Observer‐based formation tracking control for leader–follower multi‐agent systems
2018; Institution of Engineering and Technology; Volume: 13; Issue: 2 Linguagem: Inglês
10.1049/iet-cta.2018.5443
ISSN1751-8652
AutoresWei Zhao, Wenwu Yu, Huaipin Zhang,
Tópico(s)Adaptive Control of Nonlinear Systems
ResumoIET Control Theory & ApplicationsVolume 13, Issue 2 p. 239-247 Research ArticleFree Access Observer-based formation tracking control for leader–follower multi-agent systems Wei Zhao, Wei Zhao Jiangsu Provincial Key Laboratory of Networked Collective Intelligence and School of Mathematics, Southeast University, Nanjing, 210096 People's Republic of ChinaSearch for more papers by this authorWenwu Yu, Corresponding Author Wenwu Yu wwyu@seu.edu.cn Jiangsu Provincial Key Laboratory of Networked Collective Intelligence and School of Mathematics, Southeast University, Nanjing, 210096 People's Republic of China Department of Electrical Engineering, Nantong University, People's Republic of ChinaSearch for more papers by this authorHuaipin Zhang, Huaipin Zhang Institute of Advanced Technology, Nanjing University of Posts and Telecommunications, Nanjing, 210023 People's Republic of ChinaSearch for more papers by this author Wei Zhao, Wei Zhao Jiangsu Provincial Key Laboratory of Networked Collective Intelligence and School of Mathematics, Southeast University, Nanjing, 210096 People's Republic of ChinaSearch for more papers by this authorWenwu Yu, Corresponding Author Wenwu Yu wwyu@seu.edu.cn Jiangsu Provincial Key Laboratory of Networked Collective Intelligence and School of Mathematics, Southeast University, Nanjing, 210096 People's Republic of China Department of Electrical Engineering, Nantong University, People's Republic of ChinaSearch for more papers by this authorHuaipin Zhang, Huaipin Zhang Institute of Advanced Technology, Nanjing University of Posts and Telecommunications, Nanjing, 210023 People's Republic of ChinaSearch for more papers by this author First published: 10 January 2019 https://doi.org/10.1049/iet-cta.2018.5443Citations: 8AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract This study investigates the formation tracking control for second-order leader–follower multi-agent systems. To estimate the velocity and acceleration of the leader, distributed observers are constructed for each follower. Based on the observed states, the authors propose a novel distributed formation tracking control protocol and analyse two communication networks with and without communication delays. Then, they prove that each agent can follow the active leader closely and form the desired spatial pattern. Finally, simulation examples are provided to show the validity of the theoretical results. 1 Introduction Recently, we have witnessed growing interests in cooperative control of multi-agent systems (MASs) due to its practical potential in various applications including rendezvous [1], flocking [2], formation control [3], unmanned air vehicles [4] and so on. Especially formation control, as a crucial issue in the multi-agent cooperative control, which aims to cooperate a group of autonomous agents to evolve in a desired spatial pattern, has attracted considerable attention [5–14]. In general, the existing results on formation control for MASs can be categorised into three basic strategies, i.e. leader–following, behaviour-based, and virtual structure. Among these control strategies, the leader–following approach is preferred in many applications due to its simplicity and scalability. So far, there have been numerous works on the leader–following formation control problem [15–20]. In practical situations, however, it is difficult to obtain all the information of MASs due to technology limitations or environment disturbances. Hence, it is interesting and challenging to seek out an effective method to address this issue. To achieve the control goal, a common approach is to design an observer for each agent to estimate the unmeasured variables [21–28]. To estimate the speed of the leader, Gustavi and Hu [23] designed an observer-based formation control protocol using onboard sensor information. Hong et al. [21] proposed a neighbour-based estimate rule and feedback control strategy for each first-order follower to estimate unmeasured velocity of an active leader. Meanwhile, Hong et al. [22] further designed a distributed observer-based controller for each second-order follower. In [25], a neural network observer and multiple sliding surface-based control law for each following robot were designed to estimate the dynamics of the leader and track the leader robot in the desired separation and bearing, respectively. In [24, 29], distributed reduced-order observer-based consensus control protocol was proposed for each agent, which can only obtain the relative output measurements. [27] presented an observer-based control framework to solve the finite-time coordinated tracking problem for general linear MASs under general communication graphs. In [28], distributed extended state observer was constructed to estimate the follower's non-linearity, disturbance, and the leader's unknown control input simultaneously. Then, distributed extended state observer-based protocol was proposed to achieve the practical time-varying formation tracking for high-order non-linear MASs systems. All the above works on observer for MASs only consider idealised network environment, which is almost impossible in practice. Due to the limited bandwidth in a communication network, the time delay is an inescapable effect factor in formation control for MASs. It is well-known that time delay may degrade the performance of MASs and even destroy the system's stability. Thus, it is critical to address the formation control problem for MASs with communication delays. Extensive efforts have been put into the challenging problem [30–40]. In [31], the authors studied the tracking control problem of a set of discrete-time heterogeneous MASs with random communication delays represented by a Markov chain. Under the common assumption that the agents can communicate with their neighbours only during a sequence of discontinuous time intervals, Qin et al. [36] studied an observer-based formation control problem for second-order non-linear MASs with a leader where the velocity of the active leader cannot be obtained in real time. Qin et al. [37] further studied the problem with inherent delayed non-linear dynamics and intermittent communications by designing observers to estimate the leader's velocity using the intermittent neighbours-based information. Ge et al. [39] addressed the problem of cluster formation control for a networked MASs in the simultaneous presence of aperiodic sampling and communication delays. Li et al. [40] considered the adaptive event-triggered formation tracking control for second-order MASs with control protocols containing time-varying delay. Motivated by the above analysis, we investigate the formation tracking control problem for leader–follower MASs with communication delays. To estimate the velocity and acceleration of the leader, we construct distributed observers for each follower. Based on the observed states, we propose a novel distributed formation tracking controller for each follower. Finally, we prove the stability of the proposed control algorithm. The contributions of this paper are as follows: The leader's acceleration is unknown to all the followers and distributed observer is designed to estimate it for each follower. However, in [22, 27, 40], the leader's acceleration was regarded as some given policy known to all the followers in the observer or the control protocol design. In [40], the leader's control input is zero, which is much easier to deal with than the case with time-varying unknown or non-zero control input. In fact, the leader's acceleration or control input is often time-varying or unknown to its followers, it is practical to estimate it using obtained information. Compared with [41–43], when designing distributed observers and formation tracking control protocols for each follower, instead of using its neighbours full state information, only partial state information is used. The case with communication delays between the followers is also studied for the formation tracking control of MASs. In [39], communication delay is piecewise linear during aperiodic sampled instants. In [40], not only the communication delay but also its first derivative is bounded. While in our paper, the delays are just bounded and no other assumptions of communication delays are made, which satisfies broad situations. The remaining of this paper is structured as follows. In Section 2, we briefly summarise the concepts in algebraic graph theory and several lemmas to be used in this paper, and then describe the problem to be investigated. In Section 3, an observer-based formation tracking control protocol is designed and then it is extended to solve the formation tracking problem with communication delays. The convergence of the leader–follower MAS with the proposed protocols with/without communication delays is proved by the approach of Lyapunov function in Section 3. Simulation results are presented in Section 4 to demonstrate the effectiveness of the theoretical results. Finally, some conclusions are drawn in Section 5. Notations: Let be the Kronecher product. and denote the column vectors of appropriate dimensions whose elements are all ones and all zeros, respectively. and () denote the identity matrix and the zero matrix, respectively. The norm of a vector is defined as . The spectral norm of a matrix is defined as , where is eigenvalue of . For a matrix , and denote its minimal and maximal eigenvalues, respectively. 2 Preliminaries and problem formulation 2.1 Preliminaries In this paper, graph theory is utilised as a helpful mathematical tool to analyse the consensus tracking for MASs. The communication topology can be expressed by a weighted graph. Let be a weighted undirected graph of N nodes with the non-empty finite set of nodes , where a set of edges , is an adjacency matrix where represents the weighted of the information channel. If there exists a communication link between and , i.e. , then and they are called neighbours; otherwise, . The set of neighbours of agent i is denoted by . A path form i to j is an edge sequence of . If there is a path from i to j, then i and j are called connected. An undirected graph G is connected if all the pairs of distinct nodes are connected. The Laplacian matrix L of graph G is defined as , where with diagonal elements . In the undirected information exchange graph G, L has a simple zero eigenvalue with an associated eigenvector , and all the other eigenvalues are positive if and only if the graph is connected. In other words, L is positive semi-definite and has the property of . For the considered MAS with a reference leader, we concern another graph associated with the system consisting of N following agents and one leader. contains an undirected subgraph G and a node with directed edges from the leader to some of the following agents, where G is used to described the interaction topology of N following agents and represents the leader. However, since the leader is assumed self-active which moves independently, then follower i can receive information from the leader while the leader needs no information from any follower, i.e. the edges between and is unidirectional in . Thus, the connection weight is taken as positive constant if agent i is connected to the leader and otherwise is taken as zero. Let the leader's adjacency matrix B be the diagonal matrix whose i th diagonal element is . For convenience, define a new matrix , which has the following property. Lemma 1.If the graph contains a directed spanning tree with root and the subgraph G is undirected, then H is a symmetric and positive definite matrix [21]. Assumption 1.The undirected communication topology among the followers is connected. Assumption 2.For the leader, there exists at least one follower that has a directed path to the leader. Remark 1.The communication topology is said to be connected if contains a directed spanning tree with root and the subgraph associated with the followers is undirected. That is to say, the connectivity of does not require that the subgraph G is connected. However, under Assumptions 1 and 2 which are needed in the following proof, the connectivity of graph is easily obtained. Lemma 2.If , and (1)where is bounded and continuous for , continuous functions , and for , , and if there exists such that for , then we have (2)where , and is defined as [44]. Lemma 3.Let S be a symmetric matrix partitioned into block form [45] (3)where and are symmetric and square. Then if and only if (4)or equivalently (5) 2.2 Formation tracking problem of leader–follower MAS Consider a group of identical agents, where an agent labelled as 0 is assigned as the leader and the agents indexed by are referred to as followers. The dynamics of the followers is represented by (6)where and are the position and velocity of follower i, respectively, and is the control input of follower i. The dynamics of the active leader is described by (7)where are the position and velocity of the leader, respectively, is the leader's acceleration, and is the only measurable output of the leader. Furthermore, it is assumed the acceleration is modelled by (8)where is unknown but bounded: and . Remark 2.In practice, is the physical quantity that represents the change rate of the leader's acceleration, which is called the urgency of the leader. If the leader's acceleration is constant, i.e. , its velocity changes evenly and the urgency is zero. A special case is that the leader's velocity is constant, which means its acceleration and urgency are zero. If the leader's acceleration is also changing uniformly, it is necessary to introduce the urgency of the leader which describes the change rate of its acceleration as non-uniformly accelerated motion is common in the real world. Thus, it is reasonable to make the assumption that is first order differentiable. In the above, is the specific change rate of the leader's urgency, which has an upper bound. For example, the urgency is a measurable indicator that reflects the uncomfort degree of passengers. When it is too large, passengers will feel uncomfortable. Thus, due to real feel, is bounded. In some cases, it is hard to measure its neighbours' velocity in real time for follower i, and the leader's acceleration may be unknown to all the followers. Thus, in this paper, it is assumed that each follower can only obtain its own state information as well as its neighbours' position information, and the acceleration of the leader is completely unknown to the followers. In this case, we will design the distributed tracking control law for each follower only using the obtained information. The desired formation tracking can be achieved if the relative state deviations between the followers and the common non-physical leader converge to the desired formation values. To be specific, we say that our system achieves the formation tracking mission if and only if (9)where represents the desired relative position of follower i to the common leader, and are positive constants. It is assumed that is first-order differentiable with respect to t and is bounded: . Remark 3.The physical meaning of the assumptions about is that the desired formation of the MAS is time-varying during the tracking process. Especially, when the first-order differential of equals zero, the desired formation is constant. In practical applications, shows the formation change rate, which is bounded due to the manoeuvring capability limits of the MAS. Meanwhile, it is practical that the desired relative position varies according to the environment. 3 Observer-based distributed formation tracking control protocols In this section, it is devoted to solve the formation tracking problem for the leader–follower MAS. Since all the followers cannot obtain the leader's acceleration and velocity information, distributed observers are adopted to estimate them throughout the process using only the measured position information. Then, observer-based distributed control schemes are designed for the followers depending on the estimation algorithms and followers' obtained information. In what follows, we consider two cases for leader–follower multi-agent formation tracking: without communication delays and with communication delays. 3.1 Formation tracking without communication delays In the formation tracking control, typical information available to the follower is the relative positions of its neighbours and the leader, i.e. and . The tracking purpose is to maintain the expected formation with respect to the leader, which is related to the leader's position and velocity. Therefore, it is a vital step to design a distributed estimation algorithm to estimate the leader's velocity and acceleration. Now, in order to estimate the acceleration of the leader for follower i, an observer is designed as (10)where is the estimation of for follower i. Meanwhile, another observer is designed to estimate the leader's velocity for follower i as follows: (11)where denotes the estimation of , and the gains and k are to be designed. Finally, a tracking control protocol based on the observers is designed for follower i as (12) Applying the formation tracking control law (12) together with the observers (10) and (11) to system (6) yields the following closed-loop system: (13)where , , , , , denote the position, velocity, relative deviation, leader's velocity estimation and acceleration estimation of the leader–follower system, respectively. Simultaneously, can be written in a compact form as (14) Since formation tracking is guaranteed by the consensus of all followers and the leader with desired deviation, our original problem can be transformed into a consensus problem. To this end, we adopt the following transformation of variables: (15) Remark 4.Compared with [15, 17], the time-varying reference trajectory is not an orbit but a second-order leader in our paper. [15, 17] combined consensus and curve extension method, while we investigate formation tracking problem of second-order MASs by consensus-based technique which is much convenient in the control design. Although the assumption of the communication topology is the same as [15], the link topology in [15] was not the leader–follower structure while leader–follower structure is used in our paper. Compared with [17], the formation tracking control is designed uniformly without considering tracking control and formation control separately. Due to the special structure of the Laplacian matrix L, one has , then (16) Applying the variable transformation to system (6) and (13), we can obtain (17) Let , we can get the following closed-loop error dynamics: (18)where Theorem 1.If the communication topology of followers is connected and there exists at least one follower accesses to the leader, then there are positive constants and , such that the formation tracking control problem is solved with the observer-based controller, i.e. (19)for the constants and depending on and m, . Proof.Consider the following parameter-dependent Lyapunov function as candidate: (20)with Lyapunov matrix (21) According to Lemma 3, we can verify that P is symmetric and positive definite. The derivative of (20) along the trajectories of (8) satisfies (22)where To verify that is positive definite, we can apply Lemma 3 and make the following blocks to facilitate the proof. Let We can easily get , thus we need only prove , i.e. (23)which is equivalent to prove the inequalities that (24)By the above second inequality, it is obtained that (25)Since H is positive definite, there exists an inverse matrix satisfying that , in which are the eigenvalues of H. Thus, inequality (25) is transformed into a simplified form, i.e. (26)which can be rewritten as (27)If we want to get , we can choose the constant gains k and that satisfy and at the same time, i.e. and . Based on and , if one get k sufficiently large, then are positive definite, and therefore of .From the facts that (28)one has (29)Let , by inequality (30) (31)which leads to (32)From the notation of and the assumptions that , , one has (33)Regarding the fact that (34) (35)As , , we can see that (36)where (37)which implies that (38)□ Remark 5.We can see the system here is uniformly ultimately bounded stable and the bound is related to m and . When , , the upper bound of the tracking error approaches zero as . That is, in the case that the reference input of the leader is zero or the formation is time invariant, the tracking performance is better. Remark 6.In the same way as Theorem 1, we can prove the leader–follower system can realise formation tracking mission even if the interaction topology of the system is time-varying according to a piecewise constant switching signal, as long as the topology is connected during the intervals, since the Lyapunov function is a common Lyapunov function for the corresponding switched system. Remark 7.Compared with the practical MASs about formation tracking in [16, 23, 25, 46], each follower in our paper realises formation tracking through interaction with its neighbours in the communication network, and its control protocol and observer is distributed which is not only influenced by a designed local leader. Thus, our control method is much robust to external environment. While in [25], the observer was designed, respectively, without considering its neighbouring robots movement. In [16], it was assumed that each robot only receives information from one intermediary leader. The tracking controller using a leader–follower approach was implemented recursively between any pair of leader–follower robots. Thus, for each robot, its control law is only related with the information of itself and its intermediary leader. In [23], each of the following robots was assigned a local leader among its neighbours (which may be the same as the global leader) such that the resulting formation structure is connected. Compared with [46], neighbours' velocity of each follower is not used to design the formation control in our paper, which reduces the communication burden largely. 3.2 Formation tracking with communication delays In the last subsection, we assume the communication between the followers is ideal and there is no communication delay for the followers to get the information that is needed. In practical applications, however, there exists communication delays inevitably when the network transmits followers' information. In this subsection, we focus on the simple case: the time delays between followers are uniform and bounded, denoted as , which can be time-varying and non-differentiable. Thus, we adopt the following algorithm with communication delays and assume the agent can obtain its own information without time-delay (39)where , represent the estimation of the leader's acceleration and velocity, respectively, and , can be obtained by the following observers: (40) Let , , , with the same transformation of variables as the above subsection, applying the control protocol (39) and the proposed observers (40) to system (6), the closed-loop system can be written as (41) Furthermore, let , a compact form is obtained as follows: (42)where Theorem 2.If the follower's communication topology is connected and there exists at least one follower accesses to the leader which moves continuously, then there are positive constant and enough large constant which satisfy and such that the formation tracking control problem with bounded communication delay between the following agents is solved with the observer-based controller, i.e. (43)for the constant depending on and m, . Proof 1.To prove the stability of the system (41), we choose the following parameter-dependent Lyapunov function with P defined as (21) (44) The derivative of (44) along the trajectories (41) satisfies (45)where Similar to the proof of , if , based on and , is positive definite as long as one gets k sufficiently large. Then, consider the derivative of : (46)Choose appropriate k, if (47)and then is satisfied.Since the leader moves continuously, we have , of which is constant. According to Lemma 2, one obtains (48)where r is the unique positive solution of (49)Consequently, as , , it is easily obtained that , which implies that (50)Thus, the conclusion follows. Remark 8.In Theorems 1 and 2, the bound of k cannot be obtained accurately and thus it may be very conservative. From the numerical simulations shown in Section 4, it can be seen that it is not necessary to take a very large gain of k. It is not to say the larger constant k is, the better the convergence rate we have. Even though k is larger, the convergence rate still can be low, which is a common shortcoming of state-feedback control. Thus, it is important to choose appropriate k which is not discussed here. 4 Numerical simulations Consider a MAS containing four following agents (denoted by 1, 2, 3, 4) and a leader (denoted by 0). Followers are assumed to be equipped with a communication system that allows them to communicate with their neighbours instantly to exchange their position information. The simulation objective is to design controllers as described in this paper for each follower such that all the followers can track the leader in the desired formation. Assume that the directed communication topology of the leader–follower system in this paper is as described in Fig. 1. The Laplacian matrix of the subgraph G is L and the leader adjacency matrix is . By Lemma 1, is positive definite and its maximal eigenvalue is . According to Theorem 1, one can get and select in the following simulations: (51) Fig. 1Open in figure viewerPowerPoint Communication topology In the simulations, we will conduct the formation tracking in two-dimensional (2D) space, i.e. . As for the formation tracking problem in 3D space, it is similar to the case in 2D space as we can chose and select the initial positions and velocities in 3D. For the leader, the initial state is selected as , , . For the followers, choose the initial state dynamics as , , , and , , , . For the MAS that the leader's acceleration and the desired relative position are time-varying, we will give simulation results in case 1 and case 2. In case 1 and case 2, the change rate of the leader's acceleration is (52)The upper bound of 's norm is . The desired time-varying relative positions between the followers and the leader are (53) Case 1: The communication network is ideal and there exists no communication delays. For the system with control law (12) which uses only relative position measurement and each follower's own velocity, Fig. 2 describes the dynamics behaviours of the followers, which shows that the formation tracking errors of the position and velocity variables are uniformly ultimately bounded, respectively. Figs. 3 and 4 describe the observers evolutions, which demonstrate the observers can estimate the leader's acceleration and velocity well after 80 s. Fig. 2Open in figure viewerPowerPoint Evolution of the formation tracking errors for case 1 (a) Error of the position variables, (b) Error of the velocity variables Fig. 3Open in figure viewerPowerPoint Trajectories of the leader's acceleration observers for case 1 Fig. 4Open in figure viewerPowerPoint Trajectories of the leader's velocity observers for case 1 Case 2: For the case with communication delays and the time delay between the followers is set as s, Fig. 5 shows the formation tracking errors of the position and velocity variables. Figs. 6 and 7 give the observers evolutions of the leader's acceleration and velocity. Fig. 5Open in figure viewerPowerPoint Evolution of the formation tracking errors for case 2: (a) Error of the position variables, (b) Error of the velocity variables Fig. 6Open in figure viewerPowerPoint Trajectories of the leader's acceleration observers for case 2 Fig. 7Open in figure viewerPowerPoint Trajectories of the leader's velocity observers for case 2 For case 1 and case 2, the formation tracking trajectories of the MASs in 2D plane are shown in Fig. 8. From Fig. 8, we can see our proposed algorithm can realise the desired time-varying formation tracking mission for MAS with or without communication delays. Fig. 8Open in figure viewerPowerPoint Formation trajectories of the system in 2D plane (a) Formation for case 1, (b) Formation for case 2 Furthermore, in order to illustrate the convergence rate is influenced by k, we select while keeping other values the same as those in the simulation of case 1 and case 2. Fig. 9 shows that the formation tracking errors of the followers are uniformly ultimately bounded after 150 s. Compared with Fig. 2 a, we can see the convergence rate with is higher than that with . This means that the gain k given in the theory needs not to be very large in applications. In practical applications, we should select appropriate value of k to guarantee the desired convergence rate. Fig. 9Open in figure viewerPowerPoint Error of the position variables for case 1 5 Conclusion In this paper, we have investigated the leader–following formation tracking problem of second-order MASs under the condition that the communication topology among the followers is undirected and connected, and at least one of the followers can receive the information of the leader. As the velocity of the active leader is hard to be measured in real time and the leader's control input may be unknown to all the followers, distributed observers are designed for the following agents to estimate them. Based on the observers, a novel distributed formation control protocol is designed and stability analysis is presented for MAS without and with time-delays in the communication channel, respectively. With the help of algebraic graph theory and Lyapunov function, it is proved that each follower can follow the active leader and form the desired formation during the tracking process. In the future, we will further consider extending communication delays investigated in this paper to the formation control problem in the body-fixed coordinates of the leader or distributed manoeuvring with the optimised guidance of the leader. 6 Acknowledgments This work was supported by the China Postdoctoral Science Foundation under grant no. 2018M632208, the National Natural Science Foundation of China under grant no. 61673107, the National Ten Thousand Talent Program for Young Top-notch Talents under grant no. 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