Finite-time consensus of nonlinear multi-agent systems via impulsive time window theory: a two-stage control strategy

This paper concentrates on the finite-time consensus problem faced by nonlinear multi-agent systems (MASs) via impulsive time window theory with a two-stage control (TSC) strategy. The TSC strategy divides the whole control period into two parts: a variable impulsive control stage and a finite-time consensus control stage. Different from general single-stage control, TSC can dynamically adjust the time periods of impulsive control and finite-time control according to practical application requirements. Variable impulsive control is also discussed in this paper. Compared with the sampling based on traditional fixed impulsive theory, impulsive sampling in the TSC strategy occurs randomly within an impulsive time window and provides much more flexibility. In addition, a switching topology scheme is introduced in this paper to strengthen the stability of MASs. Finally, two numerical simulation examples (one leaderless case and one leader-following case) are used for the theoretical analysis.


Introduction
In recent decades, a tremendous amount of research has focused on the distributed collaborative evolution of multi-agent systems (MASs) such as unmanned aerial vehicles (UAVs), sensor networks, flocking [1][2][3][4][5]. Recently, with intensive study in this field, the topic of consensus problems has gradually attracted the attention of researchers [6][7][8][9].
However, the high communication cost produced by continuous control was not considered in [6][7][8][9]. Therefore, impulsive control has adopted appropriate control protocols to overcome this shortcoming [10]. As crucial and efficient control theory techniques, impulsive control schemes have been studied in MASs. This type of strategy only allows each node to transmit information at impulsive instants, which dramatically reduces the network's communication load. Recently, numer-ous meaningful results regarding MASs consensus via impulsive control were extensively investigated in [11][12][13][14]. Guan et al. [13] proposed three types of impulsive protocols with a fixed topology, a switching topology, and external disturbances to reach consensus on continuous-time MAS networks. Liu et al. [14] presented a distributed impulsive consensus protocol with input saturation to guarantee asymptotically dynamic consensus.
The impulsive control methods proposed in [10][11][12][13][14] are performed under a fixed impulsive interval. In other words, the impulsive instants and impulsive intervals must be defined in advance, which makes the consensus condition stricter. Nevertheless, the information exchange between agents cannot precisely occur at fixed impulsive sampling times because of external interference and practical system constraints. For example, impulsive sampling should occur at a fixed time of t n . Nevertheless, practical impulsivity may occur during an impulsive time window (t n − R, t n + R) of radius R. In other words, the aforementioned conditions may lead to an incorrect impulsive consensus in the system. To obtain more flexible impulsive instants, some valuable methods for variable impulsive control with impulsive inputs constrained in impulsive time windows were proposed in [15,16]. In recent years, the application of variable impulsive control in MASs has been discussed in [17][18][19]. Zhang et al. [17,18] discussed the variable impulsive control with leaderfollowing consensus for MASs with directed network topologies. Ma et al. [19] proposed a variable impulsive technique in an undirected interaction topology for the consensus of MASs.
Much research on impulsive control has been investigated. These studies have mainly concentrated on the final state of the given system and ignored the time required by the system to reach consensus. In other words, an MAS can enter an equilibrium state when t → ∞. Compared with asymptotic convergence, finite-time consensus improves the convergence rate of the system. Therefore, the combination of impulsive control and finite-time consensus theory considers both communication cost and convergence rate parameters [20][21][22][23]. Li [22] presented a consensus method for MASs within finite time via impulsive control under disturbances. Tian and Li [23] discussed the finitetime consensus of second-order MASs with impulsive effects. However, the previous study considered only fixed impulsive control. Therefore, the combination of variable impulsive control and finite-time consensus relaxes the restriction of the consensus condition and improves the flexibility of the system.
Last but not least, many current studies focus on fixed topology networks [24][25][26]. However, the communication topologies between agents change due to external disturbances or link failures. Therefore, switching topology is also discussed in this paper. In addition, the current research on the combination of impulsive control and finite-time consensus adopts fullstage control, which means that both control methods act from the beginning of the process. However, due to the increasing complexity and diversification of the application scenarios of MASs, the convergence state of full-stage control is relatively singular with poor flexibility. Inspired by the related works, our paper concentrates on the finite-time consensus of nonlinear MASs via impulsive time window theory and a two-stage control (TSC) strategy. Knowledge about finite-time stability theory, algebraic graph theory, and impulsive differential equations is used to achieve MAS consensus within finite time. The primary contributions of this paper are as follows.
(1) Unlike previous research, in this study, TSC is adopted for nonlinear MASs. TSC divides the whole control period into two parts: a variable impulsive control stage and a finite-time consensus control stage. This strategy can adjust the convergence period dynamically based on different practical application scenarios. Because impulsive control is a noncontinuous control strategy, the real-time update requirement is not strict. If the communication network conditions are not good, then the impulsive control period can be appropriately extended to reduce continuous control. On the other hand, if an MAS needs to obtain a faster convergence rate and the network conditions are good, the number of impulsive samples can be appropriately reduced. Therefore, the advantage of the proposed TSC technique in this paper is that it provides more options for practical applications. (2) Variable impulsive control technology is introduced in this paper. Compared with the fixed impulsive theory methods in existing studies, the variable impulsive strategy does not need to preset the impulsive sampling interval. Furthermore, impulsive instants can occur any time within the impulsive time window to increase the system flex-ibility. This method improves the anti-interference performance of the MASs and relaxes the restriction of the consensus condition. (3) Due to the unreliability of an actual communication network, the communication links between agents may be interrupted. Therefore, this paper also considers switching topology criteria to improve the stability of MAS networks.
This paper proceeds as follows. Algebraic graph theory is introduced in Sect. 1. We formulate some related lemmas, definitions, and assumptions in Sect. 2. In Sect. 3, a rigorous mathematical proof is made for MASs in the leaderless case and leader-following case with switching topology to reach consensus states. Some numerical simulations are shown in Sect. 4. Lastly, the conclusion is discussed.
Notations R is the set of real numbers. R m is a Euclidean space. R m×k is the real matrix set of m × k. I m is expressed as an identity matrix. N + is defined as a positive integer set. The symbol · represents the Euclidean norm operation. ⊗ denotes the Kronecker product. λ max (K ), λ min (K ), and K T define the largest and smallest eigenvalues and the transpose of any matrix K , respectively.

Algebraic graph theory
In this section, algebraic graph theory information is provided. A directed graph G = (ϑ, E , A ) describes the network structure of a MAS, where ϑ = (ν 1 , ν 2 , . . . , ν N ) is the set of agents in the network, and E ⊆ ϑ × ϑ is a link set. Assume that A = a i j ∈ R N ×N represents the relationships among all agents. If agent i can communicate with agent j, then a i j > 0; otherwise, a i j = 0, and we let a ii = 0. Moreover, G is a directed balanced graph if a i j = a ji > 0. N i = { j|(ν i , ν j ) ∈ E } depicts all neighbor agents of agent i. The number of edges connected to each node is represented by a degree matrix D = diag (d 1 , d 2 , . . . , d N ). Furthermore, the Laplacian matrix is expressed as Since the topology graph that describes the agents varies over time, a directed switching topology is considered as follows. Here, we define graphs G g , g = {1, 2, 3, . . . , ψ} , ψ ∈ N + , where g represents all possible topologies. The switch signal is denoted as σ (t) : [t 0 , +∞) → g, which is piecewise constant. Notably, G σ (t) ∈ G g = G 1 , G 2 , G 3 , . . . , G ψ , ψ ≥ 1 and G σ (t) is fixed for t ∈ t n , t n+1 ). Therefore, for each possible graph G σ (t) , the aforementioned matrix is redefined as follows:

Problem statement
The classical model of a nonlinear MAS is considered in this section. We propose a dynamic equation with N agents, which is expressed bẏ where x(t) ∈ R m denotes the state vector, and T is defined as the state of the MAS network. The control protocol of the network is indicated by u i (t). Ξ ∈ R m×m represents a constant matrix. A continuous nonlinear differentiable function is described by Assumption 1 There is a directed spanning tree in G σ (t) , and a leader acts as the root node. Moreover, for all possible topologies G g , the directed graph G σ (t) ∈ G g is strongly connected.

Assumption 2
Assume that ϕ(·) is a continuous nonlinear function that satisfies the Lipschitz condition; then, where Γ is a known positive parameter.
Lemma 4 [29] Assume that two constants ς ∈ (0, 1) and α > 0 exist and that the continuous positive definite function V : U → R satisfies: Then, the settling time that depends on the initial condition x 0 ∈ U is expressed as the following inequality: where U and ϑ are the open neighborhoods of the origin, and ϑ ⊆ U = R m . V (x) is radially unbounded; hence, the origin is globally finite-time stable.

Leaderless consensus
An effective TSC protocol without a leader is considered in this subsection as follows: where δ(t − t n ) is an impulse function, and b n is an impulsive control gain. For simplicity, let sign(y)|y| η = sign|y| η and sign(y i ) = [sign(y i1 ), sign(y i2 ), . . . , sign (y im )] T . The impulsive sequence satisfies 0 < t 0 < t 1 < · · · < t n−1 < t n < · · · and lim n→∞ t n = +∞, and The control parameter of the finite-time protocol is defined as , which plays an essential role in the smoothness of the consensus curve if y > 0. For the first control stage, combining controller (3) and system (1), the impulsive control system can be defined as

Definition 2
The upper right-hand Dini derivative can be described as

Assumption 3
The first stage has the following rule Remark 1 As shown in Fig. 1, the two endpoints of the nth variable impulsive time window are composed of τ l n and τ r n . t n is the sampling time of the nth impulsive window and follows the criterion of Assumption 3. T 1 is the dividing point between the first controlling stage and the second controlling stage; T 1 simultaneously serves as the endpoint of the first controlling part and the start point of the second controlling part. Moreover, T 1 is independent of the initial states of the agents, and it is determined by the number of impulsive intervals set in advance.

Assumption 4 Supposing that the switching graph G σ (t)
is connected and balanced, letx be the average state of all agents. Then, one obtainŝ If the graph is balanced, With Eq. (6), the Kronecker product is adopted to rewrite Eq. (4) into matrix form.
Theorem 1 Let Assumptions 1-4 hold and let λ 2 (L σ (t) ) represent the algebraic connectivity of the Laplacian matrix L σ (t) . Suppose that λ A and λ B describe the maximum eigenvalues of (Ξ T + Ξ) and Ξ , respectively.
≤ 0, and the system satisfies the condition 2λ B + 2Γ ≤ 2 , then the leaderless case of MASs can reach average consensus.
Proof In the first stage t ∈ [t 0 , T 1 ), the Lyapunov function is defined as When t = t n , we can obtain the derivation of the Lyapunov function as According to Assumption 2, Then, we can easily obtain that When t = t n , When t ∈ [t 0 , τ l 1 ), through (12), we have By (12), we can subdivide the impulsive time window.
When t = t 1 , by inequality (13), one has For t ∈ [t 1 , τ r 1 ), it follows from (15) and (16) that Therefore, when t ∈ [τ r 1 , τ l 2 ), the following can be derived from condition (17): Similar to the above derivation process, when n = 2, for t ∈ [τ l 2 , τ l 3 ), In general, for t ∈ [τ l n , τ l n+1 ), from the inequality One can obtain let c = 1, t ≥ t n , 0, t < t n . When t ω = T 1 , ω ∈ N + , inequality (21) further yields (22) where t ω is the number of impulsive sampling occurrences in the first stage, and T 1 is a random value in [τ l ω , τ l ω+1 ). We can set the size of ω according to the actual application scenario, and the number of impulsive instants determines T 1 . Furthermore, T 1 is the initial time of the finite-time stage and the time elapsed during the entire impulsive period. V (T 1 ) is the initial state of the finite-time control stage.
The error system for the finite-time control stage can be defined as follows: According to Theorem 1, for the second stage t ∈ [T 1 , T 2 ), the Lyapunov function is expressed as By Lemmas 1, 2, and 3, one obtains Thus, from Lemma 4, the time for MASs (1) to reach consensus is T 2 is the whole convergence time of the impulsive control stage and finite-time stage. It can be derived from the above inequality that V (t) ≡ 0 for all t ≥ T 2 . Thus, the leaderless case of the MASs exhibits global finite-time stability. The proof is completed.

Remark 2
Different from most studies, this paper divides the control protocol into two stages. The TSC strategy is more flexible so that it can be adjusted in different scenes according to the actual application scenarios. For example, in a case with abundant communication resources, if the MASs need to reach consensus faster, then the number of impulsive occurrences can be reduced to achieve this goal. In another scenario, if the communication network conditions are not good, the requirement regarding the convergence speed of the system is not strict. Then, the impulsive control period can be appropriately extended. With fullstage control, the consensus of the MASs is not flexible. Thus, in general, the TSC strategy provides more options according to different scenarios.

Leader-following consensus
A dynamic model of the leader node 0 is considered in this subsection: Supposing that x 0 (t) ∈ R m represents the state of node 0, the switching interaction topology for a leader is defined byḠ σ (t) .

Remark 3
In the leaderless case, each node in the network can communicate with other nodes, and the state values of all agents can eventually converge to an average value. In contrast, the leader-following consensus case can ignore the information of follower nodes. Namely, all follower nodes directly or indirectly communicate with the leader node. Finally, all agents will be in the same state as that of the leader node. Furthermore, we can choose different control schemes according to the requirements of practical applications. Therefore, the consensus of MASs with a leader is discussed in this subsection.
Consider an effective TSC protocol with a leader as follows: Let q i > 0 represent the relationship between leader node 0 and a follower node, and let the remaining parameters be identical to those in the leaderless case.

Definition 3
For any initial state, the consensus of MASs with a leader can be realized within finite time, and the following is satisfied: . . . , ξ im (t)] T . Similar to (7), for the first control stage, we can rewrite the error system of condition (28) into a matrix from: where H σ (t) = L σ (t) + Q, which describes the struc- [q 1 , q 2 , . . . , q N ] as a diagonal matrix. If Assumptions 1-4 hold, let λ min (H σ (t) ) represent the smallest eigenvalue of matrix H σ (t) . Let λ A and λ B denote the maximum eigenvalues of (Ξ T + Ξ) and Ξ , respectively. 0 < θ n < 1 represent the eigenvalue of

Theorem 2
Assuming that there is a known parameter ζ > 1 such that (τ l n+1 − τ l n )(λ A + 2Γ ) + ln(θ n ζ ) ≤ 0, and the system satisfies the condition 2λ B +2Γ ≤ 2 , then the consensus of MASs with a leader can be achieved within finite time.
Proof The Lyapunov function is chosen as The proof process for the first stage is similar to that of the leaderless case; then, for t ∈ [τ l n , τ l n+1 ), we can obtain where c = 1, t ≥ t n , 0, t < t n . When t ω = T 1 , ω ∈ N + , inequality (32) can further yield Compared with (23), the error equation of the second stage can be rewritten aṡ By Lemmas 1, 2, and 3, the proof process is identical to that of (24), and by taking the derivative of V (t), for simplicity, we have Therefore, from Lemma 4, the settling time of the leaderfollowing consensus problem is Then, this result shows that V (t) ≡ 0 for all t ≥ T 2 . Thus, MAS consensus for the leader-following case is reached.

Remark 4
If the impulsive interval τ n = t n+1 − t n is a positive constant, we can say that the MASs based on fixed impulsive control can realize consensus within finite time. Supposing that Assumptions 1-4 are satisfied, Theorem 2 of this paper can be rewritten as the same consensus criterion in Theorem 1 of [11]; that is, (t n+1 − t n )(λ A + 2Γ ) + ln(θ n ζ ) ≤ 0. The proof process for the second control stage corresponds to Theorem 2.

Remark 5
Although the TSC problem has been discussed in [20], MAS consensus via variable impulsive control under switching topology was not considered. Additionally, different from that of the fixed-time impulsive control strategies in [20][21][22], and [23], the impulsive interval τ n = t n+1 − t n is a fixed positive constant for any n. For the variable impulsive control strategy, by Theorem 1 and Theorem 2, the impulsive time window can be adjusted within a maximum range = τ l n+1 − τ l n , where are the components of the free time window and impulsive time window. Consequently, the t n impulsive instants can more flexibly occur within the maximum impulsive time window, which enables the system to handle more complex situations in practical systems, such as time delays and temporary interruptions of communication links.

Simulation
Two simulation examples with and without a leader are given in this section to prove the effectiveness and reliability of our control protocol in (3) and (28). Subsequently, we employ Chua's circuit [30], which can be utilized to describe the dynamic behaviors of the MASs: Fig. 2 Interaction topologies of G 1 , G 2 and G 3 where x i (t) denotes the state of the ith agent. μ 1 and μ 2 are known constants, and the function φ(·) is shown as follows: where π 1 < π 2 < 0 are two known parameters.

Example 1
In the leaderless consensus case, we consider three interaction topologies G 1 , G 2 , and G 3 with four nodes for the MASs, as indicated in Fig. 2. Then, as detailed in Fig. 2, the degree matrices of the topology graphs G 1 , G 2 , and G 3 are expressed as Hence, by L σ (t) = D σ (t) − A σ (t) , the Laplacian matrix of graphs G 1 , G 2 , and G 3 can be attained as follows.
From Theorem 1 and the given known parameters, let b n = −0.4, ζ = 1.03, and the step length be 0.0001. By simple matrix computation, we have ρ n = 0.5200. The maximum upper bound can be attained such that τ l n+1 − τ l n ≤ − ln(ρ n ζ ) λ A + 2Γ = − ln(1.03ρ n ) 30.81666 , n ≥ 1. For the convenience of analysis and calculation, we choose = 0.02. As illustrated in Fig. 3, impulsive instants can occur within the   Fig. 4. We choose 5 and 10 impulsive instances; then, T 1 ∈ [0.1, 0.12) and T 1 ∈ [0.2, 0.22) are random values. As demarcation points, the last impulsive instants of case 1 and case 2 can be described as t 5 = T 1 = 0.1049s and t 10 = T 1 = 0.2013s, respectively. Therefore, the results satisfy the condition range of T 1 . For the finite-time control stage, from Theorem 1, let = 40, υ = 2, and η = 0.7. As shown in Fig. 4, the settling time T 2 can be estimated to be approximately 0.2s and 0.25s, while the consensus error for each node can converge to zero for all t ≥ 0.2s and t ≥ 0.25s. It can be said that MASs (1) in the leaderless case can achieve average consensus under control protocol (3) with switching topology. By comparing Figs. 4 and 5, we observe that the convergence rate can be dynamically adjusted by setting different impulsive sampling sizes. The convergence time in Fig. 5 may be shorter than that of Fig. 4, but the full-stage control process has only one convergence state, and the number of impulsive samples cannot be changed. In Fig. 6, purely variable impulsive control can alleviate the communication burden between the nodes. However, such asymptotic consensus is difficult to achieve in the real world, as shown in Fig. 6. Meanwhile, purely finite-time  Fig. 7. Therefore, the control scheme of the leaderless case proposed in this study is efficient and realistic, as shown in Fig. 4.

Example 2
In the leader-following consensus case, three interaction topologies with a leader node and four follower nodes are shown in Fig. 8. The 0 node is a leader, and the remaining nodes are the follower agents. The degree matrices of the topology graphsḠ 1 ,Ḡ 2 , andḠ 3 are expressed as Then, through H σ (t) = L σ (t) + Q, we can easily obtain the pinning gain matrices of graphsḠ 1 ,Ḡ 2 , andḠ 3 as follows: The given known parameters are identical to those above. By simple calculation, we can infer that θ n = 0.9000; then, the maximum upper bound of = τ l n+1 − τ l n can be derived as τ l n+1 −τ l n ≤ − ln(θ n ζ )  Relationships between t n and the impulsive time window for the leader-following cases n ≥ 1. For convenience, the maximum upper bound is defined as = 0.0024. As shown in Fig. 9, impulsive instants can occur within the impulsive time window.
The initial position state values of the leader agent 0 and four follower agent nodes are randomly selected from the interval [−3, 3]. In Fig.10, the numbers of impulsive instants are set to 8 and 12, i.e., T 1 ∈ [0.0192, 0.0216) and T 1 ∈ [0.0288, 0.0312) are random values. The last impulsive instants of case 1 and case 2 can be expressed as t 8 = T 1 = 0.0207s and t 12 = T 1 = 0.0299s, respectively. Hence, the results satisfy the condition range of T 1 . In the second control stage, from Theorem 2, let = 40, υ = 2, and η = 0.7. From Fig. 10, the settling time T 2 can be estimated to be approximately 0.12s. In other words, all follower nodes will follow the leader node for all t ≥ 0.12s. Overall, consensus can be reached for the leader-following case under protocol (28) with switching topology. As indi- Error values obtained with a leader for any initial state via full-stage control cated in Fig. 12, the variable impulsive control strategy must converge to zero only when t → ∞. Furthermore, purely finite-time continuous control can obtain a faster convergence speed while increasing the communication cost, as displayed in Fig. 13. By comparing Figs. 10 and 11, we observe that the full-stage control strategy has only one convergence state, and it cannot dynamically adjust the convergence rate like TSC. Therefore, the consensus technique illustrated in Fig. 10 in this investigation is necessary.
Remark 6 From Theorems 1 and 2, a smaller ρ n value helps obtain a larger . Through simple calculation, Fig. 4  shows a greater impulsive interval than Fig. 10. A larger impulsive interval has a smaller convergence speed. Comparing Figs. 4 with 10, the consensus rate in Fig. 4 is smaller than that in Fig. 10 because ρ n < θ n . In addition, if ρ n = 1, the maximum upper bound becomes negative. Thus, the control protocol cannot affect the consensus of the MASs. Therefore, we should choose 0 < ρ n < 1 to obtain a suitable range.

Conclusion
This paper concentrates on the finite-time consensus of nonlinear MASs via impulsive time window theory according to a TSC strategy. TSC divides the whole control period into two parts: a variable impulsive control stage and a finitetime consensus control stage. Unlike general single-stage control studies, this research provides a new investigation perspective for application scenarios under different communication network conditions. Variable impulsive control is also discussed in this paper. Compared with that of traditional fixed impulsive theory, impulsive sampling in the TSC strategy occurs randomly within the impulsive time window, which provides much more flexibility. To over-come the unreliability of an actual network, a switching topology scheme is introduced to improve the system's stability. Finally, two numerical simulation examples (a leaderless case and a leader-following case) demonstrate the theoretical analysis. In future research, we will concentrate on studying the proposed multistage control strategy in actual applications.
Funding This work was supported in part by the National Natural Science Foundation of China under Grant Nos. 61876200, 62072066, and 62006031 and in part by the Natural Science Foundation Project of Chongqing Science and Technology Commission under Grant Nos. cstc2019jcyj-ms-xmX0545.

Data availability statement
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflict of interest
The authors declare that they have no conflicts of interest.