Event-triggered leader-following formation control of general linear multi-agent systems with distributed inﬁnite input time delays

. By employing event-triggered control technique, this paper investigates the leader-following formation control problem of general linear multi-agent systems with distributed inﬁnite input time delays. To decrease computing costs, a novel event-triggered formation protocol taking into consideration of the distributed inﬁnite time delays between agents is put forward. Under the designed triggering function and triggering condition, a sufﬁcient condition on leader-following formation is obtained, and then the Zeno-behavior of triggering time sequences is excluded for the concerned closed-loop system. The continuous update of controller for each agent is avoided. Finally, the correctness and the effectiveness of these theoretical results are demonstrated by two numerical examples.


Introduction
Distributed cooperative control of multi-agent systems (MASs) [4,11,21] has been widely concerned owing to its many advantages including cost reduction, higher reliability, efficiency and flexibility, and so on.As one of the most important problems in cooperative control of MASs, formation control has been intensively studied due to its broad application in various areas, such as formation navigation of autonomous vehicles, attitude tracking of spacecrafts, collaborative load transport of robots, cooperative surveillance of aircrafts, just to name a few.The objective of formation control is to design distributed control protocol for each agent so that all agents can achieve and retain some given geometric pattern [5,23,31].
To achieve distributed formation, communication among agents plays a key role.For MASs, exchanging information among agents, time delays occur [14,20].Time delays may result in performance degradation and even instability of the closed-loop system.Therefore, formation problem of MASs featuring time delays has long attracted interest; see [10,13,16] and the references therein.
All of the aforementioned formation control of MASs with time delays focused on bounded delays.However, in formation control problems, the current state of the system may be related to all the previous history.Moreover, MASs may have different delay intervals so that a unified model (such as the pantograph equation in physics) or a complete influence of the whole past of the state is required.Consequently, it is necessary to consider MASs with distributed infinite delay in formation problems, and modeling of such distributed infinite delays is needed.The distributed infinite delays have been utilized to model many practical applications, i.e., remote control of mechatronic systems [24], HIV-1 infection passes [2], population dynamic systems [9], traffic flow dynamics [18], neural networks [25], and so on.More recently, the consensus of MASs with distributed infinite time delays starts to receive increasing attention [27,28], which is the basis of formation problem of MASs.
On the other hand, the key of implement of information exchange among the agents is to choice an appropriate control mechanism.Due to the superiority of reducing the controller update and maintaining a satisfactory control performance of the closed-loop system, the event-triggered control (ETC) mechanism [7,26,29] has attracted persistent attention to explore the formation control problem of MASs with time delays; see [3,6,12,17].For instance, researchers in [3] designed a distributed event-triggered formation scheme based on complex-valued Laplacian to study MASs with time delays.In consideration of time delays, event-triggered control strategies [12,17] were proposed to cope with formation control problem of second-order MASs.It is worth noting that only bounded time delays were considered in these aforementioned event-triggered formation results.Nevertheless, as far as the authors' knowledge, the event-triggered formation control of MASs with distributed infinite time delays still remains open.
Motivated by the aforementioned discussions, our work aims at designing an eventtriggered controller for the formation control of general linear MASs with distributed infinite input time delays.The main contributions of this paper are summarized: (i) The formation protocol considering distributed infinite input time delays is developed, which not only has practical advantages, but also include bounded distributed input delays as special cases.(ii) The ETC approach is applied to deal with the leader-following formation control of general linear MASs.The result shows that continuous update of controller is avoided, which means the improvement of the usage of a limited bandwidth Nonlinear Anal.Model.Control, 28(4):760-779, 2023 resource.Moreover, the Zeno-behavior of the event-triggered time sequence is excluded.(iii) A sufficient condition for achieving formation is presented, and the convergence rate is estimated.To make the theoretical results more convincing, a practical example is introduced, i.e., the leader-following formation of nonholonomic vehicles of unicycle type.
Notations.R n and R n×m stand for the set of n-dimensional real column vectors and n × m-dimensional real matrices, respectively.Given a matrix A, λ i (A) denotes the ith eigenvalues of matrix

Preliminaries
Consider the following general linear multi-agent systems: Leader: where A ∈ R n×n , B ∈ R n×m .x 0 (t) ∈ R n represents the state of agent 0. x i (t) ∈ R n and u i (t) ∈ R m denote the state and control input of agent i, respectively.
The communication topology among the N followers with one leader is represented by a digraph (directed graph) G = ( V , E, Ã) with a set of nodes V = {0, 1, 2, . . ., N, }.Ã = [a ij ] N ×N is the adjacency matrix of followers, where a ij > 0 if and only if (j, i) ∈ E, and a ij = 0, otherwise.B = diag{b 1 , . . ., b N } is the leader adjacency matrix associated with G, where b i > 0 if node 0 is a in-neighbor of agent i, and b i = 0 otherwise.The degree matrix is given by D The Laplacian is defined as L = D − Ã, and then T be the desired leader-following formation.The MASs (1) is said to achieve the desired leader-following formation h if for i = 1, . . ., N , lim t→∞ (x i (t) − h i ) − (x 0 (t) − h 0 ) = 0 holds with any given initial conditions.
k is called to be triggering time instant for each agent i, which is defined as follows: where is the triggering function with β 1 > 0, β 2 > 0, λ > 0, and the measurement error , and e(t) = [e T 1 (t), . . ., e T N (t)] T .In terms of ( 1) and ( 2), one has Then, according to Kronecker product of matrix [8], it holds Nonlinear Anal.Model.Control, 28(4):760-779, 2023 Definition 2. The event-triggering time sequence The communication graph G has a spanning tree with the leader as the root.
Remark 2. (See [32].)Based on Assumption 2, for any given positive definite matrix M = M T , a unique positive definite matrix P = P T can be found to ensure the algebraic Riccati equation below holds: As a result, according to linear optimal control theory, the eigenvalues λ i (A − BB T P ) are positive.
Thus, owing to Assumptions 1 and 2, Re Hence, for t t 0 , we can take these positive constants W and satisfying

Main results
A sufficient condition on leader-following formation control of MASs ( 1) is obtained in this section, and the Zeno-behavior of {t i k } is excluded for any agent.In order to reading conveniently, let We have the following lemma.
Then one has Case 2. Z = W ξ(t 0 ) , which indicates that .
https://www.journals.vu.lt/nonlinear-analysisThen one has The contradictions in ( 13) and ( 14) show that inequality ( 12) is valid for any ρ > 1. Letting ρ → 1, the inequality below can be acquired: which indicates that the leader-following formation is reached and the convergence rate can be estimated by λ.
The proof is completed.
Theorem 2. With the same conditions in Theorem 1, there does not exist Zeno-behavior for triggering time sequence {t i k }. Proof.Denote Thus, For t ∈ [t i k , t i k+1 ), computing the upper right-hand Dini derivative of e i (t) , we get Nonlinear Anal.Model.Control, 28(4):760-779, 2023 Substituting ( 16) into (17) yields . Using ( 10) and ( 15), it holds where https://www.journals.vu.lt/nonlinear-analysisBy ( 18) and e i (t i k ) = 0 one has When the triggering function ( 4) crosses zero, the next event will be triggered, i.e., Hence, Letting T i k = t i k+1 − t i k , we can obtain that and then From ( 19) one can derive that for any agent i, inf k {T i k } > 0, which means that there does not exist Zeno-behavior for any agent.
The proof is completed.

Simulation results
The theoretical results are illustrated through two numerical examples.The simulation is performed by MATLAB software, and the pseudocode of the event-triggered formation control strategy is shown as follows: Input: The initial states x0(t0), xi(t0), the desired formation h, the delay kernels ki(τ ), other parameters, like eventtriggered parameters β1, β2 and the control matrixes F , Q. Output: The error of states, the event-triggering instants, and the evolvement of agents.BEGIN t0 = 0; for t = 0 : : .
Event-triggering instants for each follower are shown in Fig. 1(a), which demonstrates that the frequency of controller update is greatly reduced and there is no Zeno-behavior.As described in Fig. 1(b), e i for followers all converge to 0. The evolution of all agents is shown in Fig. 1(c), which indicates the accomplishment of the desired formation.
Example 2. Suppose MASs include one leader and six followers of nonholonomic vehicles of unicycle type.For i = 0, 1, 2, . . ., 6, each vehicle has the follwing kinematic model: where [ xi , ȳi ] T denotes the Cartesian coordinates of the center of mass, ϑ i represents the heading angle in the inertial frame.ν i and ω i denote the linear velocity and angular velocity, respectively.By linearizing the kinematic model for each vehicle [1,22] the leader-following MASs can be described by ( 1) with the following system parameter matrices [6,15,30]:

One can verify that
The delay kernels are given as: The desired formation is described by h It is clear that A(h i −h 0 ) = 0 for any i, thus, Q can be chosen as 0.
Letting M = 0.01 diag{1, 1, 1, 1}, by solving the Riccati equation ( 6) we have 0.0458 0.1000 0.0000 0.0000 0.1000 0.4583 0.0000 0.0000 0.0000 0.0000 0.0458 0.1000 0.0000 0.0000 0.1000 0.4583  However, we find that the number of triggering time instants for MASs is very large if λ = 9.0 • 10 −5 and β 1 ∈ (0, 1.6376 • 10 −5 ) by the above calculation.In fact, the desired formation can still be reached for some large values of β 1 and λ.For example, choosing λ = 0.1, β 1 = 0.3, and β 2 = 1, the simulation results are shown in Fig. 3, from which we can also obtain the desired formation and exclude Zeno-behavior.Thus, it should be noted that 0 < λ < v and β 1 ∈ (0, y) is only a sufficient condition in Theorem 1.How to release the conservative condition is interested for us in future study.

Conclusion
The leader-following formation for general linear MASs with distributed infinite input time delays is explored based on the ETC approach.Considering the distributed infinite time delays between agents, a novel event-triggered formation control protocol is put forward.Utilizing inequality technique, leader-following formation is achieved without Zeno-behavior.It should be noted that this paper only deals with fixed formation for a classic linear MASs model with fixed communication topology, which are too strict https://www.journals.vu.lt/nonlinear-analysis in practical application.In practice, MASs may be subject to unmatched nonlinear dynamics, external disturbance and uncertainty, while communication topology may be switching.In addition, the fixed formation is not conducive for agents to adjust relative positions in time while avoiding obstacles and to expand or contract the movement scales.Therefore, the time-varying formation of nonlinear MASs considering external disturbance and uncertainty with switching topology will be investigated in further study.

Example 1 .
Suppose MASs consist of five follower agents with one leader.Let the connectivity weights a 12 = a 14 = a 21 = a 25 = a 32 = a 45 = a 53 , b 4 = b 5 = 1.The other weights are all equal to zero.Obviously, Assumption 1 is satisfied, and
A, and Re λ i (•) be the real part of the ith eigenvalue.A T represents the transpose of A. A > 0 implies that A is a positive definite matrix.A ⊗ B means Kronecker product of matrices A and B. • represents the Euclidean norm for vectors or the induced 1-norm for matrix.diag{a 1 , . . ., a n } denotes a diagonal matrix, where a i , i = 1, . . ., n, are its diagonal elements.