigated. Different from some existing results, completed consensus can be reached in this paper among heterogeneous multiagent network instead of bounded-consensus. First, a novel distributed control protocol is proposed, and some general consensus criterions are derived formultiagent systemswithout leader. Second, a leaderwith unknown but bounded input for the heterogeneous multiagent network is considered; aperiodically intermittent communications among followers are considered to avoid channel blocking in this case. Finally, two simulation examples are presented to verify the effectiveness of the main results.


Introduction
In recent years, distributed coordinated control of multiagent systems has been widely studied due to its easy implementation, strong robustness, and high self-organizability.There were many applications of multiagent systems in the field of robotic systems, UAVs (unmanned air vehicles), wireless sensor networks, and so on [1][2][3][4].As a hot topic of multiagent systems, consensus has attracted great attention in systems and control theory.Many papers have been concerned with consensus problem of multiagent systems, in which, consensus can be reached among multiagent systems via sufficient local information exchanges between agents and their neighbors; one can see [5][6][7][8][9][10][11] and references therein.
There were two kinds of consensus named leaderless consensus and leader-following consensus, respectively.It is called leaderless consensus problem if there is no specified leader in the multiagent systems; it is called leader-following consensus problem otherwise.Both leaderless consensus and leader-following consensus have gotten many results recently.For example, distributed leaderless consensus algorithms for networked Euler-Lagrange systems have been investigated in [12].Reference [13] has studied leaderless consensus problem of a group of mobile agents interconnected by a star-like topology.On the other hand, leader-following consensus of multiagent systems has been considered in [14], in which both fixed and switching topologies have been considered.Based on -matrix strategies, pinning-controlled leader-following consensus in multiagent systems has been discussed in [15].More results about leaderless consensus or leader-following consensus can be seen in [16][17][18][19][20] and references therein.In general, the leader-following consensus problem could be translated into the stability problem of error systems.There were many theoretical tools and results about the analysis of the stability of dynamical systems [21][22][23][24].To deal with the leaderless consensus problem, a virtual leader is often introduced or some matrix analysis techniques are used.
Note that the majority of above publications were concerned with identical systems.However, heterogeneous dynamic networks widely existed in real world, in which, each agent may have different parameters.A heterogeneous multiagent network cannot force consensus by static linear controllers.Thus, more results about heterogeneous dynamic networks were concerned with quasi-consensus (also named bounded-consensus).For example, [25] investigates the bounded-consensus problem for cooperative heterogeneous agents with nonlinear dynamics in a directed communication network.Quasi-synchronization of heterogeneous dynamic networks via distributed impulsive control have been studied in [26].More recently, quasi-synchronization is investigated for heterogeneous complex networks with switching sequentially disconnected topology in [27].To the best of our knowledge, few literature works have appeared concerning the problems of complete consensus among heterogeneous multiagent systems, which provides the motivation of the current study.Note that nonlinearity is inescapable in real world systems [28][29][30].The dynamics analysis of nonlinear systems has gotten many results [31][32][33][34][35][36], even to the nonlinear fractional order systems [37,38].Thus, nonlinear multiagent systems would be considered in this paper.
In this paper, we investigated both leaderless and leaderfollowing consensus in a heterogenous network environment.The contributions of this paper are as follows: First, based a novel discontinuous distributed control protocol, the leaderless consensus has been reached under some simple conditions.Then, under the analysis method, the leader-following consensus also has been studied, in which case, the communication among followers is aperiodically intermittent.Most of the existing results about consensus for the heterogeneous multiagent systems have adopted the adaptive strategy; one can see [39][40][41][42] and references therein.Compared with some existing results, the control protocol in this paper is static, which may be easier to implement.
The rest of the paper is organized as follows: In Section 2, we introduce some definitions and some lemmas which are necessary for presenting our results in the following.The main results about leaderless consensus and leader-following consensus of heterogenous multiagents are presented in Section 3.Then, some examples are given to demonstrate the effectiveness of our results in Section 4. Conclusions are finally drawn in Section 5.
Notations.In this paper, R  denotes the set of all dimensional column vectors.Let   be -dimensional identity matrix.For any ,   be its transpose, and   () and   () be the maximum and minimum eigenvalues of , respectively.⊗ denotes Kronecker product.

Preliminaries
Considering the following heterogeneous multiagent system consisting of  agents, the dynamical behavior of th agent is described as where To get our main results, some preliminaries of graph would be given.Let G = {Δ, , } be a undirected graph of order , where Δ = {V 1 , V 2 , . . ., V  },  ⊆ Δ × Δ denote the set of nodes and edges, respectively.For any ,   = 0.   = (V  , V  ) ∈  is an edge from  to , where  ̸ = , which means that V  can receive message from V  .V  is a neighbor of V  if   ∈ .The set of all neighbors of V  can be denoted as N  = {V  :   ∈ ,  = 1, 2, . . ., }. = {  } ∈ R × denotes weighted adjacency matrix, where   is weight which is satisfied   =   ̸ = 0 if   ∈  and   = 0 otherwise.We assume   = 0 for all  = 1, 2, . . ., .
Assumption 3. The topological structure is undirected in this paper, and the undirected communication graph is connected.
By some simple derivations, one can get the following error dynamic equation for  = 1, 2, . . .,  − 1: where =1 l   () and thus the error equation can be rewritten as The following assumption needs to be satisfied in this subsection.
Mathematical Problems in Engineering
Note that condition (21) in Theorem 5 is difficult to check due to its infinity.The following results could be checked easily based on a higher request for the intermittent communication.
Corollary 10.Under Assumptions 1, 3, 6, and 7, consensus of the heterogeneous multiagent system could be achieved if there exist positive definite diagonal matrix  and constants  > 0,  > 0 such that ( 18)- (20) and the following condition holds: Proof.It is obviously that ( 29) implies (21) based on the definition of notations c and T. Consequently, the result can be obtained.
Corollary 11.Under Assumptions 1, 3, 6, and 7, consensus of the heterogeneous multiagent system could be achieved if there exist positive definite diagonal matrix  and constants  > 0,  > 0 such that ( 18)-( 20) and the following condition holds:

Numerical Simulations
In this section, two examples are given to check our theorem results above.

Leaderless Consensus.
In this subsection, we have a multiagent system consisting of four heterogeneous agents.The Laplacian matrix  is given as Without control, the trajectories of these four agents are shown as Figures 1-4 with corresponding initial values.It is obviously that they have different dynamics, the first agent has a chaotic behavior, the second agent may have a convergent trajectory, the third agent has a periodic orbit, etc.According to the simulation, we have  2 = 5.7084,  3 = 2.5655,  4 = 7.6434, and based on the analysis of Section 3.1, the L could be obtained as In order to reach consensus for the heterogeneous multiagent system, let  = 10.The   can be selected as  3 .To solve ( 9) and (10), we set  = 20; then, by using MATLAB LMI Toolbox one has  = 9.8989 and  = 0.3136 3 .By some simple computations about (8), d can be selected as d1 = 0.6, d2 = 0.3, d3 = 0.8.Then, all conditions of Theorem 5 can be satisfied.Figures 5 and 6 give the simulations for the state variables   () and error variables   () =  +1, () −  1 (), respectively.

Conclusion
The leaderless consensus and leader-follower consensus of heterogeneous multiagent network have been studied in this paper.By utilizing a discontinuous communication protocol, leaderless consensus criterions formed as LMIs have been derived at first.Then, an unknown leader has been considered; under our discontinuous control protocol, the consensus could be obtained based on some conditions.The results in this paper could be applied to tracking control problem for an unknown target.In the leader-follower case, the communication among followers was aperiodically intermittent, which could prevent blocking the ways of signal transmission.Finally, simulation results have also been given to check the obtained theory results.Noting that this paper just considered an undirected topology of the network, future works include the study of directed structure and even switching topologies, which are more suitable to the real world.Time-delays are also inescapable in real world systems [16,[43][44][45]; however, this paper has not considered the timedelay, which would be a significant topic in our future work.

3 Figure 11 : 3 Figure 12 :
Figure 11: Time evolution of state variables of leader and three followers with control.