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

Marine Engineering College, Dalian Maritime University, Dalian 116026, China 2 e State Key Laboratory of Turbulence and Complex Systems, Intelligent Biomimetic Design Lab, College of Engineering, Peking University, Beijing 100871, China Institute of Ocean Research, Peking University, Beijing 100871, China Peng Cheng Laboratory, Shenzhen 518000, China Department of Electrical Engineering and Automation, Aalto University, Espoo 02150, Finland


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][2][3] and theoretical challenges of physical constraints [4][5][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][8][9][10]. e 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].
erefore, most existing results on formation control mainly rely on the ideal hypothesis [13][14][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][17][18][19][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]. e 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 timedependent 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 secondorder 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. e remainder of this paper is organized as follows. Preliminary definitions and problem formulation are given in Section 2. 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.

Preliminaries.
Let R and R N×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 A T stand for its Euclidean norm, ∞-norm, and transpose, respectively. Let 1 N and 0 N be the N dimension column vectors with all entries being 1 and 0, respectively. Let a matrix diag a 1 , a 2 , . . . , a N represent a diagonal matrix whose diagonal entries are a 1 , a 2 , . . . , a N .
} is a set of nodes, E � V × V stands for a set of edges, and A � [a ij ] ∈ R N×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. erefore, agent j is called agent i's inneighbor. In addition, N i � j ∈ V | (j, i) ∈ E is applied to describe the in-neighbor set of agent i. Particularly, edge (i, j) links with the element a ij of a weighted adjacency matrix A, a ij > 0 if and only if (i, j) ∈ E; otherwise, a ij � 0. We use d i � N j�1 a ij to denote the in-degree of i-th agent and define L � D − A as Laplacian matrix of G, where D � diag d 1 , d 2 , . . . , d N . Subsequently, we can list the eigenvalues of L in a descending order as λ N ≥ · · · ≥ λ 2 ≥ λ 1 � 0, where λ N is the spectral radius of L.
We use the two lemmas listed below to facilitate analysis in this paper.

Lemma 1.
For any given x, y ∈ R and a > 0, the following two properties exist: xy ≤ a 2 x 2 + 1 2a y 2 ; Lemma 2 (see [30]). Given a directed graph G, which is composed of a spanning tree, a vector ξ � [ξ 1 , ξ 2 , . . . , ξ N ] T > 0 satisfies N i�1 ξ i � 1 and ξ T L � 0 N , where ξ denotes the left eigenvector corresponding to zero eigenvalues of the Laplacian matrix L. Furthermore, Lemma 3 (see [29]). e linear matrix inequalities are equivalent to either one of the conditions listed as below: 2 Complexity

Problem Formulation.
Given an MAS consisting of N (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 x i (t). To be specific, the initial positions of all agents are set to satisfy the condition shown as Here, each agent has only two neighbors, that is, in front of or behind it.
e dynamics of agent i are described as where x i and u i stand for the scalar state and the control input of agent i, respectively. Based on counterclockwise at time t, y i (t) ∈ R are presented as the angular distance from agent i to agent i + . Along with (5) and (6), it yields to where

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 law u i (t, y i (t)), 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 * . at is, y * � d.
A way-point control protocol based on sampled date was designed in [8]: 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 (t i 0 , t i 1 , . . . , t i k , . . .) to represent the event instants of agent i, such that the state of agent i at the kth event instant is described as y i (t i k ). 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 agent i as where represents the last event instant of agent i − and y i (t i k ) denotes the observer angular distance.
From (10), agent i's controller only updates at its own event sequence (t i 0 , t i 1 , . . . , t i k , . . .). For simplicity, let y i (t) � y i (t i k ); consequently, the control law (10) can be represented as where Replacing (2) and (5) into (7), the closed-loop form of agent i is rearranged by using δ i as Figure 1: Agents distributed on a prescribed circle. where According to the control law (5) and MAS (1), the circle formation control for first-order MASs is solvable by eorem 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 where δ i is the i th elements of δ � [δ 1 , δ 2 , . . . , δ N ] T ≜ L T d δ, Υ is the same diagonal matrix as in Lemma 2, and c i 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. A Lyapunov function candidate is taken into consideration
where Θ is the same diagonal matrix as Lemma 2, such that L d Θ + ΘL T d is semipositive definite. Accordingly, _ V(t) ≤ 0 and _ V(t) � 0 if and only if the circle formation problem can be solved. e derivative of the Lyapunov function (16) along trajectories of the system is derived as After enforcing the event condition (15), we get (17) is rewritten as Because 0 < 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 Lemma 2, we get Combining (10) and (15), all conditions result in where c ∈ R remains constant. In addition, N i�1 y i � 2π, ∀t ≥ 0 always satisfies and N i�1 d i � 2π. Together with y i (t) � d i δ i (t), we obtain c � 1. More exactly, lim t⟶∞ y(t) � d, which indicates that the designed circle formation can be solved in first-order MASs.
For agent i, the period ‖e i (t)‖/c i δ i , increasing from 0 to σ/‖ΥL T d ‖, is regarded as the event interval between t i k+1 and t i k . Define m � argmax i∈V ‖c i δ i ‖; therefore, agent m stands for the maximum norm of c i δ i among all the agents, which implies 4 Complexity From (20), the time ‖e m (t)‖/‖c m δ m ‖ attaining σ/‖ΥL T d ‖ is longer than �� N √ ‖e(t)‖/‖Υδ‖. at is, τ m > τ, where τ m represents positive interval (t m k+1 − t m k ) lower bound and τ is the time e(t)/Υδ increasing from 0 to σ/ �� N √ ‖ΥL T d ‖. ereby, the time derivative of ‖e(t)‖/‖Υδ‖ is d dt Using β to replace ‖e(t)‖/‖Υδ‖, it yields to we know that there exists the interval t which satisfies α(τ, 0) � σ/‖ΥL T d ‖, such that the event interval between instants t k and t k+1 is lower bounded. By solving the difference equation (22), it yields to Calculated from (23), we obtain τ � σ/(( , which derives to agent m' lower bound of interval between two event instants: From τ m > 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).

Circle Formation Control for Second-Order MASs
Given a second-order MAS with N agents where x i ∈ R is the angular state of agent i, v i (t) ∈ R is its angular velocity state, and u i ∈ R denotes its control input. Similar to transformation of (8), we obtain where e definition of the circle formation problem for second-order MASs 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 law u i (t, y i (t), ζ i (t)), 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 � 0 N are satisfied. Let ) 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 ) and the positive control gain φ is determined in sequel.

Theorem 2. e Lyapunov function (33) satisfies system (25) when the condition
holds simultaneously, where ξ max is considered as the maximum element of vector ξ in Lemma 2 and o denotes a positive constant.
Proof. When the Lyapunov 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 From δ(t) and Lemma 2, we can observe that the desired circle formation is achieved by rδ T

We define a positive constant o as
Since L d Θ + ΘL T d can be taken as positive (semipositive) definite with a single zero eigenvalue, there exists a unitary matrix where S denotes diagonal matrix with diag s 1 , s 2 , . . . , s N . In addition, s i is the eigenvalue of L d Θ + ΘL T d associated with eigenvector p i . Note that assuming s 1 � 0, the corresponding eigenvector From (37), we observe that o � 0 if and only if L T d Pς ≠ 0, ς T ς � 1 and ς 2 � · · · � ς N � 0, which indicates that ς 1 � 1 or ς 1 � −1. Without loss of generality, ς 1 � 1 can be used for further analysis. In that case, we have L T d Pς � L T d [p 1 , p 2 , . . . , p N ][1, 0, . . . , 0] T � L T d p 1 � L T d 1 N � 0, which violates the assumptions L T d Pς ≠ 0. us, o > 0 when the circle formation is not achieved. From o, we conclude that Also, according to From (39), we draw a conclusion that r ≥ (2k 2 1 /o])ξ i . e Lyapunov function (33) is semipositive definite and if the circle formation problem is solvable, it equals to 0. erefore, this candidate Lyapunov function (33) satisfies the second-order MAS (25).

□ 6 Complexity
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 where 0 < σ < 1, k, k 1 , k 2 , and ] stand for positive constants and δ i (t) and θ i (t) are the i-th element of vector δ(t) and θ(t), respectively.
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 □ Taking time derivative of Lyapunov function (33) along all trajectories of system (29), we get where min )). From τ g > 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). e proof is complete.

Simulation Examples
Considering an MAS (7) consisting of six agents, the desired distances between two adjacent agents are set to d � [π/8, tπ/2n, q3π/8h, π/2x , 7π/3C, ; π/6] T , 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 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. erefore, the sampling periods h in real-time control is chosen as 0.2 s.

Simulation of the First-Order MAS.
By the permitted range 0 < 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 ‖y i (t i k ) − y i (t)‖.  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 h avg � 0.6059 from Figure 2, and the result indicates that our method can reduce the amount of control update for firstorder MASs.

Simulation of the Second-Order MAS.
We choose coefficients σ � 0.9, φ � 1.2, k 2 � 0.5, k 1 � 2, k � 0.8, o � 2, and ] � 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 ‖y i (t i k ) − y i (t)‖, ‖ζ i (t i k ) − ζ i (t)‖. 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 h avg � 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 eventtriggered 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.

Conclusion
is 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. en, 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.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.