Research Article | Open Access
Peng Xu, Guangming Xie, Jin Tao, Minyi Xu, Quan Zhou, "Observer-Based Event-Triggered Circle Formation Control for First- and Second-Order Multiagent Systems", Complexity, vol. 2020, Article ID 4715315, 12 pages, 2020. https://doi.org/10.1155/2020/4715315
Observer-Based Event-Triggered Circle Formation Control for First- and Second-Order Multiagent Systems
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.
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 . 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 . The results lead to a guaranteed performance with a fixed sampling rate requirements concerning the optimizing schedules and sampling rates. In , 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 . In , 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.  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 .
Different from previous studies, especially , 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 . In comparison to the literature, we have three main contributions: (i) different from  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 , 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 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.
2. Preliminaries and Problem Formulation
Let and denote a set of real numbers and a real matrix, respectively. For a finite set , let denote the number of its elements. For a vector or a matrix A, let , , and stand for its Euclidean norm, -norm, and transpose, respectively. Let and be the N dimension column vectors with all entries being 1 and 0, respectively. Let a matrix represent a diagonal matrix whose diagonal entries are .
Let be a directed graph, in which is a set of nodes, stands for a set of edges, and denotes a weighted adjacency matrix. In the directed graph , for all , and for edge , 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, is applied to describe the in-neighbor set of agent i. Particularly, edge links with the element of a weighted adjacency matrix , if and only if ; otherwise, . We use to denote the in-degree of i-th agent and define as matrix of , where . Subsequently, we can list the eigenvalues of in a descending order as , where is the spectral radius of .
We use the two lemmas listed below to facilitate analysis in this paper.
Lemma 1. For any given and , the following two properties exist:
Lemma 2 (see ). Given a directed graph , which is composed of a spanning tree, a vector satisfies and , where ξ denotes the left eigenvector corresponding to zero eigenvalues of the matrix . Furthermore, is semipositive definite, where . After taking square root of each elements of , we get ; consequently, .
Lemma 3 (see ). The linear matrix inequalitiesare equivalent to either one of the conditions listed as below:
2.2. Problem Formulation
Given an MAS consisting of N () 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 at an angle of . 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. Let represent the two neighbors of the mobile agent i, where
The dynamics of agent i are described aswhere and stand for the scalar state and the control input of agent i, respectively.
Then, define a vector to determine a desired circle formation, where stands for the desired angular distance between agent i and agent . If d satisfies and , the desired circle formation is 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 law can be designed, so that under any initial condition (4), the solution to system (7) converges to the equilibrium point . That is, .
A way-point control protocol based on sampled date was designed in :It has been proved that the continuous update control law (9) can move all agents to their equilibrium point 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 to represent the event instants of agent i, such that the state of agent i at the kth event instant is described as . 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 aswhere represents the last event instant of agent and denotes the observer angular distance.
From (10), agent i’s controller only updates at its own event sequence . For simplicity, let ; consequently, the control law (10) can be represented aswhere , .
Replacing (2) and (5) into (7), the closed-loop form of agent i is rearranged by using asDefine ; then, a compact form of system dynamics iswhere , , andAccording 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 , the circle formation problem can be solved when the event-triggered condition is designed aswhere is the ith elements of , is the same diagonal matrix as in Lemma 2, and is the ith diagonal element of matrix . Moreover, in system (7), there exists at least one agent that the next interevent interval is strictly positive under the event-triggered condition (15).
Proof. A function candidate is taken into considerationwhere is the same diagonal matrix as Lemma 2, such that is semipositive definite.
Accordingly, and if and only if the circle formation problem can be solved. The derivative of the function (16) along trajectories of the system is derived asAfter enforcing the event condition (15), we get . Thus, (17) is rewritten asBecause , we can get and 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 getCombining (10) and (15), all conditions result in , where remains constant. In addition, , always satisfies and . Together with , we obtain . More exactly, , which indicates that the designed circle formation can be solved in first-order MASs.
For agent i, the period , increasing from 0 to , is regarded as the event interval between and . Define ; therefore, agent m stands for the maximum norm of among all the agents, which impliesFrom (20), the time attaining is longer than . That is, , where represents positive interval lower bound and τ is the time increasing from 0 to . Thereby, the time derivative of isUsing β to replace , it yields to . Here, , where is the solution of , . According towe know that there exists the interval t which satisfies , such that the event interval between instants and is lower bounded. By solving the difference equation (22), it yields toCalculated from (23), we obtain , where τ is the time from 0 to , which derives to agent m’ lower bound of interval between two event instants:From , we conclude that in system (7), there exists at least one agent , 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 with N agentswhere is the angular state of agent i, is its angular velocity state, and denotes its control input.
Similar to transformation of (8), we obtainwhere and always holds.
The definition of the circle formation problem for second-order 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 , such that the solution to system (25) converges to the equilibrium points under any initial condition (4). Namely, and are satisfied.
Let and stand for the observer angular velocity distance of agent i and , respectively. From , the control law for agent i can be designed aswhere , and the positive control gain φ is determined in sequel.
For simplicity, we haveSubstituting (8), (26), and (27) into (25), the closed-loop system is written aswhere , , and is the same matrix as in (14).
Define ; we then have the coordinate transformation asConsequently, system (29) is rearranged asGiving and , system (31) can be rewritten asBefore designing any event-triggered conditions, the function candidate is given aswhere r, , and ν are positive constants.
Proof. When the 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 toFrom and Lemma 2, we can observe that the desired circle formation is achieved by , and semipositive definite matrix .
We define a positive constant o asSince can be taken as positive (semipositive) definite with a single zero eigenvalue, there exists a unitary matrix such that , where S denotes diagonal matrix with . In addition, is the eigenvalue of associated with eigenvector . Note that assuming , the corresponding eigenvector . By defining , we getFrom (37), we observe that if and only if , and , which indicates that or . Without loss of generality, can be used for further analysis. In that case, we have , which violates the assumptions . Thus, when the circle formation is not achieved. From o, we conclude that . Therefore, we haveAlso, according to Lemma 3, is semipositive definite when andFrom (39), we draw a conclusion that . The function (33) is semipositive definite and if the circle formation problem is solvable, it equals to 0. Therefore, this candidate 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 , the circle formation problem can be solved when the event-triggered condition is designed aswhere , k, , , and ν stand for positive constants and and are the i-th element of vector and , respectively.
Proof. By combining (33) and (36), we haveTaking time derivative of function (33) along all trajectories of system (29), we getThen, (43) can be classified into two parts. The former part of (43) is computed byBy and condition (41) and (44) impliesAccording to Lemma 1, the latter part of (43) is rearranged intoCombining (45) and (46), (43) can be written asIn view of the event-triggered function (40) satisfying , we conclude thatFrom (48), we summarize that the function (33) is negative unless the circle formation for the second-order MAS is achievable. Moreover, we have , that is and . Thus, we get , , . Note that and . Then, , , always satisfies and . We conclude that , . To be more precise, .
The result shows that the circle formation can be achieved by all mobile agents. Furthermore, we illustrate the interevent times are positive lower bounded.
Using event-triggered condition (40), we obtain that the interval is the time that increases from 0 to .
Here, we use to represent the interval. Then, is longer or equal to the time increasing from 0 to , where .
Let denote the time from 0 to. From the analysis, the -th event of agent i occurs after the time . Similar to the definition of m, we define , which follows
Referring to the first-order method, the time derivative of iswhere hasand isCombining (50) and (51), (49) is rearranged intowhere .