Finite‐time consensus of leader‐following non‐linear multi‐agent systems via event‐triggered impulsive control

Funding information National Natural Science Foundation of China, Grant/Award Numbers: 61973092, 61374081; National Natural Science Foundation of ChinaGuangdong Joint Fund, Grant/Award Number: 2019A1515012104 Abstract This work is concerned with the finite-time consensus of leader-following non-linear multiagent systems by means of distributed event-triggered impulsive control. The finite-time consensus protocol is first put forward based on event-triggered impulsive strategies, where the impulsive instants are determined by the proposed event-triggered condition. For the event-triggered condition, it not only determines the impulsive instants but also effects the update time of the finite-time control. Moreover, compared with the existing finite-time controllers, the controller designed in this work does not contain any sign function, thereby overcoming the chattering phenomenon. In addition, without finding Zeno behaviour, the feasibility of the proposed control protocol is demonstrated. Lastly, simulations are employed to demonstrate the effectiveness of the proposed control schemes.


INTRODUCTION
With advance of artificial intelligence (AI) and computer technology at a furious space, lots of multi-agent technologies could be seen everywhere in our daily life [1][2][3][4]. Consensus problem on multi-agent systems (MASs) has been a popular object among them and has been widely applied into many fields including swarm-based computing [5], formation control of advanced unmanned aerial vehicles (UAVs) [6], group consensus of robots [7], and so on. Thus, a great deal of researches have been published about consensus problem. In the meantime, various kinds of control methods served consensus problem have been vigorously developed, which consist of intermittent control [8,9], feedback control [10,11], sliding mode control [12], and so on [13][14][15][16][17].
As far as we know, impulsive control as a class of discontinuous-time control has attracted a considerable amount of attention owing to the real applications in many areas [18][19][20][21][22]. Compared with the continuous-time control, impulsive control has a number of merits, which can reduce the amount of data transmission and ameliorate robustness of the system. Moreover, the cost of control can be reduced because of smaller control gain. In [18], the MASs with stochastically switching topologies have been investigated by means of impulsive control. In [21], by using quantized relative state measurements, the authors designed a impulsive controller to deal with the fixed-time quantized consensus problem of leader-following non-linear multiagent systems (LNMASs), which cut down the operational costs of the systems. Besides, the impulsive pinning control scheme was proposed to exponential consensus problem of stochastic LNMASs with time-varying delays in [22]. In a word, impulsive control can deal with the problem that can not be solved by continuous control.
However, in the existing results, most designed impulsive controllers are based on the time-triggered mechanism and are periodical [23][24][25]. Inevitably, in order to guarantee the consensus rate of the controlled system, the time-triggered impulsive frequency may be designed so high that it leads to unnecessary consumption of energy resources [26,27]. Hence, for the sake of saving resources and achieving the expected performance, an appropriate candidate is the event-triggered impulsive control method. With regard to the event-triggered impulsive control approach, it has more merits than other time-triggered control approach. This method not only significantly reduces the update number of control but also effectively improves the utilization of the limited bandwidth resource. Therefore, many scholars have been interested in resorting to the event-triggered impulsive control approach to deal with MASs [28][29][30][31][32]. With the help of a distributed event-triggered impulsive control protocol, the authors achieved the consensus of leader-following MASs in [29]. It should be noted that the impulsive instants relied on the event triggering instants. In [31], the memristive neural networks with time-varying delays realized the quasisynchronization via state feedback and event-triggered impulsive strategies, which the update time of the state feedback control input and working time of impulsive controller are determined by the event-triggered instants. In [32], by means of the event-based impulsive control method, the exponential stabilization of continuous-time systems has been investigated. Also, the proposed strategy has been applied to the synchronization of memristive neural networks.
It is worth mentioning that most of the recent studies relating to event-triggered consensus focus only on the asymptotic convergence, which means that the convergence can be achieved within infinite time. In practical applications, however, the equipments need to complete all planed tasks in the finite time. Besides, the service life of equipment is limited. Therefore, the finite-time control is of great significance to accelerate convergence rate and demonstrate better robustness. Taking into account these advantages, many researches about the finite-time control have been established in recent years [33][34][35][36][37]. Based on the distributed event-triggered control, the authors in [35] verified that linear MASs can realize the finite-time consensus, which the consensus speed can be further adjusted. In [37], the distributed finite-time event-driven strategy was put forward for MASs with single-integrator model. The setting time depending on the initial state and the event-triggering threshold could be estimated. To the best of our knowledge, few works address the finite-time consensus problem of the LNMASs by means of event-triggered impulsive control.
As everyone knows, traditional finite-time control protocol always inevitably includes the sign function, which maybe result in the chattering phenomenon [34,36]. Because these phenomenon of chattering may damage the equipment and thereby cause economic loss, it is imperative to design a new finitetime consensus controller without the sign function. In [38], the authors proposed the non-chattering control to ensure the fixed-time synchronization of complex networks with impulsive effects, which avoid the chattering influence. Besides, in order to avert chattering phenomenon, the authors in [39] designed the continuous control to guarantee the fixed-time synchronization of stochastic complex networks. Although some results of the finite-time controller without sign function are obtained in many existing papers [38,39], the finite-time consensus of LNMASs via event-triggered impulsive control without sign function has never been addressed in overt literature. With these in mind, how to design an efficient distributed event-triggered impulsive control strategy without sign function could achieve the finite-time consensus of LNMASs, which provides the motivation of this work.
The above discussions motivate to focus on the finite-time consensus of LNMASs with distributed event-triggered impulsive control in this work. The main contributions of this work can be summarized in three aspects as follows: • By employing the distributed event-triggered impulsive control strategy, the finite-time consensus for LNMASs is inves-tigated. Based on Lyapunov stability theory of impulsive differential and finite-time control method, some sufficient conditions are derived for reaching finite-time consensus and the settling time depending on the initial state is estimated. • Based on the designed controller, event-triggered condition determines the impulsive instants and finite-time control instants. Namely, the update time of the finite-time control and the working time of the impulsive controller are decided by the event-triggered instants. Besides, the Zeno behaviour is guaranteed to be avoided owing to derive a lower bound from inter-event time. • Different from the traditional finite-time event-triggered controllers in [34] and [36], the new control protocol does not contain sign function, which can avoid chattering phenomenon.
This work is structured as follows: In Section 2, some preliminaries and system formulation are presented. The proposed controller is required to achieve the finite-time consensus of LNMASs with event-triggered impulsive mechanism is further developed in Section 3. We provide some examples to demonstrate the effectiveness of analytical results in Section 4. Finally, the conclusions are drawn in Section 5.
Notations: Hopefully, ℝ stands for the real number set, ℝ N denotes the set of N-dimensional Euclidean space, and ℝ + represents the set of nonnegative real numbers. The set of positive integers are denoted by ℕ + . 1 = (1, 1, … , 1) T denotes an N-dimensional column vector with all elements being 1, and where the superscript T stands for transposition. |x i | is the absolute value of x i . P < 0 (P > 0) denotes a negative (positive) definite matrix. 2 (P ), N (P ), and P T denote the minimum non-zero eigenvalue, the maximum eigenvalue and the transpose of matrix P, respectively. diag{⋯} represents a diagonal matrix. ∅ stands for the empty set. For :

Algebraic graph theory
Briefly, we will introduce the basic algebraic graph theory in this subsection. As a tool, graph theory plays an important role in information exchange among agents in the multi-agent systems. In this work, an undirected graph = ( , ℰ, ) with N agents is considered by us, where the vertex set = {0, 1, … , N } represents the corresponding agent and ℰ ⊆ × stands for the set of edges. Besides,  = [a i j ] N ×N denotes the weighted adjacency matrix of undirected graph , which its elements can defined as: 10 , a 20 , … , a N 0 } stand for the leader adjacency matrix used for describing whether the followers have an exchange of information with the leader in graph̄consisting of N + 1 agents. The elements a i0 > 0 of  represent the leader connecting to the ith agent, otherwise, a i0 = 0.

Some lemmas
In this subsection, we list some lemmas, which are useful for the theoretical analysis in this article.

Lemma 2 ([41]
). Owing to a connected undirected graph , the properties of the Laplacian matrix  of graph are presented in the following words.
Remark 1. As we know, both adjacency matrix  and Laplacian matrix  of the connected undirected graph are symmetric. In this work, we consider the graph̄consisting of the followers and a leader, which the followers are undirected connection with each other and the leader at least has one directed path to the followers. Namely, the graph̄is a directed graph. So, according to the above-related graph theory and previous result [33], the matrix  of graph̄is positive definite and symmetric.
and V (t ) ≡ 0, ∀t ≥ , with the settling time given by

Lemma 4 ([43]
). Assume that there exist three constants c > 0, 0 set and m is a positive integer, and a continuous, non-negative function V (t ) such that the Dini derivative of V (t ) satisfies the conditions as follows: Then the following inequality holds: where is a constant which represents the settling time.
where P 11 is a symmetric matrix, then the following statements are equivalent: (1) P < 0,

Problem formulation
Suppose that the LNMASs consist of N followers and a leader, the communication topology is a connected graph which will be introduced later. Firstly, the dynamics of the follower agents can be described as followṡ where Then, consider the dynamics of the leader is defined aṡ where x 0 (t ) ∈ ℝ is the state of leader 0, g : ℝ × ℝ + → ℝ is the continuous non-linear function.
Remark 2. Note that a great deal of physical systems have the characteristic of non-linearity in nature. Compared with the existing results in [29] and [35], the models we studied include the non-linear part which is more suitable to the practical application.
Throughout the whole article, the following assumption and definition are put forward before starting the main results.

Assumption 1.
For any x i , x j ∈ ℝ, there exists a known and positive constant such that (1) and (2) are said to achieve leader-following finite-time consensus, if there exists a constant > 0 which depends on any initial state x i (0) and x 0 (0) such that

Definition 1. The NMASs
Here, is called the settling time.

MAIN RESULTS
In this section, the event-triggered impulsive control method is considered to investigate the finite-time consensus of LNMASs (1). In order to achieve consensus of the LNMASs in finite time, we propose the following control protocol for agent i: where 1 , 2 denote the positive constants; > 0, , are positive odd integers satisfying < . ∈ ℝ stands for the impulsive strength which satisfies 0 < < 1. l ∈ ℕ + . The sequence of event triggering instants {t i k } is defined as 0 = t i 0 < t i 1 < ⋯ < t i k−1 < t i k < ⋯, which are decided by the subsequent events, and (⋅) represents the Dirac function.
Remark 3. Different from the control protocol in [29], [34] and [37], both finite-time controller and event-triggered impulsive control are considered in this work. Inspired by [31], we design an event-triggered impulsive control to achieve the leader-following finite-time consensus in this work. It is noteworthy that the update time of the finite-time control and the working time of the impulsive controller are determined by the event-triggered instants. The impulsive control only works at t k , and the impulsive instants are relied on the triggered condition. Therefore, the event-triggered time sequence is also impulsive instant sequence. Based on the designed control scheme, the distributed event-triggered scheme of control loop is shown as Figure 1.   [34] and [37], the controller (3) in this work does not contain the sign function. So far as anyone can tell, the sign function always gives rise to a chattering phenomenon which has an effect on systems and control signals. In the existing results on studying the finite-time problems, sign function is an essential part of the controller. In our study, the finite-time consensus will be investigated by designing a controller without sign function.
We can obtain i (k) = 1 if the event of agent i is triggered at t k , then there exists an appropriate k such that Therefore, the dynamics of follower agents can be further obtained from (1) and (3) where the state of agent i is changed from Then, we define the tracking error as and thusx For developing our event-triggered strategy, we consider the combined measurement function to be defined as Obviously, one can be obtained that q(t ) = (q 1 (t ), q 2 (t ), … , q N (t )) T = x(t ) ∈ ℝ N .
We assume that at least one agent's controller is triggered at time t k . So, motivated by [ [29], Remark 3], the event-triggered matrice Λ = diag( 1 (k), 2 (k), … , N (k)) is used to express the triggered case of each agent in the system. Therefore, the control law can be further described as where . On the basis of the designed controller u i (t ) and the combined measurement function q i (t ), the new measurement error function can be defined as Further, u(t ) can be rewritten as where Remark 5. As we know, traditional measurement error functions in [29] and [36] are defined as where t k is the event time of agent i. In this work, the construction of the new measurement error function e i (t ) is decided by the control input u a (t ) and the combined measurement function q i . Hence, the constructed measurement error function can simplify the process of the following theoretical proof.
Combining (1) with (2), and substituting (9) into (1), the track error dynamics (5) of LNMASs with event-triggered impulsive control and finite-time control can be depicted as the following form: where . Without loss of generality, assuming thatx i (t ) is right-hand continuous at t = t k , for instant, . Next, we present a theoretical result to guarantee that the LNMASs (1) and (2) can achieve consensus within finite time by means of the control protocol (3), where the the triggering instant t k is decided by Algorithm 1.

Theorem 1. Under all the aforementioned assumptions and the control law (3), if there exist constants
where 2 () is the minimum non-zero eigenvalue of matrix , h i (t ) = |e i (t )| − 1 2 2 |q i (t )| denotes the event-triggered function, and the triggered impulsive and finite-time control instant t k is determined by the following event-triggered condition: then, the LNMASs (1) and (2) with the controller (3) can realize the finite-time consensus in the settling time , and the settling time bounded as follows: Proof. Define the Lyapunov candidate function as From Lemma 2, it can be obtained that For t ∈ [t k , t k+1 ), k ∈ ℕ + , calculating D + V (t ) with respect to the trajectories of (10), it yields that Based on the Assumption 1, one can obtain that Then, according to Lemma 1 and condition (11) (12), we have By Lemma 3, it is obvious that On the other hand, when t = t k , k ∈ ℕ + , according to Lemma 5, (10) and conditions (13) and (14), From Lemma 4, one obtains Evidently, the error system can be found to tend to zero in the finite time, and the settling time is estimated as follows: As consequence, based on the above discussions, the LNMASs (1) and (2) can realize consensus within finite time by means of event-triggered impulsive control. This completes the proof.
In order to get rid of the Zeno behaviour, the following Theorem is verified that the inter-execution time intervals t i k+1 − t i k has a lower bound. For this bound, it is a strictly positive constant. □ Theorem 2. Consider the tracking error dynamics of LNMAS (10) under the control rule (3), the impulsive instant t k is triggered by the condition (15). For any initial condition, there exists a strictly positive constant which is the lower bound of inter-execution time intervals t i k+1 − t i k as follows. Namely, the system (10) is free of the Zeno behaviour.
According to (6), (8), (10) and the event-triggered condition (15), it can be obtained that Based on (8), substituting the first equation of (10) into (27), we can further get that LetΓ(t ) = ( 1 + 2 + Γ(t )) 2 , and Ψ i (t ) satisfies the bound Ψ i (t ) ≤ Γ(t, Γ 0 ), where Γ(t, Γ 0 ) is the solution ofΓ(t ) with the initial value being Γ(0, Γ 0 ) = Γ 0 By computing, we can further obtain that where is the lower bound of inter-execution time intervals t i k+1 − t i k . According to the event-triggered condition in Theorem 1, Γ( , 0) = 1 2 2 need to be satisfied when next event is triggered. So we can have Based on the above discussions, the lower bound of interexecution time interval is verified to exist and it satisfies t i k+1 − t i k ≥ > 0. Therefore, the Zeno behaviour is excluded and the proof is completed.
When we remove f (x i (t ), t ) and g(x 0 (t ), t ), the systems become the leader-following linear multi-agent systems. The track error dynamics (10) can be reduced to the following form as □ Then, we can easily obtain the following corollary. (3), if there exist constants 1 > 0,

Corollary 1. Under the control law
where 2 () is the minimum non-zero eigenvalue of matrix , h i (t ) = |e i (t )| − 1 2 2 |q i (t )| denotes the event-triggered function, and the triggered impulsive and finite-time control instant t k is determined by the following event-triggered condition: then, the tracking error systems (30) with the controller (3) can realize the finite-time consensus in the settling time , and the settling time bounded The communication topologȳof a multi-agent system as follows: Proof. Similar proof method with that of Theorem 1, so we omit it. Therefore, the leader-following linear MASs can realize finitetime consensus by means of event-triggered impulsive control. This completes the proof. □

NUMERICAL SIMULATION
In this section, we will provide one example to validate the effectiveness of the proposed control schemes. It is assumed that the LNMASs consist of four followers and a leader, and the communication topologȳis shown in Figure 3.
From the Figure 3, the Laplacian matrix  and the leader adjacency matrix  are obtained as follows: Considering the LNMASs with 4 follower agents and a leader agent are described by: where non-linear characteristics are f (x i (t ), t ) = 0.5x i (t ) + 0.6 cos(x i (t )), g(x 0 (t ), t ) = 0.5x 0 (t ) + 1.5 sin(x 0 (t )), and u i (t ) is in the form of (3). Set the initial states of the followers as x(0) = (1.5 0.8 0.1 − 2) T . The initial state of the leader is chosen as x 0 (0) = 0.3. Figure 4 shows that the agents and the leader do not reach an agreement in the absence of the controller.
According to (3), the parameters of the controller are chosen as 1 = 1, 2 = 1.2, = 3∕5. By simple calculation, it can be obtained that = 0.5, which satisfies the condition (13) of Theorem 1. Besides, the condition (11) in Theorem 1 implies that = 0.5 should be satisfied. The numerical results of the system under controller (3) are depicted in Figures 5-8. According to (16), the settling time can be calculated that = 1.42. In Figures 5 and 6 under protocol (3), we can see that the setting time is approximate t = 0.2, which proves the effectiveness and feasibility of Theorem 1. According to the event-triggered   Figure 7.
To further illustrate the superiority of our strategy, we give two comparison simulations. Firstly, in [34], being short of event-triggered impulsive control, the state trajectories of LNMASs only with finite-time strategy is depicted in Figure 9. Comparing with Figure 9, it can be obviously found that the LNMASs controlled only by finite-time strategy need the bigger control gains to achieve consensus, which could result in the higher control cost.
On the other hand, we simulate Figure 10 based on [29]. Comparing with the Figure 10, it shows the state trajectories of LNMASs only with event-triggered impulsive control, while the systems under our approach achieve consensus with a faster speed.
In a word, based on comparing the above results, the LNMASs controlled by (3) can achieve consensus within the finite time. Besides, the control cost can be cut down because of small control gains.

CONCLUSION
This work studied the finite-time consensus of LNMASs by means of the distributed event-triggered impulsive controller. The designed controller does not include the sign function, which eliminates the chattering phenomenon. Several sufficient conditions are derived by the event-triggered condition, which determines the impulsive instants and update time of finite-time control. Also, the event-triggered rules have been demonstrated to perform well and can get rid of the Zeno behaviour. Numerical examples have been presented to show the effectiveness of the main results. As we know, the distributed event-triggered impulsive mechanism is a popular research. Meantime, the stochastic discretetime network is also a hotspot. How to realize finite-time consensus of stochastic discrete-time network via distributed eventtriggered impulsive scheme is our future research interest.