# Multiple Dynamic Targets Encirclement Control of Multiagent Systems

Authors: Wenguang Zhang; Jizhen Liu; Deliang Zeng
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper develops the distributed encirclement control problem of multiagent systems, in which each agent tracks multiple targets, each target can be tracked by one agent, and the numbers of the agents and the targets are the same or not. Firstly, an encirclement control protocol is proposed for multiagent systems, and this protocol contains some estimators. Secondly, some conditions are derived, under which multiagent systems can achieve encirclement control by circular formation. Finally, numerical simulations are provided to illustrate the obtained results.


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## Body

## 1. Introduction

Distributed coordination control of multiagent systems has attracted a great number of researchers from different backgrounds, such as physics, biology, control theory, robotics, and computer [1–17]. Multiagent systems arise in wide areas, including movement of flocks of birds or schools of fish, molecular conformation problems, cooperative control of unmanned aerial vehicles, formation control of mobile robots, and power systems. For instance, Olfati-Saber and Murray [2] presented two consensus protocols to solve agreement problems in a network of continuous-time and discrete-time integrator agents and investigated a systematical framework of consensus problem in networks of agents with a simple scalar continuous-time integrator in three cases. Lin and Jia [3–5] studied consensus problems for first-order or second-order multiagent systems with time varying communication delays and switching topology. In [6, 7], H∞ consensus problems were, respectively, investigated for the first-order and high-order multiagent systems, and they gave the conditions of satisfying H∞ based on linear matrix inequality. In [8], the constrained consensus problem of multiagent systems in dynamically changing unbalanced networks with communication delays has been studied. It has been shown that the error auxiliary vanishes as time evolves and the linear main body has an exponential convergence rate to a vector as a separate system.In some situations, encirclement control for multiple targets can be studied in a distributed manner. However, the work on this problem is rare currently. In [9, 10], they only considered the fixed targets. In [9], a group of unmanned aerial vehicles surrounding one target by using decentralized nonlinear model predictive control was studied. In [10], Chen et al. used the leader-follower framework to make the followers surround the stationary leaders with a fixed communication graph. A multiagent cooperative control problem in which agents move collectively to surround multiple targets was studied in [11], and the proposed control law works not only for stationary targets but also for dynamic ones. But in that paper, it is assumed that the numbers of the agents and the targets are the same.This paper will focus on the study of the distributed encirclement control and tracking problems of multiple dynamic targets by graph theory. We suppose that each agent tracks multiple targets and each target only can be tracked by one agent. Firstly, we design a control protocol including some estimators. Secondly, the required conditions to realize encirclement are proposed by Lyapunov theory. Finally, we prove this theory to be effective by the simulation.The rest of this paper is organized as follows. In Section2, we introduce some basic notations and some concepts in graph theory. In Section 3, the model to be researched is formulated and a distributed encirclement control protocol is proposed. In Section 4, the main results are stated and derived. In Section 5, numerical simulations are provided to demonstrate the effectiveness of the obtained theoretical results. In Section 6, we conclude this paper.


## 2. Notations and Preliminaries

LetG(V,E,A) be an undirected graph, where V={s1,s2,…,sn} is the set of nodes and E∈V×V is the set of edges. The node indexes belong to a finite index set I={1,2,…,n} and Ni={sj∈V(si,sj)∈E} is defined as the neighbourhood set of si. A=[aij]∈Rn×n is a symmetric weighted adjacency matrix, where the element aij represents the weight from node si to node sj. When sj∈Ni, then aij>0, or else aij=0. In the undirected graph, any (si,sj)∈E⇔(sj,si)∈E. The graph Laplacian with the diagraph is defined as L=[lij], where lii=∑j=1n‍aij and lij=-aij,i≠j. If there is a path from every node to every other node, the graph is said to be connected and undirected.Lemma 1 (see [18]).
If the undirected graphG is connected, then its Laplacian L satisfies the following:(1)
Zero is a simple eigenvalue ofL, and 1n is the corresponding eigenvector, and L1n=0.(2)
The remainingn-1 eigenvalues of L all have positive real parts. And L is a symmetric matrix and the eigenvalues 0=λmin=λ1≤λ2⋯≤λn=λmax.Lemma 2 (see [19]).
Suppose there is a positive definite Lyapunov functionV(x,t) defined on U×R+, where U∈U0 is the neighbourhood of the origin. There are positive real constants c>0 and 0<α<1, such that V˙(x,t)+cVα(x,t) is negative semidefinite on U. Then, V(x,t) is locally finite-time convergent with a settling time(1)T≤V1-αx0tc1-α.


## 3. Model and Problem Description

The multiagent systems under consideration comprisen agents and m targets. Each agent is regarded as a node in an undirected graph G. Each edge (sj,si) corresponds to an available information path from agent j to i. Moreover, each agent updates its current state based on the information received from its neighbors. We suppose that the dynamic of the ith agent is(2)y˙it=uit,i∈I=1,2,…,n,where yi(t)∈R2 denotes the position and ui(t)∈R2 is the control input of ith agent at time t.To simplify the analysis, we will consider the dynamics in polar coordinate system corresponding to system (2):(3)l˙it=vit,θ˙it=ωit,i∈I,where li(t)∈R and θi(t)∈R, respectively, denote the radius and angle of the ith agent in the polar coordinate system which regards the geometric center P=1/m∑i=1mrit as the origin. ri(t) represents the position of the ith target at time t. Obviously, yi(t)=pi(t)+[litcosθit,litsinθit]T, where pi(t) denotes the estimated value of the distance from the ith agent to the geometric center P.We say the control protocolui(t) can solve the distributed encirclement problems of system (2) if the states of agents satisfy(4)limt→∞⁡yit-1m∑k=1mrkt-kmaxi∈I⁡rit-1m∑k=1mrkt=0,limt→∞⁡θit-θjt-2πi-jn=0,i∈I,where k>1.Assumption 3.
Theith agent can track ni≥1 targets, and each target can only be tracked by one agent.
Letr¯it=1/ni∑k=1nirikt and Ψit=n/mnir¯it; then 1/n∑i=1nΨit=1/m∑i=1mrit. To solve the distributed encirclement control problem of system (3), we can first estimate the distributed center of the targets and then drive the agents encirclement and track the targets. The following control protocol is(5)vit=-k1sign⁡lit-kρit,ωit=∑j∈Nitaijθjt-θit-2πj-in,i∈I,where ρi(t) denotes the estimated value of maximum distance from r¯i(t) to the distributed center for the ith agent and k1>0 represents the control parameter.
The estimator of theith agent corresponding to the distributed center positions ρi(t) is given as follows:(6)φ˙it=α∑j∈Nitsign⁡pjt-pit,pit=φit+Ψit,i∈I,where φi(t) is the intermediate variable with φi0=0 and α>0 is the control parameter.Assumption 4.
The speed of all targets has the common upper bound as the definition in [17]; then there exists β>0 such that r˙it≤β,i∈{1,2,…,m}.
Noting thatdit=r¯it-pit,i∈I, the estimator of the ith agent corresponding to ρi(t) is rewritten as follows:(7)ρ˙it=-k2sign⁡ρit-maxj∈I⁡djt,i∈I,where k2>0 represents the control parameter.
To simplify analysis, we transform the original system (3)–(5) into an equivalent system. Let θ^i(t)=θi(t)-2iπ/n; then the closed-loop system (3)–(5) is given as follows:(8)l˙it=-k1sign⁡lit-kρit,θ^˙it=∑j∈Nitaijθ^jt-θ^it,i∈I.
Letθ^=[θ^1(t),θ^2(t),…,θ^n(t)]; then θ^˙(t)=-Lθ^. So the control protocol corresponding to system (2) is(9)uit=vitcos⁡θit-litωitsin⁡θitvitsin⁡θit+litωitcos⁡θit+p˙it,i∈I.


## 4. Main Results

Lemma 5.
Considering estimator (6), if α>n2β and Assumptions 3 and 4 are both satisfied, there must exist T1>0 such that limt→T1⁡pi(t)-1/m∑k=1mrk(t)=0 for any i∈I. That is to say, the estimate pi(t) corresponding to distributed center for all agents will converge to the distributed center of targets in finite time.Proof.
Letpt=p1Tt,p2Tt,…,pnTtT. We define the Lyapunov function as follows:(10)V1t=12∑i=1npit-1n∑k=1npktTpit-1n∑k=1npkt;then the derivative of V1(t) along the trajectories of system (6) is given by(11)V˙1t=∑i=1npit-1n∑k=1npktTp˙it-1n∑k=1np˙kt=∑i=1npit-1n∑k=1npktTα∑j∈Nitsign⁡pjt-pit+Ψ˙it-1n∑k=1np˙kt.For(12)α∑i=1npit-1n∑k=1npktT∑j∈Nitsign⁡pjt-pit=α2∑i=1n∑j∈Nitpit-pjtTsign⁡pjt-pit≤-α2∑i=1n∑j∈Nitpit-pjt,Ψ˙it=nmnir¯˙it=nm∑k=1nir˙ikt≤nmniβ≤nβ,we have(13)∑i=1npit-1n∑k=1npktTΨ˙it≤nβ∑i=1npit-1n∑k=1npkt≤nβ∑j=1,j≠inmaxi=1,2,…,n⁡pit-pjt=n2βmaxi,j=1,2,…,n⁡pit-pjt≤n2β2∑i=1n∑j∈Nitpit-pjt.For(14)∑i=1npit-1n∑k=1npktT-1n∑k=1np˙kt=∑i=1npit-∑k=1npkt-1n∑k=1np˙kt=0,we get(15)V˙1t≤-α2∑i=1n∑j∈Nitpit-pjt+n2β2∑i=1n∑j∈Nitpit-pjt=n2β2-α2∑i=1n∑j∈Nitpit-pjt.
Letm(t)=maxi,j∈I⁡{pit-pjt}. For pi(t)-1/n∑k=1npk(t)≤1/n∑k=1npi(t)-pk(t)≤mt, we obtain V˙1t≤n/2mt2. Furthermore, for ∑i=1n∑j∈Ni(t)pi(t)-pj(t)≥mt, we have(16)V˙1t+α2-n2β2n2V1t1/2≤n2β2-α2mt+n2α2-n2β2n2mt=0.Therefore, from Lemma 2, there exists T1>0 such that limt→T1⁡pi(t)-1/m∑k=1mrk(t)=0, and we get limt→T1⁡pi(t)-pj(t)=0.Lemma 6.
Considering estimator (7), one supposes that Assumptions 3 and 4 are both satisfied. If k2>2β, estimator (7) must be steady in finite time.Proof.
We define the Lyapunov function as follows:(17)V2t=∑i=1nɛisign⁡ɛi,where ɛi=ρit-maxj∈I⁡djt. Then, the derivative of V2(t) along the trajectories of system (7) is given by(18)V˙2t=∑i=1nsign⁡ɛiɛ˙i=∑i=1nsign⁡ɛi-k2sign⁡ρit-maxj∈I⁡djt-maxj∈I⁡d˙jt.
Letdt=maxj∈I⁡djt=maxj∈I⁡r¯jt-pjt. According to Lemma 5, there exists T1>0 such that pit=1/m∑k=1mrkt when t>T1.
Then, we have(19)d˙t=maxj∈I⁡d˙jt=maxj∈I⁡1nj∑k=1njr˙jkt-1m∑k=1mr˙kt≤maxj∈I⁡1nj∑k=1njr˙jkt+1m∑k=1mr˙kt≤2β.Hence,(20)V˙2t≤∑i=1nsign⁡ɛi-k2sign⁡ɛi+d˙t≤∑i=1nsign⁡ɛi-k2sign⁡ɛi+2β=∑i=1n-k2sign⁡ɛi+2β.According to the condition k2>2β, we conclude that system (7) must be steady in finite time.Lemma 7.
Considering the first equation of system (8), one supposes that Assumptions 3 and 4 are both satisfied. If k2>2β and k1>2kβ, this system can be steady in finite time.Proof.
It is easy to get the conclusion according to Lemma 4 in [1], so we ignore the proof here.Theorem 8.
Considering system (2), if the network topology of multiagent systems is connected and Assumptions 3 and 4 are both satisfied, letting α>n2β, k1>2kβ, and k2>2β, then protocol (9) can solve the distributed encirclement control problem of system (2).Proof.
From Lemma5, there exists T1>0 such that limt→T1⁡pi(t)-1/m∑k=1mrk(t)=0. From Lemma 7, there exists T2>T1 such that limt→T2⁡yi(t)-1/m∑k=1mrk(t)-kmaxj∈I⁡rj(t)-1/m∑k=1mrk(t)=0. As graph G is connected, the Laplacian L of G has nonnegative real part. For θ^˙t=-Lθ^t, then we have limt→∞⁡θ^i(t)-θ^j(t)=0, so limt→∞⁡θit-θjt-2π(i-j)/n=0.


## 5. Simulation Results and Analysis

In this section, the results of simulation by Matlab prove the effectiveness of the theoretical results obtained. The dynamic targets are(21)r1t=t+1-sin⁡t-t+10-cos⁡t,r2t=t+10-sin⁡t-t+2-cos⁡t,r3t=t+2-sin⁡t-t+3-cos⁡t,r4t=t+8-sin⁡t-t+4-cos⁡t,r5t=t+4-sin⁡t-t+5-cos⁡t,r6t=t+6-sin⁡t-t+7-cos⁡t.The corresponding fixed network topology of multiagent systems with4 nodes is shown in Figure 1. Let k=2, β=22, α=33, k1=10, and k2=10, and the initial conditions are (9,9), (10,9), (-6,2), and (1,-5). Figure 2 represents the trajectories of the agents and targets of multiagent systems and it shows that the multiagent systems with fixed topology can encircle the multiple targets in the form of circular formation.Figure 1
The fixed network topology of multiagent systems.Figure 2
The trajectories of the agents and targets of multiagent systems with fixed topology.Figure3 shows the five kinds of topologies of multiagent systems with 4 nodes and these topologies can realize the random switch obeyed uniform distribution among them. Let k=2, β=22, α=33, k1=10, and k2=10, and the initial conditions are (1,9), (10,1), (2,2), and (1,3). Figure 4 represents the trajectories of the agents and targets of multiagent systems and it shows that the multiagent systems with switching topology randomly can encircle the multiple targets in the circular formation.Figure 3
The five kinds of topologies of multiagent systems.Figure 4
The trajectory of the agents and targets of multiagent systems with switching topology randomly.


## 6. Conclusion

In this paper, we investigate the distributed encirclement of multiagent systems with multiple dynamic targets with the assumption that each agent can track multiple targets, each target only can be tracked by one agent, and the numbers of the agents and the targets are the same or not. The encirclement and tracking method in circular formation is proposed. Considering that each agent can only get partial information of targets, the target state estimators which can estimate the average position of targets are designed. In finite time, every agent’s motion radius is locally converged to circular formation radius of system within a settling time. All agents can maintain the formation which can be updated in real time according to the change of targets’ state. By Lyapunov function, it is proved that every agent can get the whole information of targets in finite time, and meanwhile it realizes the circular formation in finite time. The simulation results illustrate that this proposed method is effective for not only multiple static targets but also multiple dynamic targets.



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