Collaboration Control of Fractional-Order Multiagent Systems with Sampling Delay

Abstract

Because of the complexity of the practical environments, many distributed multiagent systems cannot be illustrated with the integer-order dynamics and can only be described with the fractional-order dynamics. In this paper, collaboration control problems of continuous-time networked fractional-order multiagent systems via sampled control and sampling delay are investigated. Firstly, the sampled-data control of multiagent systems with fractional-order derivative operator is analyzed in a directed weighted network ignoring sampling delay. Then, the collaborative control of fractional-order multiagent systems with sampled data and sampling delay is studied in a directed and symmetrical network. Many sufficient conditions for reaching consensus with sampled data and sampling delay are obtained. Some numerical simulations are presented to illustrate the utility of our theoretical results.

1. Introduction

In recent years, consensus problems in distributed networked multiagent systems have attracted increasing attention of more and more researches including control theory, mathematics, biology, physics, computer science, and robotics. The applications of multiagent systems are extensive, ranging from multiple space-craft alignment, heading direction in flocking behavior, distributed computation, and rendezvous of multiple vehicles. Based on certain quantities of interest, collaboration control problems of agent systems have been studied by many researchers and many important results have been achieved in much literature [1–8].

With the development of digital sensors and controllers, in many cases that the system itself is a continuous process, the synthesis of control law can only use the data sampled at the discrete sampling instants. Compare to continuous-time systems with continuous-time controller, continuous-time systems via sampled control have many advantages, such as flexibility, robustness, and low cost. Therefore, sampled control for continuous-time system is more coincident with applications in our real life. Robots, vehicles, airplanes, satellites, and almost all of the modern artificial products are controlled by digital controller where continuous signals are transferred into discrete ones. For consensus problems of continuous-time multiagent systems via sampled control, some interesting results about consensus problem for multiagent system have been reported [9–15]. However, in the many real applications, we always want to find how large the sampling period should be chosen to guarantee that the system runs well. This requires us to look for an upper bound of sampling period. Moreover, sampling delay of the system cannot be ignored and sometimes may play a key role in the stability analysis of the networks. Therefore, we will also study the case when sampling delay exists.

The important results of the above literature focus the consensus problems of multiagent systems with integer-order dynamical equation. In the complex environment, many phenomena cannot be explained by the framework of integer-order dynamics, for example, the synchronized motion of agents in complex environments such as macromolecule fluids and porous media [16–18]. Under these circumstances, many dynamic characteristics of natural phenomena can only be described in the dynamics of fractional-order (noninteger order) behavior, for example: flocking movement and food searching by means of the individual secretions and microbial, submarine underwater robots in the bottom of the sea with a large number of microorganisms and viscous substances, unmanned aerial vehicles running in the complex space environment [19, 20]. Cao et al. [21, 22] studied distribution coordination of multiagent systems with fractional-order dynamics firstly and gave the relationship between the number of individuals and the fractional order in the stable multiagent systems. However, to the best of our knowledge, there are few researches done on the coordination control of fractional-order multiagent systems via sampled data.

In this paper, we investigate the consensus of fractional-order multiagent systems (FOMAS) with sampled-data control. Because few methods are presented to analyze the fractional-order systems with sampling delay, the problems of the fractional-order systems with sampling delay and sampled data will become more difficult. The main innovation of this paper lies in the study on the distributed coordination of FOMAS with sampled data and sampling delay. The rest of the paper is organized as follows. In Section 2, we recall some basic definitions about fractional calculus. In Section 3, some preliminaries about graph theory, fractional-order coordination model of multiagent systems are shown out. A distributed coordination algorithm for FOMAS with sampled data control is studied in Section 4. Section 5 presents the consensus of FOMAS with sampled data and sampling delay. In Section 6, numerical examples are simulated to verify the theoretical analysis. Conclusions are finally drawn in Section 7.

2. Fractional Calculus

Fractional derivatives provide an excellent instrument to describe the memory and hereditary features of various materials and processes. Fractional calculus also appears in the control of dynamical systems, when the controlled system and the controller are described by a fractional-order differential equation. This is the main advantage of fractional derivatives in comparison with classical integer-order models, in which such effects are in fact neglected. The advantages of fractional-order derivatives become evident in modeling mechanical and electrical characteristics of real materials, as well as in many fields to describe the rheological properties of rocks.

Fractional operator plays an important role in modern science, which is used as a generalization of integration and differentiation with noninteger order fundamental operator _𝑎𝐷^𝑝_𝑡, where a and t are the limits of the operation and 𝑝∈𝑅. The continuous integrodifferential operator is defined as _𝑎𝐷^𝑝_𝑡=𝑑^𝑝𝑑𝑡^𝑝∫,𝑝>0,1,𝑝=0,_𝑡𝑎(𝑑𝜃)^−𝑝,𝑝<0. (1)

Three definitions most frequently used for the general fractional operators are the Grünwald-Letnikov (GL) definition, the Riemann-Liouville (RL), and the Caputo definition [16–18]. The GL definition is given by ^𝐺_𝑎𝐷^𝑝_𝑡𝑓(𝑡)=lim_ℎ→0ℎ^{−𝑝[(𝑡−𝑎)/ℎ]}_𝑘=0(−1)^𝑘𝑝𝑘𝑓(𝑡−𝑘ℎ), (2) where [⋅] means the integer part, and 𝑝𝑘 is fractional binomial coefficients. The RL definition is given as ^𝑅_𝑎𝐷^𝑝_𝑡𝑓(𝑡)=1Γ𝑑𝑛−𝑝𝑑𝑡^𝑛𝑛𝑡_𝑎𝑓(𝜃)(𝑡−𝜃)^{1+𝑝−𝑛}𝑑𝜃, (3) for (𝑛−1<𝑝<𝑛) and where Γ(⋅) is the Gamma function. The Caputo definition can be written as ^𝐶_𝑎𝐷^𝑝_𝑡𝑓(𝑡)=1Γ𝑝−𝑛^𝑡_𝑎𝑓^(𝑛)(𝜃)(𝑡−𝜃)^{1+𝑝−𝑛}𝑑𝜃. (4) The initial conditions for the fractional order differential equations with the Caputo derivatives are in the same form as that for the integer-order differential equations. In this paper, a simple notation 𝑓^(𝑝) is used to replace _𝑎𝐷^𝑝_𝑡𝑓(𝑡).

3. Problem Statement

Assume that multiagent system consists of 𝑛 autonomous agents then connected relations between the agents constitute a network topology 𝒢. Assume 𝒢={𝑉,𝐸,𝐴} represents a directed weighted graph, in which 𝑉={𝑣₁,𝑣₂,…,𝑣_𝑛} represents a collection of 𝑛 nodes, and its set of edges is 𝐸⊆𝑉×𝑉. The node indexes belong to a finite index set 𝐼={1,2,…,𝑛}, with adjacency matrix 𝐴=[𝑎_𝑖𝑙]∈𝑅^𝑛×𝑛 with weighted adjacency elements 𝑎_𝑖𝑙≥0. An edge of the weighted diagraph 𝒢 is denoted by 𝑒_𝑖𝑙=(𝑣_𝑖,𝑣_𝑙)∈𝐸. We assume that the adjacency element 𝑎_𝑖𝑙>0 when 𝑒_𝑖𝑙∈𝐸; otherwise, 𝑎_𝑖𝑙=0. The set of neighbors of a node 𝑖 is denoted by 𝑁_𝑖={𝑙∈𝐼∶𝑎_𝑖𝑙>0}.

Let 𝒢 be a weighted digraph without self-loops, that is, let 𝑎_𝑖𝑖=0, and matrix 𝐷=diag{𝑑₁,𝑑₂,…,𝑑_𝑛} be the diagonal matrix with the diagonal elements 𝑑_𝑖=∑^𝑛_𝑙=1𝑎_𝑖𝑙 representing the sum of the elements in the 𝑖th row of matrix 𝐴. The Laplacian matrix of the weighted digraph 𝒢 is defined as 𝐿=𝐷−𝐴. For two nodes 𝑖 and 𝑙, there is subscript set {𝑙₁,𝑙₂,…𝑙_𝑘} satisfying 𝑎_𝑖𝑙1>0,𝑎_𝑙1𝑙2>0,…,𝑎_𝑙𝑘𝑙>0, and then there is a directed linked path between node 𝑖 and node 𝑙 which is used for the information transmission, also we can say that node 𝑖 can receive the information from node 𝑙. If node 𝑖 can find a path to reach any node of the graph, then node 𝑖 is globally reachable from every other node in the digraph. In this paper, the directed graph and directed symmetrical graph for fractional-order multiagent systems will be considered.

Lemma 1 (see [5]). 0 is a simple eigenvalue of Laplacian matrix 𝐿, and 𝑋₀=𝐶[1,1,…,1]^𝑇 is corresponding right eigenvector, that is, 𝐿𝑋₀=0, if and only if the digraph 𝒢=(𝑉,𝐸,𝐴) has a globally reachable node.

Given that the dynamics of multiagent systems indicated with fractional derivative in the complex environments, the fractional dynamical equations are defined as 𝑥_𝑖^(𝛼)=𝑢_𝑖(𝑡),𝑖=1,…,𝑛, (5) where 𝑥_𝑖(𝑡)∈𝑅 and 𝑢_𝑖(𝑡)∈𝑅 represent the 𝑖th agent’s state and control input, respectively, and  𝑥_𝑖^(𝛼) represents the 𝛼(𝛼>0) order fractional derivative. Assume that the following control protocols are used in multiagent systems: 𝑢_𝑖(𝑡)=−𝛾_{𝑙∈𝑁𝑖}𝑎_𝑖𝑙𝑥_𝑖(𝑡)−𝑥_𝑙(𝑡),𝑖=1,…,𝑛, (6) where 𝑎_𝑖𝑙 represents the (𝑖,𝑙) elements of adjacency matrix 𝐴, 𝛾>0 is control gain, and 𝑁_𝑖 represents the neighbors collection of the 𝑖th agent.

Suppose that for any initial value of the system, the states of autonomous agents meet lim_𝑡→∞(𝑥_𝑖(𝑡)−𝑥_𝑙(𝑡))=0, for 𝑖, 𝑙∈𝐼, and then multiagent systems asymptotically reach consensus.

4. Sampled Control of FOMAS

Suppose that the sampling period is ℎ; then the discrete-time dynamics of multiagent systems with fractional derivative can be rewritten as 𝑥_𝑖(𝑘+1)=𝛼𝑥_𝑖(𝑘)+ℎ^𝛼𝑢_𝑖(𝑘),𝑖=1,…,𝑛, (7) we have 𝑋(𝑘+1)=Ψ𝑋(𝑘), (8) where 𝑋(𝑘)=[𝑥₁(𝑘),𝑥₂(𝑘),…,𝑥_𝑛(𝑘)]^𝑇, Ψ=𝛼𝐼_𝑛−ℎ^𝛼𝛾𝐿, and 𝐼_𝑛 is a unit matrix with 𝑛-dimensions. If the norm of matrix Ψ is satisfying ||Ψ||<1, the fractional-order discrete-time multiagent system (8) will asymptotically reach consensus.

Theorem 2. Suppose that multiagent systems are composed of 𝑛 independent agents, whose connection network topology is directed, and there is a global reachable node. Then fractional-order multiagent system (7) with sampled data can asymptotically reach consensus, if 𝛼<1 and ℎ<min𝛼Re(𝜆_𝑖)𝛾𝜆²_𝑖+1𝛾𝜆_𝑖1−𝛼²Im𝜆_𝑖𝜆_𝑖^21/𝛼,𝑖=2,…,𝑛, (9) where 𝜆_𝑖 is the eigenvalue of the Laplacian matrix 𝐿.

Proof. Since the spectral radius 𝜌 of the matrix Ψ is satisfying 𝜌≤||Ψ||, we will require the spectral radius 𝜌<1. By the definition 𝜌(Ψ)=max{|𝜂_𝑖|,𝑖=1,…,𝑛} where 𝜂_𝑖 is the characteristic value of the matrix Ψ, we should calculate the characteristic value 𝜂_𝑖 of the matrix Ψ with |𝜂_𝑖|<1.
For any matrix 𝐿, there exists a unitary matrix 𝑃 satisfying Λ=𝑃𝐿𝑃^𝐻, (10) where conjugate matrix 𝑃^𝐻=𝑃⁻¹, Λ is an upper triangular matrix, and its diagonal elements 𝜆₁,…,𝜆_𝑛 are the eigenvalues of matrix 𝐿. From 𝑃Ψ𝑃^𝐻=𝛼𝐼_𝑛−ℎ^𝛼𝛾Λ, we can obtain the characteristic values of matrix Ψ being 𝜂_𝑖=𝛼−ℎ^𝛼𝛾𝜆_𝑖,𝑖=1,…,𝑛. (11) Suppose that the connection network topology of FOMAS is directed, and there is a global reachable node; then Rank(𝐿)=𝑛−1 and 𝜆=0 is a single eigenvalue of Laplacian matrix 𝐿. Suppose 𝜆₁=0, we have 𝜂₁=𝛼−ℎ^𝛼𝛾𝜆₁=𝛼. From the stability requirement of the system (7), we obtain 𝛼<1.
For other characteristic values of Laplacian matrix 𝐿 with 𝜆_𝑖≠0(𝑖≠1), the corresponding characteristic values of matrix Ψ are being 𝜂_𝑖=𝛼−ℎ^𝛼𝛾𝜆_𝑖, 𝑖=2,…,𝑛. Let 𝜆=Re(𝜆_𝑖)+𝑗Im(𝜆_𝑖) (where 𝑗 is complex number unit), and then the corresponding characteristic values, for 𝑖=2,…,𝑛, are 𝜂_𝑖=𝛼−ℎ^𝛼𝛾Re𝜆_𝑖+𝑗Im𝜆_𝑖. (12)
From |𝜂_𝑖|<1, we can obtain 𝛼²−2𝛼ℎ^𝛼𝛾Re𝜆_𝑖+ℎ^2𝛼𝛾²𝜆²_𝑖<1. (13) It has ℎ^𝛼<𝛼Re𝜆_𝑖𝛾𝜆²_𝑖+1𝛾𝜆_𝑖1−𝛼²Im𝜆_𝑖𝜆_𝑖²,𝑖=2,…,𝑛. (14)
Then, the condition of the fractional-order multiagent system (7) is obtained. The proof is finished.

Corollary 3. Suppose that multiagent systems are composed of 𝑛 independent agents, whose connection network topology is directed and symmetrical, and there is a global reachable node. Then FOMAS (7) can asymptotically reach consensus via sampled data, if 𝛼<1 and ℎ<𝛼+1𝛾𝜆_𝑛^1/𝛼, (15) where 𝜆_𝑛 is the maximum eigenvalue of the Laplacian matrix 𝐿.

Corollary 4. Suppose multiagent systems are composed of 𝑛 independent agents, whose connection network topology is directed and symmetrical, and there is a global reachable node. Then multiagent system (7) with 𝛼=1 can asymptotically reach consensus via sampled data, if 2ℎ<𝛾𝜆_𝑛, (16) where 𝜆_𝑛 is the maximum eigenvalue of the Laplacian matrix 𝐿.

Remark 5. The system of Corollary 4 for fractional-order 𝛼=1 becomes the first-order multiagent system. The consensus condition obtained in Corollary 4 is same as that in [9].

5. Consensus of FOMAS with Sampled Data and Sampling Delay

In the practical application, the sampled-data transferring will result in the communication delays. The sampling delays will affect the control features of the system and sometimes may play a key role in the stability analysis of the network. In this section, we will study the consensus of multiagent systems with sampled data and sampling delay.

Suppose that land the sampling period is ℎ, the sampling delay is 𝜏. The sampled control protocols are used in multiagent systems as follows: 𝑢_𝑖(𝑡)=−𝛾∑_{𝑙∈𝑁𝑖}𝑎_𝑖𝑙𝑥_𝑖(𝑘−1)−𝑥_𝑙(𝑘−1),𝑡∈[𝑘ℎ,𝑘ℎ+𝜏),−𝛾∑_{𝑙∈𝑁𝑖}𝑎_𝑖𝑙𝑥_𝑖(𝑘)−𝑥_𝑙(𝑘),𝑡∈[𝑘ℎ+𝜏,𝑘ℎ+ℎ). (17)

Based on (7) and (17), the dynamics of FOMAS with sampled data and sampling delay can be rewritten as 𝑋()𝑋(𝑘)𝑋(𝑘)𝑋(),𝑘+1=Φ𝑘−1 (18) where 𝑋(𝑘)=𝑥₁(𝑘),𝑥₂(𝑘),…,𝑥_𝑛(𝑘)^𝑇,Φ=𝛼𝐼_𝑛−𝛾ℎ^𝛼−ℎ^𝛼𝜏𝐿−𝛾ℎ^𝛼𝐼𝜏𝐿_𝑛0, (19) and 𝐼_𝑛 is a unit matrix with 𝑛-dimensions. If the norm of matrix Φ is satisfying ‖Φ‖<1, the discrete-time FOMAS (18) will asymptotically reach consensus.

Lemma 6 (see [23, Hermite-Biehler Theorem]). Suppose the polynomial 𝑝(𝑠)=𝑝₀+𝑝₁𝑠+⋯+𝑝_𝑛𝑠^𝑛. (20) Substituting 𝑠=𝑖𝜔 into the polynomial 𝑝(𝑠) yields 𝑝(𝜔)=𝑚(𝜔)+𝑖𝑛(𝜔). (21) Then, the polynomial 𝑝(𝑠) is Hurwitz stability if and only if the related pair 𝑚(𝜔) and 𝑛(𝜔) is interlaced, and 𝑚(0)𝑛^′(0)−𝑚^′(0)𝑛(0)>0.

Theorem 7. Suppose that multiagent systems are composed of 𝑛 independent agents, whose connection network topology is directed and symmetrical, and there is a global reachable node. Then FOMAS (18) with sampled data and sampling delay can asymptotically reach consensus, if 𝛼<1 and 1ℎ<𝛾𝜏𝜆_𝑛^1/𝛼1,𝜏>(),3+𝛼ℎ<1+𝛼𝛾𝜆_𝑛(1−2𝜏)^1/𝛼1,𝜏≤(),3+𝛼 (22) where 𝜆_𝑛 is the maximum eigenvalue of the Laplacian matrix 𝐿.

Proof. In order to prove the asymptotical consensus of discrete-time systems (18), the spectral radius 𝜌 of the matrix Φ should be satisfied with 𝜌(Φ)<1. Since 𝜌(Φ)=max{|𝜈_𝑖|,𝑖=1,…,𝑛} where 𝜈_𝑖 is the characteristic value of the matrix Φ, we should calculate the characteristic value 𝜈_𝑖 of the matrix Φ with |𝜈_𝑖|<1.
Suppose that the network topology of FOMAS is directed and symmetrical, and there is a global reachable node; then Rank(𝐿)=𝑛−1 and 𝜆=0 is a single eigenvalue of Laplacian matrix 𝐿. There exists an orthogonal matrix 𝑃 satisfying Λ=𝑃𝐿𝑃^𝑇 where 𝑃^𝑇=𝑃⁻¹, Λ is a diagonal matrix, and its diagonal elements 𝜆₁,…,𝜆_𝑛 are the characteristic values of matrix 𝐿. Without loss of generality, we suppose 0=𝜆₁<𝜆₂≤⋯≤𝜆_𝑛. The characteristic equation of matrix Φ will be calculated as follows: det𝜈𝐼_2𝑛×2𝑛−Φ=det𝜈𝐼_𝑛−𝛼𝐼_𝑛+𝛾ℎ^𝛼−ℎ^𝛼𝜏𝐿𝛾ℎ^𝛼𝜏𝐿−𝐼_𝑛𝜈𝐼_𝑛=det𝜈²𝐼_𝑛−𝜈𝛼𝐼_𝑛−𝛾ℎ^𝛼−ℎ^𝛼𝜏𝐿+𝛾ℎ^𝛼𝜏𝐿=Π^𝑛_𝑖=1𝜈²−𝜈𝛼+𝜈ℎ^𝛼−𝜈ℎ^𝛼𝜏+ℎ^𝛼𝜏𝛾𝜆_𝑖. (23)
Let 𝑎(𝜈)=𝜈²−𝜈𝛼+(𝜈ℎ^𝛼−𝜈ℎ^𝛼𝜏+ℎ^𝛼𝜏)𝛾𝜆_𝑖. When 𝜆₁=0 and 𝑎(𝜈)=𝜈²−𝜈𝛼, we can obtain the characteristic values 𝜈₁=0 and 𝜈₂=𝛼. From the stability requirement of the system (18), we obtain 𝛼<1.
When 𝜆_𝑖>0 (𝑖=2,…,𝑛), applying the bilinear transform 𝜈=(𝑠+1)/(𝑠−1), 𝑎(𝜈) can be converted into 𝑏(𝑠) as follows: 𝑏(𝑠)=1−𝛼+ℎ^𝛼𝛾𝜆_𝑖𝑠²+21−ℎ^𝛼𝛾𝜏𝜆_𝑖𝑠+1+𝛼−ℎ^𝛼𝛾𝜆_𝑖+2ℎ^𝛼𝜏𝛾𝜆_𝑖. (24)
Let 𝑠=𝑗𝜔, and the we have 𝑏(𝜔)=𝑚(𝜔)+𝑗𝑛(𝜔), (25) where 𝑚(𝜔)=−(1−𝛼+ℎ^𝛼𝛾𝜆_𝑖)𝜔²+(1+𝛼−ℎ^𝛼𝛾𝜆_𝑖+2ℎ^𝛼𝜏𝛾𝜆_𝑖) and 𝑛(𝜔)=2(1−ℎ^𝛼𝛾𝜏𝜆_𝑖)𝜔. Applying Lemma 6, we have the following.
(1) By means of 𝑚(0)𝑛^′(0)−𝑚^′(0)𝑛(0)=2(1+𝛼−ℎ^𝛼𝛾𝜆_𝑖+2ℎ^𝛼𝜏𝛾𝜆_𝑖)(1−ℎ^𝛼𝛾𝜏𝜆_𝑖)>0, we have 1−ℎ^𝛼𝛾𝜏𝜆_𝑖>0,1+𝛼−ℎ^𝛼𝛾𝜆_𝑖+2ℎ^𝛼𝜏𝛾𝜆_𝑖>0. (26)(27)
Based on (26), it has ℎ^𝛼<1𝛾𝜏𝜆_𝑖. (28) Discussing (27), if 𝜏≥0.5, then (27) comes into existence. If 𝜏<0.5, then it require is ℎ^𝛼<1+𝛼𝛾𝜆_𝑖().1−2𝜏 (29)
(2) When 2(1−ℎ^𝛼𝛾𝜏𝜆_𝑖)≠0 from (26), the solution of 𝑛(𝜔)=0 is 0. Since 𝛼<1 we have 1−𝛼+ℎ^𝛼𝛾𝜆_𝑖>0 and 1+𝛼−ℎ^𝛼𝛾𝜆_𝑖+2ℎ^𝛼𝜏𝛾𝜆_𝑖>0 from (27); then the solutions of 𝑚(𝜔)=0 are ±(1+𝛼−ℎ^𝛼𝛾𝜆_𝑖+2ℎ^𝛼𝜏𝛾𝜆_𝑖)/(1−𝛼+ℎ^𝛼𝛾𝜆_𝑖). Therefore, the solutions of the related pair 𝑚(𝜔) and 𝑛(𝜔) are interlaced.
By comparing the right parts of (28) and (29), we can obtain that when 𝜏>1/(3+𝛼), 1/𝛾𝜏𝜆_𝑖<(1+𝛼)/𝛾𝜆_𝑖(1−2𝜏); otherwise, when 𝜏≤1/(3+𝛼), 1/𝛾𝜏𝜆_𝑖≥(1+𝛼)/𝛾𝜆_𝑖(1−2𝜏). Then, the consensus conditions of the multiagent system are obtained. The proof is finished.

Remark 8. Suppose that the fractional-order 𝛼=1; we can get the consensus condition ℎ<1/𝛾𝜏𝜆_𝑛 when 𝜏>1/4, and ℎ<2/𝛾𝜆_𝑛(1−2𝜏) when 𝜏≤1/4, where 𝜆_𝑛 is the maximum eigenvalue of matrix 𝐿.

Remark 9. Suppose that the communication delay 𝜏=0, we can get 𝜏<1/(3+𝛼); therefore, ℎ<((1+𝛼)/𝛾𝜆_𝑛)^1/𝛼 where 𝜆_𝑛 is the maximum eigenvalue of matrix 𝐿. This result is the same as the consensus condition in Corollary 3.

6. Simulations

Suppose that the system is composed of four fractional-order dynamical agents (Figure 1). The connection weights between individuals are 𝑎₂₁=𝑎₁₂=0.7, 𝑎₄₂=𝑎₂₄=0.8, 𝑎₃₁=𝑎₁₃=0.9, and 𝑎₁₄=𝑎₄₁=1. Through the network topology of the system, we can get the adjacency matrix 𝐴=.00.70.910.7000.80.900010.800 (30)

Figure 1 

Network topology of the multiagent systems.

Suppose that the order of the fractional multiagent dynamics is 𝛼=0.8 and the system control gain is 𝛾=1; then we can obtain the relationship between the sampling period and the upper bound of sampling delays (Figure 2) from the conditions in Theorem 7. In order to make the system meet the condition of reaching consensus, we can set the sampling period according to the sampling delay of the system or decide the stable fields of sampling delays by means of the sampling period. Suppose the order of the fractional multiagent dynamics is 𝛼=0.8 and the control gain 𝛾=1; then we can obtain that the upper bound of sampling period ℎ_max=1.0993 s corresponding delay 𝜏=0.2632 s from Figure 2.

Figure 2 

Relationship between the communication delay and the upper bound of sampling period.

Simulation 1. Assume the sampling delay of multiagent system is 𝜏=0.25 s, we can obtain the upper bound of sampling period is 1.0272 s. In computer simulation, selecting the sampling period ℎ=0.90 s, the consensus can be asymptotically reached (Figure 3) through fractional-order coordination algorithm.

Figure 3 

Movement trajectories of the multiagent systems with delay 0.25 s and sampling period 0.90 s.

Simulation 2. Assume the sampled delay of multiagent system is 𝜏=0.49 s, the upper bound of sampling period is 0.5054 in Figure 2. In computer simulation, selecting the sampling period ℎ=0.50 s, the consensus can be asymptotically reached much more slowly with increasing of time delay (Figure 4).

Figure 4 

Movement trajectories of the multiagent systems with delay 0.49 s and sampling period 0.50 s.

Simulation 3. Assume the sampling delay of multiagent system is 𝜏=0.50 s, the upper bound of sampling period is 0.4928 in Figure 2. In computer simulation, selecting the sampling period ℎ=0.50 s, the movement trajectories of the multiagent systems will be asymptotically diverged and the consensus cannot be reached (Figure 5) through fractional-order coordination algorithm.

Figure 5 

Movement trajectories of the multiagent systems with delay 0.50 s and sampling period 0.50 s.

Simulation 4. Let time delay continue increasing, suppose the sampling delay is 𝜏=0.80 s, the upper bound of sampling period will be 0.2739 in Figure 2. In computer simulation, selecting the sampling period ℎ=0.25 s, the movement trajectories of the multiagent systems will be asymptotically converged and the consensus can be reached (Figure 6). Although the sampling period is less than the sampling delay, the consensus can be still achieved under the condition of Theorem 7.

Figure 6 

Movement trajectories of the multiagent systems with delay 0.80 s and sampling period 0.25 s.

7. Conclusions

This paper studies distributed coordination of fractional-order multiagent system with sampled control and sampling delay. By applying the stability theory of discrete-time domain, sampled-data control of FOMAS with directed network topology is investigated, and the upper bound of the sampling period is obtained. Based on the Hermite-Biehler Theorem, the collaborative control of fractional-order multiagent systems with sampled data and sampling delay is studied. The relations between sampling delay and sampled period are obtained to ensure the consensus of FOMAS. Research of the robust stability of FOMAS will be carried out in the following work.

Disclosure

This paper has not been published previously and is not under consideration of publication elsewhere.

Acknowledgments

This research is supported in part by the State Key Development Program for Basic Research of China (Grant no. 2012CB720003), the National Natural Science Foundation of China (Grant nos. 61273152, 61127007, 61170161, and 61174085), the Natural Science Foundation of Shandong Province of China (no. ZR2011FM017) and the Science Foundation of China postdoctoral (under Grant no. 2011M500215).