Consensus of Fractional-Order Multiagent Systems with Double Integrator under Switching Topologies

Due to the complexity of the practical environments, many systems can only be described with the fractional-order dynamics. In this paper, the consensus of fractional-order multiagent systems with double integrator under switching topologies is investigated. By applying Mittag-Leffler function, Laplace transform, and dwell time technique, a sufficient condition on consensus is obtained. Finally, a numerical simulation is presented to illustrate the effectiveness of the theoretical result.


Introduction
In recent years, cooperative control of multiagent systems has received compelling attention due to its wide applications such as minisatellites [1], unmanned vehicle [2], deployment [3], and rendezvous [4].A significant and fundamental problem for cooperative control of multiagent systems (MASs) is consensus, which has broad applications in so many fields such as multivehicle systems [5], swarms, and flocks [6].It was found that many MASs can be explained in the framework of fractional-order dynamics and the distributed control of fractional-order multiagent systems is becoming an important issue.
Consensus of FMASs was early discussed by Cao and Ren in [7][8][9] and then attracted many researchers' attention.In [10], by using the matrix theory, graph theory and the frequency-domain analysis method, the control algorithm with absolute damping and communication delay was proposed to discuss the distributed formation control of FMASs.In [11], a distributed control law with a constant reference state was given, which ensured the consensus of FMASs with a time-varying reference state.In [12], the fractional Lyapunov direct method and Mittag-Leffler stability were used to study the consensus of nonlinear FMASs with directed topology, some sufficient conditions were given to guarantee the consensus of nonlinear FMASs with/without a virtual leader.In [13], based on the connectivity of the graph and Riccati equation, the control gain matrix was designed and a sufficient condition on leader-following consensus of FMASs with general linear models was obtained.In [14,15], the authors studied the leader-following consensus of FMASs with nonlinear dynamics by Lyapunov direct method, respectively.In [16], based on the properties of Mittag-Leffler function, matrix theory, and stability theory of fractionalorder differential equations, some sufficient conditions on consensus were derived to guarantee the consensuses of linear and nonlinear FMASs for any bounded input time delay.In [17], a distributed observer-type protocol was proposed to study the consensus of FMASs by applying the properties of the Kronecker product, Mittag-Leffler function, and Laplace transform.
Most of the existing literatures about FMASs consensus focused on the fixed topology.In fact, when the multiagent system moves harmoniously, communications between agents are usually intermittent due to external disturbances, limited communication range, or the instability of wireless communication itself.The consensus of FMASs under switching topologies is a more important and challenging problem.In this paper, the consensus of FMASs with double integrator under switching topologies is investigated.By using Mittag-Leffler function, Laplace transform, and dwell time technique, a sufficient condition on consensus is presented to guarantee the consensus of the system.
The switching topologies of information exchange between  agents are described by G  = (V, E  ,   ),  ∈ Λ, where V = {1, 2, . . ., } and E  ⊂ V × V are the sets of vertices and edges of the graph G  , respectively.The adjacency matrices   = (

Caputo Fractional Derivative. The Caputo fractional derivative is defined as follows:
where  > 0 is the fractional order,  is any given natural number, and Γ(⋅) is the Gamma function defined as Γ() = ∫ ∞ 0  −1  − .A simple notion   () will be used to denote      () if no confusion has arisen.
Let L{⋅} denote the Laplace transform of a function.Then we have where () is the Laplace transform of ().
Similar to the exponential function frequently used in the solutions of integer-order systems, a function frequently used in the solutions of fractional-order systems is the Mittag-Leffler function.
Definition 1 (see [18]).The two-parameter function of the Mittag-Leffler type is defined by where  > 0,  > 0, and  is a complex number.

System Model Description.
Consider the following fractional-order double-integrator multiagent systems with  agents: where 0 <  ≤ To study the consensus of multiagent systems (5), the following consensus control protocol will be used: where  > 0 is the control parameter to be designed later.

Main Results
Let Q  be the set over which there is a directed spanning tree for any G  ∈ Q  and Q  be the set over which there is no directed spanning tree for any =V+1   ( 0 , ).Based on Lemma 3.1 in [17], with a mild revision, we have the following.

Lemma 3. All the eigenvalues of matrix L𝑝 have positive real parts if and only if
Lemma 4 (see [19]).
In addition, if  is a stable matrix, then there are some real constants  ≥ 1,  > 0 such that       , (  )      ≤  − .

A Numerical Example and Simulation
Consider the fractional-order multiagent systems with  = 0.8.The switching topologies are shown in Figure 1.
Let  * = 10,  = 3, and the initial states be and V 5 (0) = 15.The control   () and switching sequence are depicted in Figure 2. The state responses   () and V  () are depicted in Figure 3, which shows the consensus is achieved asymptotically.Remark 7. From the above simulation, it is found that the condition ( 18) is conservative.But within this frame of this paper, the condition (18) can hardly be improved.Sharpening the condition may require a new method.We leave this for further investigation.

Conclusions
The consensus problem of fractional-order multiagent systems with double integrator under switching topologies is investigated, where the fractional order satisfies 0 <  ≤ 1. Consensus algorithms with a control parameter are designed.Based on the Mittag-Leffler function, Laplace transform, and dwell time technique, a sufficient condition is obtained, which ensures that the consensus of fractional-order multiagent systems can achieved asymptotically.The consensus problem of fractional-order multiagent systems with jointly connected topologies will be discussed in future study.
in the graph G  and  and  are called the initial and terminal nodes, respectively, which means that node  can receive information from node .   =   −   for every .The set of neighbors of node  is denoted by 1, Δ = {1, 2, . . ., },   (), V  () ∈ , and   () ∈  are the states and control input of the th agent, respectively.Assume that the initial states of each agent  are denoted by   ( 0 ) and V  ( 0 ).