# Observer-Based Event-Triggered Circle Formation Control for First- and Second-Order Multiagent Systems

Authors: Peng Xu; Guangming Xie; Jin Tao; Minyi Xu; Quan Zhou
Journal: Complexity
 (2020)

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## Abstract

This paper proposes an observer-based event-triggered algorithm to solve circle formation control problems for both first- and second-order multiagent systems, where the communication topology is modeled by a spanning tree-based directed graph with limited resources. In particular, the observation-based event-triggering mechanism is used to reduce the update frequency of the controller, and the triggering time depends on the norm of the state function and the trigger threshold of measurement errors. The analysis shows that sufficient conditions are established for achieving the desired circle formation, while there exists at least one agent for which the next interevent interval is strictly positive. Numerical simulations of both first- and second-order multiagent systems are also given to demonstrate the effectiveness of the proposed control laws.


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## Body

## 1. Introduction

In recent years, many research efforts have been devoted to controlling of multiagent systems (MASs) due to both its practical potentials in a variety of applications [1–3] and theoretical challenges of physical constraints [4–6]. As a significant problem in cooperative control for MASs, formation control, aiming to guide multiple agents to form and maintain predetermined geometries, has attracted considerable interests for its extensive applications in different areas [7–10]. The main focus has been devoted to the design of a distributed formation control framework, especially concerning the robustness against both external disturbances and internal uncertainties [11, 12], as well as the increased number of agents. Moreover, for MASs subjected to aperiodic sampling and communication delays, the problem of cluster formation control was addressed in [10]. Therefore, most existing results on formation control mainly rely on the ideal hypothesis [13–15], e.g., each agent is modeled as having unlimited communication capabilities, unlimited power, and unlimited processing capabilities, which allows arbitrary information to exchange pattern. However, as far as we know, few studies dealt with the limited capacity of communication and the power constraints of agents.In order to save energy and bandwidth, event-triggered control methods have been presented in [16–20]. One of the most distinct characters of event-triggered control is that control actions only update when specific events occur, which lead to ease the trade-offs among actuator effort, communication, and computation. Moreover, according to the triggered methods, event-triggered control can be mainly divided into state-dependent triggering and time-dependent triggering. A simple state event-triggered schedule based on the feedback control was studied in [16]. The results lead to a guaranteed performance with a fixed sampling rate requirements concerning the optimizing schedules and sampling rates. In [17], under conditions of decreasing thresholds of the measurement errors exponentially, a time-dependent triggered method was designed to guarantee all agents asymptotic converge to a ball centered at the average consensus. Different from most of the existing fixed threshold parameters, the threshold parameter in the improved event-triggered condition is dynamically adjusted by a dynamic rule in [18]. In [21], by proposing a pull-based event-triggered control strategy, a circle formation control problem for first-order MASs with directed topologies was studied. Further, Wen et al. [22] combined event-triggered protocols to solve circle formation problems of first-order MASs. Also, Wen et al. and Xu et al. [23, 24] have investigated a combination algorithm based on quantized communication technology, where the problem of MASs with a limitation of communication was addressed. Given the above reviews, it is noteworthy to mention that most of the existing results on event-triggered control are to prevent the case of Zeno behavior [25, 26], so that within a finite time interval, an infinite number of samplings generate. Typically, the event trigger interval having a strictly positive lower bound is a sufficient condition to exclude Zeno behavior [27].Different from previous studies, especially [23], which paid attention to the effect of qualitative communication for event-triggered control, the main objective of this paper is to provide an observer-based event-triggered method to solve circle formation problems for both first- and second-order MASs through a set of directed graphs. In our studies, each agent observes the distance from the counterclockwise direction to its nearest neighbor and the counterpart from the clockwise direction through communication, which is similar to Pioneer 3-DX [28]. In comparison to the literature, we have three main contributions: (i) different from [22] concerning the first-order model, combining with a distributed asynchronous event-triggered control algorithm, a more concise form of the event-triggered condition is designed to solve circle formation problems for both first- and second-order dynamics MASs; (ii) different from taking a complex-coordinate system transformation method in [29], the proposed strategy allows for a reduction of the number of control actions without significantly degrading performance using the simple-coordinate system transformation; and (iii) the resulting asynchronous model achieves the desired equilibrium points asymptotically while at least one agent with a positive next event interval exists, i.e., no trajectory generates in a finite time interval.The remainder of this paper is organized as follows. Preliminary definitions and problem formulation are given in Section2. In Section 3, a distributed circle formation control law for first-order MASs and the rigorous analysis of its performance are presented. Section 4 uses the event-triggered rule to address a distributed circle formation problem for second-order MASs. Section 5 discusses the simulation results, and Section 6 concludes the paper.


## 2. Preliminaries and Problem Formulation

### 2.1. Preliminaries

Letℝ and ℝN×N denote a set of real numbers and a N×N real matrix, respectively. For a finite set S, let S denote the number of its elements. For a vector or a matrix A, let A, A∞, and AT stand for its Euclidean norm, ∞-norm, and transpose, respectively. Let 1N and 0N be the N dimension column vectors with all entries being 1 and 0, respectively. Let a matrix diaga1,a2,…,aN represent a diagonal matrix whose diagonal entries are a1,a2,…,aN.LetG=V,ℰ,A be a directed graph, in which V=1,2,…,N is a set of nodes, ℰ=V×V stands for a set of edges, and A=aij∈ℝN×N denotes a weighted adjacency matrix. In the directed graph G, i,i∉ℰ for all i∈V, and for edge j,i∈ℰ, it starts from node j and ends up with node i. It is known that agent i can perceive state information from agent j. Therefore, agent j is called agent i’s in-neighbor. In addition, Ni=j∈V|j,i∈ℰ is applied to describe the in-neighbor set of agent i. Particularly, edge i,j links with the element aij of a weighted adjacency matrix A, aij>0 if and only if i,j∈ℰ; otherwise, aij=0. We use di=∑j=1Naij to denote the in-degree of i-th agent and define ℒ=D−A as Laplacian matrix of G, where D=diagd1,d2,…,dN. Subsequently, we can list the eigenvalues of ℒ in a descending order as λN≥⋯≥λ2≥λ1=0, where λN is the spectral radius of ℒ.We use the two lemmas listed below to facilitate analysis in this paper.Lemma 1.
For any givenx,y∈ℝ and a>0, the following two properties exist:(1)xy≤a2x2+12ay2;x2+y2≤x+y2,ifxy≥0.Lemma 2 (see [30]).
Given a directed graphG, which is composed of a spanning tree, a vector ξ=ξ1,ξ2,…,ξNT>0 satisfies ∑i=1Nξi=1 and ξTℒ=0N, where ξ denotes the left eigenvector corresponding to zero eigenvalues of the Laplacian matrix ℒ. Furthermore, ℒTΘ+ΘℒT is semipositive definite, where Θ=diagξ1,ξ2,…,ξN. After taking square root of each elements of Θ, we get ϒ=diagγ1,γ2,…,γN; consequently, γ=ξ,i=1,…,N.Lemma 3 (see [29]).
The linear matrix inequalities(2)QxSxSTxRx≥0,are equivalent to either one of the conditions listed as below:(3)Qx>0,Rx−STxQ−1xSx≥0,Rx>0,Qx−SxR−1xSTx≥0.


### 2.2. Problem Formulation

Given an MAS consisting ofN (N≥2) mobile agents, each agent is initially on a specific circle, and no pair of agents occupies the same position at the same time, as shown in Figure 1. For simplicity, we mark the agents counterclockwise and measure the position of the agent i,i∈1,2,…,N at an angle of xit. To be specific, the initial positions of all agents are set to satisfy the condition shown as(4)0≤x10<⋯<xi0<xi+10<⋯<xN0<2π.Figure 1
Agents distributed on a prescribed circle.Here, each agent has only two neighbors, that is, in front of or behind it. LetNi=i+,i− represent the two neighbors of the mobile agent i, where(5)i+=i+1,wheni=1,2,…,N−1,1,wheni=N,(6)i−=i−1,wheni=2,3,…,N,N,wheni=1.The dynamics of agenti are described as(7)x˙it=uit,i∈V,where xi and ui stand for the scalar state and the control input of agent i, respectively.Based on counterclockwise at timet, yit∈ℝ are presented as the angular distance from agent i to agent i+. Along with (5) and (6), it yields to(8)yit=xi+t−xit+2π,wheni=N,xi+t−xit,wheni=1,2,…,N−1,where yt=y1t,y2t,…,yNtT∈ℝN, and ∑i=1Nyit=2π always holds.Then, define a vectord=d1,d2,…di…,dNT to determine a desired circle formation, where di∈ℝ stands for the desired angular distance between agent i and agent i+. If d satisfies di>0 and ∑i=1Ndi=2π, the desired circle formation is achievable for MASs.


## 2.1. Preliminaries

Letℝ and ℝN×N denote a set of real numbers and a N×N real matrix, respectively. For a finite set S, let S denote the number of its elements. For a vector or a matrix A, let A, A∞, and AT stand for its Euclidean norm, ∞-norm, and transpose, respectively. Let 1N and 0N be the N dimension column vectors with all entries being 1 and 0, respectively. Let a matrix diaga1,a2,…,aN represent a diagonal matrix whose diagonal entries are a1,a2,…,aN.LetG=V,ℰ,A be a directed graph, in which V=1,2,…,N is a set of nodes, ℰ=V×V stands for a set of edges, and A=aij∈ℝN×N denotes a weighted adjacency matrix. In the directed graph G, i,i∉ℰ for all i∈V, and for edge j,i∈ℰ, it starts from node j and ends up with node i. It is known that agent i can perceive state information from agent j. Therefore, agent j is called agent i’s in-neighbor. In addition, Ni=j∈V|j,i∈ℰ is applied to describe the in-neighbor set of agent i. Particularly, edge i,j links with the element aij of a weighted adjacency matrix A, aij>0 if and only if i,j∈ℰ; otherwise, aij=0. We use di=∑j=1Naij to denote the in-degree of i-th agent and define ℒ=D−A as Laplacian matrix of G, where D=diagd1,d2,…,dN. Subsequently, we can list the eigenvalues of ℒ in a descending order as λN≥⋯≥λ2≥λ1=0, where λN is the spectral radius of ℒ.We use the two lemmas listed below to facilitate analysis in this paper.Lemma 1.
For any givenx,y∈ℝ and a>0, the following two properties exist:(1)xy≤a2x2+12ay2;x2+y2≤x+y2,ifxy≥0.Lemma 2 (see [30]).
Given a directed graphG, which is composed of a spanning tree, a vector ξ=ξ1,ξ2,…,ξNT>0 satisfies ∑i=1Nξi=1 and ξTℒ=0N, where ξ denotes the left eigenvector corresponding to zero eigenvalues of the Laplacian matrix ℒ. Furthermore, ℒTΘ+ΘℒT is semipositive definite, where Θ=diagξ1,ξ2,…,ξN. After taking square root of each elements of Θ, we get ϒ=diagγ1,γ2,…,γN; consequently, γ=ξ,i=1,…,N.Lemma 3 (see [29]).
The linear matrix inequalities(2)QxSxSTxRx≥0,are equivalent to either one of the conditions listed as below:(3)Qx>0,Rx−STxQ−1xSx≥0,Rx>0,Qx−SxR−1xSTx≥0.


## 2.2. Problem Formulation

Given an MAS consisting ofN (N≥2) mobile agents, each agent is initially on a specific circle, and no pair of agents occupies the same position at the same time, as shown in Figure 1. For simplicity, we mark the agents counterclockwise and measure the position of the agent i,i∈1,2,…,N at an angle of xit. To be specific, the initial positions of all agents are set to satisfy the condition shown as(4)0≤x10<⋯<xi0<xi+10<⋯<xN0<2π.Figure 1
Agents distributed on a prescribed circle.Here, each agent has only two neighbors, that is, in front of or behind it. LetNi=i+,i− represent the two neighbors of the mobile agent i, where(5)i+=i+1,wheni=1,2,…,N−1,1,wheni=N,(6)i−=i−1,wheni=2,3,…,N,N,wheni=1.The dynamics of agenti are described as(7)x˙it=uit,i∈V,where xi and ui stand for the scalar state and the control input of agent i, respectively.Based on counterclockwise at timet, yit∈ℝ are presented as the angular distance from agent i to agent i+. Along with (5) and (6), it yields to(8)yit=xi+t−xit+2π,wheni=N,xi+t−xit,wheni=1,2,…,N−1,where yt=y1t,y2t,…,yNtT∈ℝN, and ∑i=1Nyit=2π always holds.Then, define a vectord=d1,d2,…di…,dNT to determine a desired circle formation, where di∈ℝ stands for the desired angular distance between agent i and agent i+. If d satisfies di>0 and ∑i=1Ndi=2π, the desired circle formation is achievable for MASs.


## 3. Circle Formation Control for First-Order MASs

In this section, we first give the definition of the circle formation problem for first-order MASs as follows.Definition 1 (circle formation problem for first-order MASs).
Given an admissible circle formation characterized by d, a distributed control lawuit,yit,i=1,2,...,N can be designed, so that under any initial condition (4), the solution to system (7) converges to the equilibrium point x∗. That is, y∗=d.
A way-point control protocol based on sampled date was designed in [8]:(9)uit=di−di+di−yit−didi+di−yi−t,t>0,i∈V.
It has been proved that the continuous update control law (9) can move all agents to their equilibrium point x∗ but can waste communication bandwidth and unnecessary transmission energy. In order to solve this issue, an observer-based event-triggered circle formation control method for first-order MASs is proposed. It is noteworthy to mention that control actions of each agent only update at the event-triggered sampling instants, where continuous communication between neighboring agents is maintained. Here, we use an increasing sequence t0i,t1i,…,tki,… to represent the event instants of agent i, such that the state of agent i at the kth event instant is described as yitki. Note that each agent has its own event sequence since all agents are triggered asynchronously.
According to event-triggered strategies, we design the distributed circle formation control law for agenti as(10)uit=di−di−+diyitki−didi−+diyi−tki−,t∈tki,tk+1i,where tki−≜argminl∈ℕ,t≥tli−t−tli− represents the last event instant of agent i− and yitki denotes the observer angular distance.
From (10), agent i’s controller only updates at its own event sequence t0i,t1i,…,tki,…. For simplicity, let y^it=yitki; consequently, the control law (10) can be represented as(11)uit=didi−di−+diδ^it−δ^i−t,t∈tki,tk+1i,where δ^it=y^it/di, δ^i−t=y^i−t/di.
Replacing (2) and (5) into (7), the closed-loop form of agent i is rearranged by using δi as(12)δ˙it=∑j∈Nidjdi+djδ^jt−δ^it,t≥0.
Defineeit=δ^it−δit; then, a compact form of system dynamics is(13)δ˙t=−LdTδt+et,t∈tki,tk+1i,where δt=δ1t,δ2t,…,δNt∈ℝN, et=e1t,e2t,…,eNt∈ℝN, and(14)Ld=d2d2+d1+dNdN+d1−d1d2+d1…0−d1dN+d1−d2d2+d1d3d3+d2+d1d2+d1…00⋮⋮⋮⋮⋮00…dNdN+dN−1+dN−2dN−1+dN−2−dN−1dN+dN−1−dNdN+d10…−dNdN+dN−1d1dN+d1+dN−1dN+dN−1.
According to the control law (5) and MAS (1), the circle formation control for first-order MASs is solvable by Theorem 1.Theorem 1.
For any admissible circle formation characterized by d, considering system (7) and the designed control law (5) on the digraph G, the circle formation problem can be solved when the event-triggered condition is designed as(15)fit=eit−σγiδ¯iϒLdT,0<σ<1,where δ¯i is the ith elements of δ¯=δ¯1,δ¯2,…,δ¯NT≜LdTδ, ϒ is the same diagonal matrix as in Lemma 2, and γi is the ith diagonal element of matrix ϒ. Moreover, in system (7), there exists at least one agent m∈V that the next interevent interval is strictly positive under the event-triggered condition (15).Proof.
ALyapunov function candidate is taken into consideration(16)Vt=14δTtLdΘ+ΘLdTδt,where Θ is the same diagonal matrix as Lemma 2, such that LdΘ+ΘLdT is semipositive definite.
Accordingly,V˙t≤0 and V˙t=0 if and only if the circle formation problem can be solved. The derivative of the Lyapunov function (16) along trajectories of the system is derived as(17)V˙t=δTtLdΘ−LdTδt+et=−δTtLdΘLdTδt−δTtLdΘLdTet≤−ϒLdTδt2+ϒLdTδtϒLdTet.
After enforcing the event condition (15), we get ϒLdTet≤ϒLdTet≤σϒLdTδ. Thus, (17) is rewritten as(18)V˙t≤ϒLdTδt2σ−1≤ϒδ¯t2σ−1.
Because0<σ<1, we can get V˙t≤0 and V˙t=0 if and only if the circle formation problem can be solved.
Next, we explain the achievement of the circle formation in detail. According to the Lemma2, we get(19)∑i=1Nξiδit+1=∑i=1Nξiδit=⋯=∑i=1Nξiδi0.
Combining (10) and (15), all conditions result in limt⟶∞δit=limt⟶∞δjt=∑i=1Nξiδi0=c, where c∈ℝ remains constant. In addition, ∑i=1Nyi=2π, ∀t≥0 always satisfies and ∑i=1Ndi=2π. Together with yit=diδit, we obtain c=1. More exactly, limt⟶∞yt=d, which indicates that the designed circle formation can be solved in first-order MASs.
For agenti, the period eit/γiδ¯i, increasing from 0 to σ/ϒLdT, is regarded as the event interval between tk+1i and tki. Define m=argmaxi∈Vγiδ¯i; therefore, agent m stands for the maximum norm of γiδ¯i among all the agents, which implies(20)emtγmδ¯m≤etγmδ¯m≤Netϒδ¯.
From (20), the time emt/γmδ¯m attaining σ/ϒLdT is longer than Net/ϒδ¯. That is, τm>τ, where τm represents positive interval tk+1m−tkm lower bound and τ is the time et/ϒδ¯ increasing from 0 to σ/NϒLdT. Thereby, the time derivative of et/ϒδ¯ is(21)ddtetϒδ¯=ddtetTet1/2δ¯Tϒϒδ¯1/2=ete˙tetϒδ¯−δ¯Tϒϒδ¯˙etϒδ¯3=−etϒ−1ϒδ¯+LdTetetϒδ¯−δ¯TϒϒLdTδ¯+LdTetetϒδ¯2ϒδ¯≤ϒ−1ϒδ¯+ϒLdTetϒδ¯+ϒLdTϒ−1ϒδ¯+ϒLdTetetϒδ¯2≤ϒ1+etϒLdTϒδ¯ϒδ¯2.
Usingβ to replace et/ϒδ¯, it yields to β˙≤ϒ−11+ϒLdTβ2. Here, β≤αt,α0, where αt,α0 is the solution of α˙t,α0=ϒ−11+ϒLdTαt,α02, α0,α0=α0. According to(22)dαϒ−11+ϒLdTαt,α02=dt,we know that there exists the interval t which satisfies ατ,0=σ/ϒLdT, such that the event interval between instants tk and tk+1 is lower bounded. By solving the difference equation (22), it yields to(23)τ=dατ,0ϒ−11+ϒLdTατ,0=σ1+σϒLdTϒ−1.
Calculated from (23), we obtain τ=σ/N+σϒLdTϒ−1, where τ is the time et/ϒδ¯ from 0 to σ/NϒLdT, which derives to agent m’ lower bound of interval between two event instants:(24)τm=σN+σϒLdTϒ−1.
Fromτm>0, we conclude that in system (7), there exists at least one agent m∈V, the next interevent interval of which is strictly positive under event-triggered condition (15).


## 4. Circle Formation Control for Second-Order MASs

Given a second-order MAS withN agents(25)v˙it=uit,x˙it=vit,i=1,2,…,N,where xi∈ℝ is the angular state of agent i, vit∈ℝ is its angular velocity state, and ui∈ℝ denotes its control input.Similar to transformation of (8), we obtain(26)ζit=vi+t−vit,i=1,2,…,N,where ζt=ζ1t,ζ2t,…,ζNt∈ℝN and ∑i=1Nζi=0 always holds.The definition of the circle formation problem for second-orderMASs is described as follows.Definition 2.
(circle formation problem for second-order MASs). Considering an admissible circle formation, which is characterized by d, we can design a distributed control lawuit,yit,ζit,i=1,2,…,N, such that the solution to system (25) converges to the equilibrium points x∗ under any initial condition (4). Namely, y∗=d and ζ=0N are satisfied.
Letζ^it=ζitki and ζ^i−=ζi−tk,i− stand for the observer angular velocity distance of agent i and i−, respectively. From [8], the control law for agent i can be designed as(27)uit=di−di+di−φy^it+ζ^it−didi+di−φy^i−t+ζ^i−t,i=1,2,…N,where ζ^it=ζitki, ζ^i−=ζi−tk,i− and the positive control gain φ is determined in sequel.
For simplicity, we have(28)ei,yt=y^it−yit,ei,ζt=ζ^it−ζit,t∈tki,tk+1i.
Substituting (8), (26), and (27) into (25), the closed-loop system is written as(29)y˙tζ˙t=0IN−φLd−φLdytζt+00−φLd−φLdeyteζt,where eyt=e1,yt,e2,yt,…,eN,yt∈ℝN, eζt=e1,ζt,e2,ζt,…,eN,ζt∈ℝN, and Ld is the same matrix as in (14).
DefineD=diagd1,d2,…,dN; we then have the coordinate transformation as(30)δt=D−1yt,θt=D−1ζt,e1t=D−1eyt,e2t=D−1eζt.
Consequently, system (29) is rearranged as(31)δ˙tθ˙t=0IN−φLdT−φLdTδtθt+00−φLdT−φLdTe1te2t.
Givingωt=δTt,θTtT and e^t=e1Tt,e2TtT, system (31) can be rewritten as(32)ω˙t=0IN−φLdT−φLdTωt+00−φLdT−φLdTe^t.
Before designing any event-triggered conditions, theLyapunov function candidate is given as(33)Vt=12ωTtrLdΘLdTk1LdΘk1ΘLdT12νLdΘ+ΘLdTωt,where r, k1, and ν are positive constants.Theorem 2.
TheLyapunov function (33) satisfies system (25) when the condition(34)r≥2k12oνξmax,holds simultaneously, where ξmax is considered as the maximum element of vector ξ in Lemma 2 and o denotes a positive constant.Proof.
When theLyapunov function equals to zero, the circle formation problem can be solved. In this case, the circle formation cannot be achieved. According to (33), it yields to(35)2Vt=rδTtLdΘLdTδt+k1δTtLdΘθt+k1θTtΘLdTδt+θTt12νLdΘ+ΘLdTθt.
Fromδ¯t and Lemma 2, we can observe that the desired circle formation is achieved by rδTtLdΘLdTδt=rδ¯TtΘδ¯t=r∑i=1Nξiδ¯it2=0, θTt1/2νLdΘ+ΘLdTθt=0 and semipositive definite matrix 1/2νLdΘ+ΘLdTν.
We define a positive constant o as(36)o≜minLdTθ≠0,θ≠0θTtLdΘ+ΘLdTθtθTtθt.
SinceLdΘ+ΘLdT can be taken as positive (semipositive) definite with a single zero eigenvalue, there exists a unitary matrix P=p1,p2,…,pN∈ℝN×N such that LdΘ+ΘLdT=PSPT, where S denotes diagonal matrix with diags1,s2,…,sN. In addition, si is the eigenvalue of LdΘ+ΘLdT associated with eigenvector pi. Note that assuming s1=0, the corresponding eigenvector p1=1N. By defining ς=ς1,ς2,…,ςN≜PTθt, we get(37)o=minLdTθ≠0,θ≠0θTtLdΘ+ΘLdTθt=minLdTθ≠0,θ≠0θTtPSPTθt=minLdTθ≠0,θ≠0ςTSς=minLdTθ≠0,θ≠0∑i=1Nsiςi2≥0.
From (37), we observe that o=0 if and only if LdTPς≠0, ςTς=1 and ς2=⋯=ςN=0, which indicates that ς1=1 or ς1=−1. Without loss of generality, ς1=1 can be used for further analysis. In that case, we have LdTPς=LdTp1,p2,…,pN1,0,…,0T=LdTp1=LdT1N=0, which violates the assumptions LdTPς≠0. Thus, o>0 when the circle formation is not achieved. From o, we conclude that oθTtθt≤θTtLdΘ+ΘLdTθt. Therefore, we have(38)Vt≥12ωTtrLdΘLdTk1LdΘk1ΘLdT12νoωt.
Also, according to Lemma3, 1/2ωTtrLdΘLdTk1LdΘk1ΘLdT1/2νoωt is semipositive definite when 1/2νo>0 and(39)rLdΘLdT−rLdΘLdT12νo−1k1ΘLdT=LdΘLdTr−Ld2k12oνΘ2LdT≥0.
From (39), we draw a conclusion that r≥2k12/oνξi. The Lyapunov function (33) is semipositive definite and if the circle formation problem is solvable, it equals to 0. Therefore, this candidate Lyapunov function (33) satisfies the second-order MAS (25).Theorem 3.
For any admissible circle formation characterized by d, taking into account system (25) and the designed control law (27) over the digraph G, the circle formation problem can be solved when the event-triggered condition is designed as(40)fit=φϒLdT2k1+ν2ke1it2+e2it2−σφk1−φkk1δ˜it2−σφν−k2−φkνθ˜it2,where 0<σ<1, k, k1, k2, and ν stand for positive constants and δ˜it and θ˜it are the i-th element of vector δ˜t and θ˜t, respectively.Letδ˜Ttθ˜TtT=I2⊗ϒδ¯Ttθ¯TtT and the conditions(41)φk1−φkk1>0,φν−k2−φkν>0,hold simultaneously; there exists at least one agent g∈N for which the next interevent interval is strictly positive under the event-triggered condition (40) in system (25).Proof.
By combining (33) and (36), we have(42)Vt=12ωTtrLdΘLdTk1LdΘk1ΘLdT12νLdΘ+ΘLdTωt=12ωTtrLdΘLdTk1LdΘk1ΘLdTνLdΘωt.
Taking time derivative ofLyapunov function (33) along all trajectories of system (29), we get(43)V˙t=12ωTtrLdΘLdTk1LdΘk1ΘLdTνLdΘω˙t=ωTtrLdΘLdTk1LdΘk1ΘLdTνLdΘ×0IN−φLdT−φLdTωt+00−φLdT−φLdTe^t=ωTt−φk1LdΘLdTrLdΘLdT−φk1LdΘLdT−φνLdΘLdTk1ΘLdT−φνLdΘLdTωt+ωTtrLdΘLdTk1LdΘk1ΘLdTνLdΘ00−φLdT−φLdTe^t.
Then, (43) can be classified into two parts. The former part of (43) is computed by(44)ωTt−φk1LdΘLdTrLdΘLdT−φk1LdΘLdT−φνLdΘLdTk1ΘLdT−φνLdΘLdTωt=−φk1δTtLdΘLdTδt+k1θTtΘLdTθt−φνθTtLdΘLdTθt+δTtr−φk1−φνLdΘLdTθt.
Byr=φk1−ν and condition (41) and (44) implies(45)−φk1δTtLdΘLdTδt+k1θTtΘLdTθt−φνθTtLdΘLdTθt+δTtr−φk1−φνLdΘLdTθt≤−φk1δTtLdΘLdTδt−φν−k2θTtΘLdΘLdTθt.
According to Lemma1, the latter part of (43) is rearranged into(46)ωTtrLdΘLdTk1LdΘk1ΘLdTνLdΘ00−φLdT−φLdTe^t≤φk1δTtLdΘLdTe1t+φk1δTtLdΘLdTe2t+φνθTtLdΘLdTe1t+φνθTtLdΘLdTe2t=φk1δ˜TtϒLdTe1t+φk1δ˜TtϒLdTe2t+φνθ˜TtϒLdTe1t+φνθ˜TtϒLdTe2t≤kk1φδ˜t2+kνφθ˜t2+φk1+νϒLdT22ke1t2+e2t2.
Combining (45) and (46), (43) can be written as(47)V˙t≤−φk1+kk1φδ˜t2+−φν+kνφθ˜t2+φk1+νϒLdT22ke1t2+e2t2.
In view of the event-triggered function (40) satisfying fit<0, we conclude that(48)φϒLdT2k1+ν2ke1t2+e2t2≤σφk1−φkk1δ˜t2+σφν−k2−φkνθ˜t2.
From (48), we summarize that the Lyapunov function (33) is negative unless the circle formation for the second-order MAS is achievable. Moreover, we have ω˜t⟶∞t=0, that is δ˜t⟶∞t=0 and θ˜t⟶∞t=0. Thus, we get limt⟶∞δit=limt⟶∞δjt=c, limt⟶∞θit=limt⟶∞θjt=l, c,l∈ℝ. Note that yt=Dδt and ζt=Dθt. Then, ∑i=1Nyi=2π, ∑i=1Nζi=0, ∀t≥0 always satisfies and ∑i=1Ndi=2π. We conclude that c=1, l=0. To be more precise, limt⟶∞yt=d,andlimt⟶∞ζt=0.
The result shows that the circle formation can be achieved by all mobile agents. Furthermore, we illustrate the interevent timestk+1i−tki are positive lower bounded.
Using event-triggered condition (40), we obtain that the interval tk+1i−tki is the time φϒLdT2k1+ν/2ke1it2+e2it2 that increases from 0 to −σφk1−φkk1δ˜it2−σφν−k2−φkνθ˜it2.
Here, we useτi to represent the interval. Then, τi is longer or equal to the time ϒLdTk1+ν/2kφe1it2+e2it2 increasing from 0 to σμminδ˜it2+θ˜it2, where μmin=minφk1−φkk1,φν−k2−φkν.
Letτ^i denote the time e1it2+e2it2/δ˜it2+θ˜it2 from 0 to2σkμmin/k1+νφϒLdT. From the analysis, the k+1-th event of agent i occurs after the time tki+τ^i. Similar to the definition of m, we define g=argmaxi∈Vδ˜it2+θ˜it2, which follows e1gt2+e2gt2/δ˜gt2+θ˜gt2≤Ne1t2+e2t2/δ˜t2+θ˜t2=Ne^t2/ω˜t2.
Referring to the first-order method, the time derivative ofe^t/ω˜t is(49)de^tdtω˜t≤e^˙tω˜t+e^tω˜˙tω˜t2,where ω˜˙t has(50)ω˜˙t=ϒ00ϒLdTθt−φLdTLdTδt+e1t−θt−e2t≤0100ω˜t+00φϒLdTϒ−1φϒLdTϒ−1ω˜+I2⊗ϒLdTe^t≤1+2φ2ϒLdTϒ−1ω˜t+2φ2ϒLdTϒ−1ϒLdTe^t≤1+2φ2ϒLdTϒ−1ω˜t+ϒLdTe^t,and e^˙t=ω˙t is(51)e^˙t=ω˙t=θtφLdTδt+e1t+φLdTθt+e2t≤1+2φ2ϒ−1ω˜t+ϒLdTe^t.
Combining (50) and (51), (49) is rearranged into(52)de^tdtω˜t≤1+2φ2ω˜t+ϒLdTetϒω˜t+e^t1+2φ2ϒLdTω˜t+ϒLdTe^tϒω˜t2≤ϒ−1χ+2φ1+ϒLdTe^tω˜t2,where χ=max1,1/ϒLdT.
Additionally, define(53)Ξ=ϒ−11+2φ2,Φ=de^tdtω˜t,for which, Φ˙≤Ξ1+ϒLdTΦ2. Here, Φ satisfies the bound Φ≤ϕt,ϕ0, where ϕt,ϕ0 is the solution of ϕ˙t,ϕ0=Ξ1+ϒLdT2, ϕ0,ϕ0=ϕ0. Using τ˜g to denote the time that ϕt,ϕ0 grows from 0 to 2σkμmin/k1+νφϒLdT, we get dϕ/1+ϒLdTϕ2Ξ=dt.
Consequently,τ^g is longer than τ˜g that satisfies(54)ϕτ˜g,01+ϒLdTϕτ˜g,0=τ˜gΞ,where ϕτ˜g,0=2σkμmin/Nk1+νγϒLdT.
Thus,τ˜g=2σkμmin/ΞϒLdTγNk1+ν+2σkμmin. From τg>τ^g>τ˜g, we conclude that in system (25), there exists at least one agent g∈N that the next interevent interval is strictly positive under event-triggered condition (40). The proof is complete.


## 5. Simulation Examples

Considering an MAS (7) consisting of six agents, the desired distances between two adjacent agents are set to d=π/8,π/2,3π/8,π/2,π/3,π/6T, and the initial values of the MAS are randomly generated to satisfy (8). In each case, we apply the same initial settings, i.e., same angular positions and same admissible circle formations. Additionally, the unique normalized positive left eigenvector of LdT with respect to eigenvalue 0 is ξ=0.06250.0.250.18750.250.16670.0833T. In order to show advantages of the event-triggered control strategy, it is worth pointing out that the event detection of all those simulations is implemented in a sampled-data fashion, which is proved to be effective, and there is at least one trigger interval with a positive bound. Therefore, the sampling periods h in real-time control is chosen as 0.2s.


### 5.1. Simulation of the First-Order MAS

By the permitted range0<σ<1, we set σ=0.9 to ensure condition (15) holds in real-time control. Simulation results are shown in Figures 2 and 3. Figure 2(a) shows the evolution difference between the event-triggered angular distance and the expected counterpart, while Figure 2(b) reveals the event sequence of each agent. Figure 3 illustrates the fluttering of the measurement error yitki−yit.Figure 2
Simulation results of circle formation control for first-order MASs. (a) The evolution ofyitki−di for i=1,2,…,N when h=0.2s. (b) The sequence of event-triggered times for i=1,2,…,N when h=0.2s.
(a)
(b)Figure 3
The evolution ofyitki−yit for i=1,2,…,N when h=0.2s.We can see from the simulation results that the desired circle formation can be asymptotically achieved by the proposed control law (10) in distributed MASs. We also calculate the average interevent time of all mobile agents havg=0.6059 from Figure 2, and the result indicates that our method can reduce the amount of control update for first-order MASs.


### 5.2. Simulation of the Second-Order MAS

We choose coefficientsσ=0.9, φ=1.2, k2=0.5, k1=2, k=0.8, o=2, and ν=2 to ensure condition (40) holds in real-time control. Simulation results are shown in Figures 4 and 5. Figure 4(a) shows the evolution difference between the event-triggered angular distance and the expected counterpart, while Figure 4(b) reveals event sequence of each agent. Figure 5 shows the fluttering of the measurement error yitki−yit, ζitki−ζit.Figure 4
Simulation results of circle formation for second-order MASs. (a) The evolution ofyitki−di for i=1,2,…,N when h=0.2s. (b) The sequence of event-triggered times for i=1,2,…,N when h=0.2s.
(a)
(b)Figure 5
Simulation results of the fluttering of the measurement error. (a) The evolution ofyitki−yit for i=1,2,…,N when h=0.2s. (b) The evolution of ζitki−ζit for i=1,2,…,N when h=0.2s.
(a)
(b)We observe from the simulation results that the desired circle formation can also be asymptotically achieved under the proposed control law (27). Similarly, we calculate the average interevent time of all mobile agents havg=0.3545 from Figure 4. We conclude that our proposed method has the advantages of reducing the amount of control update for the second-order MASs.To sum up, compared with traditional time-planning controls in the same simulation environment, the event-triggered control-based formation control strategies proposed in this paper show the effectiveness for both first- and second-order MASs. For the second-order MAS, the average interevent time of all mobile agents is less than the first-order MAS case due to its more complicated modeling. However, from an application point of view, the proposed control strategy for second-order MASs is much more accurate than first-order MASs.


## 5.1. Simulation of the First-Order MAS

By the permitted range0<σ<1, we set σ=0.9 to ensure condition (15) holds in real-time control. Simulation results are shown in Figures 2 and 3. Figure 2(a) shows the evolution difference between the event-triggered angular distance and the expected counterpart, while Figure 2(b) reveals the event sequence of each agent. Figure 3 illustrates the fluttering of the measurement error yitki−yit.Figure 2
Simulation results of circle formation control for first-order MASs. (a) The evolution ofyitki−di for i=1,2,…,N when h=0.2s. (b) The sequence of event-triggered times for i=1,2,…,N when h=0.2s.
(a)
(b)Figure 3
The evolution ofyitki−yit for i=1,2,…,N when h=0.2s.We can see from the simulation results that the desired circle formation can be asymptotically achieved by the proposed control law (10) in distributed MASs. We also calculate the average interevent time of all mobile agents havg=0.6059 from Figure 2, and the result indicates that our method can reduce the amount of control update for first-order MASs.


## 5.2. Simulation of the Second-Order MAS

We choose coefficientsσ=0.9, φ=1.2, k2=0.5, k1=2, k=0.8, o=2, and ν=2 to ensure condition (40) holds in real-time control. Simulation results are shown in Figures 4 and 5. Figure 4(a) shows the evolution difference between the event-triggered angular distance and the expected counterpart, while Figure 4(b) reveals event sequence of each agent. Figure 5 shows the fluttering of the measurement error yitki−yit, ζitki−ζit.Figure 4
Simulation results of circle formation for second-order MASs. (a) The evolution ofyitki−di for i=1,2,…,N when h=0.2s. (b) The sequence of event-triggered times for i=1,2,…,N when h=0.2s.
(a)
(b)Figure 5
Simulation results of the fluttering of the measurement error. (a) The evolution ofyitki−yit for i=1,2,…,N when h=0.2s. (b) The evolution of ζitki−ζit for i=1,2,…,N when h=0.2s.
(a)
(b)We observe from the simulation results that the desired circle formation can also be asymptotically achieved under the proposed control law (27). Similarly, we calculate the average interevent time of all mobile agents havg=0.3545 from Figure 4. We conclude that our proposed method has the advantages of reducing the amount of control update for the second-order MASs.To sum up, compared with traditional time-planning controls in the same simulation environment, the event-triggered control-based formation control strategies proposed in this paper show the effectiveness for both first- and second-order MASs. For the second-order MAS, the average interevent time of all mobile agents is less than the first-order MAS case due to its more complicated modeling. However, from an application point of view, the proposed control strategy for second-order MASs is much more accurate than first-order MASs.


## 6. Conclusion

This paper investigated the circle formation control problem for both first- and second-order MASs under unbalanced directed networks with limited resource constraints. We first designed the observer-based event-triggered algorithm to reduce dependence on resources, in which, when the value of the event-triggered condition exceeds zero, the agent’s controller will update its states simultaneously. Moreover, it is a fundamental and practical aspect to observe neighbor information on a regular or better basis continuously. Then, we proved that if there is a spanning tree in the underlying graph, the MASs can achieve the desired circle formation by the proposed control laws, and there is at least one agent whose the next interevent interval is strictly positive. At last, we gave numerical simulation examples to illustrate that the proposed event-triggered circle formation control strategies are effective for both first- and second-order MASs. For future work, we will extend our research to more practical operations, e.g., considering the effect of time delays in communication networks, input saturation constraints, and weak links.



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*Source: 4715315-2020-03-24.xml*