# Consensus Control of Second-Order Multiagent Systems with Particle Swarm Optimization Algorithm

Authors: Xiongfeng Deng; Xiuxia Sun; Ri Liu; Shuguang Liu
Journal: Journal of Control Science and Engineering
 (2018)

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## Abstract

This paper considers the consensus problem of second-order multiagent systems. Firstly, an improved consensus control protocol is proposed. Then, the convergence of the proposed control protocol is analyzed by applying Lyapunov stability theory. In order to improve the control effect of a given system, the particle swarm optimization (PSO) algorithm is introduced and an improved PSO algorithm is proposed. Additionally, a mapping relationship with agents and the individuals of PSO algorithm is designed. Finally, two simulation examples are provided to illustrate the effectiveness of proposed control protocol and the control effect of PSO algorithm.


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## Body

## 1. Introduction

The cooperative control problem of multiagent systems has been attracting outstanding attention in the past few years. The main reason is its widespread application in various fields, such as robot systems [1], sensor networks [2], and unmanned aerial vehicle (UAV) systems [3]. The basic problem of the cooperative control of multiagent systems is consensus, which is to design a suitable control law such that the output of all agents can achieve synchronization.In existing literature, the consensus problem of multiagent systems has been widely researched. In [4], an observer-based distributed output feedback was considered to solve the consensus tracking problem of multiagent systems. In [5, 6], the iterative learning control was applied to deal with the consensus problem of multiagent systems, while the same problem was studied by using linear quadratic regulator (LQR) in [7, 8]. Moreover, the cooperative tracking problem of nonlinear multiagent systems was analyzed in [9, 10], and the consensus control problem of a class of third-order nonlinear multiagent was studied in [11]. Furthermore, distributed control method [12], impulsive control method [13], and adaptive fuzzy output feedback control approach [14] were also applied to solve the consensus problem of multiagent systems.To improve the control effect, some optimization algorithms, such as genetic algorithm [15], differential evolution algorithm [16], and PSO algorithm [17], have been considered by researchers. As an optimization method, PSO algorithm has the advantages of good search performance, simple implementation, and few adjustment parameters [18]. Due to these advantages, PSO algorithm has gained increasing interest since it was first presented. Currently, there are many results on the applications of the PSO algorithm. For example, to analyze how different algorithmic parameters in a distributed implementation affect the total evaluation and resulting fitness, a distributed PSO algorithm for a class of multirobot systems with the obstacle surroundings was researched in [19]. The nondeterministic navigation problem of UAVs was solved by using the PSO algorithm in [20], where a neighborhood control scheme was designed to eliminate the inherent weakness of PSO algorithm. In [21], the PSO algorithm was applied to deal with the three-dimensional path planning problem of UAVs.Different from the above research results, a clustering routing algorithm with nonlinear dynamic adaptive PSO algorithm for wireless sensor networks was proposed in [22], where the whole sensor field was positioned via the improved nonlinear dynamic adaptive inertia weight and evaluation of fitness function. In [23], the PSO algorithm was applied to deal with the space trajectory optimization problem. Additionally, the result that the PSO algorithm was an efficient, reliable, and accurate method for determining optimal space trajectory was demonstrated. Furthermore, the relevant parameters of PSO algorithm are still prespecified with fixed value in some papers [19–21].Motivated by these facts, we divert our attention to the consensus problem of a class of second-order multiagent systems in this work. The main contributions are outlined in two aspects. On the one hand, an improved control protocol for the given multiagent systems is proposed, and then the convergence is analyzed by the designed Lyapunov function. On the other hand, the PSO algorithm is introduced to improve control effect, where two adaptive laws are designed to adjust the updating laws of PSO algorithm. Also, a new mapping method is applied to solve the mapping problem of the agents’ states and the individuals of PSO algorithm. Finally, some simulation examples are provided to illustrate the validity of our results.The remainder of this paper is organized as follows. In Section2, some preliminaries are briefly given. In Section 3, the consensus control problem of multiagent systems is analyzed. The fundamental theory of PSO algorithm and the mapping relationship with agents are introduced in Section 4, and simulation examples are provided in Section 5. Finally, some conclusions are drawn in Section 6.


## 2. Preliminaries

LetG={V,E,A} denote an undirected graph which consists of n nodes, where V={v1,⋯,vN} is the set of vertices and E={(vi,vj):vi,vj∈V}⊆V×V is the set of edges. The weighted adjacency matrix is denoted by A=[aij]∈Rn×n, where aij>0 if and only if i≠j and aij=0 otherwise. A path between vi and vj is a sequence of distinct vertices vi,vi1,⋯,vl, where (vi,vi1)∈E. An undirected graph is connected if there exists a path from any node i to any other node j. Multiagent systems consist of n agents. Each agent can be described as a node and the exchange information among agents can be expressed as an edge with definite weight in a generalized graph. Therefore, the problem of multiagent systems can be solved through the graph theory.For the sake of disscusion below, some difinitions and lemmas are given as follows.Definition 1.
The consensus of a multiagent systems is said to be achieved if, for any initial condition, there existlimt→∞xi(t)=xj(t) and limt→∞vi(t)=vj(t) for i,j∈1,⋯,n and i≠j.Lemma 2 (see [24]).
Letσ=[σ1,⋯,σm]T∈Rm, ρ=[ρ1,⋯,ρm]T∈Rm, and C=[cij]∈Rm×m; if C is a symmetric matrix, that is, cij=cji, then we have(1)12∑i=1m∑j=1mcijσi-σjφρi-ρj=∑i=1m∑j=1mcijσiφρi-ρjwhere φ(·) represents an odd function.


## 3. Consensus Analysis of Multiagent Systems

Considering a general second-order multiagent systems with undirected graph, the dynamics of theith agent are described as(2)x˙it=vitv˙it=uitwhere xi(t)∈R, vi(t)∈R and ui(t)∈R are the position, velocity, and control input of agent i, respectively; i∈{1,⋯,n} and n is the number of agents.In this paper, a suitable control protocolui(t) needs to be chosen for the multiagent systems (2) to achieve consensus. Inspired by the results of [25, 26], an improved control protocol is given as(3)uit=-α∑j=1naijxit-xjt3-β∑j=1naijvit-vjt3and the general form of (3) is described as(4)uit=-∑j=1naijαφxit-xjt+βφvit-vjtwhere α and β are positive constants; aij is the (i,j) element of A and φ(·) is an odd function.For the convenience of analysis, the time variablet will be ignored from here on.According to (4), (2) is rewritten as(5)x˙i=viv˙i=-∑j=1naijαφxi-xj+βφvi-vjHence, we have the following Theorem.Theorem 3.
Consider the multiagent systems (2) with the consensus protocol (4), and let the exchange information topology graph of agents be undirected and connected; then the consensus can be achieved, i.e., xi(t)=xj(t) and vi(t)=vj(t) as t→∞ for i,j∈1,⋯,n and i≠j.Proof.
Letxij=xi-xj, and then have(6)x˙ij=x˙i-x˙j=vi-vjv˙i=-∑j=1naijαφxij+βφvi-vj
Design the Lyapunov function candidate(7)V=V1+V2with(8)V1=14α∑i=1n∑j=1naijφ2xij(9)V2=12∑i=1nvi2
From (8) and (9), it is obvious that V1≥0 and V2≥0; then V≥0 can be obtained. Taking the derivative of function Vi(i=1,2), we get(10)V˙1=12α∑i=1n∑j=1naijvi-vjφxij(11)V˙2=-α∑i=1n∑j=1naijviφxij-β∑i=1n∑j=1naijviφvi-vj
Considering Lemma2, we have(12)12α∑i=1n∑j=1naijvi-vjφxij=α∑i=1n∑j=1naijviφxij(13)β∑i=1n∑j=1naijviφvi-vj=12β∑i=1n∑j=1naijvi-vjφvi-vj
From (12) and (13), one gets(14)V˙=V˙1+V˙2=-12∑i=1n∑j=1naijβvi-vjφvi-vj
Due to the assumption thatG is undirected and connected and φ(·) is an odd function, it is obtained that xφ(x)>0. Hence, the following result is held:(15)V˙=-12β∑i=1n∑j=1naijvi-vj2φ2xij-12∑i=1n∑j=1naijβvi-vjφvi-vj≤0
LetΓ=xij,vi∣V˙=0. From (14), V˙=0 hints that vi=vj(i≠j). Furthermore, v˙i=v˙j(i≠j) can be deduced. Therefore, there exists ξ∈R such that v∈span(1×ξ), where v=[v1,⋯,vn]T, 1=[1,⋯,1]T, and span(·) represents the vector space of v.
Combiningvi=vj(i≠j) and (7), then we have(16)v˙i=-∑j=1naijαφxij
In addition,(17)1×ξTv˙=1×ξT-∑j=1naijαφxij=-ξ∑i=1n∑j=1naijαφxij
In view of the exchange information topology graphG is undirected, and there exists φ(xij)=-φ(xji); we have(18)ξ∑i=1n∑j=1naijαφxji+ξ∑i=1n∑j=1naijαφxij=2ξ∑i=1n∑j=1naijαφxij=0
From (16) and (18), 1×ξTv˙=0 can be obtained, indicating that v˙ is orthogonal to 1×ξ. Hence, v˙=0 is deduced, and then we have(19)-∑j=1naijαφxij=0and(20)-∑i=1nxi∑j=1naijαφxij=0which hints that(21)-12∑i=1n∑j=1naijαxi-xjφxij=0
From (21), we have aijα(xi-xj)φ(xij)=0, ∀i≠j; then xi=xj, ∀i≠j is obtained. Thereby, the results that xi=xj and vi=vj for i,j∈1,⋯,n and i≠j are obtained. The proof is completed.In order to improve the control effect, the PSO algorithm will be introduced in the following disscusion.


## 4. PSO Algorithm and the Mapping Relationship with Agents

### 4.1. PSO Algorithm

As an optimization method, PSO algorithm seeks the optimum solution of an optimization problem via selecting some particles. The characteristics of each particle include position, velocity, and fitness value. The fitness value is calculated by the fitness function, where the merit of the particle is described by the fitness valule. In the solution space, each paritcle’s position and velocity are updated by the individual extremumPbest and the population extremum Gbest, where Pbest and Gbest are updated by comparing the fitness value of the new particle.Considering aD dimension search space, let X=(X1,⋯,Xm) be a population with m particles. The position and velocity of the jth particle are defined as Xj=(xj1,⋯,xjD)T and Vj=(vj1,⋯,vjD)T, respectively. The extremum of individual and population are defined as Pbest,j=(Pbest,j1,⋯,Pbest,jD)T and Gbest=(Gbest,1,⋯,Gbest,D)T, respectively. The velocity and position updating laws of the jth particle at the (k+1)th iteration are given as(22)Vjdk+1=ωVjdk+κ1γ1Pbest,jdk-Xjdk+κ2γ2Gbest,dk-Xjdk(23)Xjdk+1=Xjdk+Vjdk+1where Xjdk and Vjdk are the dth position and velocity component of the jth individual, respectively; ω∈(0,1] is inertia weight; κi(i=1,2)≥0 are acceleration factors; γi(i=1,2)∈[0,1] are random numbers between 0 and 1; Pbest,jd is the dth best position component of the jth individual and Gbest,d is the dth best position of the entire population.In this paper, the updating laws (22) and (23) are improved and the results are given as(24)Vjdk+1=ωVjdk+κ1γ1′Pbest,jdk-Xjdk+κ2γ2′Gbest,dk-Xjdk(25)Xjdk+1=Xjdk-ΔVjdk+1sgn⁡ΔVjdk+1where(26)γ1′=Pbest,jkPbest,jk2+Gbestk2(27)γ2′=GbestkPbest,jk2+Gbestk2(28)ΔVjdk+1=Vjdk-Vjdk+1Remark 4.
From (26) and (27), note that the attributes of γ1′ and γ2′ are the same as those of γ1 and γ2. However, compared with γ1 and γ2, γ1′ and γ2′ not only overcome the uncertainty of random variation but also have the capacity of adaptive adjustment. In addition, due to ΔV being introduced in the updating law (25), the smoothness of the convergence result can be guaranteed in the end of the algorithm.
As an important part of PSO algorithm, the fitness function in this paper is designed as(29)fjk=∑d=1Dxjdk-xjdk-1+vjdk-vjdk-1where D=n is the dimension of each particle; fjk is the fitness value of the jth particle in the kth iteration; xjd0 and vjd0 are the initial states of agents and k∈(1,⋯,K) represents the kth iteration.
Lettmax and Δt be the simulation time and simulation step, respectively. Hence, the maximum number of iterations K is defined as(30)K=tmaxΔt


### 4.2. Mapping Relationship Design

For a multiagent systems withn agents, each agent is considered as a particle. Let xi and vi be the position and velocity of the ith agent, respectively. Then the sets of position and velocity are expressed as XMAS=(x1,⋯,xn) and VMAS=(v1,⋯,vn), respectively. In the PSO algorithm, let XPSO,j=(xj1,⋯,xjD) and VPSO,j=(vj1,⋯,vjD) be the position and velocity of the jth particle, respectively. Consequently, the mapping relationship between particles and agents is shown in Figure 1.Figure 1
Mapping scheme.The operation process of PSO algorithm is given as follows:Step 1.
Initialize the population and the position and velocity of all agents.Step 2.
Assign the position and velocityxi and vi to each particle.Step 3.
Run system model and calculate the fitness function (29).Step 4.
Judge the termination codmitions and updateXjdk and Vjdk according to (24) and (25).Step 5.
Repeat Steps2 and 4.The flowchart is shown in Figure2.Figure 2
The flowchart of optimization.


## 4.1. PSO Algorithm

As an optimization method, PSO algorithm seeks the optimum solution of an optimization problem via selecting some particles. The characteristics of each particle include position, velocity, and fitness value. The fitness value is calculated by the fitness function, where the merit of the particle is described by the fitness valule. In the solution space, each paritcle’s position and velocity are updated by the individual extremumPbest and the population extremum Gbest, where Pbest and Gbest are updated by comparing the fitness value of the new particle.Considering aD dimension search space, let X=(X1,⋯,Xm) be a population with m particles. The position and velocity of the jth particle are defined as Xj=(xj1,⋯,xjD)T and Vj=(vj1,⋯,vjD)T, respectively. The extremum of individual and population are defined as Pbest,j=(Pbest,j1,⋯,Pbest,jD)T and Gbest=(Gbest,1,⋯,Gbest,D)T, respectively. The velocity and position updating laws of the jth particle at the (k+1)th iteration are given as(22)Vjdk+1=ωVjdk+κ1γ1Pbest,jdk-Xjdk+κ2γ2Gbest,dk-Xjdk(23)Xjdk+1=Xjdk+Vjdk+1where Xjdk and Vjdk are the dth position and velocity component of the jth individual, respectively; ω∈(0,1] is inertia weight; κi(i=1,2)≥0 are acceleration factors; γi(i=1,2)∈[0,1] are random numbers between 0 and 1; Pbest,jd is the dth best position component of the jth individual and Gbest,d is the dth best position of the entire population.In this paper, the updating laws (22) and (23) are improved and the results are given as(24)Vjdk+1=ωVjdk+κ1γ1′Pbest,jdk-Xjdk+κ2γ2′Gbest,dk-Xjdk(25)Xjdk+1=Xjdk-ΔVjdk+1sgn⁡ΔVjdk+1where(26)γ1′=Pbest,jkPbest,jk2+Gbestk2(27)γ2′=GbestkPbest,jk2+Gbestk2(28)ΔVjdk+1=Vjdk-Vjdk+1Remark 4.
From (26) and (27), note that the attributes of γ1′ and γ2′ are the same as those of γ1 and γ2. However, compared with γ1 and γ2, γ1′ and γ2′ not only overcome the uncertainty of random variation but also have the capacity of adaptive adjustment. In addition, due to ΔV being introduced in the updating law (25), the smoothness of the convergence result can be guaranteed in the end of the algorithm.
As an important part of PSO algorithm, the fitness function in this paper is designed as(29)fjk=∑d=1Dxjdk-xjdk-1+vjdk-vjdk-1where D=n is the dimension of each particle; fjk is the fitness value of the jth particle in the kth iteration; xjd0 and vjd0 are the initial states of agents and k∈(1,⋯,K) represents the kth iteration.
Lettmax and Δt be the simulation time and simulation step, respectively. Hence, the maximum number of iterations K is defined as(30)K=tmaxΔt


## 4.2. Mapping Relationship Design

For a multiagent systems withn agents, each agent is considered as a particle. Let xi and vi be the position and velocity of the ith agent, respectively. Then the sets of position and velocity are expressed as XMAS=(x1,⋯,xn) and VMAS=(v1,⋯,vn), respectively. In the PSO algorithm, let XPSO,j=(xj1,⋯,xjD) and VPSO,j=(vj1,⋯,vjD) be the position and velocity of the jth particle, respectively. Consequently, the mapping relationship between particles and agents is shown in Figure 1.Figure 1
Mapping scheme.The operation process of PSO algorithm is given as follows:Step 1.
Initialize the population and the position and velocity of all agents.Step 2.
Assign the position and velocityxi and vi to each particle.Step 3.
Run system model and calculate the fitness function (29).Step 4.
Judge the termination codmitions and updateXjdk and Vjdk according to (24) and (25).Step 5.
Repeat Steps2 and 4.The flowchart is shown in Figure2.Figure 2
The flowchart of optimization.


## 5. Simulation Analysis

Considering the second-order multiagent systems (2) with four agents, the exchange information topology graph G is shown in Figure 3.Figure 3
Exchange information topology graph.The weighted adjacency matrixA is(31)A=0110100110010110The parameter settings are shown in Table1.Table 1
List of parameters.
multi-agent systems PSO algorithm n = 4 m = 100 α = 2.5, β=2.0 ω = 0.2 t m a x = 15 s, Δt=0.01s κ 1 = 0.7, κ2=1.5 x 0 = [ 2.5,5 , 3.6,4 ] T  ( m ) λ 1 = 0.4, λ2=0.5 v 0 = [ 3.0,3.7,1.8,1.4 ] T  ( m / s ) K = 1500In addition, the updating laws (26), (27), and (28) are used for (24) and (25), respectively. According to the number of agents, it can be deducted that the dimension D=8. To illustrate the validity of proposed control protocol (3) and the control effect of PSO algorithm, the following simulation examples are provided.Example 5 (comparison the control protocol (3) and the standard PSO algorithm).
In this example, we consider the consensus of multiagent systems (2) with control protocol (3) (scheme 1) and the standard PSO algorithm (22) and (23) (scheme 2); the simualtion results are shown in Figures 4–6.
The consensus of position and velocity with the control protocol (3) are shown in Figure 4. It can be found that the control effects are not good from Figure 4. Applying the standard PSO algorithm, the comparison results are drawn in Figures 5 and 6. Compared with control protocol (3), the consensus achieving time and the overshoot have been greatly improved by using the PSO algorithm.Figure 4
States consensus with control protocol (3).Figure 5
Position comparison results with standard PSO algorithm and control protocol (3).Figure 6
Velocity comparison results with PSO algorithm and control protocol (3).Example 6 (comparison the standard PSO algorithm and the improved PSO algorithm).
In this example, the consensus of the multiagent systems (2) with the standard PSO algorithm (22) and (23) (scheme 2) and the improved PSO algorithm (24) and (25) (scheme 3) is considered; the simualtion results are shown in Figures 7 and 8.
From Figures7 and 8, it can be seen that the output of all agents have been further improved. Compared with the srtandard PSO alorithm, the improved PSO algorithm not only reduces the consensus achieving time and overshoot but also makes the curves smoother. In general, in the ability to improve the control effect of the multiagent systems, the improved PSO algorithm proposed in this paper is superior to the other two proposed approaches.Figure 7
Position comparison results with standard PSO algorithm and improved PSO algorithm.Figure 8
Velocity comparison results with standard PSO algorithm and improved PSO algorithm.


## 6. Conclusion

In this paper, we discussed the consensus problem of a class of second-order multiagent systems with the PSO algorithm. An improved consensus protocol was designed, and then the convergence was analyzed by the Lyapunov stability theory. The PSO algorithm was introduced to improve the control effect of the multiagent systems. And then an improved PSO algorithm was proposed. In addition, we designed a mapping relationship with the agents and the individuals of PSO algorithm. Finally, the simulation results showed that the consensus with the proposed control protocol can be achieved. Also, the control effect can be improved by using PSO algorithm; and the better control effect can be obtained by utilizing the improved PSO algorithm.



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