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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 leader-following 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 the multiagent 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.
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 , synchronization of dynamical networks , distributed filtering , and load balancing . 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 , switching topology , asynchronous algorithms [4, 7], nonlinear algorithms [8, 9], quantized data [4, 10], noisy communication channel , 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.  considered a leader-following consensus problem and discussed the convergence properties of the leader-follower systems. Cao and Ren  studied a leader-following consensus problem with reduced interaction for both first- and second-order multiagent systems. Su et al.  studied a flocking algorithm with a virtual leader. Zhu and Cheng  considered leader-following consensus of second-order agents with multiple time-varying delays. Meng et al.  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 continuous-time systems together with discrete sampling. Therefore, it is significant to design the distributed control for continuous-time multiagent systems based on sampled information. There are a few reports [20–25] 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 , orbital transfer of satellite , and optimal control of economic systems . 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 [29–31], 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 , we designed impulsive velocity-control for multiagent systems with fixed topology to achieve consensus. In , 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  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.
2. Problem Formulation
Let and denote the set of real numbers and complex numbers, respectively. For , are the eigenvalues of , represent the spectral radius of . The identity matrix of order is denoted as (or simply if no confusion arises). For , and are the real and imaginary part of , respectively. is the column vector. denotes the matrix with all elements equal to zero.
Let be a directed graph (digraph) with the set of nodes , the set of edges , and the weighted adjacency matrix . In the digraph , node represents the agent , and an edge in is denoted by an ordered pair . 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 . All elements of adjacency matrix are nonnegative. For , , and assume that , . A directed path in a digraph is an ordered sequence of agents such that any ordered pair of vertices appearing consecutively in the sequence is an edge of the digraph, that is, , for any . 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. is a directed spanning tree of , if is a directed tree and . The Laplacian matrix of is defined as
Given a matrix , the digraph (without self-link) of denotes by , which is the digraph with node set such that there is an edge in from to if and only if . The matrix is nonnegative, that is, , if all element of is non-negative. The matrix , , denote . The non-negative matrix is row stochastic if all of its row sum are equal to 1. The row stochastic matrix is called indecomposable and aperiodic (SIA) if , where is some column vector.
Consider that a multiagent system consists of identical agents indexed by , which is described by where , , are the position and velocity states of the agent , respectively. is a control input for . The static leader for the system (2) is a static agent represented by , where . 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 , 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 for any initial state.
3. 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 , and the connections between agents and leader are described by , if and only if the agent can obtain information from the leader, otherwise, .
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 , the sampling time sequence satisfies ( is sampled period) and , are the control gain to be determined, and is the Dirac impulsive function.
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.
Proof. Let , for and note that , system (5) can be rewritten as follows:
From (7), one has
From (8), one has
Then, the evolution of , under impulsive control (4) can be described as follows:
Let and . Then, the multiagent system (2) achieves leader-following consensus, if and only if , .
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.
Lemma 4 (bilinear transformation theorem ). Polynomial (of degree ) is Schur stable if and only if is Hurwitz stable, where
For complex polynomial , let where , and is the imaginary unit.
Next theorem will show what kind of interaction topology can reach leader-following consensus and how to determine the control gains , and sampling period .
Proof. Let the be an eigenvalue of matrix . Then,
Then, we only need to prove that polynomials for are Schur stable.
Let Let where . Then, according to Lemma 4, polynomials , for , are Hurwitz stable if and only if polynomials for are Schur stable.
It can be proved by Lemma 5 that is Hurwitz stable if and only if (14) is satisfied. Therefore, if and only if (14) is satisfied. The proof is thus completed.
Remark 7. It can be observed from the inequality (14) that the real and imaginary part of the eigenvalues of , the sampling period , and two control gains and play important roles in achieving consensus. and , for , are necessary conditions for leader-following consensus. It is easy to see that the critical value of increases as decreases.
Remark 8. Let with , and the Laplace matrix is Note that where is an invertible matrix. , for imply that has a simple eigenvalue , and all the other eigenvalues have positive real parts. This implies that the graph contains a spanning tree. The root of the spanning tree is the leader.
Remark 9. How to choose a suitable control gain and when the sampling period is given. According to Theorem 6, is a necessary condition for consensus. Therefore, one can choose from , and then compute Then, one can choose from .
4. 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 , where and the connections between agents and leader at time are described by , if and only if the agent can obtain information from the leader at time ; otherwise, .
In order to achieve leader-following consensus under switching topology, the impulsive control input is designed as where . Let with and Let denotes the Laplace matrix of . Equivalently, the multiagent system (2) with the impulsive controller (23) can be rewritten as follows: where .
Remark 10. Note that the communication among agents only occurs at sampling times. This implies that interation graph does not contain any edges where .
Similar to the discussion in Section 4, one has Let , , where .
It is easy to know that the network (2) achieves leader-following consensus, if and , for some , .
From (26), one has
Let and ; then, where
Before moving on, the following lemmas are needed.
Lemma 11 (see ). Let be a positive integer, and let be non-negative matrices with positive diagonal entries; then, , where can be specified from matrices , .
Lemma 12 (see ). Let be a finite set of SIA matrices with the property that for each sequence of positive length, the matrix product is SIA. Then, for each infinite sequence , there exists a column vector such that
Lemma 13 (see ). Suppose that is a row stochastic matrix with positive diagonal elements. If the digraph has a directed spanning tree, then is SIA.
Lemma 14. Let be non-negative matrix, where . If is a Laplace matrix of a digraph , which has a directed spanning tree, then is a row stochastic matrix and the digraph of contains a directed spanning tree.
Proof. It is easy to check the non-negative matrix . Then, is a row stochastic matrix. Let denote the digraph of . Then, the Laplace matrix of is
Let , denote the eigenvalues of .
Let be an eigenvalue of matrix ; then, one has Let . Then, Therefore, from (34), only if .
When , Thus, when , the solutions of are and . On the other hand, if contains a spanning tree, only has one simple eigenvalue equal to zero. Therefore, only has one simple eigenvalue equal to zero, which implies that the digraph of has a spanning tree. The proof is completed.
Theorem 15. If there exists a positive integer , the union of across contains a directed spanning tree, for any non-negative integer , and where ; then, the multiagent system (2) achieves the leader-following consensus.
where is defined in (29). From (1), one has , for . Then, the following statements are satisfied: (i) is nonnegative if and only if ; (ii) is nonnegative if and only if ; (iii) is nonnegative if and only if
(iv) is nonnegative if and only if
If , then and . Note that and . Then, the following four statements are satisfied when . (i) If , then we have
(ii) If , and
then we have
(iii) If , and
then we have
According to the previous discussion, (38) is satisfied, if , and
Equation (39) is satisfied, if , and
Then, (38) and (39) are satisfied if (36) holds. This implies that , , , and are nonnegative. Then, is also nonnegative. Note that
and . Then, , is a row stochastic matrix. Then, and are also a row stochastic matrix.
Note that where .
The union of across , for any non-negative integer contains a directed spanning tree. This implies that the digraph with the Laplace matrix also contains a directed spanning tree.
By Lemma 14, from (49), the digraph of contains a spanning tree.
According to Lemma 11, one has for some . This implies that the digraph of also contains a spanning tree. It follows from Lemma 13 that is SIA. By Lemma 12, The proof is thus completed.
Remark 16. In this remark, we also show how to choose a suitable control gain and when the sampling period is given. According to Theorem 15, is also required. Similar to Remark 9, one can choose from , and then compute Then, one can choose from .
5. Illustrative Examples
In this section, two illustrative numerical examples will be given to demonstrate the correctness of theoretical analysis.
5.1. Fix Topology
The communication topology is described as in Figure 1. The Laplacian matrix and matrix are given as follows: The eigenvalues of are , . Let , ; 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 . But it cannot be achieved when (as shown in Figure 3).
5.2. Switching Topology
In this subsection, the network topology switches from a set as shown in Figure 4. The corresponding Laplacian matrices of and matrices are , , , and . Assume that , , , , and , , , , . Note that the union graph of , , , and , and .
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.
This work was supported in part by the National Natural Science Foundation of China under Grants 61073026, 61170031, 61272069, and 61073025.
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