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

Authors: Yanping Gao; Bo Liu; Min Zuo; Tongqiang Jiang; Junyan Yu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

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.


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## Body

## 1. 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) [1–10], second-order dynamics (or double integrators) [11–19], and high-order dynamics [20, 21]. Consensus problems of second-order continuous-time multiagent systems are studied in [12–14], where the information transmission among agents is intermittent. In [15–17], only the partial state of second-order agent can be measured, and thus, some static or dynamic output feedback controllers are designed. In [20–22], 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 [23–28], 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 H∞ 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 [23–25, 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 Section2, 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.Notations. Let I or In∈ℝn×n be an identity matrix, and 1n=[1,…,1]T∈ℝn; for symmetric matrices A and B, A<B (resp., A≤B) means that A-B is a negative definite (resp., negative semidefinite) matrix; ⊗ denotes the Kronecker product operator, and C⊗D=[cijD], where C=[cij] and D=[dij] are two matrices.


## 2. Preliminaries

### 2.1. Graph Theory

Some basic definitions in graph theory [32] are first introduced.A directed graph𝒢 consists of a vertex set 𝒱(𝒢) and an edge set ℰ(𝒢), where 𝒱(𝒢)={v1,…,vn} and ℰ(𝒢)⊂{(vj,vi):vj,vi∈𝒱(𝒢)}. For edge (vj,vi), vj is called the parent vertex of vi and vi is called the child vertex of vj. The set of neighbors of vertex vi is defined by N(𝒢,vi)={vj:(vj,vi)∈ℰ(𝒢)andj≠i}, and the associated index set is denoted by N(𝒢,i)={j:vj∈N(𝒢,vi)}. A (directed) path from vi1 to vik is a sequence, vi1,…,vik, of distinct vertices such that (vij,vij+1)∈ℰ(𝒢),j=1,…,k-1. A directed graph 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. A subgraph 𝒢s of 𝒢 is a graph such that 𝒱(𝒢s)⊂𝒱(𝒢) and ℰ(𝒢s)⊂ℰ(𝒢). 𝒢s is said to be a spanning subgraph if 𝒱(𝒢s)=𝒱(𝒢). For any vi,vj∈𝒱(𝒢s), if (vi,vj)∈ℰ(𝒢s)⇔(vi,vj)∈ℰ(𝒢), then 𝒢s is said to be an induced subgraph of 𝒢 and 𝒢s is also said to be induced by 𝒱(𝒢s). 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 graph𝒢(F) is a directed graph 𝒢 plus a nonnegative matrix F=[fij]∈ℝn×n, where fij>0⇔(vj,vi)∈ℰ(𝒢), and fij is called the weight of edge (vj,vi). The Laplacian matrix L=[lij]∈ℝn×n of 𝒢(f) is defined as
(1)lij={-fij,i≠j,∑s=1,s≠infis,i=j.The Laplacian matrix of𝒢(F) has the following properties.Lemma 1 (see [2]).
(i)
Zero is an eigenvalue ofL, and 1n is the associated right eigenvector.
(ii)
Zero is an algebraically simple eigenvalue ofL, and all the other eigenvalues are with positive real parts if and only if 𝒢(F) has a spanning tree.


### 2.2. Model

Consider a group of agents with the following general continuous-time dynamics:(2)x˙i(t)=Axi(t)+Bui(t),yi(t)=Cxi(t),i=1,2,…,n,
where xi∈ℝm, ui∈ℝp and yi∈ℝq, are the state, control input, and output of agent i, respectively, and A, B, C are constant matrices.Givenui, i=1,…,n, we say that ui or multiagent system (2) solves a consensus problem asymptotically if lim⁡t→∞(xi(t)-xj(t))=0, i,j=1,2,…,n, for any initial states.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 timest0,t1,…, where 0≤t0<t1<t2<⋯. Let Tk=tk+1-tk, and assume that Tk≤d, k=0,1,…; namely, all discrete time intervals have a common upper bound. Note that all Tk may not be equal.In the case that the (relative) state of each agent can be measured directly, we consider the following control input:(3)ui(t)=K∑j∈N(𝒢,i)‍aij(xj(tk)-xi(tk)),2222t∈[tk,tk+1),i=1,2,…,n,
where K is the controller gain to be designed.In some times, the state of each agent cannot be measured directly, and thus, we also consider the following static output feedback controller:(4)ui(t)=K∑j∈N(𝒢,i)‍aij(yj(tk)-yi(tk)),22222t∈[tk,tk+1),i=1,2,…,n.Remark 2.
Obviously, ifA is Hurwitz stable, then consensus can be reached for K=0. Hence, we assume that A is not Hurwitz stable in this paper.


## 2.1. Graph Theory

Some basic definitions in graph theory [32] are first introduced.A directed graph𝒢 consists of a vertex set 𝒱(𝒢) and an edge set ℰ(𝒢), where 𝒱(𝒢)={v1,…,vn} and ℰ(𝒢)⊂{(vj,vi):vj,vi∈𝒱(𝒢)}. For edge (vj,vi), vj is called the parent vertex of vi and vi is called the child vertex of vj. The set of neighbors of vertex vi is defined by N(𝒢,vi)={vj:(vj,vi)∈ℰ(𝒢)andj≠i}, and the associated index set is denoted by N(𝒢,i)={j:vj∈N(𝒢,vi)}. A (directed) path from vi1 to vik is a sequence, vi1,…,vik, of distinct vertices such that (vij,vij+1)∈ℰ(𝒢),j=1,…,k-1. A directed graph 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. A subgraph 𝒢s of 𝒢 is a graph such that 𝒱(𝒢s)⊂𝒱(𝒢) and ℰ(𝒢s)⊂ℰ(𝒢). 𝒢s is said to be a spanning subgraph if 𝒱(𝒢s)=𝒱(𝒢). For any vi,vj∈𝒱(𝒢s), if (vi,vj)∈ℰ(𝒢s)⇔(vi,vj)∈ℰ(𝒢), then 𝒢s is said to be an induced subgraph of 𝒢 and 𝒢s is also said to be induced by 𝒱(𝒢s). 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 graph𝒢(F) is a directed graph 𝒢 plus a nonnegative matrix F=[fij]∈ℝn×n, where fij>0⇔(vj,vi)∈ℰ(𝒢), and fij is called the weight of edge (vj,vi). The Laplacian matrix L=[lij]∈ℝn×n of 𝒢(f) is defined as
(1)lij={-fij,i≠j,∑s=1,s≠infis,i=j.The Laplacian matrix of𝒢(F) has the following properties.Lemma 1 (see [2]).
(i)
Zero is an eigenvalue ofL, and 1n is the associated right eigenvector.
(ii)
Zero is an algebraically simple eigenvalue ofL, and all the other eigenvalues are with positive real parts if and only if 𝒢(F) has a spanning tree.


## 2.2. Model

Consider a group of agents with the following general continuous-time dynamics:(2)x˙i(t)=Axi(t)+Bui(t),yi(t)=Cxi(t),i=1,2,…,n,
where xi∈ℝm, ui∈ℝp and yi∈ℝq, are the state, control input, and output of agent i, respectively, and A, B, C are constant matrices.Givenui, i=1,…,n, we say that ui or multiagent system (2) solves a consensus problem asymptotically if lim⁡t→∞(xi(t)-xj(t))=0, i,j=1,2,…,n, for any initial states.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 timest0,t1,…, where 0≤t0<t1<t2<⋯. Let Tk=tk+1-tk, and assume that Tk≤d, k=0,1,…; namely, all discrete time intervals have a common upper bound. Note that all Tk may not be equal.In the case that the (relative) state of each agent can be measured directly, we consider the following control input:(3)ui(t)=K∑j∈N(𝒢,i)‍aij(xj(tk)-xi(tk)),2222t∈[tk,tk+1),i=1,2,…,n,
where K is the controller gain to be designed.In some times, the state of each agent cannot be measured directly, and thus, we also consider the following static output feedback controller:(4)ui(t)=K∑j∈N(𝒢,i)‍aij(yj(tk)-yi(tk)),22222t∈[tk,tk+1),i=1,2,…,n.Remark 2.
Obviously, ifA is Hurwitz stable, then consensus can be reached for K=0. Hence, we assume that A is not Hurwitz stable in this paper.


## 3. 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.LetX(t)=(x1T(t),x2T(t),…,xnT(t))T, then multiagent system (2) under controller (3) can be written as
(5)X˙(t)=(In⊗A)X(t)-(L⊗(BK))X(tk),222222222t∈[tk,tk+1),k=0,1,….By Lemma1, there exists an invertible matrix U, the first column of which is 1n, such that
(6)U-1LU=(0α0H),
where H∈ℝ(n-1)×(n-1). Let δ(t)=(U-1⊗Im)X(t); then
(7)δ˙1(t)=Aδ1(t)-(α⊗(BK))δ~(tk),(8)δ~˙(t)=(In-1⊗A)δ~(t)-(H⊗(BK))δ~(tk),
where δ(t)=(δ1(t)T,δ~(t)T)T, δ1(t)∈ℛm.By the previous state transformation, it is easy to obtain the following lemma.Lemma 3.
Controller (3) solves a consensus problem asymptotically if and only if system (8) is asymptotically stable.Proof.
Sufficiency. Let U=[1n,U1], where U1∈ℛn×(n-1). By X(t)=(U⊗Im)δ(t), we have X(t)=1n⊗δ1(t)+(U1⊗Im)δ~(t). Clearly, if system (8) is asymptotically stable, namely, lim⁡t→∞δ~(t)=0, then lim⁡t→∞(xi(t)-xj(t))=0, i,j=1,2,…,n.
Necessity. Let U-1=(u2,U2T)T, where U2∈ℛ(n-1)×n. From U-1U=In, we have u2T1n=1, U21n=0. Since consensus is reached, there exists ϕ(t)∈ℛm such that lim⁡t→∞(xi(t)-ϕ(t))=0, i=1,2,…,n. By δ(t)=(U-1⊗Im)X(t),
(9)δ(t)-(U-1⊗Im)(1n⊗ϕ(t))=(U-1⊗Im)(X(t)-1n⊗ϕ(t))⇒δ(t)-(u2T1nU21n)⊗ϕ(t)=(U-1⊗Im)(X(t)-1n⊗ϕ(t))⇒δ(t)-(10)⊗ϕ(t)=(U-1⊗Im)(X(t)-1n⊗ϕ(t))⇒(δ1(t)-ϕ(t)δ~(t))=(U-1⊗Im)(X(t)-1n⊗ϕ(t))⇒lim⁡t→∞(δ1(t)-ϕ(t)δ~(t))=(u2T⊗ImU2⊗Im)lim⁡t→∞(X(t)-1n⊗ϕ(t)).
By lim⁡t→∞(X(t)-1n⊗ϕ(t))=0, it is easy to obtain lim⁡t→∞δ~(t)=0, which means that system (8) is asymptotically stable.By Lemma3, 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.Letτ(t)=t-tk, and let t∈[tk,tk+1); then tk=t-τ(t), and system (8) can be rewritten as
(10)δ~˙(t)=(In-1⊗A)δ~(t)-(H⊗(BK))δ~(t-τ(t))=(In-1⊗A-H⊗(BK))δ~(t)+(H⊗(BK))(δ~(t)-δ~(t-τ(t)))=A~δ~(t)+(H⊗(BK))ξ(t),
where A~=In-1⊗A-H⊗(BK), ξ(t)=δ~(t)-δ~(t-τ(t)).ByTk≤d, we have 0≤τ(t)≤d, k=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(A,B) is stabilizable and the topology graph has a spanning tree. Controller (3) solves a consensus problem asymptotically if there exist positive definite matrices P, R such that K and d satisfy the following linear matrix inequality:
(11)(PA~+A~TPP(H⊗(BK))A~TR*-1dR(H⊗(BK))TR**-1dR).Proof.
Consider the following Lyapunov-Krasovskii functional for system (10):
(12)V(t)=δ~T(t)Pδ~(t)+∫t-dt‍(s-t+d)δ~˙T(s)Rδ~˙(s)ds,
where P>0, R>0, and then
(13)V˙(t)=2δ~T(t)Pδ~˙(t)+(-∫t-dt‍δ~˙T(s)Rδ~˙(s)ds+dδ~˙T(t)Rδ~˙(t)).
By Lemma  4 in [34],
(14)∫t-dtδ~˙T(s)Rδ~˙(s)ds≥1d(δ~(t)-δ~(t-τ(t)))TR(δ~(t)-δ~(t-τ(t))),
and thus,
(15)V˙(t)≤2δ~T(t)Pδ~˙(t)-1d(δ~(t)-δ~(t-τ(t)))TR(δ~(t)-δ~(t-τ(t)))+dδ~˙T(t)Rδ~˙(t)=(δ~T(t),ξT(t))Q(δ~T(t),ξT(t))T,
where
(16)Q=(PA~+A~TPP(H⊗(BK))*-1dR)+d(A~T(H⊗(BK))T)R(A~H⊗(BK)).
By Schur complement, Q<0 if and only if (11) is satisfied. Hence, V˙(t)<0; namely, system (10) is asymptotically stable. By Lemma 3, consensus is reached.Remark 5.
By [24], if (A,B) is stabilizable and the topology graph has a spanning tree, then there exists K such that A~ is Hurwitz stable; namely, there exists P>0 such that PA~+A~TP<0. Obviously, if d>0 is small enough, then (11) must be satisfied. Hence, by (11), we can find K and d which ensure consensus.Theorem4 shows that if (A,B) is stabilizable and the topology graph has a spanning tree, then there exists controller gain K and discrete times t0,t1,…, where tk+1-tk≤d, k=0,1,…, such that consensus is reached. Moreover, K and d 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:
(17)δ^˙(t)=A^δ^(t)+(H⊗(BKC))ζ(t),
where A^=In-1⊗A-H⊗(BKC), ζ(t)=δ^(t)-δ^(t-τ(t)). By analyzing the stability of system (17), we obtain the following result.Theorem 6.
Assume that(A,B,C) is stabilizable and detectable and the topology graph has a spanning tree. Controller (4) solves a consensus problem asymptotically if there exist positive definite matrices P,R such that K and d satisfy the following LMI:
(18)(PA^+A^TPP(H⊗(BKC))A^TR*-1dR(H⊗(BKC))TR**-1dR).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.


## 4. Simulations

Consider the system of four agents, where the topology among four agents is shown in Figure1. The dynamics of each agent are
(19)x˙i=Axi+Bui,i=1,2,3,4,
where xi=(xi1,xi2,xi3)T, and
(20)A=(10221110-2),B=(121).
Since rank⁡(B,AB,A2B)=3, (A,B) is stabilizable. It is easy to verify that A~ is Hurwitz stable for K=(-4,5,2). By using the feasp solver in Matlab LMI Toolbox, (11) is feasible for d=0.0625 and it is infeasible for d=0.0626, which means that for K=(-4,5,2), the maximum d satisfying (11) is d=0.0625. By Theorem 4, for K=(-4,5,2), consensus can be reached if the maximum discrete time interval is not larger than 0.0625.Figure 1
Topology.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.Figure 2
State trajectoriesx11, x21, x31, x41 during time interval [0,3].Figure 3
State trajectoriesx12, x22, x32, x42 during time interval [0,3].Figure 4
State trajectoriesx13, x23, x33, x43 during time interval [0,3].Figure 5
State trajectoriesx11, x21, x31, x41 during time interval [7.5,9.5].Figure 6
State trajectoriesx12, x22, x32, x42 during time interval [7.5,9.5].Figure 7
State trajectoriesx13, x23, x33, x43 during time interval [7.5,9.5].


## 5. Conclusion

This paper has studied the consensus problem of continuous-time 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 continuous-time 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.



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