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The swarm stability problem of high-order linear time-invariant (LTI) singular multiagent systems with directed graph topology is investigated extensively. Consensus of multiagent systems can be regarded as a specific case of swarm stability problem. Necessary and sufficient conditions for both swarm stability and consensus are presented. These conditions depend on the graph topology and generalized inverse theory, the dynamics of agents, and interaction among the neighbours. Several examples to illustrate the effectiveness of theoretical results are given.

Recently the study of multiagent systems has attracted considerable attention from biologists (see [

Consensus of multiagent systems can be regarded as a specific case of swarm stability problem; our results shows that consensus achievement is a specific case of swarm stability, which will be called asymptotic swarm stability. Consensus problems for multiagent systems have been studied by lots of researchers. Vicsek et al. [

Recently, more attention has been paid to the consensus problem for the singular multiagent systems. It is well known that singular systems can better describe physical systems than normal ones (see [

Compared with the existing works related to consensus, the current paper is characterized with the following novel features. First the current paper focuses on swarm stability problem of high-order singular multiagent systems; our results show that consensus achievement is a specific case of swarm stability, which will be called asymptotic swarm stability. Second the main contribution of this paper is the presentation of necessary and sufficient conditions for the swarm stability of LTI singular systems with a general high-order model. The model in [

This paper is organised as follows. In Section

For any given two matrices

For any given two matrices

For the regular singular system

the system

the system

Let

Some

If

if

For any given

For

If

For

A directed weighted graph

Let

if graph

The systems which the current paper studies are regular and impulse-free.

The regularity condition could guarantee that the system has a unique solution. The regularity condition is deemed as a kind of singular system (see [

This section introduces our multiagent model. Suppose that the multiagent system consists of

Now we take a typical example, and we also consider the dynamics of the

For the Laplacian matrix

For an LTI singular system

Define the auxiliary vectors

As a result of (

For the system (

System (

By Lemmas

From the structure of the solution we can see that

Consider the system (

A pole of the system, or a finite eigenvalue of the matrix pair

The matrix pair

For a clear expression, the LTI singular systems is said to be stable if

Consider LTI singular multiagent systems with

Consider that LTI singular multiagent systems achieve full state consensus, that is. for

The swarm stability discussed in this paper is LTI singular systems, and the definition of swarm stability and asymptotic swarm stability seem similar to [

Besides consensus achievement, some phenomena investigated by the researchers, for example, flocking and formation keeping, also require a system to be swarm stable. A bird flock, vehicle platoon, or robot crew may navigate to anywhere, but the distances among its members should not go unbounded. The states of agents in a swarm stable system might still oscillate or even diverge. Swarm stability is a kind of nonequilibrium stability [

The major purpose of this section is to present a necessary and sufficient condition for the swarm stability of high-order LTI singular swarm systems. For this end, several preparations are needed.

For multiagent systems (

Without loss of generality, consider an agent

Lemma

For multiagent System (

For the LTI singular systems (

For singular multi agent system (

From the proof of Corollary

For LTI singular multi agent system (

Suppose that the system is swarm stable. If

Suppose that the system is stable. Assume that the system is swarm unstable. There may exist a pair of agents

If the graph

Owing to the definition of

This section will establish some necessary and sufficient conditions guaranteeing that the system has a consensus property.

For swarm system (

all the

If

Owing to Lemma

For system (

For swarm system (

all the

If

The proof is the same as that Theorem

For swarm system (

the graph topology

each slow subsystem eigenvalue of

If each slow subsystem eigenvalue of

For system (

For swarm system (

the graph topology

each slow subsystem eigenvalue of

For system (

In this section, numerical instances will be exhibited to illustrate the theoretical results in the previous section, three graphs

A network with three nodes and four edges, whose topology is shown in Figure

Direction interaction topology graph

Figure

For a network with five nodes and six edges, whose topology is shown in Figure

Direction interaction topology graph

For a network with five nodes and nine edges, whose topology is shown in Figure

Direction interaction topology graph

Trajectories of three agents in the first instance with non-admissible bounded initial

Relative motions in the first instance with non-admissible bounded initial

Trajectories of five agents in the second instance with admissible bounded initial

Trajectories of five agents in the third instance with admissible bounded initial

Relative motions in the third instance with admissible bounded initial

For a network with five nodes and nine edges, whose topology is shown in Figure

Trajectories of five agents in the forth instance with admissible bounded initial

For a network with three nodes and four edges, whose topology is shown in Figure

In this example,

Trajectories of three agent’s

The swarm stability problem of high-order LTI singular multiagent systems was solved through a method based on generalized inverse theory. Consensus is regarded as a specific type of swarm stability. The model considered in this paper is more general than the one in [