# Consensus of Delayed Fractional-Order Multiagent Systems Based on State-Derivative Feedback

Authors: Jun Liu; Kaiyu Qin; Wei Chen; Ping Li
Journal: Complexity
 (2018)

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## Abstract

A state-derivative feedback (SDF) is added into the designed control protocol in the existing paper to enhance the robustness of a fractional-order multiagent system (FMS) against nonuniform time delays in this paper. By applying the graph theory and the frequency-domain analysis theory, consensus conditions are derived to make the delayed FMS based on state-derivative feedback reach consensus. Compared with the consensus control protocol designed in the existing paper, the proposed SDF control protocol with nonuniform time delays can make the FMS with SDF and nonuniform time delays tolerate longer time delays, which means that the convergence speed of states of the delayed FMS with SDF is accelerated indirectly. Finally, the corresponding results of simulation are given to verify the feasibility of our theoretical results.


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## Body

## 1. Introduction

It is well known that the distributed coordination control of multiagent systems has received extensive research attention in various fields including robotics and physics. In the distributed coordination control, it is a critical problem for us to design control laws with the information of states of the agents and their neighbors to insure that multiple agents can agree on certain quantities of interest and this problem is often referred to as the consensus problem [1]. With the development of technologies such as computers, networks, and communications, consensus of multiagent systems has gradually shown enormous potential applications in the field of swarming [2], flocking [3], formation control [4], unmanned air vehicles [5], and distributed sensor networks [6].With the development of traditional integer-order derivatives and integrals, the concept of fractional calculus has long been proposed. The earliest concept of fractional calculus could be probably traced back to the 17th century [7]. Generally, different from the integer-order derivatives and integrals, the essential characteristic or behavior of an object could be better revealed by the orders of fractional calculus [8]. With the development of fractional-order derivatives and integrals, its applications have been considered by many scholars. The authors in [9] studied the fractional-order derivatives and integrals to establish the stress-strain relationships of viscoelastic materials. The authors in [10] simulated the fractional-order dynamical characteristics of self-similar protein. In [11, 12], the proportional-integral differential (PID) controllers whose dynamics were fractional-order dynamics were proposed and the performance of the fractional-order PID controllers was superior to that of the classical integer-order ones. Moreover, it has been stated in [13] that fractional derivatives were excellent tools for representing the memories and hereditary effects of all manner of materials and processes.Although fractional-order derivatives and integrals have been studied for a long time, their applications in multiagent systems have just attracted the attention of researchers in recent years. As far as we know, the consensus problem of FMSs was first investigated in [8]. Since then, many research results have been continuously springing up about consensus problems of FMSs [13–17]. The consensus problem of FMSs with a reference state was studied in [13, 14]. The consensus of FMSs about event-triggered control was investigated in [15]. In particular, the authors in [16] studied consensus control protocols for heterogeneous FMSs, which were composed of two different kinds of agents. In [17], the tracking of consensus for the FMSs based on the control method of sliding mode was investigated. Because time delays are ubiquitous in a FMS, some research results begin to take the impact of delays into account. The authors in [18] derived a consensus condition to guarantee the consensus of FMSs whose input delays were identical. In [19, 20], the authors successively studied the FMS with communication delays, whose homogeneous dynamics and heterogeneous dynamics were used to illustrate the agents of a system. In [21], the two types of maximum tolerable delay were obtained to insure reaching consensus for a FMS, whose nonuniform time delays contained up to nn−1 different values when the FMS consisted of n agents. In [22, 23], a distributed consensus protocol based on the delayed state-derivative feedback (SDF) was designed to improve the robustness against communication delays, which were identical. Hitherto, there are few research works done on the improvement for consensus performance of the FMSs with nonuniform time delays and the consensus of FMSs with SDF and nonuniform time delays.Hence, we shall study the impact of time delays on the consensus of FMSs based on SDF and how to enhance robustness of the delayed FMSs to nonuniform time delays. First of all, a control protocol based on delayed SDF is designed and the closed-loop dynamics are built by applying graph theory and matrix theory tools. Then, by employing the Laplace transform of the Caputo derivative, the transfer function matrix of the delayed FMS based on SDF is derived and the consensus problems of the delayed FMS based on SDF are transformed into the distribution problems of the eigenvalues of the transfer function matrix in the complex plane, that is, the stability problems of the delayed FMS based on SDF. Finally, the two types of maximum tolerable delay are obtained to guarantee consensus for the delayed FMS based on SDF.The main contributions of this article are as follows. First, we consider the fractional-order dynamics. The fractional-order dynamics can better reveal the essential characteristic or behavior of an object in a complex environment. Second, we consider the nonuniform time delays which contain symmetric and asymmetric time delays, and obtain the two types of maximum tolerable delay. Third, we can determine the range of fractional-orderα to improve the robustness of the FMS with nonuniform time delays by using a graphical method. Finally, we add a SDF into the designed control protocol to enhance the robustness of a FMS against nonuniform time delays.Compared to the previous research work, the following merits exist in this paper. Firstly, unlike the results in [21], this paper will enhance the consensus performance of the FMS with nonuniform time delays in [21] under the same conditions. Secondly, compared with research on the consensus of integer-order systems based on SDF in [1, 24], this paper mainly studies the consensus of FMSs based on SDF. Finally, although the consensus of delayed FMSs based on SDF in [22, 23] was investigated, all the time delays in [22, 23] were uniform time-delays, which contain the same value. However, this paper considers nonuniform time-delays, which contain up to nn−1 different values when the FMS consists of n agents.The main contents of this paper are as follows. Section2 introduces some basic preliminaries about graph theory, fractional operator, and its Laplace transform. A control protocol based on delayed SDF is designed and the closed-loop dynamics is built in Section 3. The convergence analysis of consensus and the sufficient conditions are obtained in Section 4. In Section 4, we also study the effect of the designed protocol with delayed SDF on the robustness of the FMS against nonuniform time delays. In Section 5, to verify the theoretical results, some examples are simulated. Finally, the conclusions are presented in Section 6.


## 2. Preliminaries

### 2.1. Graph Theory

LetGV,ℰ,A be an interacted graph with the node set V=v1,v2,v3,…,vn, the edge set ℰ⊆V×V, and a weighted adjacency matrix A=aik∈ℝn×n. The node indices belong to a finite index set ℐ=1,2,…,n. An edge eik=vk,vi depicts that node vi can receive information from node vk, which means aik>0, otherwise aik=0. Besides, we assume aii=0 for i∈ℐ. Define Ni=k∈ℐ,k≠i as the subscript set of neighbours of node vi. If a graph describes all the edges eik∈ℰ to satisfy aik=aki≥0, then the graph is called an undirected graph; if there exists any aik≠aki, then the graph is called a directed graph. A directed path is a sequence of edges in a directed graph with the form v1,v2,v2,v3,v3,v4,…, where vi∈V, and if there is a path from every node to every other node, the graph is said to be strongly connected. A spanning tree exists in a directed graph, which means there is a node such that every other node has a directed path to this node. The out-degree of node vi is defined as degoutvi=∑k=1Maik, and the Laplacian matrix of the interaction graph is L=Δ−A∈ℝn×n, where Δ≜diagdegoutv1,degoutv2,…,degoutvi,…,degoutvn. For some graphs such as G1, G2,…,GM, and graph G composed of the same nodes, the L of graph G is the sum of the other graphs’ Laplacian matrix if the edge set of graph G is the sum of that of the other graphs G1,G2,…,GM, that is, L=∑m=1mLm.Lemma 1 (see [25]).
If graphG is an undirected connected graph, then its Laplacian matrix L has a zero eigenvalue and the other eigenvalues are positive real numbers.Lemma 2 (see [25]).
If graphG is a directed graph and has a spanning tree, then its Laplacian matrix L has a zero eigenvalue and the other eigenvalues have a positive real part.


### 2.2. Fractional Operator

There are several common fractional operator definitions such as the Caputo operator and Grunwald-Letnikov operator. This paper will use the Caputo operator to analyze the asymptotic consensus properties because it is widely used in engineering and its physical meaning is easy to understand. The derivative of the Caputo operator offt is defined as follows:
(1)0CDtαft=1Γu−α∫0tfuηt−ηα−u+1dη,where α is the order of the derivative of Caputo operator, u−1<α≤u, u∈Z+, and Γ⋅ is given by
(2)Γσ=∫0∞e−ttσ−1dt,where σ is an arbitrary real number.


### 2.3. Laplace Transform

In order to facilitate the development of the subsequent results, we letfαt replace 0CDtαft, and let Fs=Lft=∫0−∞e−stftdt, then the Laplace transform of the Caputo derivative is obtained:
(3)Lfαt=sαFs−sα−1f0−,α∈0,1,sαFs−sα−1f0−−sα−2f′0−,α∈1,2,where f0−=limt→0−ft and f′0−=limt→0−f′t.


## 2.1. Graph Theory

LetGV,ℰ,A be an interacted graph with the node set V=v1,v2,v3,…,vn, the edge set ℰ⊆V×V, and a weighted adjacency matrix A=aik∈ℝn×n. The node indices belong to a finite index set ℐ=1,2,…,n. An edge eik=vk,vi depicts that node vi can receive information from node vk, which means aik>0, otherwise aik=0. Besides, we assume aii=0 for i∈ℐ. Define Ni=k∈ℐ,k≠i as the subscript set of neighbours of node vi. If a graph describes all the edges eik∈ℰ to satisfy aik=aki≥0, then the graph is called an undirected graph; if there exists any aik≠aki, then the graph is called a directed graph. A directed path is a sequence of edges in a directed graph with the form v1,v2,v2,v3,v3,v4,…, where vi∈V, and if there is a path from every node to every other node, the graph is said to be strongly connected. A spanning tree exists in a directed graph, which means there is a node such that every other node has a directed path to this node. The out-degree of node vi is defined as degoutvi=∑k=1Maik, and the Laplacian matrix of the interaction graph is L=Δ−A∈ℝn×n, where Δ≜diagdegoutv1,degoutv2,…,degoutvi,…,degoutvn. For some graphs such as G1, G2,…,GM, and graph G composed of the same nodes, the L of graph G is the sum of the other graphs’ Laplacian matrix if the edge set of graph G is the sum of that of the other graphs G1,G2,…,GM, that is, L=∑m=1mLm.Lemma 1 (see [25]).
If graphG is an undirected connected graph, then its Laplacian matrix L has a zero eigenvalue and the other eigenvalues are positive real numbers.Lemma 2 (see [25]).
If graphG is a directed graph and has a spanning tree, then its Laplacian matrix L has a zero eigenvalue and the other eigenvalues have a positive real part.


## 2.2. Fractional Operator

There are several common fractional operator definitions such as the Caputo operator and Grunwald-Letnikov operator. This paper will use the Caputo operator to analyze the asymptotic consensus properties because it is widely used in engineering and its physical meaning is easy to understand. The derivative of the Caputo operator offt is defined as follows:
(1)0CDtαft=1Γu−α∫0tfuηt−ηα−u+1dη,where α is the order of the derivative of Caputo operator, u−1<α≤u, u∈Z+, and Γ⋅ is given by
(2)Γσ=∫0∞e−ttσ−1dt,where σ is an arbitrary real number.


## 2.3. Laplace Transform

In order to facilitate the development of the subsequent results, we letfαt replace 0CDtαft, and let Fs=Lft=∫0−∞e−stftdt, then the Laplace transform of the Caputo derivative is obtained:
(3)Lfαt=sαFs−sα−1f0−,α∈0,1,sαFs−sα−1f0−−sα−2f′0−,α∈1,2,where f0−=limt→0−ft and f′0−=limt→0−f′t.


## 3. Problem Formulation

Assume that a FMS is made up ofn agents, each of which is considered as a node in graph G. GV,ℰ,A represents the communication topology of the FMS. The dynamic model of agent i is given as follows:
(4)xiαt=uit,i∈ℐ,where the ith agent’s state is denoted by xit∈ℝ, the α order Caputo derivative of xit is denoted by xiαtα∈0,1, and the control input is denoted by uit∈ℝ.Definition 1.
The FMS in (4) can achieve consensus when the states of all agents satisfy
(5)limt→+∞xit−xkt=0,for ∀i, k∈ℐ.Authors in [21] studied the consensus problems of a FMS with nonuniform time delays, and the distributed control protocol is designed by
(6)uit=∑k∈Niaikxkt−τik−xit−τik,i,k∈ℐ,where aik denotes the element of A, Ni denotes the subscript set of neighbours of the agent i, and τik is the time delay it takes agent i to receive the information of state of the agent k. If τik=τki holds for all i,k∈ℐ, the time delays are said to be symmetric. Otherwise, the time delays are said to be asymmetric.In [21], it has been illustrated that the consensus of the FMS in (4) with nonuniform time delays can be achieved by the protocol in (6) when all the τik<τ¯, which is called the maximum tolerable delay. Moreover, the FMS in (4) cannot achieve consensus by the protocol in (6) when all the τik>τ¯. Motivated by the method in [1, 24], we shall use the information xkt−τik+γxkαt−τik and xit−τik+γxiαt−τik, respectively, instead of xkt−τik and xit−τik to reduce the impact of time delays on consensus, where γ denotes the intensity of the delayed SDF. In addition, we also assume that τm∈τik:i,k∈ℐm=1,2,…,M denote M different time delays. Finally, the control protocol in (6) can be rewritten to
(7)uit=∑k∈Niaikxkt−τm−xit−τm+γxkαt−τm−xiαt−τm.Defineφt=x1t,x2t,…,xntT. By the protocol in (7), the closed-loop dynamics of the FMS in (4) with SDF and nonuniform time delays can be written as
(8)φαt=−∑m=1MLmφt−τm−γ∑m=1MLmφαt−τm,where φαt represents the Caputo derivative of φt with α order and Lm represents the Laplacian matrix of a subgraph, which is associated with the time delay τm.


## 4. Consensus Convergence Analysis

### 4.1. Case 1: Symmetric Time Delays over Undirected Topology

#### 4.1.1. Main Results of Case 1

Theorem 1.
Consider a FMS with SDF and symmetric time-delays over a connected and undirected graphG. By the distributed control protocol in (7), the FMS in (8) with SDF and symmetric time delays can asymptotically achieve consensus if all τm<τ¯, and the FMS in (8) with SDF and symmetric time delays cannot achieve consensus if all τm>τ¯.
(9)τ¯=π2−α2ω¯+1ω¯arctanγω¯αsinπα/21+γω¯αcosπα/2,where
(10)ω¯=γλn2cosπα/2+λn1+γ2cos2πα/2λn2−γ2λn21−γ2λn21/α,γ∈0,1λn,and λn is the maximum eigenvalue of L.Proof 1.
Here, the dynamic performance of the FMS in (8) with SDF and symmetric time delays is studied, so it is not necessary for us to consider the impact of the initial state. Taking the Laplace transform to the FMS in (8) with SDF and symmetric time delays, we have
(11)sαInΨs−sα−1Inφ0−=−∑m=1MLme−sτm+γ∑m=1MLmsαe−sτmΨs,where Ψs is the Laplace transform of φt, φ0− is the initial value of φt, and In∈ℝn×n is the unit matrix.From (11), we have the characteristic equation of the FMS in (8) with SDF and symmetric time delays:
(12)detsαIn+γ∑m=1MLmsαe−sτm+∑m=1MLme−sτm=0.The roots of (12) are called the eigenvalues of the FMS in (8) with SDF and symmetric time delays. First of all, we assume that the FMS in (8) with SDF and symmetric time delays is stable and can reach consensus when τm=0. Then, it is easy to obtain that as τm increases continuously from zero, the eigenvalues of the FMS in (8) with SDF and symmetric time delays in the complex plane will change continuously from the LH (left half-plane) to the RH (right half-plane). Once the trajectories of these eigenvalues reach the RH through the imaginary axis, the FMS in (8) with SDF and symmetric time delays will no longer be stable, which results in the failure of the consensus condition. So, it is essential for us to consider the time delay τ¯ when the nonzero eigenvalues of the FMS in (8) with SDF and symmetric time delays appear on the imaginary axis for the first time, and the time delay τ¯, which is known as maximum tolerable delay, will become the critical point of stability of the FMS in (8) with SDF and symmetric time delays. Now set s=−jω and it is the imaginary eigenvalue of the FMS in (8) with SDF and symmetric time delays, u∈ℂn is the corresponding eigenvector, u=1, and let uH be the conjugate transpose of u, then the following equation can be obtained:
(13)−jωαIn+γ∑m=1MLm−jωαejωτm+∑m=1MLmejωτmu=0.Since all the roots of (12) appear in the form of conjugate pairs, it is only necessary to study the case ω>0. Let the left side of (13) be multiplied by uH, then we have the following series of equations:
(14)uH−jωαIn+γ∑m=1MLm−jωαejωτm+∑m=1MLmejωτmu=0,uHu−jωα+uHγ∑m=1MLm−jωαejωτm+∑m=1MLmejωτmu=0,∑m=1MuHLmuejωτmγ−jωα+1=−uHu−jωα,∑m=1MuHLmuuHuejωτm=−−jωαγ−jωα+1=−ωα−jα1+γ−jαωα=−ωαcos−π/2+jsin−π/2α1+e−jπα/2γωα=−ωαej−πα/21+γωαcosπα/2−jγωαsinπα/2=ωαejπ2−α/21+γωαcosπα/22+γωαsinπα/22e−jarctanγωαsinπα/2/1+γωαcosπα/2.Then, we define(15)Fω≜∑m=1Mamejωτm=ωαejπ2−α/21+γωαcosπα/22+γωαsinπα/22e−jarctanγωαsinπα/2/1+γωαcosπα/2,where am=uHLmu/uHu.According to (15) and Lemma 1, we can get
(16)Fω=∑m=1Mamejωτm≤∑m=1Mam=uHLuuHu≤λn.SinceFω=ωα/1+γωαcosπα/22+γωαsinπα/22, we have ωα/1+γωαcosπα/22+γωαsinπα/22≤λn, which leads to ω≤ω¯=γλn2cosπα/2+λn1+γ2cos2πα/2λn2−γ2λn2/1−γ2λn21/α.Next, we need to discuss the principal value of the argument ofFω. Based on (15), we know that
(17)argFω=π2−α2+arctanγωαsinπα/21+γωαcosπα/2≜θω,where θω∈π/2+arctanγω,π.If we suppose thatδ=sinπα/2 and σ=cosπα/2, then it is easy to arrive at δ2+σ2=1 and we get
(18)τω≜θωω=π2−α2ω⏟Γ1ω+1ωarctanγωαsinπα/21+γωαcosπα/2⏟Γ2ω=π2−α2ω⏟Γ1ω+1ωarctanγωαδ1+γωασ⏟Γ2ω.According to (18), we can calculate the first derivative of τω about ω:
(19)Γω≜dτωdω=Γ1′ω+Γ2′ω,where
(20)Γ1′ω=−π2−α2ω2<0,and
(21)Γ2′ω=αγδωα−11+γσωα−αγ2δσω2α−1ω1+γσωα2+γ2δ2ω2α−arctanγδωα/1+γσωαω2=αγδωα1+γσωα−αγ2δσω2αω21+γσωα2+γ2δ2ω2α−arctanγδωα/1+γσωαω2=αγδωαω21+γσωα2+γ2δ2ω2α−arctanγδωα/1+γωασω2=αγδωα/1+γσωα2+γ2δ2ω2α−arctanγδωα/1+γσωαω2.If we assume that there is a functionZω established by
(22)Zω=arctanγδωα1+γσωα−αγδωα1+γσωα2+γ2δ2ω2α,then the first derivative of Zω is as follows:
(23)Z′ω=αγδωα−11+γσωα−αγ2δσω2α−11+γσωα2+γ2δ2ω2α−α2γδωα−11+γσωα2+γ2δ2ω2α−21+γσωααγσωα−1+2αγ2δ2ω2α−1αγδωα1+γσωα2+γ2δ2ω2α2=αγδωα−11+γσωα2+γ2δ2ω2α−α2γδωα−11+γσωα2+γ2δ2ω2α+21+γσωααγσωα−1+2αγ2δ2ω2α−1αγδωα1+γσωα2+γ2δ2ω2α2=α1−αγδωα−11+2γσωα+γ2ω2α+2αγσωα−1+2αγ2ω2α−1αγδωα1+γσωα2+γ2δ2ω2α2=αγδωα−11−α+21−αγσωα+1−αγ2ω2α+2αγσωα+2αγ2ω2α1+γσωα2+γ2δ2ω2α2=αγδωα−11−α+2γσωα+1+αγ2ω2α1+γσωα2+γ2δ2ω2α2>0.SinceZ′ω>0, Zω is an increasing function. It is very convenient to obtain that Zω>Z0=0 when ω>0. Since Zω>0, Γ2′ω<0, which means Γω=Γ1′ω+Γ2′ω<0. Since Γω<0, τω is a decreasing function of ω, and when ω≤ω¯, there is
(24)τ¯=τω¯≤τω.On the other hand, when allτm<τ¯, the following inequality can be obtained:
(25)τω=θωω=arg∑m=1Mamejωτmω≤maxωτmω<ωτ¯ω=τ¯.The contradiction between the inequality in (25) and the inequality in (24) is obvious. Accordingly, when all τm are less than τ¯, we can avoid the eigenvalues of the FMS in (8) with SDF and symmetric time delays crossing the imaginary axis to reach the unstable RH, and the FMS in (8) with SDF and symmetric time delays can reach consensus; when all τm are equal to τ¯, s=−jω¯ is an imaginary eigenvalue of the FMS in (8) with SDF and symmetric time delays, whose corresponding eigenvector uω¯ makes ∑m=1Mam=λn hold; when all τm are more than τ¯, there must exist at least one eigenvalue of the FMS in (8) with SDF and symmetric time delays in the RH, and the states of the FMS in (8) with SDF and symmetric time delays will no longer converge and the FMS in (8) with SDF and symmetric time delays cannot reach consensus.Remark 1.
From Theorem1, we can get τ¯>0 and ω¯>0. γ∈0,1/λn is implied in (9). So it is necessary for achieving consensus of the FMS in (8) with symmetric time delays that γ∈0,1/λn.Corollary 1.
Consider a FMS with SDF and symmetric time delays over a connected and undirected graphG. When α=1, by the distributed control protocol in (7), the FMS in (8) with SDF and symmetric time delays can asymptotically achieve consensus if all τm<τ¯, and the FMS in (8) with SDF and symmetric time delays cannot achieve consensus if all τm>τ¯.
(26)τ¯=π2ω¯+1ω¯arctanγω¯,where ω¯=λn1−γ2λn2/1−γ2λn2, γ∈0,1/λn, and λn is the maximum eigenvalue of L.


#### 4.1.2. Robustness Analysis for Case 1

According to Theorem1, for the given FMS in (4), if applying the control protocol in (6), that is, the control protocol in (7) with γ=0, we obtain
(27)τ¯γ=0=π2−α2λn1/α≜τup1.If applying the control protocol in (7), we obtain
(28)τ¯0<γ<1/λn=π2−α2ω¯+1ω¯arctanγω¯αsinπα/21+γω¯αcosπα/2≜τup2,where ω¯=γλn2cosπα/2+λn1+γ2cos2πα/2λn2−γ2λn2/1−γ2λn21/α.In order to show the effect of the protocol in (7) on the robustness against symmetric time delays, we are supposed to further determine the range of α to insure τup2>τup1, which means
(29)τup≜τup2−τup1>0.Obviously, the inequality in (29) contains multiple parameters. However, for a fix undirected interconnection topology, we can determine the range of α and find a proper value of γ to improve the robustness of the FMS with symmetric time delays. Since it is very difficult for us to solve the inequality in (29) by the analytic method, we shall analyze it by the graphical method and let λn=3.5229. Figure 1 is the three-dimensional diagram of τup with respect to parameters α and γ, and it is easy for us to find proper parameters α,γ to ensure τup>0 by the graphical method. According to Figure 1, we can find that α∈α∗,1 when γ changes from 0 to 1/λn and τup>0, and α∗ which is decided by the inequality in (29) is easy to obtain in Figure 1 when the value of γ is determined.Figure 1
The three-dimensional diagram ofτup with respect to parameters α and γ in case 1.In particular, if we assumeγ=0.1, then the relationship between τup and α can be shown in Figure 2. According to Figure 2, it is obvious that α∈0.854,1α∗≜0.854 when τup>0 and γ=0.1.Figure 2
The relationship betweenτup and α when γ=0.1 in case 1.


### 4.2. Case 2: Asymmetric Time Delays over Directed Topology

#### 4.2.1. Main Results of Case 2

Theorem 2.
Consider a FMS with SDF and asymmetric time delays over a directed interconnection graphG that has a spanning tree. By the distributed control protocol in (7), the FMS in (8) with SDF and asymmetric time delays can asymptotically achieve consensus if all τm<τ¯, and the FMS in (8) with SDF and asymmetric time delays cannot achieve consensus if all τm>τ¯.
(30)τ¯=minλi≠0π2−α/2+arctanγωi¯αsinπα/2/1+γωi¯αcosπα/2−argλiω¯i,where ω¯i=γλi¯2cosπα/2+λi¯1+γ2cos2πα/2λi¯2−γ2λi¯2/1−γ2λi¯21/α, γ∈0,γ∗γ∗=minλi≠01/λi, λi¯ is the λi which makes τ¯ minimized, and λi is the ith eigenvalue of L.Proof 2.
Let one apply the above frequency-domain proof method, which has been used in proving Theorem1. Suppose that s=−jω≠0 is the eigenvalue of the FMS in (8) with SDF and asymmetric time delays on the imaginary axis, u∈ℂn is the corresponding eigenvector, and u=1. According to Lemma 2, one can get
(31)Ba≜∑m=1Mamejωτm=−−jωαγ−jωα+1=ωα1+γωαcosπα/22+γωαsinπα/22ejπ2−α/2+arctanγωαsinπα/2/1+γωαcosπα/2.Taking the modulus of both sides of (31) and regarding ω as the function of Ba, we can get
(32)ωBa=γBa2cosπα/2+Ba1+γ2cos2πα/2Ba2−γ2Ba21−γ2Ba21/α,where ωBa is an increasing function of Ba.Calculating the principal value of the argument of (31) on both sides separately, and we have
(33)argBa=π2−α2+arctanγωαsinπα/21+γωαcosπα/2.According to the definition ofBa in (31), we have
(34)argBa≤arg∑m=1Mam+maxωτm,and it yields that
(35)maxωτm≥π2−α2+arctanγωαsinπα/21+γωαcosπα/2−arg∑m=1Mam⏟ε.Since∑m=1Mam=uHLu/uHu, the possible values of ∑m=1Mam must be nonzero eigenvalues of L, that is, ∑m=1Mam=λiλi≠0. So when γ<minλi≠01/λi and Ba≤λi, we have ωBa≤ωλi¯=ω¯i=γλi¯2cosπα/2+λi¯1+γ2cos2πα/2λi¯2−γ2λi¯2/1−γ2λi¯21/α. If we let all τm<τ¯, we can obtain
(36)maxωτm<ω¯iτ¯=minλi≠0π2−α/2+arctanγωi¯αsinπα/2/1+γωi¯αcosπα/2−argλiω¯iω¯i≤ε=π2−α2+arctanγωαsinπα/21+γωαcosπα/2−arg∑m=1Mam.The contradiction between the inequality in (36) and the inequality in (35) is obvious. Accordingly, when all τm<τ¯, the eigenvalues of the FMS in (8) with SDF and asymmetric time delays cannot reach or cross the imaginary axis, then the FMS in (8) with SDF and asymmetric time delays will remain stable and the FMS in (8) with SDF and asymmetric time delays can reach consensus. On the other hand, when all τm>τ¯, there must exist at least one eigenvalue of the FMS in (8) with SDF and asymmetric time delays in the RH, then the states of the FMS in (8) with SDF and asymmetric time delays will no longer converge and the FMS in (8) with SDF and asymmetric time delays cannot reach consensus.Remark 2.
From Theorem2, let one get τ¯>0 and ω¯i>0. γ∈0,γ∗γ∗=minλi≠01/λi is implied in (30). So it is necessary for achieving consensus of the FMS in (8) with SDF and asymmetric time delays that γ∈0,γ∗γ∗=minλi≠01/λi.Corollary 2.
Consider a FMS with SDF and asymmetric time delays over a directed interconnection graphG with a spanning tree. When α=1, by the distributed control protocol in (7), the FMS in (8) with SDF and asymmetric time delays can asymptotically achieve consensus if all τm<τ¯, and the FMS in (8) with SDF and asymmetric time delays cannot achieve consensus if all τm>τ¯.
(37)τ¯=minλi≠0π/2+arctanγωi¯−argλiω¯i,where ω¯i=λi¯1−γ2λi¯2/1−γ2λi¯2, γ∈0,γ∗γ∗=minλi≠01/λi, λi¯ is the λi which makes τ¯ minimized, and λi is the ith eigenvalue of L.


#### 4.2.2. Robustness Analysis for Case 2

According to Theorem2, for the given FMS in (4), if applying the control protocol in (6), that is, the control protocol in (7) with γ=0, we obtain
(38)τ¯γ=0=minλi≠0π2−α/2−argλiλi1/α≜τup1.When we apply the control protocol in (7), we obtain
(39)τ¯0<γ<γ∗=minλi≠0π2−α/2+arctanγωi¯αsinπα/2/1+γωi¯αcosπα/2−argλiω¯i≜τup2,where ω¯i=γλi¯2cosπα/2+λi¯1+γ2cos2πα/2λi¯2−γ2λi¯2/1−γ2λi¯21/α, and γ∗=min∣λi∣≠01/λi.In order to show the effect of the protocol in (7) on the robustness against asymmetric time delays, we are supposed to further determine the range of α to insure τup2>τup1, which means
(40)τup≜τup2−τup1>0.Obviously, the inequality in (40) contains multiple parameters. However, for a fix directed interconnection graph G that has a spanning tree, we can determine the range of α and find a proper value of γ to improve the robustness of the FMS with asymmetric time delays. Since it is very difficult for us to solve the inequality in (40) by the analytic method, we shall analyze it by the graphical method and let γ∗=0.6967. Figure 3 is the three-dimensional diagram of τup with respect to parameters α and γ, it is easy for us to find proper parameters α,γ to ensure τup>0 by the graphical method. According to Figure 3, we can find that α∈α∗,1 when γ changes from 0 to γ∗ and τup>0, and α∗ which is decided by the inequality in (40) is easy to obtain in Figure 3 when the value of γ is determined.Figure 3
The three-dimensional diagram ofτup with respect to parameters α and γ in case 2.In particular, if we assumeγ=0.1, then the relationship between τup and α can be shown in Figure 4. According to Figure 4, it is obvious that α∈0.7503,1α∗≜0.7503 when τup>0 and γ=0.1.Figure 4
The relationship betweenτup and α when γ=0.1 in case 2.


## 4.1. Case 1: Symmetric Time Delays over Undirected Topology

### 4.1.1. Main Results of Case 1

Theorem 1.
Consider a FMS with SDF and symmetric time-delays over a connected and undirected graphG. By the distributed control protocol in (7), the FMS in (8) with SDF and symmetric time delays can asymptotically achieve consensus if all τm<τ¯, and the FMS in (8) with SDF and symmetric time delays cannot achieve consensus if all τm>τ¯.
(9)τ¯=π2−α2ω¯+1ω¯arctanγω¯αsinπα/21+γω¯αcosπα/2,where
(10)ω¯=γλn2cosπα/2+λn1+γ2cos2πα/2λn2−γ2λn21−γ2λn21/α,γ∈0,1λn,and λn is the maximum eigenvalue of L.Proof 1.
Here, the dynamic performance of the FMS in (8) with SDF and symmetric time delays is studied, so it is not necessary for us to consider the impact of the initial state. Taking the Laplace transform to the FMS in (8) with SDF and symmetric time delays, we have
(11)sαInΨs−sα−1Inφ0−=−∑m=1MLme−sτm+γ∑m=1MLmsαe−sτmΨs,where Ψs is the Laplace transform of φt, φ0− is the initial value of φt, and In∈ℝn×n is the unit matrix.From (11), we have the characteristic equation of the FMS in (8) with SDF and symmetric time delays:
(12)detsαIn+γ∑m=1MLmsαe−sτm+∑m=1MLme−sτm=0.The roots of (12) are called the eigenvalues of the FMS in (8) with SDF and symmetric time delays. First of all, we assume that the FMS in (8) with SDF and symmetric time delays is stable and can reach consensus when τm=0. Then, it is easy to obtain that as τm increases continuously from zero, the eigenvalues of the FMS in (8) with SDF and symmetric time delays in the complex plane will change continuously from the LH (left half-plane) to the RH (right half-plane). Once the trajectories of these eigenvalues reach the RH through the imaginary axis, the FMS in (8) with SDF and symmetric time delays will no longer be stable, which results in the failure of the consensus condition. So, it is essential for us to consider the time delay τ¯ when the nonzero eigenvalues of the FMS in (8) with SDF and symmetric time delays appear on the imaginary axis for the first time, and the time delay τ¯, which is known as maximum tolerable delay, will become the critical point of stability of the FMS in (8) with SDF and symmetric time delays. Now set s=−jω and it is the imaginary eigenvalue of the FMS in (8) with SDF and symmetric time delays, u∈ℂn is the corresponding eigenvector, u=1, and let uH be the conjugate transpose of u, then the following equation can be obtained:
(13)−jωαIn+γ∑m=1MLm−jωαejωτm+∑m=1MLmejωτmu=0.Since all the roots of (12) appear in the form of conjugate pairs, it is only necessary to study the case ω>0. Let the left side of (13) be multiplied by uH, then we have the following series of equations:
(14)uH−jωαIn+γ∑m=1MLm−jωαejωτm+∑m=1MLmejωτmu=0,uHu−jωα+uHγ∑m=1MLm−jωαejωτm+∑m=1MLmejωτmu=0,∑m=1MuHLmuejωτmγ−jωα+1=−uHu−jωα,∑m=1MuHLmuuHuejωτm=−−jωαγ−jωα+1=−ωα−jα1+γ−jαωα=−ωαcos−π/2+jsin−π/2α1+e−jπα/2γωα=−ωαej−πα/21+γωαcosπα/2−jγωαsinπα/2=ωαejπ2−α/21+γωαcosπα/22+γωαsinπα/22e−jarctanγωαsinπα/2/1+γωαcosπα/2.Then, we define(15)Fω≜∑m=1Mamejωτm=ωαejπ2−α/21+γωαcosπα/22+γωαsinπα/22e−jarctanγωαsinπα/2/1+γωαcosπα/2,where am=uHLmu/uHu.According to (15) and Lemma 1, we can get
(16)Fω=∑m=1Mamejωτm≤∑m=1Mam=uHLuuHu≤λn.SinceFω=ωα/1+γωαcosπα/22+γωαsinπα/22, we have ωα/1+γωαcosπα/22+γωαsinπα/22≤λn, which leads to ω≤ω¯=γλn2cosπα/2+λn1+γ2cos2πα/2λn2−γ2λn2/1−γ2λn21/α.Next, we need to discuss the principal value of the argument ofFω. Based on (15), we know that
(17)argFω=π2−α2+arctanγωαsinπα/21+γωαcosπα/2≜θω,where θω∈π/2+arctanγω,π.If we suppose thatδ=sinπα/2 and σ=cosπα/2, then it is easy to arrive at δ2+σ2=1 and we get
(18)τω≜θωω=π2−α2ω⏟Γ1ω+1ωarctanγωαsinπα/21+γωαcosπα/2⏟Γ2ω=π2−α2ω⏟Γ1ω+1ωarctanγωαδ1+γωασ⏟Γ2ω.According to (18), we can calculate the first derivative of τω about ω:
(19)Γω≜dτωdω=Γ1′ω+Γ2′ω,where
(20)Γ1′ω=−π2−α2ω2<0,and
(21)Γ2′ω=αγδωα−11+γσωα−αγ2δσω2α−1ω1+γσωα2+γ2δ2ω2α−arctanγδωα/1+γσωαω2=αγδωα1+γσωα−αγ2δσω2αω21+γσωα2+γ2δ2ω2α−arctanγδωα/1+γσωαω2=αγδωαω21+γσωα2+γ2δ2ω2α−arctanγδωα/1+γωασω2=αγδωα/1+γσωα2+γ2δ2ω2α−arctanγδωα/1+γσωαω2.If we assume that there is a functionZω established by
(22)Zω=arctanγδωα1+γσωα−αγδωα1+γσωα2+γ2δ2ω2α,then the first derivative of Zω is as follows:
(23)Z′ω=αγδωα−11+γσωα−αγ2δσω2α−11+γσωα2+γ2δ2ω2α−α2γδωα−11+γσωα2+γ2δ2ω2α−21+γσωααγσωα−1+2αγ2δ2ω2α−1αγδωα1+γσωα2+γ2δ2ω2α2=αγδωα−11+γσωα2+γ2δ2ω2α−α2γδωα−11+γσωα2+γ2δ2ω2α+21+γσωααγσωα−1+2αγ2δ2ω2α−1αγδωα1+γσωα2+γ2δ2ω2α2=α1−αγδωα−11+2γσωα+γ2ω2α+2αγσωα−1+2αγ2ω2α−1αγδωα1+γσωα2+γ2δ2ω2α2=αγδωα−11−α+21−αγσωα+1−αγ2ω2α+2αγσωα+2αγ2ω2α1+γσωα2+γ2δ2ω2α2=αγδωα−11−α+2γσωα+1+αγ2ω2α1+γσωα2+γ2δ2ω2α2>0.SinceZ′ω>0, Zω is an increasing function. It is very convenient to obtain that Zω>Z0=0 when ω>0. Since Zω>0, Γ2′ω<0, which means Γω=Γ1′ω+Γ2′ω<0. Since Γω<0, τω is a decreasing function of ω, and when ω≤ω¯, there is
(24)τ¯=τω¯≤τω.On the other hand, when allτm<τ¯, the following inequality can be obtained:
(25)τω=θωω=arg∑m=1Mamejωτmω≤maxωτmω<ωτ¯ω=τ¯.The contradiction between the inequality in (25) and the inequality in (24) is obvious. Accordingly, when all τm are less than τ¯, we can avoid the eigenvalues of the FMS in (8) with SDF and symmetric time delays crossing the imaginary axis to reach the unstable RH, and the FMS in (8) with SDF and symmetric time delays can reach consensus; when all τm are equal to τ¯, s=−jω¯ is an imaginary eigenvalue of the FMS in (8) with SDF and symmetric time delays, whose corresponding eigenvector uω¯ makes ∑m=1Mam=λn hold; when all τm are more than τ¯, there must exist at least one eigenvalue of the FMS in (8) with SDF and symmetric time delays in the RH, and the states of the FMS in (8) with SDF and symmetric time delays will no longer converge and the FMS in (8) with SDF and symmetric time delays cannot reach consensus.Remark 1.
From Theorem1, we can get τ¯>0 and ω¯>0. γ∈0,1/λn is implied in (9). So it is necessary for achieving consensus of the FMS in (8) with symmetric time delays that γ∈0,1/λn.Corollary 1.
Consider a FMS with SDF and symmetric time delays over a connected and undirected graphG. When α=1, by the distributed control protocol in (7), the FMS in (8) with SDF and symmetric time delays can asymptotically achieve consensus if all τm<τ¯, and the FMS in (8) with SDF and symmetric time delays cannot achieve consensus if all τm>τ¯.
(26)τ¯=π2ω¯+1ω¯arctanγω¯,where ω¯=λn1−γ2λn2/1−γ2λn2, γ∈0,1/λn, and λn is the maximum eigenvalue of L.


### 4.1.2. Robustness Analysis for Case 1

According to Theorem1, for the given FMS in (4), if applying the control protocol in (6), that is, the control protocol in (7) with γ=0, we obtain
(27)τ¯γ=0=π2−α2λn1/α≜τup1.If applying the control protocol in (7), we obtain
(28)τ¯0<γ<1/λn=π2−α2ω¯+1ω¯arctanγω¯αsinπα/21+γω¯αcosπα/2≜τup2,where ω¯=γλn2cosπα/2+λn1+γ2cos2πα/2λn2−γ2λn2/1−γ2λn21/α.In order to show the effect of the protocol in (7) on the robustness against symmetric time delays, we are supposed to further determine the range of α to insure τup2>τup1, which means
(29)τup≜τup2−τup1>0.Obviously, the inequality in (29) contains multiple parameters. However, for a fix undirected interconnection topology, we can determine the range of α and find a proper value of γ to improve the robustness of the FMS with symmetric time delays. Since it is very difficult for us to solve the inequality in (29) by the analytic method, we shall analyze it by the graphical method and let λn=3.5229. Figure 1 is the three-dimensional diagram of τup with respect to parameters α and γ, and it is easy for us to find proper parameters α,γ to ensure τup>0 by the graphical method. According to Figure 1, we can find that α∈α∗,1 when γ changes from 0 to 1/λn and τup>0, and α∗ which is decided by the inequality in (29) is easy to obtain in Figure 1 when the value of γ is determined.Figure 1
The three-dimensional diagram ofτup with respect to parameters α and γ in case 1.In particular, if we assumeγ=0.1, then the relationship between τup and α can be shown in Figure 2. According to Figure 2, it is obvious that α∈0.854,1α∗≜0.854 when τup>0 and γ=0.1.Figure 2
The relationship betweenτup and α when γ=0.1 in case 1.


## 4.1.1. Main Results of Case 1

Theorem 1.
Consider a FMS with SDF and symmetric time-delays over a connected and undirected graphG. By the distributed control protocol in (7), the FMS in (8) with SDF and symmetric time delays can asymptotically achieve consensus if all τm<τ¯, and the FMS in (8) with SDF and symmetric time delays cannot achieve consensus if all τm>τ¯.
(9)τ¯=π2−α2ω¯+1ω¯arctanγω¯αsinπα/21+γω¯αcosπα/2,where
(10)ω¯=γλn2cosπα/2+λn1+γ2cos2πα/2λn2−γ2λn21−γ2λn21/α,γ∈0,1λn,and λn is the maximum eigenvalue of L.Proof 1.
Here, the dynamic performance of the FMS in (8) with SDF and symmetric time delays is studied, so it is not necessary for us to consider the impact of the initial state. Taking the Laplace transform to the FMS in (8) with SDF and symmetric time delays, we have
(11)sαInΨs−sα−1Inφ0−=−∑m=1MLme−sτm+γ∑m=1MLmsαe−sτmΨs,where Ψs is the Laplace transform of φt, φ0− is the initial value of φt, and In∈ℝn×n is the unit matrix.From (11), we have the characteristic equation of the FMS in (8) with SDF and symmetric time delays:
(12)detsαIn+γ∑m=1MLmsαe−sτm+∑m=1MLme−sτm=0.The roots of (12) are called the eigenvalues of the FMS in (8) with SDF and symmetric time delays. First of all, we assume that the FMS in (8) with SDF and symmetric time delays is stable and can reach consensus when τm=0. Then, it is easy to obtain that as τm increases continuously from zero, the eigenvalues of the FMS in (8) with SDF and symmetric time delays in the complex plane will change continuously from the LH (left half-plane) to the RH (right half-plane). Once the trajectories of these eigenvalues reach the RH through the imaginary axis, the FMS in (8) with SDF and symmetric time delays will no longer be stable, which results in the failure of the consensus condition. So, it is essential for us to consider the time delay τ¯ when the nonzero eigenvalues of the FMS in (8) with SDF and symmetric time delays appear on the imaginary axis for the first time, and the time delay τ¯, which is known as maximum tolerable delay, will become the critical point of stability of the FMS in (8) with SDF and symmetric time delays. Now set s=−jω and it is the imaginary eigenvalue of the FMS in (8) with SDF and symmetric time delays, u∈ℂn is the corresponding eigenvector, u=1, and let uH be the conjugate transpose of u, then the following equation can be obtained:
(13)−jωαIn+γ∑m=1MLm−jωαejωτm+∑m=1MLmejωτmu=0.Since all the roots of (12) appear in the form of conjugate pairs, it is only necessary to study the case ω>0. Let the left side of (13) be multiplied by uH, then we have the following series of equations:
(14)uH−jωαIn+γ∑m=1MLm−jωαejωτm+∑m=1MLmejωτmu=0,uHu−jωα+uHγ∑m=1MLm−jωαejωτm+∑m=1MLmejωτmu=0,∑m=1MuHLmuejωτmγ−jωα+1=−uHu−jωα,∑m=1MuHLmuuHuejωτm=−−jωαγ−jωα+1=−ωα−jα1+γ−jαωα=−ωαcos−π/2+jsin−π/2α1+e−jπα/2γωα=−ωαej−πα/21+γωαcosπα/2−jγωαsinπα/2=ωαejπ2−α/21+γωαcosπα/22+γωαsinπα/22e−jarctanγωαsinπα/2/1+γωαcosπα/2.Then, we define(15)Fω≜∑m=1Mamejωτm=ωαejπ2−α/21+γωαcosπα/22+γωαsinπα/22e−jarctanγωαsinπα/2/1+γωαcosπα/2,where am=uHLmu/uHu.According to (15) and Lemma 1, we can get
(16)Fω=∑m=1Mamejωτm≤∑m=1Mam=uHLuuHu≤λn.SinceFω=ωα/1+γωαcosπα/22+γωαsinπα/22, we have ωα/1+γωαcosπα/22+γωαsinπα/22≤λn, which leads to ω≤ω¯=γλn2cosπα/2+λn1+γ2cos2πα/2λn2−γ2λn2/1−γ2λn21/α.Next, we need to discuss the principal value of the argument ofFω. Based on (15), we know that
(17)argFω=π2−α2+arctanγωαsinπα/21+γωαcosπα/2≜θω,where θω∈π/2+arctanγω,π.If we suppose thatδ=sinπα/2 and σ=cosπα/2, then it is easy to arrive at δ2+σ2=1 and we get
(18)τω≜θωω=π2−α2ω⏟Γ1ω+1ωarctanγωαsinπα/21+γωαcosπα/2⏟Γ2ω=π2−α2ω⏟Γ1ω+1ωarctanγωαδ1+γωασ⏟Γ2ω.According to (18), we can calculate the first derivative of τω about ω:
(19)Γω≜dτωdω=Γ1′ω+Γ2′ω,where
(20)Γ1′ω=−π2−α2ω2<0,and
(21)Γ2′ω=αγδωα−11+γσωα−αγ2δσω2α−1ω1+γσωα2+γ2δ2ω2α−arctanγδωα/1+γσωαω2=αγδωα1+γσωα−αγ2δσω2αω21+γσωα2+γ2δ2ω2α−arctanγδωα/1+γσωαω2=αγδωαω21+γσωα2+γ2δ2ω2α−arctanγδωα/1+γωασω2=αγδωα/1+γσωα2+γ2δ2ω2α−arctanγδωα/1+γσωαω2.If we assume that there is a functionZω established by
(22)Zω=arctanγδωα1+γσωα−αγδωα1+γσωα2+γ2δ2ω2α,then the first derivative of Zω is as follows:
(23)Z′ω=αγδωα−11+γσωα−αγ2δσω2α−11+γσωα2+γ2δ2ω2α−α2γδωα−11+γσωα2+γ2δ2ω2α−21+γσωααγσωα−1+2αγ2δ2ω2α−1αγδωα1+γσωα2+γ2δ2ω2α2=αγδωα−11+γσωα2+γ2δ2ω2α−α2γδωα−11+γσωα2+γ2δ2ω2α+21+γσωααγσωα−1+2αγ2δ2ω2α−1αγδωα1+γσωα2+γ2δ2ω2α2=α1−αγδωα−11+2γσωα+γ2ω2α+2αγσωα−1+2αγ2ω2α−1αγδωα1+γσωα2+γ2δ2ω2α2=αγδωα−11−α+21−αγσωα+1−αγ2ω2α+2αγσωα+2αγ2ω2α1+γσωα2+γ2δ2ω2α2=αγδωα−11−α+2γσωα+1+αγ2ω2α1+γσωα2+γ2δ2ω2α2>0.SinceZ′ω>0, Zω is an increasing function. It is very convenient to obtain that Zω>Z0=0 when ω>0. Since Zω>0, Γ2′ω<0, which means Γω=Γ1′ω+Γ2′ω<0. Since Γω<0, τω is a decreasing function of ω, and when ω≤ω¯, there is
(24)τ¯=τω¯≤τω.On the other hand, when allτm<τ¯, the following inequality can be obtained:
(25)τω=θωω=arg∑m=1Mamejωτmω≤maxωτmω<ωτ¯ω=τ¯.The contradiction between the inequality in (25) and the inequality in (24) is obvious. Accordingly, when all τm are less than τ¯, we can avoid the eigenvalues of the FMS in (8) with SDF and symmetric time delays crossing the imaginary axis to reach the unstable RH, and the FMS in (8) with SDF and symmetric time delays can reach consensus; when all τm are equal to τ¯, s=−jω¯ is an imaginary eigenvalue of the FMS in (8) with SDF and symmetric time delays, whose corresponding eigenvector uω¯ makes ∑m=1Mam=λn hold; when all τm are more than τ¯, there must exist at least one eigenvalue of the FMS in (8) with SDF and symmetric time delays in the RH, and the states of the FMS in (8) with SDF and symmetric time delays will no longer converge and the FMS in (8) with SDF and symmetric time delays cannot reach consensus.Remark 1.
From Theorem1, we can get τ¯>0 and ω¯>0. γ∈0,1/λn is implied in (9). So it is necessary for achieving consensus of the FMS in (8) with symmetric time delays that γ∈0,1/λn.Corollary 1.
Consider a FMS with SDF and symmetric time delays over a connected and undirected graphG. When α=1, by the distributed control protocol in (7), the FMS in (8) with SDF and symmetric time delays can asymptotically achieve consensus if all τm<τ¯, and the FMS in (8) with SDF and symmetric time delays cannot achieve consensus if all τm>τ¯.
(26)τ¯=π2ω¯+1ω¯arctanγω¯,where ω¯=λn1−γ2λn2/1−γ2λn2, γ∈0,1/λn, and λn is the maximum eigenvalue of L.


## 4.1.2. Robustness Analysis for Case 1

According to Theorem1, for the given FMS in (4), if applying the control protocol in (6), that is, the control protocol in (7) with γ=0, we obtain
(27)τ¯γ=0=π2−α2λn1/α≜τup1.If applying the control protocol in (7), we obtain
(28)τ¯0<γ<1/λn=π2−α2ω¯+1ω¯arctanγω¯αsinπα/21+γω¯αcosπα/2≜τup2,where ω¯=γλn2cosπα/2+λn1+γ2cos2πα/2λn2−γ2λn2/1−γ2λn21/α.In order to show the effect of the protocol in (7) on the robustness against symmetric time delays, we are supposed to further determine the range of α to insure τup2>τup1, which means
(29)τup≜τup2−τup1>0.Obviously, the inequality in (29) contains multiple parameters. However, for a fix undirected interconnection topology, we can determine the range of α and find a proper value of γ to improve the robustness of the FMS with symmetric time delays. Since it is very difficult for us to solve the inequality in (29) by the analytic method, we shall analyze it by the graphical method and let λn=3.5229. Figure 1 is the three-dimensional diagram of τup with respect to parameters α and γ, and it is easy for us to find proper parameters α,γ to ensure τup>0 by the graphical method. According to Figure 1, we can find that α∈α∗,1 when γ changes from 0 to 1/λn and τup>0, and α∗ which is decided by the inequality in (29) is easy to obtain in Figure 1 when the value of γ is determined.Figure 1
The three-dimensional diagram ofτup with respect to parameters α and γ in case 1.In particular, if we assumeγ=0.1, then the relationship between τup and α can be shown in Figure 2. According to Figure 2, it is obvious that α∈0.854,1α∗≜0.854 when τup>0 and γ=0.1.Figure 2
The relationship betweenτup and α when γ=0.1 in case 1.


## 4.2. Case 2: Asymmetric Time Delays over Directed Topology

### 4.2.1. Main Results of Case 2

Theorem 2.
Consider a FMS with SDF and asymmetric time delays over a directed interconnection graphG that has a spanning tree. By the distributed control protocol in (7), the FMS in (8) with SDF and asymmetric time delays can asymptotically achieve consensus if all τm<τ¯, and the FMS in (8) with SDF and asymmetric time delays cannot achieve consensus if all τm>τ¯.
(30)τ¯=minλi≠0π2−α/2+arctanγωi¯αsinπα/2/1+γωi¯αcosπα/2−argλiω¯i,where ω¯i=γλi¯2cosπα/2+λi¯1+γ2cos2πα/2λi¯2−γ2λi¯2/1−γ2λi¯21/α, γ∈0,γ∗γ∗=minλi≠01/λi, λi¯ is the λi which makes τ¯ minimized, and λi is the ith eigenvalue of L.Proof 2.
Let one apply the above frequency-domain proof method, which has been used in proving Theorem1. Suppose that s=−jω≠0 is the eigenvalue of the FMS in (8) with SDF and asymmetric time delays on the imaginary axis, u∈ℂn is the corresponding eigenvector, and u=1. According to Lemma 2, one can get
(31)Ba≜∑m=1Mamejωτm=−−jωαγ−jωα+1=ωα1+γωαcosπα/22+γωαsinπα/22ejπ2−α/2+arctanγωαsinπα/2/1+γωαcosπα/2.Taking the modulus of both sides of (31) and regarding ω as the function of Ba, we can get
(32)ωBa=γBa2cosπα/2+Ba1+γ2cos2πα/2Ba2−γ2Ba21−γ2Ba21/α,where ωBa is an increasing function of Ba.Calculating the principal value of the argument of (31) on both sides separately, and we have
(33)argBa=π2−α2+arctanγωαsinπα/21+γωαcosπα/2.According to the definition ofBa in (31), we have
(34)argBa≤arg∑m=1Mam+maxωτm,and it yields that
(35)maxωτm≥π2−α2+arctanγωαsinπα/21+γωαcosπα/2−arg∑m=1Mam⏟ε.Since∑m=1Mam=uHLu/uHu, the possible values of ∑m=1Mam must be nonzero eigenvalues of L, that is, ∑m=1Mam=λiλi≠0. So when γ<minλi≠01/λi and Ba≤λi, we have ωBa≤ωλi¯=ω¯i=γλi¯2cosπα/2+λi¯1+γ2cos2πα/2λi¯2−γ2λi¯2/1−γ2λi¯21/α. If we let all τm<τ¯, we can obtain
(36)maxωτm<ω¯iτ¯=minλi≠0π2−α/2+arctanγωi¯αsinπα/2/1+γωi¯αcosπα/2−argλiω¯iω¯i≤ε=π2−α2+arctanγωαsinπα/21+γωαcosπα/2−arg∑m=1Mam.The contradiction between the inequality in (36) and the inequality in (35) is obvious. Accordingly, when all τm<τ¯, the eigenvalues of the FMS in (8) with SDF and asymmetric time delays cannot reach or cross the imaginary axis, then the FMS in (8) with SDF and asymmetric time delays will remain stable and the FMS in (8) with SDF and asymmetric time delays can reach consensus. On the other hand, when all τm>τ¯, there must exist at least one eigenvalue of the FMS in (8) with SDF and asymmetric time delays in the RH, then the states of the FMS in (8) with SDF and asymmetric time delays will no longer converge and the FMS in (8) with SDF and asymmetric time delays cannot reach consensus.Remark 2.
From Theorem2, let one get τ¯>0 and ω¯i>0. γ∈0,γ∗γ∗=minλi≠01/λi is implied in (30). So it is necessary for achieving consensus of the FMS in (8) with SDF and asymmetric time delays that γ∈0,γ∗γ∗=minλi≠01/λi.Corollary 2.
Consider a FMS with SDF and asymmetric time delays over a directed interconnection graphG with a spanning tree. When α=1, by the distributed control protocol in (7), the FMS in (8) with SDF and asymmetric time delays can asymptotically achieve consensus if all τm<τ¯, and the FMS in (8) with SDF and asymmetric time delays cannot achieve consensus if all τm>τ¯.
(37)τ¯=minλi≠0π/2+arctanγωi¯−argλiω¯i,where ω¯i=λi¯1−γ2λi¯2/1−γ2λi¯2, γ∈0,γ∗γ∗=minλi≠01/λi, λi¯ is the λi which makes τ¯ minimized, and λi is the ith eigenvalue of L.


### 4.2.2. Robustness Analysis for Case 2

According to Theorem2, for the given FMS in (4), if applying the control protocol in (6), that is, the control protocol in (7) with γ=0, we obtain
(38)τ¯γ=0=minλi≠0π2−α/2−argλiλi1/α≜τup1.When we apply the control protocol in (7), we obtain
(39)τ¯0<γ<γ∗=minλi≠0π2−α/2+arctanγωi¯αsinπα/2/1+γωi¯αcosπα/2−argλiω¯i≜τup2,where ω¯i=γλi¯2cosπα/2+λi¯1+γ2cos2πα/2λi¯2−γ2λi¯2/1−γ2λi¯21/α, and γ∗=min∣λi∣≠01/λi.In order to show the effect of the protocol in (7) on the robustness against asymmetric time delays, we are supposed to further determine the range of α to insure τup2>τup1, which means
(40)τup≜τup2−τup1>0.Obviously, the inequality in (40) contains multiple parameters. However, for a fix directed interconnection graph G that has a spanning tree, we can determine the range of α and find a proper value of γ to improve the robustness of the FMS with asymmetric time delays. Since it is very difficult for us to solve the inequality in (40) by the analytic method, we shall analyze it by the graphical method and let γ∗=0.6967. Figure 3 is the three-dimensional diagram of τup with respect to parameters α and γ, it is easy for us to find proper parameters α,γ to ensure τup>0 by the graphical method. According to Figure 3, we can find that α∈α∗,1 when γ changes from 0 to γ∗ and τup>0, and α∗ which is decided by the inequality in (40) is easy to obtain in Figure 3 when the value of γ is determined.Figure 3
The three-dimensional diagram ofτup with respect to parameters α and γ in case 2.In particular, if we assumeγ=0.1, then the relationship between τup and α can be shown in Figure 4. According to Figure 4, it is obvious that α∈0.7503,1α∗≜0.7503 when τup>0 and γ=0.1.Figure 4
The relationship betweenτup and α when γ=0.1 in case 2.


## 4.2.1. Main Results of Case 2

Theorem 2.
Consider a FMS with SDF and asymmetric time delays over a directed interconnection graphG that has a spanning tree. By the distributed control protocol in (7), the FMS in (8) with SDF and asymmetric time delays can asymptotically achieve consensus if all τm<τ¯, and the FMS in (8) with SDF and asymmetric time delays cannot achieve consensus if all τm>τ¯.
(30)τ¯=minλi≠0π2−α/2+arctanγωi¯αsinπα/2/1+γωi¯αcosπα/2−argλiω¯i,where ω¯i=γλi¯2cosπα/2+λi¯1+γ2cos2πα/2λi¯2−γ2λi¯2/1−γ2λi¯21/α, γ∈0,γ∗γ∗=minλi≠01/λi, λi¯ is the λi which makes τ¯ minimized, and λi is the ith eigenvalue of L.Proof 2.
Let one apply the above frequency-domain proof method, which has been used in proving Theorem1. Suppose that s=−jω≠0 is the eigenvalue of the FMS in (8) with SDF and asymmetric time delays on the imaginary axis, u∈ℂn is the corresponding eigenvector, and u=1. According to Lemma 2, one can get
(31)Ba≜∑m=1Mamejωτm=−−jωαγ−jωα+1=ωα1+γωαcosπα/22+γωαsinπα/22ejπ2−α/2+arctanγωαsinπα/2/1+γωαcosπα/2.Taking the modulus of both sides of (31) and regarding ω as the function of Ba, we can get
(32)ωBa=γBa2cosπα/2+Ba1+γ2cos2πα/2Ba2−γ2Ba21−γ2Ba21/α,where ωBa is an increasing function of Ba.Calculating the principal value of the argument of (31) on both sides separately, and we have
(33)argBa=π2−α2+arctanγωαsinπα/21+γωαcosπα/2.According to the definition ofBa in (31), we have
(34)argBa≤arg∑m=1Mam+maxωτm,and it yields that
(35)maxωτm≥π2−α2+arctanγωαsinπα/21+γωαcosπα/2−arg∑m=1Mam⏟ε.Since∑m=1Mam=uHLu/uHu, the possible values of ∑m=1Mam must be nonzero eigenvalues of L, that is, ∑m=1Mam=λiλi≠0. So when γ<minλi≠01/λi and Ba≤λi, we have ωBa≤ωλi¯=ω¯i=γλi¯2cosπα/2+λi¯1+γ2cos2πα/2λi¯2−γ2λi¯2/1−γ2λi¯21/α. If we let all τm<τ¯, we can obtain
(36)maxωτm<ω¯iτ¯=minλi≠0π2−α/2+arctanγωi¯αsinπα/2/1+γωi¯αcosπα/2−argλiω¯iω¯i≤ε=π2−α2+arctanγωαsinπα/21+γωαcosπα/2−arg∑m=1Mam.The contradiction between the inequality in (36) and the inequality in (35) is obvious. Accordingly, when all τm<τ¯, the eigenvalues of the FMS in (8) with SDF and asymmetric time delays cannot reach or cross the imaginary axis, then the FMS in (8) with SDF and asymmetric time delays will remain stable and the FMS in (8) with SDF and asymmetric time delays can reach consensus. On the other hand, when all τm>τ¯, there must exist at least one eigenvalue of the FMS in (8) with SDF and asymmetric time delays in the RH, then the states of the FMS in (8) with SDF and asymmetric time delays will no longer converge and the FMS in (8) with SDF and asymmetric time delays cannot reach consensus.Remark 2.
From Theorem2, let one get τ¯>0 and ω¯i>0. γ∈0,γ∗γ∗=minλi≠01/λi is implied in (30). So it is necessary for achieving consensus of the FMS in (8) with SDF and asymmetric time delays that γ∈0,γ∗γ∗=minλi≠01/λi.Corollary 2.
Consider a FMS with SDF and asymmetric time delays over a directed interconnection graphG with a spanning tree. When α=1, by the distributed control protocol in (7), the FMS in (8) with SDF and asymmetric time delays can asymptotically achieve consensus if all τm<τ¯, and the FMS in (8) with SDF and asymmetric time delays cannot achieve consensus if all τm>τ¯.
(37)τ¯=minλi≠0π/2+arctanγωi¯−argλiω¯i,where ω¯i=λi¯1−γ2λi¯2/1−γ2λi¯2, γ∈0,γ∗γ∗=minλi≠01/λi, λi¯ is the λi which makes τ¯ minimized, and λi is the ith eigenvalue of L.


## 4.2.2. Robustness Analysis for Case 2

According to Theorem2, for the given FMS in (4), if applying the control protocol in (6), that is, the control protocol in (7) with γ=0, we obtain
(38)τ¯γ=0=minλi≠0π2−α/2−argλiλi1/α≜τup1.When we apply the control protocol in (7), we obtain
(39)τ¯0<γ<γ∗=minλi≠0π2−α/2+arctanγωi¯αsinπα/2/1+γωi¯αcosπα/2−argλiω¯i≜τup2,where ω¯i=γλi¯2cosπα/2+λi¯1+γ2cos2πα/2λi¯2−γ2λi¯2/1−γ2λi¯21/α, and γ∗=min∣λi∣≠01/λi.In order to show the effect of the protocol in (7) on the robustness against asymmetric time delays, we are supposed to further determine the range of α to insure τup2>τup1, which means
(40)τup≜τup2−τup1>0.Obviously, the inequality in (40) contains multiple parameters. However, for a fix directed interconnection graph G that has a spanning tree, we can determine the range of α and find a proper value of γ to improve the robustness of the FMS with asymmetric time delays. Since it is very difficult for us to solve the inequality in (40) by the analytic method, we shall analyze it by the graphical method and let γ∗=0.6967. Figure 3 is the three-dimensional diagram of τup with respect to parameters α and γ, it is easy for us to find proper parameters α,γ to ensure τup>0 by the graphical method. According to Figure 3, we can find that α∈α∗,1 when γ changes from 0 to γ∗ and τup>0, and α∗ which is decided by the inequality in (40) is easy to obtain in Figure 3 when the value of γ is determined.Figure 3
The three-dimensional diagram ofτup with respect to parameters α and γ in case 2.In particular, if we assumeγ=0.1, then the relationship between τup and α can be shown in Figure 4. According to Figure 4, it is obvious that α∈0.7503,1α∗≜0.7503 when τup>0 and γ=0.1.Figure 4
The relationship betweenτup and α when γ=0.1 in case 2.


## 5. Simulation Results

It is necessary for us to compare the conclusions of this paper with those of [21], so the simulation conditions of this paper must be consistent with those of [21].


### 5.1. Example 1: Simulations for Case 1

Consider a FMS described by (4) with four agents, whose communication topology is given in Figure 5. The L which is associated with this topology graph is
(41)L=2.6−0.7−0.9−1−0.71.50−0.8−0.900.90−1−0.801.8,and its four eigenvalues are 0, 0.8771, 2.4000, and 3.5229, respectively. From the four eigenvalues of L, we can get λn=3.5229, which leads to γ<1/λn=0.2838.Figure 5
The communication topology in example 1.According robustness analysis for case 1,α∈0.854,1α∗≜0.854 when τup=τup2−τup1>0 and γ=0.1. According to Theorem 1 and Remark 1, it is easy to obtain that the maximum tolerable delay of the FMS in (4) by the protocol in (6), that is, the protocol in (7) with γ=0 is τup1=τ¯=0.4264s and the maximum tolerable delay of the FMS in (4) by the protocol in (7) with γ=0.1 is τup2=τ¯=0.4474s when α=0.9. Now let us suppose that the initial states of the FMS (4) are taken as x1t=0=−6.4, x2t=0=−3.2, x3t=0=3.2, and x4t=0=6.4. Here, a set of symmetric time delays is used to simulate τ12=τ21=0.430s, τ13=τ31=0.435s, τ14=τ41=0.440s, and τ24=τ42=0.445s.It is obvious that all symmetric time delaysτm are greater than τup1 and less than τup2.Figures6 and 7 show the trajectories of xit with the symmetric time delays by applying the different control protocols when τup1<allτm<τup2. From these simulation results, it is obvious that the given FMS in (4) by the control protocol in (6) diverges and cannot reach consensus, whereas the FMS in (4) applying the SDF control protocol in (7) converges to the same states and can reach consensus. Hence, the introduced SDF control protocol can enhance the robustness of the FMS in (4) to symmetric time delays.Figure 6
The trajectories ofxit applying the protocol in (6) under symmetric time delays when all τm>τup1.Figure 7
The trajectories ofxit applying the protocol in (7) under symmetric time delays when all τm<τup2.On the other hand, under the same conditions, we suppose thatτ12=τ21=0.45s, τ13=τ31=0.46s, τ14=τ41=0.47s, and τ24=τ42=0.48s. Figure 8 shows the trajectories of xit applying the protocol in (7) under symmetric time delays when all τm>τup2. It is clear that the FMS in (4) cannot reach consensus.Figure 8
The trajectories ofxit applying the protocol in (7) under symmetric time delays when all τm>τup2.


### 5.2. Example 2: Simulations for Case 2

Consider a FMS described by (4) with four agents, whose communication topology with a spanning tree is given in Figure 9. The L which is associated with this topology graph is
(42)L=100−1−0.70.700−0.900.900−0.800.8,and its four eigenvalues are 0, 0.9000, 1.2500 + 0.7053i, and 1.2500 − 0.7053i, respectively. From the four eigenvalues of L, we can get γ∗=minλi≠01/λi=0.6967, which leads to γ<γ∗=0.6967.Figure 9
The connected interaction topology in example 2.According robustness analysis for case 2,α∈0.7503,1α∗≜0.7503 when τup=τup2−τup1>0 and γ=0.1. According to Theorem 2 and Remark 2, it is also easy to obtain the maximum tolerable delay of the FMS in (4) by the protocol in (6), that is, the protocol in (7) with γ=0 is τup1=τ¯=0.8126s and the maximum delay of the FMS in (4) by the protocol in (7) with γ=0.1 is τup2=τ¯=0.8751s when α=0.9. Now let us suppose that the initial states of the FMS in (4) are taken as x1t=0=−6.4, x2t=0=−3.2, x3t=0=3.2, and x4t=0=6.4. Here, a set of asymmetric time delays is used to simulate τ14=0.83s, τ21=0.84s, τ31=0.85s, and τ42=0.86s.It is obvious that all asymmetric time delaysτm are greater than τup1 and less than τup2.Figures10 and 11 show the trajectories of xit with the asymmetric time delays by applying the different control protocols when τup1<allτm<τup2. From these simulation results, it is obvious that the given FMS in (4) by the control protocol in (6) diverges and cannot reach consensus, whereas the FMS in (4) applying the SDF control protocol in (7) converges to the same states and can reach consensus. Hence, the introduced SDF control protocol can enhance the robustness of the FMS in (4) to asymmetric time delays.Figure 10
The trajectories ofxit applying the protocol in (6) under asymmetric time delays when all τm>τup1.Figure 11
The trajectories ofxit applying the protocol in (7) under asymmetric time delays when all τm<τup2.On the other hand, under the same conditions, we suppose thatτ14=0.88s, τ21=0.89s, τ31=0.90s, and τ42=0.91s. Figure 12 shows the trajectories of xit applying the protocol in (7) under asymmetric time delays when all τm>τup2. It is clear that the FMS in (4) cannot reach consensus.Figure 12
The trajectories ofxit applying the protocol in (7) under asymmetric time delays when all τm>τup2.


## 5.1. Example 1: Simulations for Case 1

Consider a FMS described by (4) with four agents, whose communication topology is given in Figure 5. The L which is associated with this topology graph is
(41)L=2.6−0.7−0.9−1−0.71.50−0.8−0.900.90−1−0.801.8,and its four eigenvalues are 0, 0.8771, 2.4000, and 3.5229, respectively. From the four eigenvalues of L, we can get λn=3.5229, which leads to γ<1/λn=0.2838.Figure 5
The communication topology in example 1.According robustness analysis for case 1,α∈0.854,1α∗≜0.854 when τup=τup2−τup1>0 and γ=0.1. According to Theorem 1 and Remark 1, it is easy to obtain that the maximum tolerable delay of the FMS in (4) by the protocol in (6), that is, the protocol in (7) with γ=0 is τup1=τ¯=0.4264s and the maximum tolerable delay of the FMS in (4) by the protocol in (7) with γ=0.1 is τup2=τ¯=0.4474s when α=0.9. Now let us suppose that the initial states of the FMS (4) are taken as x1t=0=−6.4, x2t=0=−3.2, x3t=0=3.2, and x4t=0=6.4. Here, a set of symmetric time delays is used to simulate τ12=τ21=0.430s, τ13=τ31=0.435s, τ14=τ41=0.440s, and τ24=τ42=0.445s.It is obvious that all symmetric time delaysτm are greater than τup1 and less than τup2.Figures6 and 7 show the trajectories of xit with the symmetric time delays by applying the different control protocols when τup1<allτm<τup2. From these simulation results, it is obvious that the given FMS in (4) by the control protocol in (6) diverges and cannot reach consensus, whereas the FMS in (4) applying the SDF control protocol in (7) converges to the same states and can reach consensus. Hence, the introduced SDF control protocol can enhance the robustness of the FMS in (4) to symmetric time delays.Figure 6
The trajectories ofxit applying the protocol in (6) under symmetric time delays when all τm>τup1.Figure 7
The trajectories ofxit applying the protocol in (7) under symmetric time delays when all τm<τup2.On the other hand, under the same conditions, we suppose thatτ12=τ21=0.45s, τ13=τ31=0.46s, τ14=τ41=0.47s, and τ24=τ42=0.48s. Figure 8 shows the trajectories of xit applying the protocol in (7) under symmetric time delays when all τm>τup2. It is clear that the FMS in (4) cannot reach consensus.Figure 8
The trajectories ofxit applying the protocol in (7) under symmetric time delays when all τm>τup2.


## 5.2. Example 2: Simulations for Case 2

Consider a FMS described by (4) with four agents, whose communication topology with a spanning tree is given in Figure 9. The L which is associated with this topology graph is
(42)L=100−1−0.70.700−0.900.900−0.800.8,and its four eigenvalues are 0, 0.9000, 1.2500 + 0.7053i, and 1.2500 − 0.7053i, respectively. From the four eigenvalues of L, we can get γ∗=minλi≠01/λi=0.6967, which leads to γ<γ∗=0.6967.Figure 9
The connected interaction topology in example 2.According robustness analysis for case 2,α∈0.7503,1α∗≜0.7503 when τup=τup2−τup1>0 and γ=0.1. According to Theorem 2 and Remark 2, it is also easy to obtain the maximum tolerable delay of the FMS in (4) by the protocol in (6), that is, the protocol in (7) with γ=0 is τup1=τ¯=0.8126s and the maximum delay of the FMS in (4) by the protocol in (7) with γ=0.1 is τup2=τ¯=0.8751s when α=0.9. Now let us suppose that the initial states of the FMS in (4) are taken as x1t=0=−6.4, x2t=0=−3.2, x3t=0=3.2, and x4t=0=6.4. Here, a set of asymmetric time delays is used to simulate τ14=0.83s, τ21=0.84s, τ31=0.85s, and τ42=0.86s.It is obvious that all asymmetric time delaysτm are greater than τup1 and less than τup2.Figures10 and 11 show the trajectories of xit with the asymmetric time delays by applying the different control protocols when τup1<allτm<τup2. From these simulation results, it is obvious that the given FMS in (4) by the control protocol in (6) diverges and cannot reach consensus, whereas the FMS in (4) applying the SDF control protocol in (7) converges to the same states and can reach consensus. Hence, the introduced SDF control protocol can enhance the robustness of the FMS in (4) to asymmetric time delays.Figure 10
The trajectories ofxit applying the protocol in (6) under asymmetric time delays when all τm>τup1.Figure 11
The trajectories ofxit applying the protocol in (7) under asymmetric time delays when all τm<τup2.On the other hand, under the same conditions, we suppose thatτ14=0.88s, τ21=0.89s, τ31=0.90s, and τ42=0.91s. Figure 12 shows the trajectories of xit applying the protocol in (7) under asymmetric time delays when all τm>τup2. It is clear that the FMS in (4) cannot reach consensus.Figure 12
The trajectories ofxit applying the protocol in (7) under asymmetric time delays when all τm>τup2.


## 6. Conclusion

In order to enhance the robustness of a FMS against nonuniform time delays, a control protocol based on SDF and nonuniform time delays is introduced in this paper. First of all, the consensus problem is investigated for the FMS with SDF and symmetric time delays over undirected topology. Then, the consensus problem is investigated for the FMS with SDF and asymmetric time delays over directed topology. By the robustness analysis, it is obvious that the control protocol-based on SDF with the appropriate intensity can enhance the robustness for the FMS to nonuniform time delays. Finally, the validity of the theoretical analysis is verified by the corresponding simulation results. In addition, inspired by [26, 27], the consensus problems or formation control problems of delayed double-integrator FMSs based on round-robin protocols or attacks will be one of the most interesting topics of our future research work.



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