Networked Convergence of Fractional-Order Multiagent Systems with a Leader and Delay

This paper investigates the convergence of fractional-order discrete-time multiagent systems with a leader and sampling delay by using Hermite-Biehler theorem and the change of bilinearity. It is shown that such system can achieve convergence depending on the sampling interval h, the fractional-order α, and the sampling delay τ and its interconnection topology. Finally, some numerical simulations are given to illustrate the results.


Introduction
Recently, more and more scholars focus on the coordinated control [1,2] of multiagent systems such as the consensus [3][4][5] and the controllability [6][7][8].However, most of the practical distribution systems are fractional order [9][10][11][12].Recently, with the development of society, fractional-order calculus theory [13][14][15][16] is widely used to study the signal processing and control, picture processing and artificial intelligence, and so on.The consensus of multiagent systems refers to the fact that agents in the system can transfer information and influence each other according to a certain protocol or algorithm, and eventually agents will tend to the consensus behavior with the evolution of the time in [17].In fact, for most of multiagent systems, there widely exist time delays as in [18].So the property of multiagent systems with time delays has always been the hot problem.In [19], the authors studied consensus of multiagent systems with heterogeneous delays and leader-following with integer-order and continuous time.In [20], the paper considered the consensus of fractionalorder multiagent systems with sampling delays without the leader.
However, for a complex environment, multiagent systems with fractional-order can be better to describe some real natural phenomena.Some basic issues of fractional-order multiagent systems with time delay, such as the convergence, are still lacking in studying.Specially, for a fractional-order multiagent system, which depends crucially on sampling interval ℎ, the fractional-order , and its interconnection topology, therefore, it is more difficult to study the convergence of the fractional-order multiagent system.
In this paper, we consider the convergence of fractionalorder discrete-time multiagent systems with a leader and sampling delay.The leader plays the role of an external input or signal to followers, and the followers update their states based on the information available from their neighbors and the leader.We will establish convergence conditions and discuss relations among sampling interval ℎ, the fractionalorder , its sampling delay , and its interconnection topology of such network.
The remainder of this paper is organized as follows.Section 2 gives the model and some preliminaries.Section 3 presents the main results, and some simulations are given in Section 4. Finally, Section 5 gives the conclusion.

Preliminaries and Problem Statement
In this section, we introduce some useful concepts and notations about the definition of fractional derivative [21], graph theory, and convergence of the multiagent systems.

Mathematical Problems in Engineering
Denote a directed graph as G = (V, E, ) consisting of a nonempty set of vertices V and E = {(, ) : ,  ∈ V} is a set of edges, where (, ) means an arc starts from  and ends by .If ,  ∈ V and (, ) ∈ E, then we say that  and  are adjacent or  is a neighbor of .We make N  = { ∈ V : (, ) ∈ E} be the neighborhood set of node . = [  ] is an adjacency matrix of graph G, where   ≥ 0 is the coupling weight between any two agents. = diag{ 1 ,  2 , . . .,   } ∈ R × is a degree matrix of G; its diagonal elements   = ∑ ∈N    ,  = 1, 2, . . ., , for the graph.Then the Laplacian of the weighted graph G is defined as The agent  is a globally reachable agent if it has paths to all of other agents.
Definition 1 (see [17]).Assume that, for arbitrary given initial values, if lim ∈ N, where   () ∈ R  is the state value of agent of the multiagent system  ( ∈ N, N presents an index set (1, 2, . . ., )),  0 () ∈ R  , and   is a constant which is changed with different .Then we have that the multiagent system is convergence.
Definition 2 (see [21] (Grunwald-Letnikov)).For any real number , the integer part written for  is [].If the function () has continuous ( + 1)-order derivative in the interval [, ] and  equals [] at last when  > 0, then let -order derivative be Consider a multiagent system is composed of +1 agents, where the first  (labeled from 1 to ) are followers and the remainder agent  + 1 (labeled 0) is leader.The fractionalorder discrete-time multiagent system with a leader and sampling time is described by where ∈ (0, 1),   ∈ R  is the state of follower  ( ∈ N, N presents an index set (1, 2, . . ., )), and  0 ∈ R  is the state of the leader.N  is the neighbor set of agent .  ≥ 0,  0 ≥ 0 represent the coupling information between followers and from the leader to the followers, respectively; otherwise,   = 0 and  0 = 0; ℎ > 0 is the sampling interval and the sampling interval is ℎ and the sampling delay is 0 <  < ℎ.
Theorem 5. Suppose system ( 4) is a symmetrical and directly weighted network and the leader is a globally reachable agent; then the state of each agent can converge to the spanning space of the leader's state, if and only if  < min{1/2, ℎ} and where   is the biggest eigenvalue of matrix ( + ).

Conclusion
In this paper, we have investigated the convergence problem of the fractional-order discrete-time multiagent system with a leader and sampling delay.We have obtained the convergence results depending on the sampling interval ℎ, the fractionalorder , and the sampling delay.

Figure 1 :
Figure 1: The trajectories of nine agents in the dynamical network.

Figure 2 :Figure 3 :
Figure 2: The trajectories of nine agents in the dynamical network.

Figure 4 :Figure 5 :
Figure 4: The trajectories of nine agents in the dynamical network.
) with 1 +  − ℎ    + 2ℎ    > 0. Therefore, system (4) is asymptotic convergence.Suppose system (4) is a symmetrical and directly weighted network and the leader is a globally reachable agent; then the state of each agent can converge to the spanning space of the leader's state, if () < 1, where  = ℎ  ( + ) and () is spectral radius of matrix .Notice from Theorem 6 that‖ ⋅ ‖ is ‖ ⋅ ‖ 2 and ‖ ⋅ ‖ 2 is 2-norm.Remark 8. Theorem 5 describes the relation of the convergence of such system and time delay, while Theorem 6 describes the relation of the convergence of such system and spectral radius of matrix .