Event‐triggered consensus control for second‐order multi‐agent systems

IET Control Theory & ApplicationsVolume 9, Issue 5 p. 667-680 Regular PapersFree Access Event-triggered consensus control for second-order multi-agent systems Duosi Xie, Duosi Xie School of Automation, Nanjing University of Science and Technology, Nanjing, Jiangsu, 210094 People's Republic of ChinaSearch for more papers by this authorShengyuan Xu, Corresponding Author Shengyuan Xu syxu@njust.edu.cn School of Automation, Nanjing University of Science and Technology, Nanjing, Jiangsu, 210094 People's Republic of ChinaSearch for more papers by this authorZe Li, Ze Li School of Mechanical and Electrical Engineering, Suzhou University of Science and Technology, Suzhou, 215000 People's Republic of ChinaSearch for more papers by this authorYun Zou, Yun Zou School of Automation, Nanjing University of Science and Technology, Nanjing, Jiangsu, 210094 People's Republic of ChinaSearch for more papers by this author Duosi Xie, Duosi Xie School of Automation, Nanjing University of Science and Technology, Nanjing, Jiangsu, 210094 People's Republic of ChinaSearch for more papers by this authorShengyuan Xu, Corresponding Author Shengyuan Xu syxu@njust.edu.cn School of Automation, Nanjing University of Science and Technology, Nanjing, Jiangsu, 210094 People's Republic of ChinaSearch for more papers by this authorZe Li, Ze Li School of Mechanical and Electrical Engineering, Suzhou University of Science and Technology, Suzhou, 215000 People's Republic of ChinaSearch for more papers by this authorYun Zou, Yun Zou School of Automation, Nanjing University of Science and Technology, Nanjing, Jiangsu, 210094 People's Republic of ChinaSearch for more papers by this author First published: 01 March 2015 https://doi.org/10.1049/iet-cta.2014.0219Citations: 63AboutSectionsPDF 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 onFacebookTwitterLinked InRedditWechat Abstract In this study, an event-triggered control strategy is proposed to achieve consensus in a multi-agent system under a directed topology. The proposed control strategy utilises a piecewise continuous control law and an event-triggering function for each agent. The control law only updates at discrete event instants computed using an event-triggering function, which depends on the states of the agents at the current and outdated event instant. This control approach is first applied to a first-order system and is further extended to a second-order system. Simulation examples are presented to illustrate the efficiency of the proposed control strategy. 1 Introduction Along with rapid development in various engineering fields, both the size and complexity of a practical system have increased substantially. One way to addresses this issue is to divide the entire system into several sub-systems. Each sub-system is able to individually complete its part of the work and thereby accomplish the whole objective as a system. This cooperation makes the system efficient, flexible and powerful and in some instances the ability of the system can also be enhanced significantly compared with a monolithic system. By considering each sub-system as an agent, collective behaviours such as flocking [1], swarming [2], formation control [3] and consensus reaching [4–6] have been studied. Various tools such as matrix theory, algebraic graph theory and control theory have been explored on controller design, stability criteria analysis and some other aspects [7, 8]. However, the cooperation among agents consumes large quantity of energy on continuous updating and frequent transmissions. With an increase in the complicity of task, the congestion of communication channel, the cost of incremental energy is becoming a major issue. The discrete control strategy associating with the sampled state [9–11] offers an efficient control strategy. For sake of saving energy, scheduling the controller to update at a variable sampling rate is more efficient and flexible compared with a fixed sampling rate. Such control method is provided by the event-triggered control strategy [12]. The sampling instants are determined by a triggering function, which is defined with respect to the measurement error. When the triggering function overpasses zero, an event is released. The control actuator will keep steady during the interval in between of two instants. This event-triggered control rule can be applied to different control strategies, such as output feedback control [13], model-based control [14], quantisation control [15] and consensus control [16–20]. Centralised event-triggered consensus control in multi-agent system was first proposed in [16]. The whole system shares one event-triggering function that triggers all the agents at the same time. Meanwhile, the centralised event-triggering function depends on the states of all the agents. It is obvious that this centralised scheme is not suitable as the number of agents increases. A decentralised consensus control strategy is therefore proposed to improve the scalability of such a control strategy [21]. The decentralised strategy gives distributed event-triggering function to each agent in the system. In the literature, most of the event-triggering consensus control strategies are applied to a first-order multi-agent system [20, 21]. The second-order consensus problem, which cannot be solved by a simple extension of the first-order case has also attached some attentions. The event-triggered control strategy has been used to solve this problem in the centralised way in [22–24]. The result in literature [18] extended the centralised event-triggering function in [24] into decentralised pattern. Both of the literatures utilised a control law that designed under an exponential relation between the coefficient of the position state and the coefficient of the velocity state. This coupling requirement may constrain the application of the control law. In [19, 20], a sampling model is purposed that the agent samples both its own and its neighbours state at the agent's event instants. Various methods are used together with the event-triggered control strategy in the consensus analysis of multi-agent systems, such as matrix analysis and the linear matrix inequality (LMI) method [25]. Different topologies have been considered as well. The multi-agent system under undirected topology is studied in [21]. The directed topology case, which is more difficult to analysis, has attracted a lot of attentions. In this paper, event-triggered control strategies are proposed for both first-order and second-order dynamics of a kind of multi-agent system under directed topology. The event-triggered strategy is decentralised that each agent has been assigned a specific event-triggering function that utilises only neighbours' information. The decentralised event-triggering function depends on the measurement error, that is, the difference between state at the current time and at the last event instant. An agent is triggered when its event-triggering function exceeds the designed threshold. Once the agent is triggered, its current state information will be sampled and sent to its neighbours and thereby the controller updates the control actuation. Furthermore, an improved event-triggering function is designed for each agent that only uses the information for the agent and the discrete information that sampled and sent by neighbours at their event instants. By applying the improved event-triggering function, the communication energy between agents can also be minimised. The paper is organised as follows: Section 2 states some preliminaries that are used in this paper. Section 3 addresses the consensus control method for the first-order multi-agent system under directed topology. Besides, the event intervals between two events have been studied in this section as well. Section 4 addresses the consensus problem for the second-order system from applying the event-triggered rule. In Section 5, the improved event-triggering functions are designed for both the first-order and the second-order systems. The effectiveness of the improved event-triggering functions is proved by using Lyapunov method. The event intervals are analysed to show that infinite accumulation of events can be avoided. Simulation results are illustrated in Section 6 and finally a conclusion is presented in Section 7. Notation. Throughout this paper, is an identity matrix and is a zero matrix. For a vector x, x⊤ denotes the transpose vector and ∥x ∥ denotes the Euclidean norm of the vector. For a matrix A, A⊤ denotes the transpose of matrix of A whereas ∥A ∥ represents the matrix 2-norm of the matrix A. denote the set of non-negative integers. 2 Preliminaries 2.1 Graph theory For a multi-agent system that consists of N agents, the topology of this system can be represented by a graph , where is the set of all the agents in the system and is the set of all edges between connected agents. A directed edge in the set ℰ is represented by (i, j), which implies that agent j can receive information from agent i. Thus agent i is a neighbour of agent j. The neighbours of agent j can be denoted as a set . A directed path from agent j to agent k in is a sequence of ordered edges (j, j + 1), (j + 1, j + 2), …, (k − 1, k) ∈ ℰ. In a directed topology, a directed spanning tree exists if there are directed paths from at least one agent to all the other agents. A directed topology is strongly connected if there always exists a directed path from agents to each of the other agent. An adjacency matrix is defined to describe the topology of the whole system. , where aii = 0, aij = 1, if (j, i) ∈ ℰ, aij = 0 otherwise. The in-degree of the agent i is defined as . For all of the agents in the system, the in-degree matrix is defined as Γ = diag{din(i)}. The Laplacian matrix of the graph is defined as . Lemma 1.For any and ϵ > 0, one has the following properties: ; (x2 + y2) ≤ (x + y) 2, if xy ≥ 0. Lemma 2 [7].If the directed graph of a multi-agent system contains a directed spanning tree, then the eigenvalues of the Laplacian matrix L satisfy . Where λi is the i th eigenvalues of the Laplacian matrix L, ∀i = 1, …, N, respectively, and ℛ(λl) denotes the real part of the λl. Lemma 3 [26].The Laplacian matrix L is irreducible if and only if the topology of the directed system is strong connected. Lemma 4 [26].If the Laplacian matrix L is irreducible, then there exists a positive vector ξ ​ = ​[ξ1, ξ2, …, ξN]⊤, such that ξ is the left eigenvector of L associating with eigenvalue 0. Moreover, L⊤ Ξ + ΞL is semi-positive definite where Ξ = diag{ξ1, ξ2, …, ξN}. By taking extractions of square root of each element of Ξ, a matrix is obtained as Λ = diag{ρ1, ρ2, …, ρN} such that Lemma 5 [27].The following LMI is equivalent to either of the following conditions: Q (x) > 0, R (x) − S (x) ⊤ Q−1 (x) S (x) ≥ 0; R (x) > 0, Q (x) − S (x) R−1 (x) S (x) ⊤ ≥ 0. 3 Consensus control for the first-order multi-agent system In this section, the event-triggered control strategy for the first-order multi-agent systems is discussed. At first, on the basis of generate control law, two piecewise continuous control laws together with centralised event-triggering function and decentralised event-triggering function are proposed to minimise the frequency of controller updating. The results will show that the agents are able to achieve consensus associating with appropriate designed event-triggering functions. The event intervals under both centralised and decentralised scenarios are analysed to guarantee positive lower bound of the intervals between two events. 3.1 Problem description Consider a first-order multi-agent system associated with N agents. The dynamics of the agents are described by (1)where is the state of the agent i, specifically, the position; xi (0) is the initial position; is the given control input. Definition 1.The multi-agent system listed above is said to achieve consensus if, for any initial condition A widely used control law for this system is This continuous updating control law can drive the system to achieve consensus while consuming unnecessary transmission energy and occupying communication channel. An event-triggered control strategy that minimise the update frequency thereby minimising the energy consumption is proposed for the system. By using the event-triggered control strategy, the controllers of agents update at discrete event instants, which are calculated by predesigned event-triggering function. 3.2 Event-triggered consensus control of the first-order multi-agent system At first, a centralised event-triggering control strategy for the first-order system (1) will be studied. Latter, on the basis of the centralised event-triggered strategy, a decentralised event-triggered strategy is proposed. 3.2.1 Centralised event-triggered control strategy From this strategy, the system only has one global event-triggering function. The event instants are indexed by, k = 0, 1, …, such that tk denotes the k th event of the system. At the event instants, all the agents in the system will synchronously send their states to neighbours and update the control law with the received states. As the controllers of agents only update at event instants, the updating energy can be reduced. By using the event-triggered strategy, the control input for agent i is designed as (2)where α is a positive constant. xi(tk), i = 1, …, N is the state of agent i at the k th event instants. In order to synthesise all the agents, we use x (t) = [x1 (t), x2 (t), …, xN(t)]⊤ to denote the states of agents. From the definition, e (t) equals to 0 when an event occurs as e (tk) ≜ x (tk) − x (tk) = 0. Before stating the centralised event-triggering function, we first define the measurement error e (t) = x (tk) − x (t). The dynamics of agents can be written into (3)For the first-order multi-agent system (1), the centralised event-triggering function is defined as (4)where Λ is the same diagonal matrix as described in Lemma 4. When fc(t) reaches zero, all the agents will be triggered. The measurement error is set to zero and it will grow until the event-triggering function overpasses zero again. Theorem 1.Considering the first-order system (1) under directed strongly connected topology, the event-triggered control law (2) is utilised and the event-triggering function (4) is enforced to satisfy fc ≤ 0. Then for any initial condition, the consensus of the system (1) can be reached. Proof.To analyse the effectiveness of the proposed strategy, a candidate Lyapunov function is considered (5)where is the diagonal matrix described in Lemma 4. From Lemma 4, L⊤ Ξ + ΞL is semi-positive definite and it equals to zero only when x = a * 1N for a is a constant. Consequently, V (t) ≥ 0 and V (t) = 0 if and only if the system has achieved consensus. Along with the trajectories of the state as described in (3), the time derivative of the Lyapunov function is Since the event-triggering function is enforced to be greater than zero, one can obtain that ∥ΛLe ∥ ≤ ∥ ΛL ∥∥ e ∥ ≤ σ ∥ ΛLx ∥. Hence the time derivative of the Lyapunov function turns into As 0 < σ < 1, one has and if and only if the consensus is achieved. The centralised event-triggered strategy is proved effective to solve the consensus problem of the system (1). □ 3.2.2 Decentralised event-triggered control strategy It is obvious that the centralised event-triggering function requires the states of all the agents. This disadvantage causes unnecessary communication cost. Hence in order to improve the performance of the event-triggered strategy, a decentralised event-triggering strategy is proposed in the following. The decentralised event-triggering strategy assigns each agent a distributed event-triggering function such that only neighbours' information are required. By applying the decentralised event-triggering function, each agent only samples its state when its triggered. The k th event of agent i is denoted by the , as denotes the event sequence of the agent. Note that since agents trigger asynchronously, each agent has their own event sequence. The measurement error of agent i is defined as . It is clear that ei(t) = 0 when . The control input of agent i is designed as (6)where denotes the last event instant of agent j. From (6), the controller of agent i will update at both its own event instants and the neighbours' event instants . As mentioned before, is the instant that agent i's event-triggering function overpasses 0, where the function is defined as follow (7) is the i th elements of vector where . According to the definition of , one has , which means that only involves local information. At the k th triggering instant of agent i, the agent will sample its state, and then update its own controller by using the newly sampled state and send the state to its neighbours. The neighbours will therefore update their controllers with the received state as well, but they will not sample their states at this moment unless their event-triggering functions exceed zero. Then combining (1) and (6) and the description of the measurement error, the dynamics of agents turns into By synthesising all the agents, we also have Theorem 2.Considering the first-order multi-agent system described by (1) under directed strongly connected topology. By using the piecewise-continuous control law (6), the system can achieve consensus, for any initial condition, if the distributed event-triggering function (7) is enforced to satisfy fi(t) ≤ 0. Proof.Since the distributed event-triggering function in enforced to satisfy fi(t)​ ≤ ​0, we have . It follows that . We also have ∥ΛLe ∥ ≤ ∥ ΛL ∥∥ e ∥ ≤ σ ∥ ΛLx ∥. Using the same Lyapunov function that used in Theorem 1, one can obtain . Consequently one can easily jump to the result that the control law (6) is capable to drive the system reaching consensus by using the decentralised event-triggering function (7). □ 3.3 Event interval analysis of the first-order multi-agent system The purpose of the event-triggered strategy is energy saving, hence the accumulation of events should be avoided. The intervals between events that determined by the centralised event-triggering function is first analysed. 3.3.1 Event intervals under the centralised strategy According the definition of the measurement error, we have and e (tk) = 0. Besides, from the mechanism of event-triggering strategy, the next event occurs when ( as defined previously). In other words, the agents will not be triggered until . Therefore the interval between two agents is the time that grows from 0 to [σ /∥ ΛL ∥]. Similar to [12], the time derivative of is By using θ to denote , one has . Hence θ satisfies the bound θ ≤ ϕ (t, ϕ0), where ϕ (t, ϕ0) is the solution of . Calculated from one can get that the interval between event instants tk and tk +1 is lower bounded by the interval τ that satisfies ϕ (τ, 0) = [σ /∥ ΛL ∥]. The solution of the above differential equation is (8)It is clear that the intervals between two events that determined by the centralised event-triggering function (4) is positive lower bounded. 3.3.2 Event intervals under the decentralised strategy Theorem 3.Consider a first-order system (1) under directed strongly connected topology. By using the piecewise-continuous control law (6) and event-triggering function (7). Then for any initial condition in , there exists at least one follower , which has strictly positive interval lower bounded by τq > 0. Proof.One can easily obtain that for agent i, the event interval between and is the period that grows from 0 to [σ /∥ ΛL ∥]. Denote . Thus agent q has the maximum norm of among all the agents. It follows that Hence the time reaches [σ /∥ ΛL ∥] is longer than spends. Thereby one has that τq ≥ τ*, where τ* is the time grows from 0 to . From (8), one can obtain It leads to that the minimal interval between two event instants of agent q is □ Remark 1.The thresholds of event-triggering functions can be different. As listed in (7), the event-triggering function of agent i is that . The derivative of the Lyapunov function (5) has . In this paper, the upper threshold of the event-triggering function of agent i is set to 0, that is, fi(t) ≤ 0. Hence we have , which guarantees the consensus achievement. Similarly, we can obtain the result if the thresholds of the event-triggering functions are different from each other. Let us choose −βi ≤ 0, to be the threshold of the event-triggering function fi(t), that is, fi(t) ≤ − βi. Then the derivative of the Lyapunov function turns into , where βmin = min {βi, i = 1, …, N}. It is clear that the system can reach consensus by utilising event-triggering functions with different non-positive thresholds. (If −βi > 0, the convergence of the system may not be guaranteed). It is worth mentioning that a negative threshold lowers the upper bound of the measurement error compare the zero threshold. Moreover, if −βi < 0, from the description of the event-triggering function, the agent i will be triggered once is smaller than βi. For the worst case, the accumulation of events may exists. Consequently the performance of the event-triggered strategy is restrained by the negative threshold. In this paper, for simplicity and for best performance of the event-triggered strategy, we set the thresholds of all the event-triggering functions fi(t) to be zero. 4 Consensus control for the second-order multi-agent system Since the agent dynamics of the second-order system is more complicated, the consensus controller design for the second-order multi-agent system is more difficult comparing to the first-order case. 4.1 Problem description Considering a second-order multi-agent system under directed strongly connected topology, the dynamics of the agents are described as (9)where , and are the position state, the velocity state and the control input of agent i, respectively. All initial values of position states and velocity states belong to . Definition 2.The consensus of the second-order multi-agent system (9) is achieved if, for any initial conditions For the second-order multi-agent system (9), the generate control model that utilising both the position and the velocity states is where α, β are independent positive constant coefficients. It is clear that this control law also requires continuous updating and communication. 4.2 Centralised event-triggered consensus control strategy of the second-order multi-agent system The segmental continuous control law for the second-order multi-agent system has a similar form to the first-order control law. In stead of continuous states, both the position and the velocity states of the agents at the last event instant are used in the event-triggered control law (10)where tk is the latest event instant of the system. xi(tk), vi(tk), i = 1, …, N are the position and velocity states of agent i at the event instant, respectively. α, β are independent positive constant coefficients. According to the proposed control law (10), the dynamics of the agent i can be written into where t ∈ [tk, tk +1). The position and the velocity measurement errors are defined separately that , . Then the uniform expression of the dynamics of the system (9) is obtained by using x (t) ⊤ = [x1 (t), x2 (t), …, xN(t)], v (t) ⊤ = [v1 (t), v2 (t), …, vN(t)] Given two vectors as y = [x⊤, v⊤]⊤ and , and then the dynamics of the system is as below (11)Similar to the definition of , we define , and . The centralised event-triggering function fc(t) for the second-order system is given by (12)where . k, k1, k2, η are positive constants. Before stating the result concerning with the event-triggered strategy, we first introduce a candidate Lyapunov function (13)where r, k1, η are positive constants. Lemma 6.The Lyapunov function (13) is qualified for the system (9) if the following condition holds: (14)where c is a positive constant; ξmax is the maximum element of the vector ξ that mentioned in Lemma 4. Proof.It is easy to obtain that the Lyappunov function equals to zero if and only if the consensus is reached. Now the case that the consensus is not reached is discussed. From (13), it follows that: It is easy to acquire that . It equals to 0 only when the position states of the agents achieve consensus. From Lemma 4, (1/2)η (L⊤ Ξ + ΞL) is semi-positive definite and v⊤ (1/2)η (L⊤ Ξ + ΞL) v equals to 0 only when the velocity states of the agents achieve consensus. Similar to Lemma 6 in [26], a positive constant c is defined such that Then one can find a diagonal matrix Ω = diag{ω1, …, ωN} associating with L⊤ Ξ + ΞL that ωi is the i th eigenvalue of L⊤ Ξ + ΞL, ∀i = 1, …, N. Then there exists a unitary matrix P = [p1, p2, …, pN] such that L⊤ Ξ + ΞL = P ΩP⊤. It is clear that pi is the eigenvector of L⊤ Ξ + ΞL associated with eigenvalue ωi. Since L⊤ Ξ + ΞL is semi-positive definite, it has one zero eigenvalue. Without lose of generality, assume ω1 = 0. It follows that p1 = 1N. By using notation ζ = [ζ1, …, ζN]⊤ ≜ P⊤ v, one has One have that c = 0 if and only if LPζ ≠ 0, ζ⊤ ζ = 1, and ζ2 = ⋯ = ζN = 0, which implies that ζ1 = 1, or −1. Without loss of generality, use ζ1 = 1 for further analysis. In that case, , which violates the condition that LPζ ≠ 0. Therefore c > 0 if the consensus is not achieved yet. From the definition of c, it follows that cv⊤ v ≤ v⊤ (L⊤ Ξ + ΞL) v. From above, the Lyapunov function has (15)From Lemma 5, is semi-positive definite if One have that if , then the Lyapunov function (15) is positive definite. From the condition that , the Lyapunov function (15) is semi-positive definite and it equals to zero if and only if the consensus is reached. From the above discuss, the candidate Lyapunov function (13) is positive if the consensus is not reached and it equals to 0 if and only if the consensus is achieved. Hence the Lyapunov function (13) is qualified for the second-order system. □ Theorem 4.Consider the second-order multi-agent system as described by (9) with piecewise-continuous control law (10). The system (9) is able to achieve consensus, for any initial condition, if the centralised event-triggering function (12) is enforced to satisfy fc(t) ≤ 0 and the following conditions are satisfied: (16) (17) Proof.The candidate Lyapunov function (13) can be represented as Hence the derivative of the Lyapunov function along the trajectory (11) is (18)The former part of (18) has (19)From choosing r = βk1 − αη and the condition (17), the expression (19) is smaller or equal to (20)Meanwhile, the latter part of (18) has (21)Therefore the derivative of the Lyapunov function has Since the centralised event-triggering function (12) is enforced to satisfy fc(t) < 0, one can obtain that . The consensus of the second-order multi-agent system (9) is achieved. □ The centralised event-triggered strategy also requires global states. In the following, a decentralised event-triggered control strategy is proposed to solve this problem. 4.3 Decentralised event-triggered consensus control law of the second-order multi-agent system The decentralised event-triggered strategy assigns each agent a distributed event-triggering function. As same as the decentralised strategy of the first-order system, the event instants of agent i are determined only by its own distributed event-triggering function. We still use to denote the k th event instant of agent i. The position measurement error and the velocity measurement are defined as and , , respectively. The decentralised control input of agent i is designed as (22)where, . and are the latest event instants of agent i and agent j, and are the position and velocity states of agent i and agent j at their latest event instant respectively. α, β are independent positive constant coefficients. According to the proposed control law (22) and the expression of measurement errors, the dynamics of the agent i can be written into Recalling the notations and , an uniform description of the system dynamic is obtained by synthesising all the agents One can obtain that the system dynamics can be represented as same as stated in (11). The distributed event-triggering function fi(t) corresponding to agent i is (23)where 0 < σ < 1, k, k1, k2, η are p