Impulsive Consensus for Leader-Following Multiagent Systems with Fixed and Switching Topology

This paper studied the consensus problem of the leader-following multiagent system. It is assumed that the state information of the leader is only available to a subset of followers, while the communication among agents occurs at sampling instant. To achieve leaderfollowing consensus, a class of distributed impulsive control based on sampling information is proposed. By using the stability theory of impulsive systems, algebraic graph theory, and stochastic matrices theory, a necessary and sufficient condition for fixed topology and sufficient condition for switching topology are obtained to guarantee the leader-following consensus of themultiagent system. It is found that leader-following consensus is critically dependent on the sampling period, control gains, and interaction graph. Finally, two numerical examples are given to illustrate the effectiveness of the proposed approach and the correctness of theoretical analysis.


Introduction
During the past several decades, the consensus problem of the multiagent system has drawn a great deal of attentions because of its broad applications in many domains, including distributed coordination [1], synchronization of dynamical networks [2], distributed filtering [3], and load balancing [4].The basic idea of consensus is to design a distributed control such that the team of agents can achieve a state agreement only by locally available information without central control stations.Consensus problem has been addressed in various situations, such as time delay [5], switching topology [6], asynchronous algorithms [4,7], nonlinear algorithms [8,9], quantized data [4,10], noisy communication channel [11], and second-order model [12,13].
Inspired by some biological systems and engineering applications, the leader-following consensus problem has received a lot of interest.The leader is a special agent whose motion is independent of all other agents and thus is followed by all other agents.It has been widely used in many applications [14,15].For the first-order multiagent systems, Jadbabaie et al. [16] considered a leader-following consensus problem and discussed the convergence properties of the leader-follower systems.Cao and Ren [17] studied a leaderfollowing consensus problem with reduced interaction for both first-and second-order multiagent systems.Su et al. [18] studied a flocking algorithm with a virtual leader.Zhu and Cheng [19] considered leader-following consensus of secondorder agents with multiple time-varying delays.Meng et al. [20] studied the leaderless and leader-following consensus algorithms with communication and input delays under a directed network by the Lyapunov theorems and the Nyquist stability criterion.
In recent years, owing to the development of digital sensors and the constraints of transmission bandwidth of networks, many control systems can be modeled by continuoustime systems together with discrete sampling.Therefore, it is significant to design the distributed control for continuoustime multiagent systems based on sampled information.There are a few reports [20][21][22][23][24][25] dealing with this problem, where the control inputs regulate the velocity of each agent continuously over the sampling period.
On the other hand, impulsive dynamical systems exhibit continuous evolutions typically described by ordinary differential equations and instantaneous state jumps or impulses.
It is also well known that the impulsive control is more efficient than one of continuous control in many situations.The examples include ecosystems management [26], orbital transfer of satellite [27], and optimal control of economic systems [28].The main idea of impulsive control is to instantaneously change the state of a system when some conditions are satisfied.During the last few decades, it has been widely applied into the synchronization problems of complex dynamical networks [29][30][31], which can be regarded as first-order multiagent systems with nonlinear dynamics.In many real-world system, agents are governed by both position and velocity dynamics.The impulsive control for second-order multiagent system was studied in [32,33], where both velocity and position are instantaneously changed by impulsive control, but position cannot change quickly in many situation.Therefore, it is more reasonable to only regulate the velocity of each agent to reach consensus [34,35].In [34], we designed impulsive velocity-control for multiagent systems with fixed topology to achieve consensus.In [35], an impulsive control was proposed in which the current position data of its neighbours and the past position data of its own state were utilised to regulate the velocity of agents.
This paper aims to investigate the consensus problem of leader-following multiagent systems by using impulsive control which only regulates the velocity of agents.Our main contributions are summarised as follows.First, a necessary and sufficient condition under fixed topology is derived, and it is found that the leader-following consensus in multiagent systems with sampling information can be reached if and only if the sampled period is bounded by critical values which depend on control gains and the interaction graph.Second, a sufficient condition under switching topology is obtained, and it is shown that the impulsive interval is restricted by an upper bound which depends on control gains, the diagonal element of the Laplacian matrix, and the connections between agents and leader.The two key difference between this paper and our earlier work [34] are that the leader-following case is taken into account and that this paper considers multiagent systems under switching topology.
The remainingpart of the paper is organized as follows.In Section 2, some necessary mathematical preliminaries are given.Main results of this paper, that is, the convergence of the distributed impulsive control under fixed and switching topology, are presented in Sections 3 and 4. In Section 5, some illustrative numerical examples are given.Concluding remarks are finally stated in Section 6.

Problem Formulation
Let R and C denote the set of real numbers and complex numbers, respectively.For  = (  ) × ∈ R × ,  1 (),  2 (), . . .,   () are the eigenvalues of , () represent the spectral radius of .The identity matrix of order  is denoted as   (or simply  if no confusion arises).For  ∈ C, Re() and Im() are the real and imaginary part of , respectively.1  = (1, 1, . . ., 1)  is the column vector.0 × denotes the × matrix with all elements equal to zero.
Let G = {V, E, A} be a directed graph (digraph) with the set of nodes V = {1, 2, . . .}, the set of edges E ∈ V × V, and the weighted adjacency matrix A = (  ) × .In the digraph G, node  represents the agent , and an edge in G is denoted by an ordered pair {, }.{, } ∈ E if and only if the agent  can directly receive information from the th agent.In this case, the th agent is the neighbor of the th agent.The set of neighbors of the th agent is denoted by N  = { ∈ V | (, ) ∈ E}.All elements of adjacency matrix are nonnegative.For ,  ∈ V,  ∈ N  ⇔   > 0, and assume that   = 0,  ∈ V.A directed path in a digraph G is an ordered sequence V 1 , V 2 , . . ., V  of agents such that any ordered pair of vertices appearing consecutively in the sequence is an edge of the digraph, that is, (V  , V +1 ) ∈ E, for any  = 1, 2, . . ., −1.A directed tree is a digraph, where there exists an agent, called the root, such that any other agent of the digraph can be reached by one and only one path starting at the root.T G = {V T , E T } is a directed spanning tree of G, if T G is a directed tree and V T = V.The Laplacian matrix (G) = (  ) × of G is defined as Given a matrix P = (  ) ∈ R × , the digraph (without self-link) of P denotes by G(P), which is the digraph with node set V = {1, 2, . . ., } such that there is an edge in G(P) from  to  if and only if   ̸ = 0.The matrix  is nonnegative, that is,  ≥ 0, if all element of  is non-negative.The matrix ,  ∈ R × ,  ≥  denote  −  ≥ 0. The nonnegative matrix  is row stochastic if all of its row sum are equal to 1.The row stochastic matrix  ∈ R × is called indecomposable and aperiodic (SIA) if lim  → ∞   = 1    , where  is some  × 1 column vector.
Consider that a multiagent system consists of  identical agents indexed by 1, 2, . . ., , which is described by where  = 1, 2, . . ., ,   () ∈ R, V  () ∈ R are the position and velocity states of the agent , respectively.  () ∈ R  is a control input for  = 1, 2, . . ., .The static leader for the system (2) is a static agent represented by  0 () =  0 , where  0 ∈ R. The edges between the agents and the leader is unidirectional; namely, there are only partial agents that can obtain information from the leader.It is also assumed that each agent can only obtain information from other agents or the leader at sampling times.This paper focuses on the problem of designing   (),  = 1, 2, . . .,  based on sampling information to make all  agents converge to a static leader.
Definition 1.The leader-following consensus of the multiagent system (2) with static leader is said to be achieved if lim for any initial state.

Leader-Following Consensus under Fixed Topology
In this section, the leader-following consensus problem under fixed topology is considered.The interaction between agents in this part is described by a fixed digraph G = {V, E, A}, and the connections between agents and leader are described by   ∈ R,   > 0 if and only if the agent  can obtain information from the leader, otherwise,   = 0.
In order to achieve the leader-following consensus of the multiagent system (2) with sampled information under fixed topology, the impulsive control for the agent , is designed as where  ∈ V, the sampling time sequence 0 are the control gain to be determined, and (⋅) is the Dirac impulsive function.
Equivalently, the multiagent system (2) with impulsive controller (4) can be rewritten as follows: where Remark 2. From (5), the control input of each agent only uses the information from its neighbors at sampling instants and are only applied at sampling instants.This is quite different from the previously mentioned works, where the control inputs are applied continuously.The velocity of the agent is instantaneously changed at sampling times.This is feasible when the operating time of the impulsive controller is much smaller than the sampled period.Lemma 3. The multiagent system (2) with impulsive control (4) achieves leader-following consensus asymptotically if and only if () < 1, where Proof.Let x () =   () −  0 (), for  ∈ V and note that ∑ ∈V    0 = 0, system (5) can be rewritten as follows: From ( 7), one has From ( 8), one has Then, the evolution of x (  ), V  (  ) under impulsive control (4) can be described as follows: Let x() = ( x1 (  ), . . ., x (  ))  and V() = (V 1 ( +  ), . . ., V  ( +  ))  .Then, the multiagent system (2) achieves leaderfollowing consensus, if and only if lim  → +∞ x() = 0, lim  → +∞ V() = 0.
Equivalently, (10) can be rewritten as follows: Therefore, it is easy to obtain the result by the stability theory of discrete-time systems.
The following lemmas and definition are needed for the subsequent development.
Next theorem will show what kind of interaction topology can reach leader-following consensus and how to determine the control gains  1 ,  2 and sampling period ℎ.Theorem 6.The multiagent system (2) with impulsive control (4) under fixed topology achieves the leader-following consensus asymptotically if and only if where   ,  = 1, 2, . . .,  are the eigenvalues of  + .
Proof.Let the  be an eigenvalue of matrix .Then, Let Then, we only need to prove that polynomials   () for  ∈ V are Schur stable. Let Let where    = 1/ Note that where is an invertible matrix.Re(  ) > 0, for  ∈ V imply that L has a simple eigenvalue 0, and all the other eigenvalues have positive real parts.This implies that the graph G contains a spanning tree.The root of the spanning tree is the leader.Remark 9. How to choose a suitable control gain  1 and  2 when the sampling period ℎ is given.According to Theorem 6,  2 < 2 is a necessary condition for consensus.Therefore, one can choose  2 from (0, 2], and then compute Then, one can choose  1 from (0, Θ).

Leader-Following Consensus under Switching Topology
In this section, the leader-following consensus under switching topology is considered.The interaction between agents at sampling time   is described by time-varying digraph G() = {V, E(), A()}, where A() = (  ()) × and the connections between agents and leader at time  are described by   (),   () > 0 if and only if the agent  can obtain information from the leader at time ; otherwise,   () = 0.
Remark 10.Note that the communication among agents only occurs at sampling times.This implies that interation graph does not contain any edges G() = 0 where  ̸ =   .
Lemma 12 (see [39]).Let Lemma 13 (see [40]).Suppose that P ∈ R × is a row stochastic matrix with positive diagonal elements.If the digraph G(P) has a directed spanning tree, then P is SIA.
be non-negative matrix, where ,  1 ,  2 > 0. If  is a Laplace matrix of a digraph G, which has a directed spanning tree, then P is a row stochastic matrix and the digraph of P contains a directed spanning tree.
Proof.It is easy to check the non-negative matrix P1 2 = 1 2 .Then, P is a row stochastic matrix.Let G(P) denote the digraph of P.Then, the Laplace matrix of G(P) is Let   ,  = 1, 2, . . .,  denote the eigenvalues of .
Thus, when   () = 0, the solutions of () = 0 are  = 0 and  =  + .On the other hand, if G contains a spanning tree,  only has one simple eigenvalue equal to zero.Therefore, (P) only has one simple eigenvalue equal to zero, which implies that the digraph of P has a spanning tree.The proof is completed.
Note that where The union of G(  ) across  ∈ [ 0 ,  0 + ], for any nonnegative integer  0 contains a directed spanning tree.This implies that the digraph with the Laplace matrix S   0 also contains a directed spanning tree.By Lemma 14, from (49), the digraph of ∑  0 + = 0 P() contains a spanning tree.
According to Lemma 11, one has for some .This implies that the digraph of ∏  0 + = 0 P() also contains a spanning tree.It follows from Lemma 13 that ∏ The proof is thus completed.
Remark 16.In this remark, we also show how to choose a suitable control gain  1 and  2 when the sampling period ℎ is given.According to Theorem 15,  2 < 2 is also required.Similar to Remark 9, one can choose  2 from (0, 2], and then compute Then, one can choose  1 from (0, Θ).

Illustrative Examples
In this section, two illustrative numerical examples will be given to demonstrate the correctness of theoretical analysis.The eigenvalues of  +  are  1 ( + ) =  2 ( + ) = 1,  3 ( + ) =  4 ( + ) = 2. Let  2 = 1, ℎ = 2; according to Theorem 6, the network can achieve leader-following consensus, if and only if Figure 2 shows that the leader-following consensus can be achieved when  1 = 0.49.But it cannot be achieved when  1 = 0.51 (as shown in Figure 3).

Switching Topology.
In this subsection, the network topology switches from a set { G1 , G2 , G3 , G4 } as shown in    Figure 5 shows that the leader-following consensus can be achieved when  1 = 0.95.

Conclusions
In this paper, the leader-following consensus problem of the multiagent system is considered.The impulsive control, which only needs sampled information and regulates the velocity of each agent at sampling times, is proposed for the leader-following consensus.Several new criteria are established for the leader-following consensus of the system under both fixed and switching topology.Illustrated examples have been given to show the effectiveness of the proposed impulsive control.