# A Novel Consensus Protocol for Two-Dimension First-Order Multiagent Systems

Authors: Yibo Zhang; Jinfeng Gao; Jia Ren
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

A novel consensus protocol for two-dimension first-order multiagent systems (MAS) is proposed. After a novel protocol for two-dimension first-order MAS is adopted; consensus and two kinds of possible results are concerned. Then consensus asymptotical conditions and ranges of key parameters are given, and convergence speed is analyzed. Numerical examples illustrate effectiveness of the proposed consensus protocol.


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## Body

## 1. Introduction

Networks have been concerned for many years, and it has been researched in many areas such as in sensor networks [1], distribute networks, or wireless networks. Multiagent systems (MAS) have been concerned for many years. It reflects relationships of individuals in a network, such as critical rules or protocols or control of each individual [2]. One common way of pattern is that all agents converge to the same value, which is called consensus or agreement.Consensus problem is applied in robots, spacecraft, and cooperative multiple vehicles, and it mainly concerns exchange ways of information. For example, reference [3] presented theoretical explanation for characters of continuous and discrete MAS with/without leaders. This classical work showed relations between connection of agents, protocols, and consensus. And [4] researched a new technique based on complex Laplacian to achieve arbitrary formation shape and global stability. Reference [5] mainly researched linear time-varying MAS, a necessary and sufficient condition of consensus ability with a set of admissible protocols. Then some applications are put up; [6] discussed a kind of autonomous scale control of multi-agent in a MAS with shape constrains. It provided a new method for choosing an appropriate geometry and discussed system effect by different parameters. Reference [7] considered MAS with leader of bounded unknown input. After two distributed discontinuous controllers were added, a sufficient condition was applied to ensure stabilization. References [8, 9] researched grouping control and coordinate control of MAS according in different conditions.Suppose there aren agents in MAS and they are connected by wireless networks, which can be described as a (directed) graph. The graph can be expressed in an adjacency matrix form called Laplacian matrix L=lik (1≤i,k≤n) that satisfies the following conditions:(1)lik≤0i≠k∑k=1nlik=0i=1,2,…,n.According to [3], the following premise is introduced.Premise. A first-order MAS should satisfy the condition that exchange topology has a spanning tree or -L has only one zero eigenvalue and all the other nonzero eigenvalues have negative real parts.The following is based on the fact that the premise is satisfied. In this work, a kind of two-dimension first-order MAS is concerned. Then a novel protocol and its consensus are proposed, and range of key parameter is given. Moreover, convergence speed is analyzed. Several simulations are shown to ensure the results.


## 2. Consensus and Convergence Speed

### 2.1. Consensus for Two-Dimension MAS

Suppose there aren agents in a plat coordinate system, which means each agent has two coordinate positions named as x and y, respectively. Dynamic equations of the ith agent (i=1,2,…,n) are shown as following:(2)x˙i=uxiy˙i=uyi.The consensus is that all agents finally converge together, with their positions converging to the same value, or it can be expressed asxi-xk→0 and yi-yk→0, whent→∞, wherei,k=1,2,…,n.Protocol of two-dimension system is introduced as(3)uxi=-∑k=1nlikxi-xk+γyi-ykuyi=-∑k=1nlikxi-xk+yi-yk.Letx=x1x2⋯xnTandy=y1y2⋯ynT. Equation (2) changes to the following form: (4)x˙y˙=uxiuyi=Γxy,whereΓ=-L-γL-L-L.Supposeμii=1,2,…,nis theith eigenvalue of -L; then(5)detλIn+L=detλIn--L=∏i=1nλ-μi.Because-L has exactly one zero eigenvalue, and all the other nonzero eigenvalues have negative real parts; without loss of generality, let μ1=0 and the other nonzero eigenvalues μi=-αi±jβi, whereαi>0 (i=2,3,…,n). If μi is real, βi=0, and βi>0 if there are conjugate complex eigenvalues.Letμ-i (i=1,2,…,2n) be theith eigenvalue of Γ; then detλI2n-Γ=∏i=12nλ-μ-i.Note that(6)detλI2n-Γ=detλIn+LγLLλIn+L=detλ2In+2λL+1-γL2.Letλ1 and λ2 be roots of the character equation λ2+2λ+1-γ=0; then there are two kinds of possibilities according to different γ.(1)
λ
1
and
λ
2
Are Real. If γ≥0, λ1=1+γ, and λ2=1-γ are real, then (6) can be decomposed to the following form:(7)detλI2n-Γ=detλ2In+2λL+1-γL2=detλIn+1+γL·detλIn+1-γL.Compared with (5), (7) turns to(8)detλI2n-Γ=detλIn+1+γL·detλIn+1-γL=∏i=1nλ-1+γμi∏i=1nλ-1-γμi.Then the following lemma is proposed.Lemma 1.
When0≤γ<1, consensus protocol (3) achieves consensus asymptotically, if and only if -L has exactly one zero eigenvalue and the other n-1 eigenvalues have negative real parts, which is equal to the condition if and only if Γ has exactly two zero eigenvalues and the other 2n-2 eigenvalues have negative real parts.Proof.
Sufficiency. If 0≤γ<1, 1-γ>0, so it is obvious that μ-1=1-γμ1=0 andReμ-i=-1-γαi<0 (i=2,3,…,n). This means that μ-i (i=1,2,…,n) have the same sign with μi (i=1,2,…,n). The same conclusion can be drawn whenμ-i=1+γμi (i=1,2,…,n) if0≤γ<1. Considering (8), the following conclusion is drawn: Γ has exactly two zero eigenvalues and the others have negative real parts, which is equal to -L having exactly one zero eigenvalue and the others have negative real parts.
Letp=p1Tp2TT be an eigenvector of Γ associated with one of zero eigenvalues, where p1,p2∈Rn; then(9)Γp=-L-γL-L-Lp1p2=0,which implies that -Lp1=0 and -Lp2=0 if0≤γ<1. Note that -L has only one linearly independent eigenvector associated with zero eigenvalue, so p1=p2. Moreover, Γ has only one linearly independent eigenvector p=p1Tp1TT; that is, zero eigenvalue ofΓhas geometric multiplicity equal to one.
All of the above show that if-L is chosen as asymptotically,Γachieves consensus asymptotically also.
Necessity (reduction to absurd). SupposeΓhaving exactly two zero eigenvalues and the other eigenvalues having negative real parts do not hold. From sufficient process, relationships between eigenvalues of Γ and -L are μ-i=1+γμi or μ-i=1-γμi, which means zero eigenvalues of Γ are double times those of -L. Suppose there are more than two zero eigenvalues of Γ, for example, four zero eigenvalues, so there are at least two zero eigenvalues of -L.According to [3], consensus cannot be achieved, which is contradicting with the premise, so there must be only two zero eigenvalues of Γ.Then suppose there is at least one nonzero eigenvalue having positive real part. Note thatΓ can be transformed to Jordan canonical form by a nonsingular matrix Q:(10)Γ=Q-1JQ.Then(11)eΓt=Q-1eJtQ=Q-11001×2n-20101×2n-202n-2×102n-2×1eJ′tQ.It can be seen that if there is at least one eigenvalue with positive real part ofΓ, when t→∞ and eJ′t→∞, then eΓt→∞, which is contradicting with the following condition: “consensus protocol (2) achieves consensus asymptotically.”So the assumption is wrong.(2)
λ
1
and
λ
2
Are Conjugate Complex Eigenvalues. If γ<0 and λ1,2=1±j-γ are one pair of conjugate complex eigenvalues, then (6) can be decomposed to the following form:(12)detλI2n-Γ=detλIn+1-j-γL·detλIn+1+j-γL=∏i=1nλ-1-j-γμi·∏i=1nλ-1+j-γμi.Then the following lemma is proposed.Lemma 2.
When-minμi≠0⁡αi2/βi2<γ<0, consensus protocol (3) achieves consensus asymptotically, if and only if -L has exactly one zero eigenvalue and the other n-1 eigenvalues have negative real parts, which is equal to the condition if and only if Γ has exactly two zero eigenvalues and the other 2n-2 eigenvalues have negative real parts.The proof is omitted for similarity with Lemma1. The differenence is range of γ, which is described as follows.Becauseμ-i=1+j-γμi or μ-i=1-j-γμi, if μ1≠0, two cases are concerned.Case 1 (all nonzero eigenvaluesμi (i=2,3,…,n) are real).
According to the premise,μi<0, then both μ-i=1+j-γμi and μ-i=1-j-γμi have negative real parts.Case 2 (there are at least one pair of conjugate complex eigenvalues).
Suppose there is only one pair of conjugate complex eigenvalues marked asμi+ and μi-. Let μi±=-α±jβ, where α>0 and β>0. Because μ-i is multiplied by two vectors 1±-γ and μi, its module is product while phase angle is the sum of them. All possible results are shown in Figure 1.Figure 1
All possible results ofμ-i=(1±j-γ)μi.In Figure1, θi denotes intersection angle between vector μi± and Y-axis and θΓ is abstract phase angle of 1±j-γ. It can be seen that whether μ-i=1±j-γμi± have negative real parts is decided by angle of μ-i. Note that phase angle of μi+ is in 90°,180° while that of μi- is in 180°,270°, and if -γ>0, then 0<θΓ=arctan-γ<90°, so phase angles of 1+j-γμi+ and 1-j-γμi- are in range of 90°,270°. In order to ensure all eigenvalues of Γ have negative real parts, phase angles of 1-j-γμi+ and 1+j-γμi- must be in range of 90°,180° and 180°,270°, respectively, which means θΓ<θi.If there are more than one pair of conjugate complex eigenvalues, it is obvious thatθΓ<θi should satisfied for all pairs of conjugate complexes, so θΓ<minβi≠0⁡θi. Note that θΓ=arctan-γ and θi=arctanαi/βi; then,(13)arctan-γ<minβi≠0arctanαiβi⟹-γ<minβi≠0αiβi⟹γ>-minβi≠0αi2βi2.So -minβi≠0⁡αi2/βi2<γ<0. And for the other real eigenvalues, αi2/βi2=∞, which means that minμi≠0⁡αi2/βi2=minβi≠0⁡αi2/βi2. Moreover, if all nonzero eigenvalues are real, -minμi≠0⁡αi2/βi2<γ<0 turns to -∞<γ<0, which is equal to γ<0; it is obvious that conditions of Case 1 and Case 2 can be rewritten in the same form as -minμi≠0⁡αi2/βi2<γ<0.


### 2.2. Convergence Speed

Convergence speed of system (1) is decided by eigenvalue with the smallest absolute value of real part. Reconsider (12); if all nonzero eigenvalues have negative real parts, then the slowest convergence speed of system (2) is decided by the eigenvalue whose real part is the nearest to Y-axis. Three cases are analyzed according to choice of γ.Case 1 (0≤γ<1).
From (7), it is obviously that the smallest eigenvalue is one of 1-λμi, so smaller 1-γ (or bigger γ) will cause slower convergence speed.Case 2 (-minμi≠0⁡αi2/βi2<γ<0 and all nonzero μi are real).
From (7), it is obvious that Reμ-i=Re1±j-γμi=μi, which means γ has no effect on convergence speed. The bigger imaginary part -γ of μ-i means bigger γ will cause larger amplitude of oscillation.Case 3 (-minμi≠0⁡αi2/βi2<γ<0 and there are conjugate complex eigenvalues of μi).
From Figure1, Reμ-i is given by Reμ-i=Re1-j-γμi+=Re1-j-γ-α+jβ=β-γ-α, so bigger β-γ-α (or smaller γ) will cause slower convergence.


## 2.1. Consensus for Two-Dimension MAS

Suppose there aren agents in a plat coordinate system, which means each agent has two coordinate positions named as x and y, respectively. Dynamic equations of the ith agent (i=1,2,…,n) are shown as following:(2)x˙i=uxiy˙i=uyi.The consensus is that all agents finally converge together, with their positions converging to the same value, or it can be expressed asxi-xk→0 and yi-yk→0, whent→∞, wherei,k=1,2,…,n.Protocol of two-dimension system is introduced as(3)uxi=-∑k=1nlikxi-xk+γyi-ykuyi=-∑k=1nlikxi-xk+yi-yk.Letx=x1x2⋯xnTandy=y1y2⋯ynT. Equation (2) changes to the following form: (4)x˙y˙=uxiuyi=Γxy,whereΓ=-L-γL-L-L.Supposeμii=1,2,…,nis theith eigenvalue of -L; then(5)detλIn+L=detλIn--L=∏i=1nλ-μi.Because-L has exactly one zero eigenvalue, and all the other nonzero eigenvalues have negative real parts; without loss of generality, let μ1=0 and the other nonzero eigenvalues μi=-αi±jβi, whereαi>0 (i=2,3,…,n). If μi is real, βi=0, and βi>0 if there are conjugate complex eigenvalues.Letμ-i (i=1,2,…,2n) be theith eigenvalue of Γ; then detλI2n-Γ=∏i=12nλ-μ-i.Note that(6)detλI2n-Γ=detλIn+LγLLλIn+L=detλ2In+2λL+1-γL2.Letλ1 and λ2 be roots of the character equation λ2+2λ+1-γ=0; then there are two kinds of possibilities according to different γ.(1)
λ
1
and
λ
2
Are Real. If γ≥0, λ1=1+γ, and λ2=1-γ are real, then (6) can be decomposed to the following form:(7)detλI2n-Γ=detλ2In+2λL+1-γL2=detλIn+1+γL·detλIn+1-γL.Compared with (5), (7) turns to(8)detλI2n-Γ=detλIn+1+γL·detλIn+1-γL=∏i=1nλ-1+γμi∏i=1nλ-1-γμi.Then the following lemma is proposed.Lemma 1.
When0≤γ<1, consensus protocol (3) achieves consensus asymptotically, if and only if -L has exactly one zero eigenvalue and the other n-1 eigenvalues have negative real parts, which is equal to the condition if and only if Γ has exactly two zero eigenvalues and the other 2n-2 eigenvalues have negative real parts.Proof.
Sufficiency. If 0≤γ<1, 1-γ>0, so it is obvious that μ-1=1-γμ1=0 andReμ-i=-1-γαi<0 (i=2,3,…,n). This means that μ-i (i=1,2,…,n) have the same sign with μi (i=1,2,…,n). The same conclusion can be drawn whenμ-i=1+γμi (i=1,2,…,n) if0≤γ<1. Considering (8), the following conclusion is drawn: Γ has exactly two zero eigenvalues and the others have negative real parts, which is equal to -L having exactly one zero eigenvalue and the others have negative real parts.
Letp=p1Tp2TT be an eigenvector of Γ associated with one of zero eigenvalues, where p1,p2∈Rn; then(9)Γp=-L-γL-L-Lp1p2=0,which implies that -Lp1=0 and -Lp2=0 if0≤γ<1. Note that -L has only one linearly independent eigenvector associated with zero eigenvalue, so p1=p2. Moreover, Γ has only one linearly independent eigenvector p=p1Tp1TT; that is, zero eigenvalue ofΓhas geometric multiplicity equal to one.
All of the above show that if-L is chosen as asymptotically,Γachieves consensus asymptotically also.
Necessity (reduction to absurd). SupposeΓhaving exactly two zero eigenvalues and the other eigenvalues having negative real parts do not hold. From sufficient process, relationships between eigenvalues of Γ and -L are μ-i=1+γμi or μ-i=1-γμi, which means zero eigenvalues of Γ are double times those of -L. Suppose there are more than two zero eigenvalues of Γ, for example, four zero eigenvalues, so there are at least two zero eigenvalues of -L.According to [3], consensus cannot be achieved, which is contradicting with the premise, so there must be only two zero eigenvalues of Γ.Then suppose there is at least one nonzero eigenvalue having positive real part. Note thatΓ can be transformed to Jordan canonical form by a nonsingular matrix Q:(10)Γ=Q-1JQ.Then(11)eΓt=Q-1eJtQ=Q-11001×2n-20101×2n-202n-2×102n-2×1eJ′tQ.It can be seen that if there is at least one eigenvalue with positive real part ofΓ, when t→∞ and eJ′t→∞, then eΓt→∞, which is contradicting with the following condition: “consensus protocol (2) achieves consensus asymptotically.”So the assumption is wrong.(2)
λ
1
and
λ
2
Are Conjugate Complex Eigenvalues. If γ<0 and λ1,2=1±j-γ are one pair of conjugate complex eigenvalues, then (6) can be decomposed to the following form:(12)detλI2n-Γ=detλIn+1-j-γL·detλIn+1+j-γL=∏i=1nλ-1-j-γμi·∏i=1nλ-1+j-γμi.Then the following lemma is proposed.Lemma 2.
When-minμi≠0⁡αi2/βi2<γ<0, consensus protocol (3) achieves consensus asymptotically, if and only if -L has exactly one zero eigenvalue and the other n-1 eigenvalues have negative real parts, which is equal to the condition if and only if Γ has exactly two zero eigenvalues and the other 2n-2 eigenvalues have negative real parts.The proof is omitted for similarity with Lemma1. The differenence is range of γ, which is described as follows.Becauseμ-i=1+j-γμi or μ-i=1-j-γμi, if μ1≠0, two cases are concerned.Case 1 (all nonzero eigenvaluesμi (i=2,3,…,n) are real).
According to the premise,μi<0, then both μ-i=1+j-γμi and μ-i=1-j-γμi have negative real parts.Case 2 (there are at least one pair of conjugate complex eigenvalues).
Suppose there is only one pair of conjugate complex eigenvalues marked asμi+ and μi-. Let μi±=-α±jβ, where α>0 and β>0. Because μ-i is multiplied by two vectors 1±-γ and μi, its module is product while phase angle is the sum of them. All possible results are shown in Figure 1.Figure 1
All possible results ofμ-i=(1±j-γ)μi.In Figure1, θi denotes intersection angle between vector μi± and Y-axis and θΓ is abstract phase angle of 1±j-γ. It can be seen that whether μ-i=1±j-γμi± have negative real parts is decided by angle of μ-i. Note that phase angle of μi+ is in 90°,180° while that of μi- is in 180°,270°, and if -γ>0, then 0<θΓ=arctan-γ<90°, so phase angles of 1+j-γμi+ and 1-j-γμi- are in range of 90°,270°. In order to ensure all eigenvalues of Γ have negative real parts, phase angles of 1-j-γμi+ and 1+j-γμi- must be in range of 90°,180° and 180°,270°, respectively, which means θΓ<θi.If there are more than one pair of conjugate complex eigenvalues, it is obvious thatθΓ<θi should satisfied for all pairs of conjugate complexes, so θΓ<minβi≠0⁡θi. Note that θΓ=arctan-γ and θi=arctanαi/βi; then,(13)arctan-γ<minβi≠0arctanαiβi⟹-γ<minβi≠0αiβi⟹γ>-minβi≠0αi2βi2.So -minβi≠0⁡αi2/βi2<γ<0. And for the other real eigenvalues, αi2/βi2=∞, which means that minμi≠0⁡αi2/βi2=minβi≠0⁡αi2/βi2. Moreover, if all nonzero eigenvalues are real, -minμi≠0⁡αi2/βi2<γ<0 turns to -∞<γ<0, which is equal to γ<0; it is obvious that conditions of Case 1 and Case 2 can be rewritten in the same form as -minμi≠0⁡αi2/βi2<γ<0.


## 2.2. Convergence Speed

Convergence speed of system (1) is decided by eigenvalue with the smallest absolute value of real part. Reconsider (12); if all nonzero eigenvalues have negative real parts, then the slowest convergence speed of system (2) is decided by the eigenvalue whose real part is the nearest to Y-axis. Three cases are analyzed according to choice of γ.Case 1 (0≤γ<1).
From (7), it is obviously that the smallest eigenvalue is one of 1-λμi, so smaller 1-γ (or bigger γ) will cause slower convergence speed.Case 2 (-minμi≠0⁡αi2/βi2<γ<0 and all nonzero μi are real).
From (7), it is obvious that Reμ-i=Re1±j-γμi=μi, which means γ has no effect on convergence speed. The bigger imaginary part -γ of μ-i means bigger γ will cause larger amplitude of oscillation.Case 3 (-minμi≠0⁡αi2/βi2<γ<0 and there are conjugate complex eigenvalues of μi).
From Figure1, Reμ-i is given by Reμ-i=Re1-j-γμi+=Re1-j-γ-α+jβ=β-γ-α, so bigger β-γ-α (or smaller γ) will cause slower convergence.


## 3. Simulation Results

In this section, examples are illustrated to prove conclusions drawn in Section2.


### 3.1. Case1

A digraph in [4] is shown in Figure 2.Figure 2
A digraph in [4].It is obvious that-L=-11000-1100000001-1, whose eigenvalues are λ1=0 and λ2,3,4=-1. Choose initial coordinate of the four agents as x0=0.6-0.4-0.20.8T and y0=-0.80.7-0.9-1T, respectively. All simulation results are shown in Figure 3 according to different γ>0.Figure 3
Simulation results with differentγ>0.
(a)
Simulation results (γ=0)
(b)
Simulation results (γ=0.5)
(c)
Simulation results (γ=1)
(d)
Simulation results (γ=1.2)From Figure3, asymptotical consensus is achieved when 0≤γ<1, while it is not achieved if γ≥1. And agents stay in fixed positions when γ=1 while they reflect in separating trend.Moreover, asymptotical consensus is achieved in less than 10 seconds whenγ=0 and in about 20 seconds when γ=0.5, which agrees with Case 1 in Section 2.2.


### 3.2. Case2

Simulation results with the same system whenγ<0 are shown in Figure 4.Figure 4
Simulation results with differentγ<0 when all eigenvalues of -L is real.
(a)
Simulation results (γ=-0.5)
(b)
Simulation results (γ=-3)If all eigenvalues of-L are real, any γ<0 can satisfy Lemma 2, so consensuses are achieved when γ=-0.5 or γ=-3. There are no obvious differences between their convergence speeds. More oscillation appears when γ=-3, which agrees with Case 2 in Section 2.2.


### 3.3. Case3

Next,-L is chosen as -21010-11010-100000, and other conditions remain the same, so eigenvalues are λ1=0, λ2=-0.2451, and one pair of conjugate complex eigenvalues λ3,4=-1.8774±j0.7449. Simulation results are shown in Figure 5.Figure 5
Simulation results with differentγ<0 when there are conjugate complex eigenvalues.
(a)
Simulation results (γ=-1)
(b)
Simulation results (γ=-3)
(c)
Simulation results (γ=-7)According to Lemma2, -minμi≠0⁡αi2/βi2=-6.3521. When -minμi≠0⁡αi2/βi2<γ=-1<0 or -minμi≠0⁡αi2/βi2<γ=-3<0, consensus is achieved, while opposite result is obtained when γ=-7<-minμi≠0⁡αi2/βi2. And the convergence speed is a little faster when γ=-1, which agrees with Case 3 in Section 2.2.


## 3.1. Case1

A digraph in [4] is shown in Figure 2.Figure 2
A digraph in [4].It is obvious that-L=-11000-1100000001-1, whose eigenvalues are λ1=0 and λ2,3,4=-1. Choose initial coordinate of the four agents as x0=0.6-0.4-0.20.8T and y0=-0.80.7-0.9-1T, respectively. All simulation results are shown in Figure 3 according to different γ>0.Figure 3
Simulation results with differentγ>0.
(a)
Simulation results (γ=0)
(b)
Simulation results (γ=0.5)
(c)
Simulation results (γ=1)
(d)
Simulation results (γ=1.2)From Figure3, asymptotical consensus is achieved when 0≤γ<1, while it is not achieved if γ≥1. And agents stay in fixed positions when γ=1 while they reflect in separating trend.Moreover, asymptotical consensus is achieved in less than 10 seconds whenγ=0 and in about 20 seconds when γ=0.5, which agrees with Case 1 in Section 2.2.


## 3.2. Case2

Simulation results with the same system whenγ<0 are shown in Figure 4.Figure 4
Simulation results with differentγ<0 when all eigenvalues of -L is real.
(a)
Simulation results (γ=-0.5)
(b)
Simulation results (γ=-3)If all eigenvalues of-L are real, any γ<0 can satisfy Lemma 2, so consensuses are achieved when γ=-0.5 or γ=-3. There are no obvious differences between their convergence speeds. More oscillation appears when γ=-3, which agrees with Case 2 in Section 2.2.


## 3.3. Case3

Next,-L is chosen as -21010-11010-100000, and other conditions remain the same, so eigenvalues are λ1=0, λ2=-0.2451, and one pair of conjugate complex eigenvalues λ3,4=-1.8774±j0.7449. Simulation results are shown in Figure 5.Figure 5
Simulation results with differentγ<0 when there are conjugate complex eigenvalues.
(a)
Simulation results (γ=-1)
(b)
Simulation results (γ=-3)
(c)
Simulation results (γ=-7)According to Lemma2, -minμi≠0⁡αi2/βi2=-6.3521. When -minμi≠0⁡αi2/βi2<γ=-1<0 or -minμi≠0⁡αi2/βi2<γ=-3<0, consensus is achieved, while opposite result is obtained when γ=-7<-minμi≠0⁡αi2/βi2. And the convergence speed is a little faster when γ=-1, which agrees with Case 3 in Section 2.2.


## 4. Conclusion

A novel consensus protocol is presented for the linear two-dimension first-order multiagent systems (MASs). Numerical examples illustrated the effectiveness of the proposed consensus protocol according to different key parameters. Simulation results ensure the relationship between range of key parameter and convergence speed.



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