Nonlinear Analysis: Hybrid Systems 31 (2019) 1–18 Contents lists available at ScienceDirect Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs Variable impulsive consensus of nonlinear multi-agent systems ∗ Tiedong Ma a,b , , Zhengle Zhang a,b , Bing Cui c a Key Laboratory of Complex System Safety and Control (Chongqing University), Ministry of Education, Chongqing 400044, China School of Automation, Chongqing University, Chongqing 400044, China c School of Automation, Beijing Institute of Technology, Beijing 100081, China b article info Article history: Received 19 December 2017 Accepted 25 July 2018 Available online xxxx Keywords: Variable impulsive control Consensus Multi-agent systems Distributed control a b s t r a c t In this paper, the consensus problem for nonlinear multi-agent systems with variable impulsive control method is studied. In order to decrease the communication wastage, a novel distributed impulsive protocol is designed to achieve consensus. Compared with the common impulsive consensus method with fixed impulsive instants, the variable impulsive consensus method proposed in this paper is more flexible and reliable in practical application. Based on Lyapunov stability theory and some inequality techniques, several novel impulsive consensus conditions are obtained to realize the consensus of multi-agent systems. Finally, some necessary simulations are performed to validate the effectiveness of theoretical results. © 2018 Elsevier Ltd. All rights reserved. 1. Introduction Consensus of multi-agent systems has been extensively investigated in recent years due to its potential application in physics science and mathematics [1–6]. Generally, there are two research directions of consensus problem: the leaderless regulation problem and the leader-following tracking problem. For the leaderless case, the distributed controllers are designed for each node (agent), so that all nodes eventually converge to an unprescribed common value, which may be a constant, or may be time-varying [7–10]. For the tracking problem, a leader node is considered and acts as a command generator that generates the desired reference trajectory and ignores information from follower nodes [11–14]. There are many approaches to realize consensus of multi-agent systems, such as robust control [15,16], feedback control [17,18], adaptive control [19,20], etc. Compared with the above control algorithms, impulsive control can reduce the transmission waste in some cases, and the state information is only transmitted at impulsive instants, which dramatically reduces the amount of synchronization information transmitted among the nodes of multi-agent systems and makes the method more efficient in a large number of real-life applications [21–26]. In the literature about the impulsive consensus of multi-agent systems, some significant topics have been discussed, including convergence speed [27,28], cooperative tracking problem [29,30], consensus schemes with switching topologies [31, 32], uncertainties [33,34] and control gain error [35–37], etc. Note that the proposed control schemes in the aforementioned literature are performed with fixed impulsive instants. However, due to the hardware constraints, real systems cannot put impulsive instants at expected time exactly. For instance, we want to impose an impulse at time instant τ , but the practical impulse maybe occur in a short time window [τ − r , τ + r ], where τ and r are the center and radius of the impulsive time window respectively. Obviously, this case is not accordant with the theoretical consensus condition for fixed impulsive ∗ Corresponding author. E-mail address: tdma@cqu.edu.cn (T. Ma). https://doi.org/10.1016/j.nahs.2018.07.004 1751-570X/© 2018 Elsevier Ltd. All rights reserved. 2 T. Ma et al. / Nonlinear Analysis: Hybrid Systems 31 (2019) 1–18 control, which causes the failure of the consensus. To overcome this problem, the variable impulsive control strategy is an effective and necessary tool, which obtains larger control region in practical application. In recent years, some pioneering works investigated the stability, stabilization and synchronization with impulsive time windows, such as stability of Cohen– Grossberg neural networks [38], hybrid neural networks [39], stochastic fuzzy delayed neural networks [40], linear delayed impulsive differential systems [41] and impulsive functional differential systems [42], stabilization and synchronization of linear systems [43], general nonlinear systems [44] and Hopfield-type neural networks [45], synchronization of coupled delayed switched neural networks [46], periodically multiple state-jumps impulsive control systems [47], comparison system approach of impulsive control system [48], sandwich control systems (cyclic control system) [49]. However, due to the complexity of distributed variable impulsive control and communication graph (network topology), there are few works combining the consensus of multi-agent systems with impulsive time windows. The impulsive consensus method with impulsive time window can relieve the shortage of fixed impulsive control, and the impulsive instant can be chosen within a fixed region (i.e., the impulsive time window). In fact, there is also another impulsive protocol (so-called odd impulsive control and its extension method) has the same function. That is to say, the consensus can be realized when the impulsive sequence {t1 , t3 , t5 , . . .} or {t2k−1 } (k = 1, 2, . . .) is determined, and there is not any restriction for the even impulsive sequence {t2 , t4 , t6 , . . .} or {t2k }. In this case, we can choose impulsive instant t2k within (t2k−1 , t2k+1 ). It is worth noting that the extension of odd impulsive control strategy to variable impulsive consensus of multi-agent systems is meaningful and significant in real application. Motivated by the above discussions, this paper studies the consensus of multi-agent systems with variable impulsive control. By designing an effective distributed impulsive controller, some novel sufficient conditions with impulsive time window and odd impulsive sequence are obtained. Due to the difficulty for computers and machines to put impulses at exact time, the existing control methods for consensus of multi-agent systems with fixed impulses are no longer practical and valid. Indeed, larger networks correspond to more distributed impulsive control nodes, and the common fixed impulsive control cannot meet the actual requirement. Thus, the variable impulsive control seems particularly necessary in the consensus scheme. In this paper, the variable impulsive consensus method with impulsive time window and odd impulsive sequence are investigated respectively. To the best of our knowledge, this is the first time to intensively explore the variable impulsive consensus with impulsive time window and odd impulsive sequence. The derived results are novel and practical for the consensus problem, and also very coincident with the real world. The main contributions of this paper can be summarized as follows. (1) The impulsive consensus of a very general class of multi-agent nonlinear systems with impulsive time window is studied for the first time. The restriction of fixed impulsive instants is changed into the interval of the centers or left endpoints of adjacent impulsive time window. (2) The odd impulsive consensus of multi-agent nonlinear systems is studied for the first time. The restriction of fixed even impulsive instants is removed. (3) Conditions for consensus are given in terms of simple algebraic inequalities, and impulsive control parameters (control gain and some impulsive instants) are analyzed and seen to depend on the graph and system parameters. This gives design guidance for selection of the controller parameters. (4) Compared with the existing fixed impulsive consensus cases, the proposed variable impulsive control schemes can allow larger control region, which is more flexible and practical in real application. The rest of the paper is organized as follows. Some preliminaries are described in the next section. In Section 3, the model of nonlinear multi-agent system is given. In Section 4, impulsive consensus of multi-agent system with impulsive time window is analyzed. The conditions of odd impulsive sequence are obtained in Section 5. In Section 6, the simulation results are presented to verify the effectiveness of consensus conditions and show the difference of several control methods. Finally, we conclude this paper in Section 7. Throughout this paper, the superscript ‘T ’ stands for the transpose for a matrix. R, Rn , Rn×m denote the real numbers, the n-dimensional Euclidean space, the set of all n × m real matrices respectively. Matrices, if not explicitly stated, are assumed to have compatible dimensions. Let N+ = {1, 2, . . .}. In is the n dimensional identity matrix. diag(d1 , . . . , dN ) indicates the diagonal matrix with diagonal elements d1 to dN . ⊗ denotes the Kronecker product. λmax (A) denotes the maximal eigenvalue of matrix A. 2. Graph theory notions Throughout this paper, the communication graph among the agents is represented by a directed graph G = (V , E , A) with a nonempty finite set of N nodes V = {v1 , . . . , vN }, a set of edges or arcs E ∈ (V × V ), and the associate adjacency matrix A = [aij ] ∈ RN ×N . An edge rooted at node j and ended at node i is denoted by (vj , vi ) and aij > 0 if (vj , vi ) ∈ E , otherwise aij = 0. We assume that there are no repeated edges and no self-loops, i.e., aii = 0, ∀i ∈ N+ . Node j is called a neighbor of node i if (vj , vi ) ∈ E . The set of neighbors of node i is denoted as Ni = {j|(vj , vi ) ∈ E }. Define the in-degree of ∑N N ×N node i is di = . The Laplacian matrix is L = D − A. In a directed graph, j=1 aij and in-degree matrix as D = diag{di } ∈ R a sequence of successive edges in the form {(vi , vk ), (vk , vl ), . . . , (vm , vj )} is a direct path from node i to node j. A digraph is said to have a spanning tree, if there is a node (called a root node), such that there is a directed path from the root to any other node in the graph. T. Ma et al. / Nonlinear Analysis: Hybrid Systems 31 (2019) 1–18 3 3. Problem description Consider a system with N agents is described by ẋi (t) = Axi (t) + ψ (xi (t)), (1) where xi ∈ R is the state of agent i, (i = 1, 2, . . . , N), ψ : R → R is a continuous nonlinear function of agent i. A ∈ R is a known matrix, all the considered agents share a common state space Rn , and xi = [xi1 , xi2 , . . . , xin ]T . Consider the leader node as n n n ẋ0 (t) = Ax0 (t) + ψ (x0 (t)), n×n (2) where x0 ∈ R is the state of leader node. Let ci > 0 as the weight of edge from leader node to node i, ci > 0 if and only if there is an edge from the leader node to node i, and C = diag{ci } ∈ Rn×n . The distributed impulsive controlled system is denoted as n ⎧ ⎪ ẋi (t) = Axi (t) + ψ (xi (t)), t ̸ = tk , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨∆xi (tk ) = xi (tk+ ) − xi (tk− ) = bk ei (tk ) ∑ = bk ( aij (xi (tk ) − xj (tk )) + ci (xi (tk ) − x0 (tk ))), t = tk , ⎪ ⎪ ⎪ ⎪ j∈Ni ⎪ ⎪ ⎩ + xi (t0 ) = xi (t0 ), t0 ≥ 0, i = 1, 2, . . . , N , k ∈ N+ , (3) where ∆xi (tk ) is the jump of the state of node i at impulse instant tk , the impulsive instant sequence {tk } satisfies 0 < t0 < t1 < t2 < · · · < tk−1 < tk < · · ·, limk→∞ tk = ∞, bk is the impulsive control gain of the multi-agent nonlinear system. Assume that xi (t) is the left continuous at tk , that is xi (tk ) = xi (tk− ). Definition 1. The global synchronization error is defined as δi (t) = xi (t) − x0 (t), δi (t) ∈ Rn . (4) Definition 2. The local neighborhood synchronization error for agent i is defined as ei (t) = ∑ aij (xi (t) − xj (t)) + ci (xi (t) − x0 (t)), (5) j∈Ni where ei (t) = [ei1 (t), ei2 (t), . . . , ein (t)]T ∈ Rn . Assumption 1. Continuous nonlinear function ψ : Rn → Rn satisfies the Lipschitz condition, ∥ψ (x1 ) − ψ (x2 )∥ ≤ l ∥x1 − x2 ∥ , (6) where l is the Lipschitz constant. Define the augmented graph as G = (V , E , A), where V = {v0 , v1 , . . . , vN }, E ∈ (V × V ) and A = [aij ] ∈ R(N +1)×(N +1) . The following assumption on the graph topology is required for cooperative synchronization problems. Assumption 2. The augmented graph G contains a spanning tree with the root node being the leader node 0. Remark 1. Assumption 2 is very mild, and it includes almost all possible graph topology, such as graph G contains a spanning tree and at least the root node can get access to the leader node, graph G is disconnected and each separated subgroup is either a single node or contains a spanning tree. From (1) and (2), the error system is obtained as { δ̇i (t) = Aδi (t) + ψ (xi (t)) − ψ (x0 (t)), t ̸= tk , ∆δi (tk ) = bk ei (tk ). (7) By using Kronecker product, the system (7) can be rewritten as { δ̇ (t) = (IN ⊗ A)δ (t) + ψ (x(t), x0 (t)), t ̸= tk , ∆δ (tk ) = bk ((L + C ) ⊗ In )δ (tk ), (8) where δ (t) = [δ1T (t), δ2T (t), . . . , δNT (t)]T , x(t) = [xT1 (t), xT2 (t), . . . , xTn (t)]T ∈ RnN , x0 (t) = 1N ⊗ x0 (t) ∈ RnN , ψ (x(t), x0 (t)) = [ψ T (x1 (t)) − ψ T (x0 (t)), ψ T (x2 (t)) − ψ T (x0 (t)), . . . , ψ T (xN (t)) − ψ T (x0 (t))]T . The aim of the cooperative tracking problem is to design the impulsive controller for all nodes i in G , such that all nodes synchronize to the leader node for any initial conditions, i.e., lim δ (t) = 0. t →∞ (9) 4 T. Ma et al. / Nonlinear Analysis: Hybrid Systems 31 (2019) 1–18 Fig. 1. The diagram of impulsive time window. For simplicity, in the rest of this paper, all variables will be denoted without the time parameter t, i.e., x: = x(t), xi : = xi (t), δ : = δ (t), δi : = δi (t). 4. Consensus of multi-agent system with impulsive time window In this section, the main results will be given to accomplish consensus of nonlinear multi-agent systems with impulsive time window. The impulse radiuses and impulse centers satisfy the following Assumption 3, and the relation of some important instants is shown in Fig. 1. The shaded area in Fig. 1 indicates the possible range of actual impulsive instants, which corresponds to the so-called impulsive time window. Assumption 3. τkl −1 < tk−1 < τkr−1 < τkl < tk < τkr < τkl +1 < tk+1 < τkr+1 , k ∈ N+ , where τkl = τk − rk and τkr = τk + rk are the left and right endpoints of the kth impulsive time window respectively. τk and rk are the center and radius of the kth time window. Theorem 1. Suppose that Assumptions 1–3 hold, if there exists a constant ξ > 1 such that (λA + 2l)(∆k+1 − µk+1 ) + ln(λk ξ ) < 0, (10) where λA and λk are the maximum eigenvalue of A + A and (bk (L + C ) + IN ) (bk (L + C ) + IN ) respectively, and 0 < ∆k = τk − τk−1 < ∞, 0 < µk = rk − rk−1 < ∞ are the impulsive centers distance and radius difference respectively. Then the consensus of multi-agent systems (1) can be realized. T T Proof. Choose the Lyapunov function as V (t) = δ T δ , for t ∈ (tk−1 , tk ], one gets D+ V (t) = δ̇ T δ + δ T δ̇ = δ T (IN ⊗ (A + AT ))δ + 2δ T ψ (x, x0 ) ≤ (λA + 2l)δ T δ = (λA + 2l)V (t). (11) This lead to V (t) ≤ V (tk+−1 ) exp((λA + 2l)(t − tk−1 )). (12) When t = tk , one can get V (tk+ ) = δ T (tk+ )δ (tk+ ) = ((bk ((L + C ) ⊗ In ) + InN )δ (tk ))T ((bk ((L + C ) ⊗ In ) + InN )δ (tk )) = δ T (tk )((bk (L + C ) + IN )T (bk (L + C ) + IN ) ⊗ In )δ (tk ) ≤ λk V (tk ). (13) Note that the last inequality of (13) uses the property of Kronecker product, i.e., λmax (Π ⊗ In ) = λmax (Π ), where Π is a square matrix. For t ∈ (t0 , τ1 − r1 ], it follows from (12) that V (t) ≤ V (t0 ) exp((λA + 2l)(t − t0 )). (14) Similarly, if t ∈ (τ1 − r1 , t1 ], from (12), one can derive V (t) ≤ V (t0 ) exp((λA + 2l)(t − t0 )). (15) T. Ma et al. / Nonlinear Analysis: Hybrid Systems 31 (2019) 1–18 5 If t ∈ (t1 , τ2 − r2 ], from (12) and (13), it yields V (t) ≤ V (t1+ ) exp((λA + 2l)(t − t1 )) ≤ λ1 V (t1 ) exp((λA + 2l)(t − t1 )) ≤ λ1 V (t0 ) exp((λA + 2l)(t1 − t0 )) exp((λA + 2l)(t − t1 )) = λ1 V (t0 ) exp((λA + 2l)(t − t0 )). (16) Therefore, for t ∈ (τ1 − r1 , τ2 − r2 ], one can get κ V (t) ≤ λ11 V (t0 ) exp((λA + 2l)(t − t0 )), (17) where κk = { 0, 1, t < tk , k ∈ N+ . t ≥ tk In general, for t ∈ (τk − rk , τk+1 − rk+1 ], one can attain κ V (t) ≤ V (t0 )λ1 λ2 · · · λk−1 λkk exp((λA + 2l)(t − t0 )). From condition (10), one can get λk exp(λA + 2l)(∆k+1 − µk+1 ) < 1 . ξ Thus, for t ∈ (τk − rk , τk+1 − rk+1 ], k ∈ N+ , it yields (18) κ V (t) ≤ V (t0 )λ1 λ2 · · · λk−1 λkk exp((λA + 2l)(t − t0 )) κk ≤ V (t0 )λ1 λ2 · · · λk−1 λk exp((λA + 2l)(τk+1 − rk+1 − t0 )) ≤ V (t0 ) exp((λA + 2l)(τ1 − r1 − t0 ))(λ1 exp((λA + 2l)(∆2 − µ2 ))) · · · (19) κk (λk−1 exp((λA + 2l)(∆k − µk )))λk exp((λA + 2l)(∆k+1 − µk+1 )) ≤ 1 ξ k−1 κ V (t0 )λkk exp((λA + 2l)(∆k+1 − µk+1 + τ1 − r1 − t0 )). κ Since V (t0 )λkk exp((λA + 2l)(∆k+1 − µk+1 + τ1 − r1 − t0 )) is a finite constant, and 1/ξ k−1 → 0 as k → ∞. Thus the consensus error δ can globally asymptotically converges to zero. The proof is completed. □ Remark 2. Let τkl = τk − rk be the left endpoint of kth impulsive time window. Then the condition (10) of Theorem 1 can be changed as (λA + 2l)(τkl +1 − τkl ) + ln(λk ξ ) < 0, (20) where τkl +1 − τkl = τk+1 − rk+1 − (τk − rk ) = ∆k+1 − µk+1 . The equivalence of conditions (10) and (20) means that as long as the distance of left endpoint of adjacent impulsive time window satisfied condition (20), the consensus of multi-agent system (1) can be realized. Remark 3. The consensus condition (11) can be transformed into the following form: 1 exp((λA + 2l)(∆k+1 − µk+1 ))λk > ξ. 1 , which includes system exp((λA +2l)(∆k+1 −µk+1 ))λk l l k+1 k and system topology connection k+1 That is to say, the consensus condition is that the constraint combination parameters λA + 2l, impulsive control gain bk , impulsive interval ∆k+1 − µ =τ −τ L + C , is greater than a finite constant ξ > 1. Furthermore, it is easy to conclude that smaller ξ can obtain larger acceptable interval ∆k+1 − µk+1 (i.e., τkl +1 − τkl ), which implies larger impulsive control region. Therefore, the constant ξ contributes to the recursive result (20), which shows the existence and necessity of ξ in the consensus condition. In the proof of Theorem 1, we discuss the interval t ∈ (τk − rk , τk+1 − rk+1 ] = (τkl , τkl +1 ], k ∈ N+ , i.e., the interval between two left endpoints of the adjacent impulsive time windows. If the interval is changed into t ∈ (τk−1 , τk ], similarly, i.e., two centers distance of the adjacent impulsive time windows, then we can obtain the following Theorem 2. Theorem 2. Suppose that Assumptions 1–3 hold, if there exists a constant ξ > 1 such that (λA + 2l)(τk+1 − τk ) + ln(λk ξ ) < 0, (21) where λA and λk have same definitions with Theorem 1. Then the consensus of multi-agent systems (1) can be achieved. Proof. Choose the Lyapunov function as V (t) = δ T δ , similar to the proof of Theorem 1, i.e., (11) and (12), for t ∈ (tk−1 , tk ], it yields V (t) ≤ V (tk+−1 ) exp((λA + 2l)(t − tk−1 )). (22) 6 T. Ma et al. / Nonlinear Analysis: Hybrid Systems 31 (2019) 1–18 Table 1 The possible cases for t ∈ [t0 , τ1 ]. t 1 ≤ τ1 t1 > τ1 t ∈ [t0 , t1 ] Case 1 t ∈ (t1 , τ1 ] t ∈ [t0 , τ1 ] Case 2 Case 3 Fig. 2. The diagram of Case 1 for t ∈ [t0 , τ1 ]. Fig. 3. The diagram of Case 2 for t ∈ [t0 , τ1 ]. When t = tk , similar to Theorem 1, i.e., (13), one can get V (tk+ ) ≤ λk V (tk ). (23) For t ∈ [t0 , τ1 ], there are three cases (see Table 1 and Figs. 2–4) to be considered. ■ Case 1. It follows from (22) that V (t) ≤ V (t0 ) exp((λA + 2l)(t − t0 )). (24) ■ Case 2. It follows from (23) and (24) that V (t) ≤ V (t1+ ) exp((λA + 2l)(t − t1 )) ≤ λ1 V (t1 ) exp((λA + 2l)(t − t1 )) ≤ λ1 V (t0 ) exp((λA + 2l)(t1 − t0 )) exp((λA + 2l)(t − t1 )) = λ1 V (t0 ) exp((λA + 2l)(t − t0 )). (25) ■ Case 3. It follows from (22) that V (t) ≤ V (t0 ) exp((λA + 2l)(t − t0 )). (26) Therefore, from (24)–(26), for t ∈ (t0 , τ1 ], one gets κ V (t) ≤ λ11 V (t0 ) exp((λA + 2l)(t − t0 )), where κk (k ∈ N+ ) has the same definition with Theorem 1. (27) T. Ma et al. / Nonlinear Analysis: Hybrid Systems 31 (2019) 1–18 7 Fig. 4. The diagram of Case 3 for t ∈ [t0 , τ1 ]. Table 2 The possible cases for t ∈ (τ1 , τ2 ]. t 1 ≤ τ1 t 2 ≤ τ2 t2 > τ2 t1 > τ1 t 2 ≤ τ2 t2 > τ2 t ∈ (τ1 , t2 ] t ∈ (t2 , τ2 ] Case 1 Case 2 ∈ (τ1 , τ2 ] ∈ (τ1 , t1 ] ∈ (t1 , t2 ] ∈ (t2 , τ2 ] Case 3 Case 4 Case 5 Case 6 t ∈ (τ1 , t1 ] t ∈ (t1 , τ2 ] Case 7 Case 8 t t t t Fig. 5. The diagram of Case 1 for t ∈ (τ1 , τ2 ]. For t ∈ (τ1 , τ2 ], there are eight cases (see Table 2 and Figs. 5–12) to be considered. ■ Case 1. It follows from (22) and (23) that V (t) ≤ V (t1+ ) exp((λA + 2l)(t − t1 )) ≤ λ1 V (t1 ) exp((λA + 2l)(t − t1 )) ≤ λ1 V (t0 ) exp((λA + 2l)(t1 − t0 )) exp((λA + 2l)(t − t1 )) = λ1 V (t0 ) exp((λA + 2l)(t − t0 )). (28) ■ Case 2. It follows from (23) and (28) that V (t) ≤ V (t2+ ) exp((λA + 2l)(t − t2 )) ≤ λ2 V (t2 ) exp((λA + 2l)(t − t2 )) ≤ λ2 λ1 V (t0 ) exp((λA + 2l)(t2 − t0 )) exp((λA + 2l)(t − t2 )) = λ1 λ2 V (t0 ) exp((λA + 2l)(t − t0 )). ■ Case 3. (29) 8 T. Ma et al. / Nonlinear Analysis: Hybrid Systems 31 (2019) 1–18 Fig. 6. The diagram of Case 2 for t ∈ (τ1 , τ2 ]. Fig. 7. The diagram of Case 3 for t ∈ (τ1 , τ2 ]. Fig. 8. The diagram of Case 4 for t ∈ (τ1 , τ2 ]. It follows from (22) and (23) that V (t) ≤ V (t1+ ) exp((λA + 2l)(t − t1 )) ≤ λ1 V (t1 ) exp((λA + 2l)(t − t1 )) ≤ λ1 V (t0 ) exp((λA + 2l)(t1 − t0 )) exp((λA + 2l)(t − t1 )) = λ1 V (t0 ) exp((λA + 2l)(t − t0 )). (30) ■ Case 4. It follows from (22) that V (t) ≤ V (t0 ) exp((λA + 2l)(t − t0 )). ■ Case 5. (31) T. Ma et al. / Nonlinear Analysis: Hybrid Systems 31 (2019) 1–18 9 Fig. 9. The diagram of Case 5 for t ∈ (τ1 , τ2 ]. Fig. 10. The diagram of Case 6 for t ∈ (τ1 , τ2 ]. It follows from (23) and (31) that V (t) ≤ V (t1+ ) exp((λA + 2l)(t − t1 )) ≤ λ1 V (t1 ) exp((λA + 2l)(t − t1 )) ≤ λ1 V (t0 ) exp((λA + 2l)(t1 − t0 )) exp((λA + 2l)(t − t1 )) = λ1 V (t0 ) exp((λA + 2l)(t − t0 )). (32) ■ Case 6. It follows from (23) and (32) that V (t) ≤ V (t2+ ) exp((λA + 2l)(t − t2 )) ≤ λ2 V (t2 ) exp((λA + 2l)(t − t2 )) ≤ λ2 λ1 V (t0 ) exp((λA + 2l)(t2 − t0 )) exp((λA + 2l)(t − t2 )) = λ1 λ2 V (t0 ) exp((λA + 2l)(t − t0 )). (33) ■ Case 7. It follows from (22) that V (t) ≤ V (t0 ) exp((λA + 2l)(t − t0 )). (34) ■ Case 8. It follows from (23) and (34) that V (t) ≤ V (t1+ ) exp((λA + 2l)(t − t1 )) ≤ λ1 V (t1 ) exp((λA + 2l)(t − t1 )) ≤ λ1 V (t0 ) exp((λA + 2l)(t1 − t0 )) exp((λA + 2l)(t − t1 )) = λ1 V (t0 ) exp((λA + 2l)(t − t0 )). (35) Based on (28)–(35), for t ∈ (τ1 , τ2 ], one gets κ κ V (t) ≤ λ11 λ22 V (t0 ) exp((λA + 2l)(t − t0 )). (36) 10 T. Ma et al. / Nonlinear Analysis: Hybrid Systems 31 (2019) 1–18 Fig. 11. The diagram of Case 7 for t ∈ (τ1 , τ2 ]. Fig. 12. The diagram of Case 8 for t ∈ (τ1 , τ2 ]. In general, for t ∈ (τk−1 , τk ], one can derive κ κ k−1 k V (t) ≤ V (t0 )λ1 λ2 · · · λk−2 λk− 1 λk exp((λA + 2l)(t − t0 )). From condition (21), one can get λk exp(λA + 2l)(τk+1 − τk ) < 1 ξ . (37) Thus, for t ∈ (τk−1 , τk ], k ∈ N+ , κ κ κ κ k−1 k V (t) ≤ V (t0 )λ1 λ2 · · · λk−2 λk− 1 λk exp((λA + 2l)(t − t0 )) k−1 k ≤ V (t0 )λ1 λ2 · · · λk−2 λk− 1 λk exp((λA + 2l)(τk − t0 )) ≤ V (t0 ) exp((λA + 2l)(τ1 − t0 ))λ1 exp((λA + 2l)(τ2 − τ1 )) · · · λk−2 exp((λA + 2l)(τk−1 − ≤ κ 1 V (t0 ) kk ξ k−2 λ κk−1 λk− 1 κk−1 τk−2 ))λk− 1 (38) κk exp((λA + 2l)(τk − τk−1 ))λk exp((λA + 2l)(τk − τk−1 )) exp((λA + 2l)(τ1 − t0 )). κ κk−1 Since V (t0 )λkk λk− 1 exp((λA + 2l)(τk − τk−1 )) exp((λA + 2l)(τ1 − t0 )) is a finite constant, and 1/ξ k−2 → 0 as k → ∞. Thus the consensus error δ can globally asymptotically converge to zero. The proof is completed. □ Remark 4. It follows from the condition (10) of Theorem 1 that both the center τk and the radius rk of the impulsive time window are needed. In fact, from equivalence of conditions (10) and (20), only the left endpoints of the adjacent impulsive time windows are necessary. Therefore, in this case, the consensus condition (20) is less conservative than (10) of Theorem 1. By comparison, it follows from the condition (21) of Theorem 2 that only the center τk of the adjacent impulsive time windows is necessary. The choice of condition (20) or (21) can depend on the practical system to be considered. 5. Consensus of multi-agent system with odd impulsive sequence In this section, we consider another variable impulsive control scheme, odd impulsive sequence control, which means that we only need to control the interval between adjacent odd impulsive instants t2k−1 , and the even impulsive instants t2k T. Ma et al. / Nonlinear Analysis: Hybrid Systems 31 (2019) 1–18 11 Fig. 13. The diagram of odd impulsive sequence. can be chosen anywhere between adjacent odd impulsive instants. The corresponding diagram of odd impulsive sequence is shown in Fig. 13. Theorem 3. Suppose that Assumptions 1 and 2 hold, if there exists a constant ξ > 1 such that (λA + 2l)(t2k+1 − t2k−1 ) + ln(λ2k−1 λ2k ξ ) < 0, (39) where λA and λk have the same definitions with Theorems 1 and 2. The impulsive interval tk − tk−1 ∈ (0, ω], and ω is the finite upper bound of impulsive interval. Then the consensus of multi-agent systems (1) can be realized. Proof. Consider a Lyapunov function as V (t) = δ T δ , similar to the proof of Theorem 2, i.e., (22), for t ∈ (tk−1 , tk ], one gets V (t) ≤ V (tk+−1 ) exp((λA + 2l)(t − tk−1 )). (40) When t = tk , similar to Theorem 2, i.e., (23), one can obtain V (tk+ ) ≤ λk V (tk ). (41) For t ∈ (t0 , t1 ], it follows from (40) that V (t) ≤ V (t0 ) exp((λA + 2l)(t − t0 )). (42) V (t1 ) ≤ V (t0 ) exp((λA + 2l)(t1 − t0 )). (43) and For t ∈ (t1 , t2 ], it follows from (40), (41) and (43) that V (t) ≤ V (t1+ ) exp((λA + 2l)(t − t1 )) ≤ λ1 V (t1 ) exp((λA + 2l)(t − t1 )) ≤ λ1 V (t0 ) exp((λA + 2l)(t1 − t0 )) exp((λA + 2l)(t − t1 )) = λ1 V (t0 ) exp((λA + 2l)(t − t0 )). (44) In general, for t ∈ (tk , tk+1 ], V (t) ≤ V (tk+ ) exp((λA + 2l)(t − tk )) ≤ λ1 λ2 . . . λk V (t0 ) exp((λA + 2l)(t − t0 )). Furthermore, it follows from (39) and (45) that (45) 12 T. Ma et al. / Nonlinear Analysis: Hybrid Systems 31 (2019) 1–18 • Case 1. When t ∈ (t2k−1 , t2k ], 2k−1 V (t) ≤ V (t0 ) ∏ λi exp((λA + 2l)(t − t0 )) i=1 2k−1 ≤ V (t0 ) ∏ λi exp((λA + 2l)(t2k − t0 )) i=1 (46) = V (t0 )λ1 λ2 exp((λA + 2l)(t3 − t1 )) · · · λ2k−3 λ2k−2 exp((λA + 2l)(t2k−1 − t2k−3 )) λ2k−1 exp((λA + 2l)(t2k − t2k−1 )) exp((λA + 2l)(t1 − t0 )) ≤ 1 ξ k−1 V (t0 ) exp((λA + 2l)(t1 − t0 ))λ2k−1 exp((λA + 2l)(t2k − t2k−1 )). • Case 2. When t ∈ (t2k , t2k+1 ], V (t) ≤ V (t0 ) 2k ∏ λi exp((λA + 2l)(t − t0 )) i=1 ≤ V (t0 ) 2k ∏ λi exp((λA + 2l)(t2k+1 − t0 )) (47) i=1 = V (t0 )λ1 λ2 exp((λA + 2l)(t3 − t1 )) · · · λ2k−1 λ2k exp((λA + 2l)(t2k+1 − t2k−1 )) exp((λA + 2l)(t1 − t0 )) ≤ ξ1k V (t0 ) exp((λA + 2l)(t1 − t0 )). Since (λA + 2l), (t1 − t0 ) and (t2k − t2k−1 ) are finite constants, and 1/ξ k → 0 as k → ∞. Thus, it follows from (46) and (47) that the consensus error δ can globally asymptotically converge to zero. The proof is completed. □ If we extend the odd impulsive sequence in Theorem 3 to the more general form, the following corollary can be obtained. Corollary 1. Suppose that Assumptions 1 and 2 hold, if there exist a bounded integer n0 > 0 and a constant ξ > 1 such that (λA + 2l)(tkn0 +1 − t(k−1)n0 +1 ) + ln(λ(k−1)n0 +1 · · · λkn0 ξ ) < 0. (48) Then the consensus of multi-agent systems (1) can be achieved. Proof. Similar to the proof of Theorem 3, for t ∈ (tk , tk+1 ], one gets V (t) ≤ V (tk+ ) exp((λA + 2l)(t − tk )) ≤ λ1 λ2 . . . λk V (t0 ) exp((λA + 2l)(t − t0 )). (49) Furthermore, it follows from (48) and (49) that • Case 1. When t ∈ (t(k−1)n0 +1 , t(k−1)n0 +2 ], (k−1)n0 +1 V (t) ≤ V (t0 ) ∏ λi exp((λA + 2l)(t − t0 )) i=1 (k−1)n0 +1 ≤ V (t0 ) ∏ λi exp((λA + 2l)(t(k−1)n0 +2 − t0 )) i=1 = V (t0 )λ1 λ2 · · · λn0 exp((λA + 2l)(tn0 +1 − t1 ))λn0 +1 λn0 +2 · · · λ2n0 exp((λA + 2l)(t2n0 +1 − tn0 +1 )) · · · λ(k−2)n0 +1 λ(k−2)n0 +2 · · · λ(k−1)n0 exp((λA + 2l)(t(k−1)n0 +1 − t(k−2)n0 +1 )) λ(k−1)n0 +1 exp((λA + 2l)(t(k−1)n0 +2 − t(k−1)n0 +1 )) exp((λA + 2l)(t1 − t0 )) ≤ 1 ξ k−1 V (t0 )λ(k−1)n0 +1 exp((λA + 2l)(t(k−1)n0 +2 − t(k−1)n0 +1 )) exp((λA + 2l)(t1 − t0 )). (50) T. Ma et al. / Nonlinear Analysis: Hybrid Systems 31 (2019) 1–18 13 Fig. 14. Topology of the communication graph G . • Case 2. When t ∈ (t(k−1)n0 +2 , t(k−1)n0 +3 ], (k−1)n0 +2 V (t) ≤ V (t0 ) ∏ λi exp((λA + 2l)(t − t0 )) i=1 (k−1)n0 +2 ≤ V (t0 ) ∏ λi exp((λA + 2l)(t(k−1)n0 +3 − t0 )) i=1 = V (t0 )λ1 λ2 · · · λn0 exp((λA + 2l)(tn0 +1 − t1 ))λn0 +1 λn0 +2 · · · λ2n0 exp((λA + 2l)(t2n0 +1 − tn0 +1 )) · · · λ(k−2)n0 +1 λ(k−2)n0 +2 · · · λ(k−1)n0 exp((λA + 2l)(t(k−1)n0 +1 − t(k−2)n0 +1 )) λ(k−1)n0 +1 λ(k−1)n0 +2 exp((λA + 2l)(t(k−1)n0 +3 − t(k−1)n0 +1 )) exp((λA + 2l)(t1 − t0 )) ≤ 1 ξ k−1 (51) V (t0 )λ(k−1)n0 +1 λ(k−1)n0 +2 exp((λA + 2l)(t(k−1)n0 +3 − t(k−1)n0 +1 )) exp((λA + 2l)(t1 − t0 )). .. . • Case n0 . When t ∈ (tkn0 , tkn0 +1 ], V (t) ≤ V (t0 ) kn0 ∏ λi exp((λA + 2l)(t − t0 )) i=1 ≤ V (t0 ) kn0 ∏ λi exp((λA + 2l)(tkn0 +1 − t0 )) i=1 (52) = V (t0 )λ1 λ2 · · · λn0 exp((λA + 2l)(tn0 +1 − t1 ))λn0 +1 λn0 +2 · · · λ2n0 exp((λA + 2l)(t2n0 +1 − tn0 +1 )) · · · λ(k−2)n0 +1 λ(k−2)n0 +2 · · · λ(k−1)n0 exp((λA + 2l)(t(k−1)n0 +1 − t(k−2)n0 +1 )) λ(k−1)n0 +1 λ(k−1)n0 +2 · · · λkn0 exp((λA + 2l)(tkn0 +1 − t(k−1)n0 +1 )) exp((λA + 2l)(t1 − t0 )) ≤ 1 ξk V (t0 ) exp((λA + 2l)(t1 − t0 )). Similar to the proof of Theorem 3, it follows from (50)–(52) that the consensus error δ can globally asymptotically converge to zero. The proof is completed. □ Remark 5. Inequality condition (48) is the extension of condition (39). As the especial cases, n0 = 1 is the situation of full impulsive sequence (correspond to the common fixed time impulsive control), n0 = 2 is the situation of odd impulsive sequence (correspond to the protocol in Theorem 3). From (39) or (48), if n0 = 2, we just need to choose odd impulsive sequence {t2k−1 } (i.e., t1 , t3 , t5 , . . . ,), and the even impulsive instant t2k can be chosen within the fixed range (i.e.,(t2k−1 , t2k+1 )). From (48), if n0 = 3, we just need to choose the impulsive sequence {t3k−2 } (i.e., t1 , t4 , t7 , . . . ,), and the impulsive instants t3k−1 (i.e., t2 , t5 , t8 , . . . ,) and t3k (i.e., t3 , t6 , t9 , . . . ,) can be chosen within fixed range (i.e.,(t3k−2 , t3k+1 )). By comparison, it concludes that larger n0 is helpful to obtain larger possible impulsive control region. On the other hand, from the term λ(k−1)n0 +1 · · · λkn0 of (48), larger n0 corresponds to more complex inequality condition. Therefore, the choice of n0 in inequality (48) should depend on the practical system to be considered. 14 T. Ma et al. / Nonlinear Analysis: Hybrid Systems 31 (2019) 1–18 Fig. 15. The relation between tk and τkl . Fig. 16. Consensus error for the result in Theorem 1. 6. Numerical examples In this section, three examples are provided to validate the effectiveness of the inequality conditions and illustrate the characteristic of several control methods. The Chua’s oscillator [50] is described as ⎧ ⎪ ⎨ẋi1 = p1 (xi2 − xi1 − g(xi1 )), ẋi2 = xi1 − xi2 + xi3 , ⎪ ⎩ ẋi3 = −p2 xi2 , where p1 and p2 are system parameters and the g(·) is the piecewise linear function, which is described as g(xi1 ) = m2 xi1 + 0.5(m1 − m2 )(|xi1 + 1| − |xi1 − 1|), where m1 < m2 < 0 are two given constants, and further one gets [ −p1 (1 + m2 ) A= 1 0 p1 −1 −p2 0 1 , 0 ] ψ (xi ) = [ ] −0.5p1 (m1 − m2 )(|xi1 + 1| − |xi1 − 1|) 0 0 . T. Ma et al. / Nonlinear Analysis: Hybrid Systems 31 (2019) 1–18 15 Fig. 17. The relation between tk and τk . Fig. 18. Consensus error for the result in Theorem 2. Let p1 = 9.21, p2 = 15.995, m1 = −1.25, m2 = −0.758. Then it yields λA = 16.5492, l = |m1 p1 | = 11.5125. Consider the multi-agent system topology as shown in Fig. 14. The main results (Theorems 1–3) will be validated based on the topology G . From Fig. 14, the adjacency matrix, in-degree matrix, Laplacian matrix and pinning gain matrix are obtained as: ⎡ 0 ⎢1 A=⎣ 0 0 1 0 0 0 0 0 0 1 ⎤ ⎡ 0 1 0⎥ ⎢0 ,D = ⎣ 1⎦ 0 0 0 0 1 0 0 0 0 1 0 ⎤ ⎡ 0 1 0⎥ ⎢−1 ,L = D − A = ⎣ 0⎦ 0 0 1 −1 1 0 0 0 0 1 −1 ⎤ ⎡ 0 1 0 ⎥ ⎢0 ,C = ⎣ −1⎦ 0 1 0 0 0 0 0 0 0 0 0 ⎤ 0 0⎥ . 0⎦ 1 Let bk = −0.6, ξ = 1.01, from the consensus Eq. (10) in Theorem 1, we can attain the evaluation of the left endpoint distance of impulsive window as τkl +1 − τkl = ∆k+1 − µk+1 < − ln(λk ξ ) λA + 2l =− ln(1.01λk ) 39.5742 . The relation between the impulsive instant tk and left endpoint τkl of kth impulsive window is shown in Fig. 15. It is apparent that the actual impulsive instant tk is greater than the left endpoint instant τkl . Furthermore, Fig. 16 shows that 16 T. Ma et al. / Nonlinear Analysis: Hybrid Systems 31 (2019) 1–18 Fig. 19. The impulsive instants tk and k. Fig. 20. Consensus error for the result in Theorem 3. all agents can synchronize the leader, and indicates that the tracking process is fast (less than 0.14 s) by using impulsive consensus protocol with impulsive time window. Now keep bk and ξ unchanged, from (21), one gets the center distance of two adjacent impulsive time window as τk+1 − τk < − ln(λk ξ ) λA + 2l =− ln(1.01λk ) 39.5742 . The relation between the impulsive instant tk and the center distance of adjacent impulsive time window τk is shown in Fig. 17, which shows red star nodes (impulsive instants tk ) are distributed at both sides of blue points (the centers of impulsive time window τk ). Fig. 18 shows that consensus of the multi-agent system can be realized in 0.14 s. Next we consider the case of the odd impulsive sequence. Keep bk and ξ unchanged, from (39) in Theorem 3, one can acquire t2k+1 − t2k−1 < − ln(λ2k−1 λ2k ξ ) λA + 2l =− ln(1.01λ2k−1 λ2k ) 39.5742 . Thus, we can get the odd impulsive sequence {t2k−1 }, which is shown with blue mark in Fig. 19. Based on the characteristics of odd impulsive sequence, the corresponding even impulsive instants t2k can be chosen within the interval (t2k−1 , t2k+1 ), which is also shown with red mark in Fig. 19. As shown in Fig. 20, all the agents can synchronize to the leader in 0.16 s. T. Ma et al. / Nonlinear Analysis: Hybrid Systems 31 (2019) 1–18 17 Throughout the paper, the initial conditions are taken as x0 = [2, 0, −1]T , x1 = [−3, 1, 2]T , x2 = [−2, −1, 2]T , x3 = [−1, −1, 1]T and x4 = [1, 1, 1]T respectively. 7. Conclusions In this paper, the consensus scheme of nonlinear multi-agent systems via variable impulsive control is studied. Based on Lyapunov stability theory and some effective inequality techniques, several novel sufficient consensus conditions with impulsive time window and odd impulsive sequence are derived respectively. 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