# Flocking for Multiagent Systems with Partial Information Exchange Based on Inclusion Principle

Authors: Hui Wei; Xuebo Chen
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper investigates the flocking problem of multi-agents with partial information exchange, which means that only part, but not all, of the agents are informed of the group objective. A distributed flocking model based on the inclusion principle is provided to simplify the design and analysis of multi-agent systems. Furthermore, to reduce the communication energy consumption, an improved flocking algorithm based on the model is proposed to achieve stable flocking for all the agents. The stability of the multi-agent system is then established, with the help of the Lyapunov stability theorem and LaSalle’s invariance principle. Especially, considering the individual heterogeneity in both nature and engineering applications, we also investigate the flocking problem of multi-agents with different sensing radiuses and equilibrium distances. Finally, two kinds of simulation results are presented to demonstrate the validity of the proposed results. This work provides an insight not only into the properties of the different species of individual flocking, but also into the theoretical framework for the engineering design of multi-agent systems considering individual heterogeneity.


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## Body

## 1. Introduction

Flocking, a common phenomenon in nature, can eventually achieve a group objective through local communication among the neighbouring agents. Examples of these agents include flocks of birds, schools of fish, groups of ants, and crowds of people. For many decades, flocking problems have received a great deal of attention from researchers in biology, social science, physics, control science, computer science, and so on [1–10], due to the emergence of swarming for a group of agents with local interactions. This phenomenon of swarming finds a broad range of applications in swarm robots [11], mobile sensor networks [12], and unmanned aircraft systems (UAS) [13–15].In 1986, Reynolds produced a computer model by simulating flocks of birds, which consists of three basic rules of cohesion, separation, and alignment [9]. Soon after this, Vicsek et al. first studied the collective behavior of bird swarms from a theoretical perspective [16–19]. For instance, Vicsek et al. [16] proposed a self-driven particle model in 1995, which mainly concentrated on the emergence of flocking, whereas Toner and Tu [17] preferred the continuum mechanics method. From the particle-based model proposed by Vicsek et al. [16], Helbing et al. [18] conducted a series of experiments to explore the mechanism of escape panic. In 2003, Jadbabaie et al. [19] studied the linear Vicsek model without noise and were the first to provide a theoretical explanation and mathematical proof of this model. In 2006, inspired by the Reynolds rules, Olfati-Saber [7] designed two typical flocking algorithms: flocking in an obstacle-free environment and flocking with obstacle avoidance. Moreover, he performed 2D flocking in the presence and absence of obstacles and 3D flocking in free space. In his algorithms, it was assumed that all the agents can obtain the information of the group objective (or virtual leader). Such an assumption, however, requires significant energy consumption during the flocking process. This makes it almost impossible to be implemented in engineering applications, such as multitarget consensus circle pursuit for multiagent systems via a distributed multiflocking method. To overcome this limitation, Su et al. [20] extended the typical flocking algorithm, considering that only partial agents are informed of the group objective, and then, numerical simulations indicated that most agents will asymptotically track the group objective, even if only a fraction of the group are informed agents. However, there are some uninformed agents that still fail to track the group objective during the flocking process. Therefore, it is necessary to improve the typical flocking algorithm further in this work.The aforementioned works have greatly promoted the research process of flocking control for multi-agent systems and established an important foundation for future researches.Recently, some studies were conducted that are closely related to the research topic of this paper. For example, Luo et al. [21] investigated the multi-target tracking problem of multi-agent systems, which, however, is still a deadlock problem. In order to overcome this problem, Pei et al. [22] proposed a local multi-flocking algorithm, which successfully resulted in the multi-target consensus pursuit for multi-agent systems. Xi et al. [23] provided a theoretical framework for the analysis of the output consensus problem of high-order linear time-invariant multi-agent systems, during the flocking process. Cao and Ren [24] presented a distributed consensus tracking algorithm for first-order kinematics and a distributed swarm tracking algorithm for second-order dynamics, which mainly concentrated on solving the distributed coordinated tracking problem. Based on the continuous-time information-weighted Kalman consensus filter (IWKCF), Luo and Li [25] proposed a distributed topology optimization scheme to decrease the communication complexity of flocking for multi-agent systems, in which each agent produces a local optimally rigid graph with its neighbouring agents. However, most of the above studies have merely focused on the homogeneous multi-agent system. Considering the individual characteristics, multi-agent systems can be divided into two types: homogeneous and heterogeneous [26]. Individual heterogeneity exists widely in both nature and engineering applications, such as multi-UAS in military confrontations, different growth backgrounds, social distancing, and economic strength in human society. Therefore, it is of great theoretical and practical significance for the research of the flocking problem of multi-agents with different equilibrium distances and sensing radiuses. Note that the equilibrium distance is proportional to the sensing radius (according to the works of [7]). For the sake of brevity, we will discuss only the different equilibrium distances later.As further research is being conducted on multi-agent systems, we can determine that the motion consensus of multi-agents is similar to the synchronization problem of complex networks in essence, which is the solution to achieve the consistency of the state of all agents [27–29]. Furthermore, the inclusion principle and its extended pair-wise decomposition can take full advantage of the interconnection relationship between subsystems, which have been widely applied in fields such as multi-area interconnected power systems [30–32], formation control of unmanned aerial vehicles [33], Petri nets [34], etc. Hence, the inclusion principle and its extended pair-wise decomposition can be chosen as a proper tool to simplify the design and analysis of flocking for multi-agent systems.In this paper, we first divide a complex swarm system into multiple pair-wise subsystems based on the inclusion principle. Then, from the Lennard-Jones potential function [35] and self-organization process [36], an improved flocking algorithm is designed for each subsystem. Thereby, flocking of multi-agents with partial information exchange is achieved via the coordinated control of the pair-wise subsystems, which means that only a part, but not all, of the agents can be informed of the group objective. Compared with the typical flocking algorithm [7] and the existing distributed topology optimization scheme [25], this work, to some extent, has decreased the calculated amount and greatly reduces the communication energy consumption during the flocking process. Furthermore, considering the individual heterogeneity in both nature and engineering applications, the multi-agent system can be divided into three different species, according to the size of the equilibrium distance. Particularly, one of the aspects we are very interested in is the collision avoidance between different species of multi-agents during the flocking process. In order to explore the internal mechanism of this interesting problem, two kinds of simulations are designed to investigate the flocking of multi-agents with different equilibrium distances, which can provide an insight into the properties of the different species of individual flocking.In brief, the main contributions of this paper are as follows: (1) A distributed flocking model based on the inclusion principle is proposed to simplify the design and analysis of flocking for multi-agent systems. (2) Based on the Lennard-Jones potential function and self-organization process, an improved flocking algorithm is designed to achieve flocking of multi-agents with partial information exchange. (3) Considering the individual heterogeneity in both nature and engineering applications, the flocking problem of multi-agents with different equilibrium distances is also investigated.An outline of the paper is organized as follows. Section2 introduces some preliminary knowledge. A distributed flocking model based on the inclusion principle is presented in Section 3. Section 4 proposes an improved flocking algorithm. In Section 5, the stability analysis of multi-agent systems is proven with the help of the Lyapunov stability theorem and LaSalle’s invariance principle. Several simulation results are provided in Section 6. Finally, the conclusions are presented in Section 7.


## 2. Preliminaries

The flocking algorithm design process proposed in this paper is based on some preliminary knowledge about graph theory, especially, the inclusion principle and its extended pairwise decomposition.


### 2.1. Graph Theory

For the description of multi-agent systems by graph theory in this paper, we have some definitions as follows [37].Definition 1.
Suppose that a network topology consists ofN vertexes. The network topology can be described by an undirected graph G=v,E, where v=1,2,…,N is a vertex set and E⊆i,j:i,j∈v,j≠i denotes the edge set with vertexes of junctions.Definition 2.
The adjacency matrixAa=aij of the undirected graph G is defined as(1)aij=1,ifi,j∈v,0,others,where Aa is symmetric, i.e. AaT=Aa.Definition 3.
The degree matrix of undirected graphG is a diagonal matrix D=DAa with diagonal elements ∑j=1naij that are row sums of Aa. The Laplacian matrix L=lij is an n×n matrix, which is defined as(2)L=D−Aa.


### 2.2. Inclusion Principle

As mentioned in Section1, the inclusion principle is a theoretical basis for pair-wise decomposition and decentralized control of complex interconnected systems [32]. In order to control multi-agent systems coordinately, we introduce some definitions and theorems as follows [30].Consider a continuous-time linear time-invariant systemS and its expanded system S˜ described by(3)S:x˙=Ax+Bu,y=Cx,S˜:x˜˙=A˜x˜+B˜u˜,y˜=C˜x˜,where x∈Rn, u∈Rm, and y∈Rl are the state, input, and output vectors of the system S, respectively. x˜t∈Rn˜, u˜t∈Rm˜, and y˜t∈Rl˜ are the state, input, and output vectors of the expanded system S˜, respectively. The matrices A, B, C and A˜, B˜, C˜ have appropriate dimensions. It is supposed that n≤n˜, m≤m˜, and l≤l˜.Definition 4.
The expanded systemS˜ includes the system S, namely, S˜⊃S. If there is a quadruplet of full-rank matrices V,U,R,S satisfying UV=In, such that for any initial condition x0∈Rn and any input u∈Rm, when x˜0=Vx0, u˜=Ru, there is xt;t0,x0,u=Ux˜t;t0,x˜0,u˜ and yxt=Sy˜x˜t for all t≥t0.Theorem 1.
If the systemS is a restriction of the expanded system S˜, namely S⊂RS˜, then there exists a triplet of full-rank matrices V,R,T such that(4)A˜V=VA,B˜R=VB,C˜V=TC,orMAV=0,MBR=0,MCV=0,where V, R, T, MA, MB, and MB will be illustrated in Theorem 2.Proof.
The detailed proof is carried out in [30].Theorem 2.
If the systemS is an aggregation of the expanded system S˜, namely, S⊂AS˜, then there exists a triplet of full-rank matrices U,Q,S such that(5)UA˜=AU,UB˜=BQ,SC˜=CU,orUMA=0,UMBR=0,SMCV=0.As mentioned above, the relationship between the systemS and expanded system S˜ is described by(6)A˜=VAU+MA,B˜=VBQ+MB,C˜=TCU+MC,where V, U, R, Q, T, and S are full-rank transformation matrices with dimensions of n˜×n, n˜×n, m˜×m, m˜×m, and l˜×l, l˜×l, respectively, and satisfy UV=In, QR=Im, and ST=Il. MA, MB, and MC are complementary matrices with dimensions of n˜×n˜, m˜×m˜, and l˜×l˜, respectively.Proof.
The detailed proof is carried out in [30].
The systemS can be decomposed into various forms. A typical decomposition is nonoverlapped; the dynamic subsystems are identified as(7)Sij:x˙i=Aiixi+Biiui,yi=Ciixi,i=1,2,…,N,where xi∈Rni, ui∈Rmi, and yi∈Rli are the state, input, and output vectors of the subsystem Si, respectively. Meanwhile, the decomposition is usually overlapped by some states, inputs, outputs, and even a subsystem between pair-wise subsystems. For example, pair-wise subsystems can be described by(8)Sij:x˙ix˙j=AiiAijAjiAjjxixj+Bii00Bjjuiuj,yiyj=Cii00Cjjxixj,i=1,2,…,N−1,j=i+1,…,N.Theorem 3.
Let(9)S˜p:x˜˙p=A˜px˜p+B˜pu˜p,y˜p=C˜px˜p,where x˜p∈Rn˜, u˜p∈Rm˜, and y˜p∈Rl˜are the state, input, and output vectors of the system S˜p, respectively. Suppose that the expanded system S˜ includes the system S, namely, S˜⊃S, and x˜p=PA−1x˜, u˜p=PB−1u˜, y˜p=PC−1y˜, where PA, PB, and PC are nonsingular permutation matrices with appropriate dimensions, such that A˜P=PA−1A˜PA, B˜P=PA−1B˜PB, and C˜P=PC−1C˜PA. Hence, S˜P⊃S satisfying UP=UPA, VP=PA−1V, RP=PB−1R, and TP=PC−1T.Proof.
The detailed proof is carried out in [30].


## 2.1. Graph Theory

For the description of multi-agent systems by graph theory in this paper, we have some definitions as follows [37].Definition 1.
Suppose that a network topology consists ofN vertexes. The network topology can be described by an undirected graph G=v,E, where v=1,2,…,N is a vertex set and E⊆i,j:i,j∈v,j≠i denotes the edge set with vertexes of junctions.Definition 2.
The adjacency matrixAa=aij of the undirected graph G is defined as(1)aij=1,ifi,j∈v,0,others,where Aa is symmetric, i.e. AaT=Aa.Definition 3.
The degree matrix of undirected graphG is a diagonal matrix D=DAa with diagonal elements ∑j=1naij that are row sums of Aa. The Laplacian matrix L=lij is an n×n matrix, which is defined as(2)L=D−Aa.


## 2.2. Inclusion Principle

As mentioned in Section1, the inclusion principle is a theoretical basis for pair-wise decomposition and decentralized control of complex interconnected systems [32]. In order to control multi-agent systems coordinately, we introduce some definitions and theorems as follows [30].Consider a continuous-time linear time-invariant systemS and its expanded system S˜ described by(3)S:x˙=Ax+Bu,y=Cx,S˜:x˜˙=A˜x˜+B˜u˜,y˜=C˜x˜,where x∈Rn, u∈Rm, and y∈Rl are the state, input, and output vectors of the system S, respectively. x˜t∈Rn˜, u˜t∈Rm˜, and y˜t∈Rl˜ are the state, input, and output vectors of the expanded system S˜, respectively. The matrices A, B, C and A˜, B˜, C˜ have appropriate dimensions. It is supposed that n≤n˜, m≤m˜, and l≤l˜.Definition 4.
The expanded systemS˜ includes the system S, namely, S˜⊃S. If there is a quadruplet of full-rank matrices V,U,R,S satisfying UV=In, such that for any initial condition x0∈Rn and any input u∈Rm, when x˜0=Vx0, u˜=Ru, there is xt;t0,x0,u=Ux˜t;t0,x˜0,u˜ and yxt=Sy˜x˜t for all t≥t0.Theorem 1.
If the systemS is a restriction of the expanded system S˜, namely S⊂RS˜, then there exists a triplet of full-rank matrices V,R,T such that(4)A˜V=VA,B˜R=VB,C˜V=TC,orMAV=0,MBR=0,MCV=0,where V, R, T, MA, MB, and MB will be illustrated in Theorem 2.Proof.
The detailed proof is carried out in [30].Theorem 2.
If the systemS is an aggregation of the expanded system S˜, namely, S⊂AS˜, then there exists a triplet of full-rank matrices U,Q,S such that(5)UA˜=AU,UB˜=BQ,SC˜=CU,orUMA=0,UMBR=0,SMCV=0.As mentioned above, the relationship between the systemS and expanded system S˜ is described by(6)A˜=VAU+MA,B˜=VBQ+MB,C˜=TCU+MC,where V, U, R, Q, T, and S are full-rank transformation matrices with dimensions of n˜×n, n˜×n, m˜×m, m˜×m, and l˜×l, l˜×l, respectively, and satisfy UV=In, QR=Im, and ST=Il. MA, MB, and MC are complementary matrices with dimensions of n˜×n˜, m˜×m˜, and l˜×l˜, respectively.Proof.
The detailed proof is carried out in [30].
The systemS can be decomposed into various forms. A typical decomposition is nonoverlapped; the dynamic subsystems are identified as(7)Sij:x˙i=Aiixi+Biiui,yi=Ciixi,i=1,2,…,N,where xi∈Rni, ui∈Rmi, and yi∈Rli are the state, input, and output vectors of the subsystem Si, respectively. Meanwhile, the decomposition is usually overlapped by some states, inputs, outputs, and even a subsystem between pair-wise subsystems. For example, pair-wise subsystems can be described by(8)Sij:x˙ix˙j=AiiAijAjiAjjxixj+Bii00Bjjuiuj,yiyj=Cii00Cjjxixj,i=1,2,…,N−1,j=i+1,…,N.Theorem 3.
Let(9)S˜p:x˜˙p=A˜px˜p+B˜pu˜p,y˜p=C˜px˜p,where x˜p∈Rn˜, u˜p∈Rm˜, and y˜p∈Rl˜are the state, input, and output vectors of the system S˜p, respectively. Suppose that the expanded system S˜ includes the system S, namely, S˜⊃S, and x˜p=PA−1x˜, u˜p=PB−1u˜, y˜p=PC−1y˜, where PA, PB, and PC are nonsingular permutation matrices with appropriate dimensions, such that A˜P=PA−1A˜PA, B˜P=PA−1B˜PB, and C˜P=PC−1C˜PA. Hence, S˜P⊃S satisfying UP=UPA, VP=PA−1V, RP=PB−1R, and TP=PC−1T.Proof.
The detailed proof is carried out in [30].


## 3. Distributed Flocking Model Based on Inclusion Principle

During the flocking process of multi-agents, the dynamics of the multi-agent system should be modeled mathematically to facilitate further investigation. This does not depend on whether the agent is natural biology (e.g., bees and ants) or a piece of engineering equipment (e.g., robots and UAVs). In this section, we assume that all agents have the same mass and size. Consider a group ofN mobile agents (or subsystems) with second-order linear dynamics, which is described by(10)Si:p˙iv˙i=0100pivi+01ui,ypiyvi=1001pivi,i=1,2,…,N,where pi,vi∈Rm (e.g., m = 2 or 3, when a flocking in a 2D or 3D space) denote the position and velocity vector of the agent i, respectively. ui∈Rm is the control input of agent i. Suppose that each agent has a limited sensing radius (or interaction range). Let r>0 denotes the sensing radius, and then, the neighbor set of the agent i is defined as(11)Ni=j∈v:pj−pi<r,where ⋅ is the Euclidean norm in Rm; each agent can acquire the information of other mobile agents within its neighbor set. Moreover, one dynamic (or static) γ−agent represents the group objective (or virtual leader) of multi-agent systems, which drives all agents to track the group objective with the following model:(12)p˙γ=vγ,v˙γ=fγpγ,vγ,where pγ,vγ,fγ∈Rm denote the position, velocity, and control input vector of the group objective, respectively. The initial state vector pairs are set to pγ0,vγ0=pd,vd. A static γ−agent means that its state is fixed, that is, pγ,vγ=pd,vd for all the way. In Section 6, we assume that the group objective always moves at a constant velocity to the fixed direction. Thereby, dynamic equation (12) of the group objective is simplified as p˙γ=vd, v˙γ=0.Considering thatpi, vi are measurable, the linear state-space model of the multi-agent system can be described by(13)S:x˙=Ax+Bu,y=Cx,where x=p1,v1,p2,v2,…,pN,vNT, u=u1,u2,…,uNT, and y=yp1,yv1,…,ypN,yvNT are the state, input, and output vectors of the system S, respectively. Meanwhile, A=A11⋯A1N⋮⋱⋮AN1⋯ANN, B=blockdiagB11,…,BNN, and C=blockdiagC11,…,CNN, in whichAij, Bii, and Cii are submatrices with dimensions ofni×nj, ni×mi, and li×ni, respectively, and i=1,2,…,N,j=1,2,…,N. According to equations (10), (11), and (13), pairwise subsystems (or neighbouring agents) can be identified by(14)Sij:p˙iv˙id˙ijp˙jv˙j=01000000000−10010000100000pividijpjvj+0010000001uiuj,ypiyviydijypjyvj=1000001000001000001000001pividijpjvj,i=1,2,…,N,∀j∈Ni,where dij=pj−pi.To illustrate the applicability of the pair-wise decomposition [30] and the improved flocking algorithm (see Section 4), we consider an interconnected multi-agent system (the number of agents N≥3) as an example. This system with its equilibrium state is shown in Figure 1(a). The vertices denote the subsystems (or agents), represented by blue spheres. The edges denote the interconnections between the neighbouring subsystems, represented by solid lines. The dashed ovals represent the pair-wise subsystems that can be decomposed, as shown in Figure 1(b). The basic interconnection coefficient eij is used to represent the subsystems’ connection states, if the subsystem Si is connected with the subsystem Sj, then eij=1;, otherwise, eij=0. From Figure 1(c), the dashed ovals, respectively, divide the pair-wise subsystem in the system S, among which the intersections, namely, intersected subsystems, are the overlapped parts between subsystems, that are also called overlapped subsystems.Figure 1
Schematic diagram of stable flocking for multiagent systems. (a) The whole system. (b) The system withN = 3. (c) The expanded pairwise subsystems.
(a)(b)(c)Based on the inclusion principle and its extended pair-wise decomposition [32], we consider the coordinated control of each pair-wise subsystem as the basic control of the multi-agent system. Then, the flocking control for the whole system is realized through the parallel coordinated control of these pair-wise subsystems, thereby simplifying the analysis and design of such complex systems. For example, when N = 3, the system (shown in Figure 1(b)) is represented by equation (13) with(15)A=A11A12A13A21A22A23A31A32A33,B=blockdiagB11,B22,B33,C=blockdiagC11,C22,C33=I.The topology can be considered as a composite of the loop structure between the subsystemsS1, S2, and S3. From Theorems 1 and 2, we apply the following expansion transformations:(16)V=blockdiagI11,I11T,I22,I22T,I33,I33T,U=0.5blockdiagI11,I11,I22,I22,I33,I33,MA=0.5×A11−A11A12−A12−A13A13−A11A11A12−A12−A13A13−A21A21A22−A2200−A21A21−A22A2200A31−A3100A33−A33A31−A3100−A33A33.Hence,(17)A˜=A110A1200A130A11A1200A130A21A220000A210A2200A31000A330A310000A33.Then, according to Theorem3, we use the permutation matrix,(18)PA=∏←k=15pAkk+1=pA12pA23pA34pA45pA56,resulting in(19)A˜p=pA−1A˜PA=A11A1200A130A21A22000000A22A230000A32A33000000A33A310A1200A13A11,where we have the desired pair-wise control structure as shown in Figure 1(c), that is, each pair-wise subsystem satisfies equation (14).According to equation (11), the equilibrium structure (or desired geometry) of flocking requires that the distance among the neighbouring agents is constant, thereby satisfying the following constraints:(20)pj−pi=de,where i=1,2,…,N,∀j∈Nit as time t⟶∞. The equilibrium distance de is a positive constant, usually de<r. Particularly, the equilibrium distance plays an important role in the desired formation (e.g., a geometric model of flocks).


## 4. Flocking Control Algorithm

From the above distributed flocking model, an improved flocking algorithm is designed to achieve stable flocking of multi-agents. Since 2006, the typical flocking algorithm, proposed by Olfati-Saber [7], has been extensively applied in fields such as mobile sensor networks and unmanned aircraft systems (UAS), etc. [12–15]. In [7], the control protocol (or input) ui for agent i consists of the following three components:(21)ui=uig+uid+uiγ,where uig and uid are the gradient-based term and velocity consensus term of the control protocol ui, respectively. uiγ is the navigational feedback to track the group objective, which is described by(22)uiγ=c1pγ−pi+c2vγ−vi,c1,c2>0.In this section, a novel distributed flocking algorithm is considered based on the Lennard-Jones potential function [35] and the self-organization process [36]. In contrast with the typical flocking algorithm [7], we simplify the interaction protocols between the neighbouring agents and develop a complex flocking of multi-agents, via a simpler pair-wise action function that only considers repulsion and attraction between the neighbouring agents (or the pair-wise subsystems). More specifically, according to (14) and (20), the control protocol ui for the agent i (or the subsystem Si) is given by the following function:(23)ui=∑j∈Nifαpj−pieij+∑j∈Niaijvj−vi+c1pγ−pi+c2vγ−vi,i∈v,where eij=pj−pi/1+εpj−pi2,ε∈0,1 is a value along the line connecting pi to pj, and aij is the element of the adjacency matrix Aa as in Definition 2 and satisfies (1). In addition, compared with the traditional artificial potential function [7, 20–25], a simpler pair-wise action function that only considers repulsion and attraction among the neighbouring agents (or the pair-wise subsystems) is presented to reflect the interaction protocols. In this way, to some extent, the calculation difficulties can be decreased during the flocking process. Thus, the pair-wise action function fαx is defined as(24)fαx=k11x−1de,x≤de,0,x>r,k2de−xex−de2/r,de<x≤r,where k1 and k2 are defined as the coefficients of the interconnection force, respectively, and especially, the pair-wise action function fαx can be drawn as in Figure 2 when de=5. Figure 2 shows that the action force between the neighbouring agents is an attraction for a long range, but repulsion for a short range. Moreover, the pair-wise potential function Fαx is described by(25)Fαx=∫dexfαsds.Figure 2
The pairwise action function with a finite cutoff.Note that the pair-wise potential functionFαpj−pi is a differential, nonnegative function satisfying the following conditions: (1) Fα reaches its unique minimum when pj−pi=de; (2) Fα⟶∞ as pj−pi⟶0; (3) Fα remains constant if pj−pi>r.In the flocking control protocol (23), it is assumed that all agents can obtain the information of the group objective, which is almost impossible and requires considerable energy consumption in practice. For the purpose of reducing the communication energy consumption, in this section, we assume that only a part, but not all, of the agents are informed of the group objective. Consequently, the control protocol (23) is modified as(26)ui=∑j∈Nifαpj−pieij+∑j∈Niaijvj−vi+hic1pγ−pi+c2vγ−vi,c1,c2>0,where the control indicator hi is utilized to achieve the flocking of multi-agents with partial information exchange. In other words, if the agent i is informed of the group objective, hi=1; otherwise, hi=0. For example, in Section 6, we assume that there are M1≤M≤N informed agents, which means that hi=1 for i=1,2,…,M, but hi=0 for i=M+1,M+2,…,N. Note that different from the work of Su et al. [20], we are more focused on selecting an appropriate ratio of the informed agents, which can not only reduce the communication energy consumption, but also enable all the agents (either informed or uninformed) to track the group objective during the flocking process.


## 5. Stability Analysis

As discussed above, an improved flocking algorithm based on the proposed distributed flocking model is proposed. In this section, we first introduce a theorem to establish a stable flocking of multi-agents and then prove this theorem via the Lyapunov stability theorem and LaSalle’s invariance principle.Theorem 4.
Consider a group ofN mobile agents (or subsystems) with dynamics (10) applying the flocking control protocol (26). Assume that the initial positions and velocities of all agents are chosen at random with the Gaussian distribution. Then, the following statements hold:(i)
No collisions occur between neighbouring agents(ii)
The velocity of all agents asymptotically become consistent(iii)
Flocking of all agents is formed asymptoticallyProof.
Definep˜i=pi−pγ, v˜i=vi−vγ as the position error vector and velocity error vector, respectively. Consequently, the error dynamic of the agent i is described by(27)S^i:p˜˙iv˜˙i=0100p˜iv˜i+01ui,yp˜iyv˜i=1001p˜iv˜i,i=1,2,…,N.
Moreover, letpij=pi−pj and p˜ij=p˜i−p˜j; clearly, p˜ij=pij. Hence, the collective potential function Vi in [7] is modified as(28)V˜ip˜ij=∑j∈v/iFαp˜ij=∑j∉Ni,j≠iFαr+∑j∈NiFαp˜ij.
Similarly, flocking control protocol (26) of the agent i can be rewritten as(29)ui=∑j∈Nifαp˜ijeij+∑j∈Niaijv˜j−v˜i−hic1p˜i+c2v˜i,c1,c2>0.
We choose an energy-like Lyapunov function as follows:(30)Qp˜,v˜=12∑i=1NUip˜+v˜iTv˜i,where(31)Uip˜=∑j=1,j≠iNFαp˜ij+hic1v˜iTv˜i=V˜ip˜ij+hic1v˜iTv˜i,p˜=p˜1,p˜2,…,p˜NT,v˜=v˜1,v˜2,…,v˜NT.
Thanks to the symmetry ofFαx and the adjacent matrix Aa in Definition 2, it follows that(32)∂Fα∂p˜ij=∂Fα∂p˜i=−∂Fα∂p˜j.Then,(33)12∑i=1NU˙ip˜=∑i=1Nv˜iT∇v˜iV˜ip˜ij+hic1v˜iTp˜i.
Consequently,(34)Q˙p˜,v˜=12∑i=1NU˙ip˜+∑i=1Nv˜iTv˜˙i=−v˜TLt+c2Ht⊗Inv˜≤0,where ⊗ is the Kronecker product notation, Ltdenotes the Laplacian matrix in Definition 3, Ht=diagh1,h2,…,hN, and In is the identity matrix with n dimensions.
Note that the graph of the adjacent matrixAa is connected, as shown in Figure 1(a). At this moment, Lt and Ht are both positive semidefinite matrices, and especially, it is clear that Lt+c2Ht is positive semidefinite as well. Hence, Q˙p˜,v˜≤0, which indicates that Qp˜,v˜=Qt is a nonincreasing function over time t. Thereby, Qt≤Q0 for all t≥0, and Q0 is the initial value of Qt. From equations (30) and (31), we conclude that c1p˜iTp˜i≤2Q0 for any agent i, which guarantees the flocking of multi-agents. Therefore, part (iii) is proven.
SinceQt>0 and Q˙t≤0, we assume that Ω=p˜T,v˜TTQt≤Q0 is an invariant set. From LaSalle’s invariance principle, the trajectories of all agents starting from Ω will converge to the largest set Ψ=p˜T,v˜TTQ˙t=0. Suppose that the number of connected subgraphs is mt. For any t>0, there is an orthogonal permutation matrix Pt such that Lt+c2Ht can be transformed into the following form:(35)L˜t+c2H˜t=PtLt+c2HtPTt=diagL1t+c2H1t,…,Lkt+c2Hkt,…,Lmt+c2Hmt,where Lkt and Hkt are the Laplacian matrix and the diagonal matrix corresponding to the kth connected subgraph, respectively, k=1,2,…,mt. The velocity error vector can also be rewritten as v˜p=Pt⊗Inv˜, consequently, we have(36)v˜TLt+c2Ht⊗Inv˜=v˜pTL˜t+c2H˜t⊗Inv˜p.
As mentioned above,L˜t+c2H˜t is positive semidefinite as well. Given (34) and (36), we conclude that Q˙t=0 if and only if v˜pTL˜t⊗Inv˜p=0 and v˜pTc2H˜t⊗Inv˜p=0, which is equivalent to v1≡⋯≡vk≡⋯≡vN. Therefore, part (ii) is proven.
Finally, we prove part (i) by contradiction. Suppose that at least two agents are colliding during the flocking process; then, we acquire(37)Qp˜,v˜=12∑i=1NUip˜+v˜iTv˜i=12∑i=1NV˜ip˜ij+hic1v˜iTv˜i+12∑i=1Nv˜iTv˜i≥12∑i=1NV˜ip˜ij≥Q0,which contradicts the condition Qt≤Q0, and this hypothesis is not valid. Therefore, part (i) is proven.


## 6. Simulation Results and Discussion

In this section, examples of numerical simulation are provided to validate the effectiveness of the improved flocking algorithm proposed in Section4.Consider a group of 150 agents moving in a two-dimensional space. The initial positions of the agents are randomly distributed in the box−50,50×−50,50, as shown in Figure 3(a). The initial velocities of the agents are randomly chosen from the box 0,2×0,2. The initial position and velocity of the group objective are set as pγ0=0,0T and vγ0=0.5,0T, respectively, and marked with a red ×. The other parameters are given in Table 1, which remain fixed throughout all simulations. It is worth noting that, for convenience, the above control parameters of the proposed algorithm are all derived from the works of Olfati-Saber [7], Su et al. [20], and Lehn [36]. Furthermore, all simulations are implemented on a platform with the following configurations: 2.20 GHz CPU, 8.00 GB RAM, Windows 10, and MATLAB R2016b.Figure 3
The flocking forN=150 agents applying control protocol (26) in 2D. The red triangles denote the positions of all agents; the heading angle of each triangle represents the velocity direction of the corresponding agent, whereas the red × denotes the position of the group objective. An undirected edge connecting two agents means that they are neighbors of each other. (a) t=0 sec. (b) t=5 sec. (c) t=10 sec. (d) t=20 sec. (e) t=30 sec. (f) t=40 sec.
(a)(b)(c)(d)(e)(f)Table 1
Parameters of the improved flocking algorithm.
ParametersNotation and valueThe equilibrium distancede=5The sensing radiusr=1.2deRepulsion coefficient of function (24)k1=1Attraction coefficient of function (24)k2=5Coefficients of control protocol (26)c1=c2=1Simulation step sizeΔt=0.1sFigure3 displays consecutive snapshots of the flocking for N=150 agents applying control protocol (26). To decrease the communication energy consumption, only 30% of the group is randomly selected as the informed agent; hence, there are 45 informed agents. Therefore, the initial state of the group is seen as highly disconnected subgroups, and fragmentation appears in this case, as shown in Figure 3(a). From Figures 3(b) to 3(d), it is clear that more and more agents are connected, but at a certain distance (where no collisions occur). Finally, the group of N=150 agents form a geometric flocking, as shown in Figure 3(e), and maintain the formation thereafter, as shown in Figure 3(f).The velocity consensus during the flocking process is shown in Figure4. The plot shows that the velocities in the x- and y-axes eventually converge to a constant set, which means that the velocities of all agents become consistent asymptotically. Therefore, the velocity consensus is achieved.Figure 4
The velocities of all agents during the flocking process. (a)x-axis direction. (b) y-axis direction.
(a)(b)The above observations are fully consistent with our predictions of Theorem4, indicated in Section 5.In the simulation stated above, we assume that all the agents are of equal mass and size and have the same sensing radiusr and equilibrium distance de. Nevertheless, for instance, due to the difference in the growth background, social status, and economic strength, etc., it is impossible for two people to be completely identical, in reality. Even for a pair of twins, their cognition and ideology will be different. Therefore, we will fully consider the individual heterogeneity in the following simulations, as we are interested in more than two different species of flocking.Considering the difference in cognitive level among individuals, we divide the group ofN=150 agents into three species, whose equilibrium distances de are set as 5, 10, and 15, respectively. Correspondingly, their sensing radiuses r are set as 6, 12, and 18, respectively (according to the work of Olfati-Saber [7]). The remaining parameters are the same as given in Table 1.Figure5 displays consecutive snapshots of the flocking for three species of agents applying the control protocol (26). The initial positions of all agents are randomly distributed in the box −70,70×−70,70. The initial velocities of all agents are randomly chosen from the box −1,1×−1,1. The initial position and velocity of the group objective are set as pγ0=−10,0Tand vγ0=0.5,0T, respectively. Similarly, there are 45 informed agents. The red, green, and blue triangles represent agents with equilibrium distances de of 5, 10, and 15, respectively. The numbers of red, green, and blue triangles are 110, 30, and 10, respectively. Hence, according to the initial state of the group shown in Figure 5(a), the lengths of links between the neighbouring agents are not equal. From Figure 5(b) to 5(e), it is clear that the group of N=150 agents eventually form an irregular flocking and maintain the formation thereafter, as can be seen in Figure 5(f). Likewise, no collisions are seen to occur during the flocking process. In this simulation, we find that the green and blue triangles asymptotically move to the periphery and eventually become distributed around the edge of the group, when the multiagent system is stable. Thereafter, the number of each three species is changed to 50, respectively, while the other parameters remain unchanged. From Figure 6, this self-organizing classification becomes more obvious at the moment. The group is centered on the group objective, and the three species of red, green, and blue agents surround the group objective from the inside to outside. In other words, based on the size of the equilibrium distance de, the three species are arranged surrounding the group objective by de from small to large. It is interesting to discuss the causes of this phenomenon.Figure 5
The flocking for three species of agents applying control protocol (26) in 2D. The triangles denote the positions of all agents; the heading angle of each triangle represents the velocity direction of the corresponding agent, whereas the red × denotes the position of the group objective. The red, green, and blue triangles represent agents whose equilibrium distance de is 5, 10, and 15, respectively. The numbers of red, green, and blue triangles are 110, 30 and 10, respectively. An undirected edge connecting two agents means that they are neighbors of each other. (a) t=0 sec. (b) t=4 sec. (c) t=18 sec. (d) t=30 sec. (e) t=36 sec. (f) t=50 sec.
(a)(b)(c)(d)(e)(f)Figure 6
The flocking for three species of agents applying control protocol (26) in 2D. The triangles denote the positions of all agents; the heading angle of each triangle represents the velocity direction of the corresponding agent, whereas the red × denotes the position of the group objective. The red, green, and blue triangles represent the agent whose equilibrium distance de is 5, 10, and 15, respectively, and the number of the three species is equal to 50. An undirected edge connecting two agents means that they are neighbors of each other. (a) t=0 sec. (b) t=5 sec. (c) t=18 sec. (d) t=25 sec. (e) t=40 sec. (f) t=50 sec.
(a)(b)(c)(d)(e)(f)Figure7 shows the pairwise action function fα with different equilibrium distances. The plot shows that, when the distance between the agent i and the agent j is equal tode, its action forcefα=0, whereas for the agent k with a larger equilibrium distance (i.e., 2de and 3de), its action forcefα>0, which can be considered as the repulsive force. Hence, the agent k will gradually move to the edge of the group until its action force is zero. This indicates that the equilibrium distance plays an important role in the flocking. Apparently, a species of agents with a small de exhibit a stronger cohesive force than the others with the large distance.Figure 7
The action force function with different equilibrium distances.It is also worth noting that there are very few agents with large equilibrium distances that are surrounded by agents with small equilibrium distances. At this point, its links appear denser because its sensing radiusr is larger than that of the neighbouring agents.To compare with the typical flocking algorithm [7] and the rigid flocking algorithm based on the IWKCF [25], a simple model is used to calculate the communication energy consumption [38], which is usually applied to wireless microsensor networks. In [38], the radio hardware energy dissipation is related to the distance d between the transmitter and receiver. In other words, for a relatively short distance, the energy dissipation is inversely proportional to d2, whereas d4 for a relatively long distance. Hence, the communication energy consumption for transmitting a l-bit message with a distance d is defined as(38)ETxl,d=ETx−elecl+ETx−ampl,d=lEelec+lεfsd2,d<dc,lEelec+lεmpd4,d≥dc,where Eelec denotes the electronic energy, which depends on factors such as the digital coding, filtering, and spreading of the signal, etc. εfs and εmp are related to the sensitivity of the receiver and the admissible bit-error rate, respectively. dc represents the cross-over distance. For the theoretical experiments in this work, these parameters are set as Eelec=50nJ/bit, εfs=10pJ/bit/m2, dc=100m, and εmp=0.0013pJ/bit/m2, which are the same as the experiments conducted in [25, 38].Figure8 shows the communication energy consumption compared with the proposed flocking algorithm, the algorithm proposed by Olfati-Saber [7] and Luo et al. [25]. The plot reveals that the communication energy consumption in this paper is significantly lower than that obtained in previous works, which has mainly benefited from the inclusion principle and improved flocking algorithm.Figure 8
Communication energy consumption comparison between the proposed flocking algorithms [7, 25].


## 7. Conclusion

In this paper, the flocking problem of multi-agents with partial information exchange was investigated based on the inclusion principle. For the purpose of reducing the communication energy consumption, an improved flocking algorithm has been proposed to achieve a stable collective behavior of the multi-agent system. The stability of the multiagent system was analyzed using Theorem4 and its relevant proof. In addition, considering the individual heterogeneity in both nature and engineering applications, we then investigated the flocking problem of multi-agents with different sensing radiuses and equilibrium distances. Finally, numerical simulations verified the effectiveness of the proposed results. Compared with the above simulation results, we found that there exists an interesting self-organizing classification during the flocking process. It is believed that this phenomenon can provide an insight into the properties of the different species of individual flocking. However, the main limitation of this paper is that the proposed flocking model is only applicable to linear systems. Therefore, future works include, but are not limited to, the following research interests: (1) considering the flocking problem of multi-agents with nonlinear dynamics, (2) applying the proposed flocking algorithm to the formation control of swarm robots.



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*Source: 6424691-2020-07-18.xml*