Consensus of Continuous-Time Multiagent Systems with General Linear Dynamics and Nonuniform Sampling

This paper studies the consensus problem of multiple agents with general linear continuous-time dynamics. It is assumed that the information transmission among agents is intermittent; namely, each agent can only obtain the information of other agents at some discrete times, where the discrete time intervals may not be equal. Some sufficient conditions for consensus in the cases of state feedback and static output feedback are established, and it is shown that if the controller gain and the upper bound of discrete time intervals satisfy certain linear matrix inequality, then consensus can be reached. Simulations are performed to validate the theoretical results.


Introduction
Consensus is one of the fundamental issues in the study of distributed control of multiagent systems, and it has wide applications in formation control of multiple robots, communication among sensor networks, cooperative control of unmanned aerial vehicles, and so forth.Much research work on consensus has been emerged, and most of the existing work focuses on the consensus problems of multiple agents with special dynamics, such as first-order dynamics (or single integrator) [1][2][3][4][5][6][7][8][9][10], second-order dynamics (or double integrators) [11][12][13][14][15][16][17][18][19], and high-order dynamics [20,21].Consensus problems of second-order continuous-time multiagent systems are studied in [12][13][14], where the information transmission among agents is intermittent.In [15][16][17], only the partial state of second-order agent can be measured, and thus, some static or dynamic output feedback controllers are designed.In [20][21][22], consensus is studied for high-order and nonlinear multiagent systems, respectively.
In recent years, the consensus of multiple agents with general linear dynamics has been paid more and more attention, such as [23][24][25][26][27][28], and the analysis of such multiagent systems is more challenging than the case of special dynamics.In [24], consensus is considered in the case of static output feedback, and it is proved that the consensus is equivalent to the Hurwitz stable or Schur stable of a constant matrix, which is determined by the topology and the system dynamics.By studying the stability of the constant matrix, it is shown that consensus can be reached for continuous-time multiagent systems if and only if the system is stabilizable and detectable and the topology has a spanning tree under some rank constraints, and a necessary condition is also provided for consensus of discrete-time multiagent systems.In [25], consensus of continuous-time and discrete-time multiagent systems under a dynamic output feedback controller, which is actually a state estimator, is investigated, respectively.By applying the result in [29], some sufficient and necessary conditions are presented in [25].In [26], the joint effect of network topology, agent dynamics, and communication data rate on consensus of discrete-time multiagent systems is analyzed, and it is shown that under perfect state feedback, consensus is reached if and only if the dynamics of each agent is stabilizable and the unstable eigenvalues of each agent satisfy some constraints.In [27], the  ∞ consensus problem of continuous-time multiagent systems is studied under dynamic output feedback by applying the robust control theory and linear matrix inequality technique.In [28], consensus of discrete-time multiagent systems under a dynamic output feedback controller, which is actually an observer-type controller, is discussed, and the discrete-time consensus region is analyzed for neurally stable agents and unstable agents, respectively.
It should be mentioned that [23][24][25]27] all study the consensus of continuous-time multiagent systems, and the information transmission among agents is continuous.However, due to the limitation of bandwidth, the cost of communication, the technique constraints, and so forth, it is possible to transmit information in the intermittent manner.In addition, sampled-data control has many favorable properties, such as flexibility, robustness, and low cost see [30] for further details.Hence, it is also necessary to study the consensus of general continuous-time multiagent systems with intermittent information transmission.To the authors' best knowledge, there is little research work reported on this problem.Based on the previous consideration, we analyze the consensus of continuous-time multiagent systems with general dynamics, where each agent can only obtain the information of other agents at discrete times.Moreover, the discrete time intervals may not be equal, which often occurs in the event-driven systems or networked control systems [31].The sufficient condition for consensus and the method to design controller gain are presented.
This paper is organized as follows.In Section 2, we present some concepts in graph theory and formulate the model to be studied.In Section 3, main results are stated.In Section 4, simulations are provided to illustrate the effectiveness of the theoretical results.Conclusion remarks are made in Section 5.
A directed graph G consists of a vertex set V(G) and an edge set E(G), where = }, and the associated index set is denoted by (G, ) = { : G is strongly connected if there is a path from every vertex to every other vertex.A directed tree is a directed graph, where every vertex except one special vertex has exactly one parent vertex, and the special vertex, called root vertex, has no parent vertices and can be connected to any other vertices via paths.
, then G  is said to be an induced subgraph of G and G  is also said to be induced by V(G  ).A spanning tree of G is a directed tree which is a spanning subgraph of G. G is said to have a spanning tree if some edges form a spanning tree of G.
A matrix is called nonnegative if each of its elements is nonnegative.A weighted directed graph G() is a directed graph G plus a nonnegative matrix  = [  ] ∈ R × , where The Laplacian matrix of G() has the following properties.
(i) Zero is an eigenvalue of , and 1  is the associated right eigenvector.
(ii) Zero is an algebraically simple eigenvalue of , and all the other eigenvalues are with positive real parts if and only if G() has a spanning tree.
In this paper, we consider the case of intermittent information transmission; namely, each agent can only obtain the information of its neighbors at some discrete times  0 ,  1 , . .., where 0 ≤  0 <  1 <  2 < ⋅ ⋅ ⋅ .Let   =  +1 −   , and assume that   ≤ ,  = 0, 1, . ..; namely, all discrete time intervals have a common upper bound.Note that all   may not be equal.
In the case that the (relative) state of each agent can be measured directly, we consider the following control input: where  is the controller gain to be designed.

Mathematical Problems in Engineering 3
In some times, the state of each agent cannot be measured directly, and thus, we also consider the following static output feedback controller: Remark 2. Obviously, if  is Hurwitz stable, then consensus can be reached for  = 0. Hence, we assume that  is not Hurwitz stable in this paper.

Main Results
In this section, we will present a sufficient condition for consensus under controllers (3) and ( 4), respectively, and the methods to design controller gains are also provided.
By Lemma 3, we will analyze the stability of system (8) by applying the input delay approach [33], which is an effective method to deal with the stability of continuous-time systems with intermittent input.
By   ≤ , we have 0 ≤ () ≤ ,  = 0, 1, . ... Obviously, the stability of system ( 8) is equivalent to that of system (10).By analyzing the stability of system (10), we obtain the following main result.Theorem 4. Assume that (, ) is stabilizable and the topology graph has a spanning tree.Controller (3) solves a consensus problem asymptotically if there exist positive definite matrices ,  such that  and  satisfy the following linear matrix inequality: Proof.Consider the following Lyapunov-Krasovskii functional for system (10): where  > 0,  > 0, and then By Lemma 4 in [34], and thus, where By Schur complement,  < 0 if and only if (11) is satisfied.Hence, V() < 0; namely, system (10) is asymptotically stable.By Lemma 3, consensus is reached.
Remark 5.By [24], if (, ) is stabilizable and the topology graph has a spanning tree, then there exists  such that Ã is Hurwitz stable; namely, there exists  > 0 such that  Ã + Ã  < 0. Obviously, if  > 0 is small enough, then (11) must be satisfied.Hence, by (11), we can find  and  which ensure consensus.Theorem 4 shows that if (, ) is stabilizable and the topology graph has a spanning tree, then there exists controller gain  and discrete times  0 ,  1 , . .., where  +1 −  ≤ ,  = 0, 1, . .., such that consensus is reached.Moreover,  and  can be obtained by (11), which can be solved easily by the feasp solver in Matlab LMI Toolbox.
Similar to the analysis in the case of state feedback, the consensus under controller (4) is equivalent to the asymptotic stability of the following system: where Â =  −1 ⊗  −  ⊗ (), () = δ() − δ( − ()).By analyzing the stability of system (17), we obtain the following result.Remark 7.Although only the synchronous case is considered, the method in our paper can be applied to study the asynchronous case; namely, the discrete times of each agent are independent of the others.

Simulations
Consider the system of four agents, where the topology among four agents is shown in Figure 1.The dynamics of each agent are where   = ( 1 ,  2 ,  3 )  , and ) .
Without loss of generality, the discrete time intervals are chosen from (0, 0, 0625] randomly.Then the state strategies of four agents during time interval [0, 3] are shown in Figures 2,  3, and 4, and the state strategies of four agents during time interval [7.5, 9.5] are shown in Figures 5, 6, and 7, which validate our theoretical results.

Conclusion
This paper has studied the consensus problem of continuoustime multiagent systems with general linear dynamics and nonuniform sampling.By applying a state transformation and the input-delay approach, the consensus under consideration is equivalent to the asymptotic stability of a continuoustime system with time-varying delay.By analyzing the asymptotic stability of the continuous-time system, it is shown that there exist a controller gain and discrete times such that consensus can be reached.Furthermore, the controller gain and the upper bound of discrete time intervals can be obtained easily by solving a linear matrix inequality.Simulations have been provided to illustrate the effectiveness of the theoretical results.