Distributed impulsive control for heterogeneous multi-agent systems based on event-triggered scheme

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Distributed impulsive control for heterogeneous multi-agent systems based on event-triggered scheme Ji Han a, Huaguang Zhang a,∗, Xiaodong Liang b, Rui Wang a a College

of Information Science and Engineering, Northeastern University, Box 134, Shenyang 110819, PR China of Electrical and Computer Engineering, College of Engineering University of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9 Canada

b Department

Received 29 January 2018; received in revised form 2 December 2018; accepted 6 January 2019 Available online xxx

Abstract In this paper, the distributed impulsive control for heterogeneous multi-agent systems based on event-triggered approach is investigated. According to whether the information transfer of the dynamic compensator is continuous or not, two different kinds of impulsive controllers are designed, respectively. Based on these two kinds of controllers, the corresponding distributed event-triggered conditions are provided, which make the impulsive instants of all agents do not need occur simultaneously. Moreover, the lower bound of impulsive intervals can also be guaranteed for all the event-triggered conditions, which means that the control schemes given in this paper can avoid the Zeno-behavior successfully. Eventually, a simulation example is proposed to support the effectiveness of the results obtained in this paper. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction During the recent years, the problem about multi-agent systems has been investigated by more and more people due to its wide applications such as flocking [1], automated highway systems [2], formation control [3], robot [4], swarming [5] and so on. As one of the basic problem for multi-agent systems, the consensus problem has attracted numerous people to ∗

Corresponding author. E-mail addresses: [email protected] (J. Han), [email protected] (H. Zhang), [email protected] (X. Liang), [email protected] (R. Wang). https://doi.org/10.1016/j.jfranklin.2019.01.055 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: J. Han, H. Zhang and X. Liang et al., Distributed impulsive control for heterogeneous multi-agent systems based on event-triggered scheme, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.01.055

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study it, see [6–11]. In [6], the consensus problem for multi-agent systems and the information flow of each agent were described by using the topology graph, and the design of topology graph that can make multi-agent systems achieve consensus was also provided. Then, as one interesting problem for consensus of multi-agent systems, how to relax the constraint condition of topology graph that can reduce the information transfer was researched by many existing works. For example, the consensus problem for multi-agent systems was studied in [9] with directed topology graph, not undirected topology graph, which means that the two-way information channel between two agents can be avoided. After that, the case of using switching topology graph was considered in [10] and [11], which provided the methods that the multi-agent systems can achieve consensus even though the topology graph was disconnected for all the time. Although literatures [6–11] have provided the effective methods to solve the consensus problem for multi-agent systems, most of them assumed that the system models of all the agents should be identical, which is difficult to realize. Therefore, considering about this problem, the consensus problem for heterogeneous multi-agent systems was pointed out and investigated by many existing works. The consensus for heterogeneous multi-agent systems with two special cases that only two different kinds of dynamic models existed and the system model of each agent was almost same except having different time delay for each agent’s controller were studied in [12] and [13], respectively. Then, considering the characteristic of heterogeneous multi-agent systems, the output consensus for heterogeneous was investigated, for example, see [14–18], which aimed to achieve the consensus of each agent’s output rather than the consensus of each agent’s state. Moreover, the case where the heterogeneous multiagent systems was considered with external disturbance was analyzed in [19] In some existing works [6–19], the controller for each agents needed the continuous update and information transfer, which may cause the congestion of information channel and waste of energy. For solving this problem, many existing papers were provided. For example, [20] discussed the digital communication which can greatly reduce the information transfer and the intermittent communication was presented in [21]. Hence, as one method to avoid this problem, the impulsive control, which makes the system be controlled by the controller only at the impulsive instants and has been used in many practical applications such as chaotic system [22] and orbital transfer of satellites [23], was investigated by many existing works. As we know, a traditional scheme of impulsive control that has been used for both normal system [24,25] and multi-agent systems [26–28] is to make impulsive time interval be fixed and predetermined, which means that the impulsive control occurs at fixed sampling period. However, in order to make system achieve consensus or stable, the fixed impulsive time interval is given by considering the worst situation, which leads to enhance the conservatism. For improving this method, the event-triggered impulsive control that makes the impulsive instants be decided by agent’s behavior and some events, and can reduce the frequency of impulsive control successfully has been investigated by many existing works, shown in [29,30]. In [29], the event-triggered impulsive control for dynamic systems was studied, which considered both the cases of linear system and nonlinear system. Furthermore, the consensus problem for homogeneous multi-agent systems utilizing clustered event-triggered impulsive control was investigated in [30]. It can be seen that the event-triggered impulsive control for normal system and multiagent systems have been investigated in some existing papers, shown in [29,30], which have provided the effective methods to make system be controlled successfully. Nevertheless, there still exist some places that can be improved: (i) Most of the existing papers about the eventPlease cite this article as: J. Han, H. Zhang and X. Liang et al., Distributed impulsive control for heterogeneous multi-agent systems based on event-triggered scheme, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.01.055

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triggered impulsive control for multi-agent systems assumed that all the agents’ system models were identical, while the case of heterogeneous multi-agent systems was few considered. (ii) Although the event-triggered scheme has been used to design the impulsive instants, all the agents shared the same impulsive instants in many existing works, which may cause high power if the number of agents is large enough. Motivated by the aforementioned issues, the consensus problem of heterogeneous multi-agent systems with impulsive scheme needs to be further studied. In this paper, the event-triggered impulsive control for heterogeneous multi-agent systems is investigated. By utilizing LMI theory and some existing results of impulsive control schemes, two kinds of controllers and corresponding event-triggered conditions are provided, respectively. The main contributions of this paper are: (i) This paper provides the schemes for solving the consensus problem of heterogeneous multi-agent systems using event-triggered impulsive control, while most existing works about impulsive control did not consider that the multi-agent systems are heterogeneous. (ii) For the two kinds of controllers given in this paper, the corresponding event-triggered conditions that the impulsive instants of the agents can be different are provided, which can avoid the case that impulsive instants for all agents should be always identical. Comparatively, most existing works cannot avoid this case, which will cause huge energy at the impulsive instants. (iii) The event-triggered schemes given in this paper combine the traditional sampling scheme with event-triggered schemes, which means that these schemes not only make the impulsive instants be driven by the designed event-triggered conditions but also provide a fixed lower bound of the impulsive time interval so as to avoid the Zeno-behavior successfully. The organization of the rest of this paper is listed as follows. The preliminaries and problem formulation are given in Section 2. In Section 3, the analysis of consensus problem for heterogeneous multi-agent systems with two kinds of controllers and the corresponding distributed event-triggered conditions is provided. The numerical example for supporting our results is given in Section 4 and a conclusion is drawn in Section 5. 2. Preliminaries and problem formulation 2.1. Notations (i) Define Rn and Rm × n as the sets of all n-dimensional vectors and all m × n-dimensional real matrices, respectively. (ii) Define  ·  as the Euclidean norm of n-dimensional vectors. (iii) Define I and O as the identity matrix and zero matrix with compatible dimensional, respectively. Moreover, In represents a n × n-dimensional identity matrix. (iv) Define rowi (A ) and coli (A) as the i-row and i-col of matrix A, respectively. Moreover, rowi, j (A ) and coli,j (A) represent [rowi (A )T rowi+1 (A )T · · · row j (A )T ]T and [coli (A ) coli+1 (A ) · · · col j (A )], respectively. (v) Define a− as the left-hand limit of a. (vi) Define Reλi (A) as the real part of the ith eigenvalue of A. (vii) Define ρ(A) as the spectral radius of A. Please cite this article as: J. Han, H. Zhang and X. Liang et al., Distributed impulsive control for heterogeneous multi-agent systems based on event-triggered scheme, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.01.055

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2.2. Algebraic graph topology Considering a system consisting of a leader and N agents, a directed graph given by g¯ = {0} ∪ g is used to describe the relationships of information exchange between them, where {0} and g are related to the leader and N agents, respectively. g = (E, V, A ), where E ⊂ V × V represents the set of edges of the graph g, V = {1, 2, . . . , N } stands for the set of vertices that represents the N agents, and A = (ai j ) ∈ RN×N is the weighted adjacency matrix associated with g and the element of it is given as follows: set ai j = 1 if the directed edge ( j, i) ∈ E, which means that agent i can get the information from agent j, and agent j is called an in-neighbor of agent i, ai j = 0 otherwise. Moreover, define Ni = { j| j ∈ V, ( j, i) ∈ E} as the set of in-neighbor index of agent i. Let matrix B = diag{b1 , b2 , . . . , bN } represent the leader adjacency matrix associated with graph g¯, where bi = 1 means that agent i can take information from leader, bi = 0 otherwise. A sequence of edges (q1 , q2 ), (q2 , q3 ), . . . , (qm−1 , qm ) in a directed graph with distinct agents qs (s = 1, 2, . . . , m) is called a directed path from agent q1 to agent qm . In this paper, assume that the network  g¯ has no self-loops or parallel edges. Set matrix D = diag{d1 , d2 , . . . , dN } with di = j∈Ni ai j , for i = 1, 2, . . . , N . Let H = B + D − A. The following results can be given. Lemma 1 [31]. If there is a directed spanning tree with the leader as the root in the graph topology g¯, Reλi (H) > 0, for i = 1, 2, . . . , N . 2.3. Problem formation Consider the heterogeneous multi-agent systems described as follows: x˙0 (t ) = A0 x0 (t ),

(1)

y0 (t ) = C0 x0 (t ),

(2)

x˙i (t ) = Ai xi (t ), t = tki ,

(3)

xi (tki ) = xi (tki− ) + ui (tki ),

(4)

yi (t ) = Ci xi (t ),

(5)

where x0 (t ) ∈ Rn0 , y0 (t) ∈ Rm represent the state and output of leader, respectively, and xi (t ) ∈ Rni , yi (t) ∈ Rm , ui (t ) ∈ Rni and tki represent the state, output, control and impulsive instant of agent i for i = 1, 2, . . . , N, respectively. Throughout this paper, the following assumptions should be satisfied. Assumption 1. The topology graph at least has a directed topology graph with the leader as the root. Assumption 2. There exists matrix i such that i A 0 = A i i , Ci i = C0 , i = 1, 2, . . . , N. Two different schemes of controllers for each agent are given as follows: Please cite this article as: J. Han, H. Zhang and X. Liang et al., Distributed impulsive control for heterogeneous multi-agent systems based on event-triggered scheme, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.01.055

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Scheme a:

⎡

z˙i (t ) = A0 zi (t ) + Fi ⎣



5

⎤ ai j (zi (t ) − z j (t )) + bi (zi (t ) − x0 (t ))⎦,

(6)

j∈Ni

ui (tki ) = Ki (xi (tki− ) − i zi (tki− )) ⎡ ⎤  tki−  i− + (I + Ki )eAi tk · e−Ai s i Fi ⎣ ai j (zi (s) − z j (s)) + bi (zi (s) − x0 (s))⎦ds, i tk−1

(7)

j∈Ni

Scheme b: z˙i (t ) = A0 zi (t ), t = tki , ⎡ zi (tki ) = zi (tki− ) + Fi ⎣



(8) ⎤ ai j (zi (tki− ) − z j (tki− )) + bi (zi (tki− ) − x0 (tki− ))⎦,

(9)

j∈Ni

u(tki ) = Ki (xi (tki− ) − i zi (tki− )),

(10)

where gain matrices Ki ∈ Rni ×ni , Fi ∈ Rn0 ×n0 and impulsive instant tki need to be designed. Remark 1. It can be easily found out that the dynamic compensators (6) and Eqs. (8)–(9) are used to transmit the information to each agent. Therefore, these two schemes represent two different cases: Scheme a represents the case that the information transfer of each agent is continuous and the control for each agent occurs at discrete-time instant; Scheme b represents the case that not only the information transfer, but also the control for each agent just occur at the discrete-time instant. Definition 1. The heterogeneous multi-agent systems (1)–(5) achieve output consensus by using dynamic compensator (6) and controller (7) (or dynamic compensator (8)–(9) and controller (10)) if limt→∞ yi (t ) − y0 (t ) = 0 for any x0 (0) and xi (0), for i = 1, 2, . . . , N . The objective of this paper is to find proper conditions that can make heterogeneous multiagent systems achieve output consensus with the control Schemes a and b, respectively. 3. Main results 3.1. Scheme a Consider the heterogeneous multi-agent systems (1)–(5) with Scheme a, the following conclusion can be given. Theorem 1. Assume that heterogeneous multi-agent systems (1)–(5) satisfy Assumptions 1 and 2, it can achieve output consensus with compensator (6) and controller (7) if there exist matrices Ki ∈ Rni ×ni , Fi ∈ Rn0 ×n0 , symmetric positive definite matrices Si ∈ Rni ×ni , P ∈ RN n0 ×N n0 and positive ti > 0 for i = 1, 2, . . . , N such that P (A˜ 0 + F˜ H˜ ) + (A˜ 0 + F˜ H˜ )T P < 0,

(11)

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Si Si (I + Ki )eAi ti

[(I + Ki )eAi ti ]T Si > 0, i = 1, 2, . . . , N, Si

(12)

where A˜ 0 = IN  A0 , H˜ = H  In0 , F˜ = diag{F1 , F2 , . . . , FN }, and the event-triggered condition is designed as follows: i tk+1 = min{tki + T , inf {t > tki + ti , | [(I + Ki )eAi (t −tk ) ε1i (tki )]T Si (I + Ki )eAi (t −tk ) ε1i (tki ) i

i

> δi (ε1i (tki ))T Si ε1i (tki )}},

(13)

where T > 0 is constant and can be chosen large enough, δ i such that 0 < δ i < 1 for i = 1, 2, . . . , N . Proof. Take ε1i (t ) = xi (t ) − i zi (t ) and ε2i (t ) = zi (t ) − x0 (t ), by using (1)-(7), we have ⎡ ⎤  ε˙1i (t ) = Ai ε1i (t ) − i Fi ⎣ ai j (ε2i (t ) − ε2 j (t )) + bi ε2i (t )⎦, t = tki , (14) j∈Ni

ε1i (tki ) = ε1i (tki− ) + ui (tki ), ⎡ ε˙2i (t ) = A0 ε2i (t ) − Fi ⎣



(15) ⎤ ai j (ε2i (t ) − ε2 j (t )) + bi ε2i (t )⎦.

(16)

j∈Ni

Obviously, if limt→∞ ε1i (t ) = 0 and limt→∞ ε2i (t ) = 0, we have limt→∞ yi (t ) − y0 (t ) = limt→∞ Ci xi (t ) − Ci i zi (t ) + Ci i zi (t ) − C0 x0 (t ) ≤ limt→∞ Ci ε1i (t ) + T T T C0 ε2i (t ) = 0. Set ε2 (t ) = [ε21 (t ) ε22 (t ) · · · ε2N (t )]T , then we have ε˙2 (t ) = (A˜ 0 + F˜ H˜ )ε2 (t ).

(17)

Since Eq. (11) holds, A˜ 0 + F˜ H˜ is Hurwitz, which means that limt→∞ ε2i (t ) = 0. For ε1i (t), we have ⎡  t  i ε1i (t ) = eAi (t −tk ) ε1i (tki ) − eAi t e−Ai s i Fi ⎣ ai j (ε2i (s) − ε2 j (s)) tki

⎤

j∈Ni

i + bi ε2i (s)⎦ ds, t ∈ (tki , tk+1 ),

(18)

i i− i− ε1i (tk+1 ) = ε1i (tk+1 ) + Ki ε1i (tk+1 ) ⎡ ⎤ i−  tk+1  i− + (I + Ki )eAi tk+1 · e−Ai s i Fi ⎣ ai j (ε2i (s) − ε2 j (s)) + bi ε2i (s)⎦ds tki

= (I + Ki )e

i− Ai (tk+1 −tki )

j∈Ni

ε1i (tki )

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+ (I + Ki )e

− (I + Ki )e = (I + Ki )e

i− Ai tk+1

i− Ai tk+1

 ·

tki

 ·

i− Ai (tk+1 −tki )

i− tk+1

tki

i− tk+1

⎡ e−Ai s i Fi ⎣ ⎡ e−Ai s i Fi ⎣



⎤ ai j (ε2i (s) − ε2 j (s)) + bi ε2i (s)⎦ds

j∈Ni



7

⎤ ai j (ε2i (s) − ε2 j (s)) + bi ε2i (s)⎦ds

j∈Ni

ε1i (tki ).

(19)

It can be easily found out that Eq. (19) can be equivalent to the following discrete-time system: αi (k + 1) = Ai (k)αi (k), ε1i (tki )

where αi (k) = and Ai (k) = (I + Ki )e αi (k)T Si αi (k). Then we have

(20) i− Ai (tk+1 −tki )

. Set Lyapunov function Vi (k) =

Vi (k) = Vi (k + 1) − Vi (k) = αiT (k + 1)Si αi (k + 1) − αiT (k)Si αi (k) = αiT (k)(ATi (k)Si Ai (k) − Si )αi (k).

(21)

By using Schur complement and condition (12), we have Vi (k) < 0 if = ti and α i (k) = 0. Meanwhile, according to event-triggered condition (13), Vi (k) < 0 when i tk+1 − tki > ti , α i (k) = 0 and satisfies condition (13). Hence Vi (k) is a monotonic decreasing function with the lower bound 0, which leads to limk→∞ Vi (k) = 0. Then, we have limk→∞ ε1i (tki ) = 0. Also, we can get i tk+1

− tki

i ε1i (t ) ≤ eAi (t −tk ) ε1i (tki ) + (t ), t ∈ (tki , tk+1 ), (22) t  where (t ) =  t i eAi (t−s) i Fi [ j∈Ni ai j (ε2i (s) − ε2 j (s)) + bi ε2i (s)]ds. By using k i limt→∞ ε2i (t ) = 0 and tk+1 − tki ≤ T , we have limt→∞ (t ) = 0. According to i tk+1 − tki ≤ T , limt→∞  (t ) = 0, limt→∞ ε1i (tki ) = 0 and condition (22), we can i get that limt→∞ ε1i (t ) = 0 when t ∈ (tki , tk+1 ). Hence, limt→∞ ε1i (t ) = 0. The proof is completed.  i

Remark 2. According to the proof of Theorem 1, it can be found out that the objective of Assumption 2 is to make the output consensus problem of systems (1)–(5) can be solved by the similar way of solving consensus problem of homogeneous systems. If systems (1)–(5) are homogeneous, limt→∞ yi (t ) − y0 (t ) = 0 can be obtained by achieving limt→∞ xi (t ) − x0 (t ) = 0. Then, set μi (t ) = xi (t ) − x0 (t ), since A0 = A1 = A2 = · · · = AN = A, systems (1)–(5) can be changed as μ˙ i (t ) = Aμi (t ), t = tki , μi (tki ) = μi (tki− ) + ui (tki ). Therefore, the original problem is shifted to achieve limt→∞ μi (t ) = 0 with the changed systems. However, since systems (1)–(5) are heterogeneous, xi (t) and x0 (t) may have different dimensions, and Ai may be different. Therefore, the aforementioned method cannot be used for heterogeneous multi-agent systems directly. Assume that Assumption 2 holds, according to Ci i = C0 , limt→∞ yi (t ) − y0 (t ) = 0 can be obtained by Please cite this article as: J. Han, H. Zhang and X. Liang et al., Distributed impulsive control for heterogeneous multi-agent systems based on event-triggered scheme, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.01.055

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achieving limt→∞ xi (t ) − zi (t ) = 0 and limt→∞ zi (t ) − x0 (t ) = 0. Moreover, according to i A0 = Ai i , ε˙1i (t ) = Ai xi (t ) − i A0 zi (t ) = Ai xi (t ) − Ai i zi (t ) = Ai (xi (t ) − i zi (t )) = Ai ε1i (t ) for t = tki . Then, the output consensus problem is shifted to achieve limt→∞ ε1i (t ) = 0 and limt→∞ ε2i (t ) = 0. Hence, the output consensus problem for systems (1)–(5) can be analyzed by a similar way of analyzing homogeneous multi-agent systems when Assumption 2 satisfies. Furthermore, as we know, giving an assumption like Assumption 2 is a normal method to solve consensus problem of heterogeneous multi-agent systems and many similar assumption conditions was given in existing papers, such as [32] and [33]. Remark 3. It can be found out that the triggered condition (13) for agent i only needs the messages of xi (t) and zi (t). Moreover, according to Eq. (6), zi (t) includes the information of zj (t) for j ∈ N j , which comes form the neighbor of agent i. Therefore, the triggered instant for each agent only depends on its neighbors and itself, which means that condition (13) is distributed. On the other hand, since the triggered condition for each agent is different, the probability of the case that the impulse for each agent occurs simultaneously is low. Furthermore, in order to avoid this case thoroughly, an improved condition can be given. Assume that the information of triggered time instant tkj for j ∈ N j can also be transferred to agent i, the improved condition can be given as follows: ⎧ i i min{tki +θ −1 T , inf {t > tki +θ −1 ti | [(I +Ki )eAi θ (t −tk ) ε1i (tki )]T Si (I +Ki )eAi θ (t −tk ) ε1i (tki ) ⎪ ⎪ ⎨ i > δi (ε1i (tki ))T Si ε1i (tki )}}, if tk+1 = tkj for j ∈ Ni and j < i, k  = 1, 2, . . . , i tk+1 = , i i i i ⎪ min{tk + T , inf {t > tk + ti | [(I + Ki )eAi (t −tk ) ε1i (tki )]T Si (I + Ki )eAi (t −tk ) ε1i (tki ) ⎪ ⎩ > δi (ε1i (tki ))T Si ε1i (tki )}}, otherwise. i where T > 0 is constant and can be chosen large enough, 1 < θ < min{ t }. By simple analT ysis, we can get that this improved condition can avoid the case of all the agents having the same impulsive instant.

3.2. Scheme b Let us consider Scheme b. In this part, we consider two different cases according to whether the impulsive time instants for all agents are the same or not. 3.2.1. Central impulsive control Assume that the impulsive time instants for all agents are the same, which means that tk1 = tk2 = · · · = tkN for i = 1, 2, . . . , N and k = 1, 2, . . .. Then the following conclusion can be obtained. Theorem 2. Consider that Assumptions 1–2 and condition tk1 = tk2 = · · · = tkN hold, the heterogeneous multi-agent systems (1)–(5) can achieve output consensus with compensator and controller (8)–(10) if there exist matrices Ki ∈ Rni ×ni , Fi ∈ Rn0 ×n0 , symmetric positive definite matrices Si ∈ Rni ×ni , P ∈ RN n0 ×N n0 for i = 1, 2, . . . , N, and positive t > 0 such that   ˜ P [(I + F˜ H˜ )eA0 t ]T P > 0, (23) ˜ P (I + F˜ H˜ )eA0 t P

Si Si (I + Ki )eAi t

[(I + Ki )eAi t ]T Si > 0, i = 1, 2, . . . , N, Si

(24)

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where H˜ , A˜ 0 and F˜ have the same definitions given in Theorem 1, and the impulsive time instant tk is designed as follows: tk+1 = min{tk + T , inf {t > tk + t | ε T (tk )RT (t )QR(t )ε(tk ) > δε T (tk )Qε(tk )}},

(25)

where T T ε(t ) = [ε1T (t ), ε2T (t )]T , ε1 (t ) = [ε11 (t ), ε12 (t ), . . . , ε1TN (t )]T , T T T T ε2 (t ) = [ε21 (t ), ε22 (t ), . . . , ε2N (t )] ,

R1 (t ) R2 (t ) ˜ , R1 (t ) = (I + K )eA(t−tk ) , R(t ) = O R3 (t )

˜ F˜ H˜ eA˜ 0 (t−tk ) , R3 (t ) = (I + F˜ H˜ )eA˜ 0 (t−tk ) , R2 (t ) = − ˜ = diag{1 , 2 , . . . , N }, K = diag{K1 , K2 , . . . , KN },  A˜ = diag{A1 , A2 , . . . , AN }, 0 < δ < 1,

 ∗ ∗ T has the same meaning as given in Theorem 1 and Q ∈ Rn ×n with n∗ = Ni=1 ni + N n0 is a symmetric positive definite matrix such that RT (tk + t )QR(tk + t ) < Q. Proof. By utilizing the proof of Theorem 1, the problem of output consensus for heterogeneous multi-agent can be turned into the asymptotic stability problem of the system described as follows: ˜

ε1 (t ) = eA(t−tk ) ε1 (tk ), t ∈ (tk , tk+1 ),

(26)

− − ε1 (tk+1 ) = R1 (tk+1 )ε1 (tk ) + R2 (tk+1 )ε2 (tk ),

(27)

˜

ε2 (t ) = eA0 (t−tk ) ε2 (tk ), t ∈ (tk , tk+1 ),

(28)

− ε2 (tk+1 ) = R3 (tk+1 )ε2 (tk ),

(29)

which means that ε(t ) = eA

∗

(t−tk )

ε(tk ), t ∈ (tk , tk+1 ),

− ε(tk+1 ) = R(tk+1 )ε(tk ),

(30) (31)

where A∗ = diag{A˜ , A˜ 0 }.  Since conditions (23) and (24) hold, ρ(R1 (tk + t )) < 1 and ρ(R2 (tk + t )) < 1, which means that ρ(R(tk + t )) < 1 and must exist the symmetric positive definite matrix Q such that RT (tk + t )QR(tk + t ) < Q. Set V (k) = ε T (tk )Qε(tk ), be similar to Theorem 1, V (k + 1) = V (k + 1) − V (k) < 0 according to conditions (23), (24) and (25). Therefore, we have limk→∞ V (k) = 0. Since Q > 0, limk→∞ ε(tk ) = 0. By utilizing the similar analysis of Theorem 1, we have limt→∞ ε(t ) = 0. The proof is completed. 3.2.2. Distributed impulsive control In this part, assume that the impulsive time instants of all agents are different. More specifically, we consider a special case that tki such that t11 = t0 , tk2 = tk1 + t1,1 , tk3 = tk2 + t1,2 , . . . , tkN = tkN−1 + t1,N−1 , Please cite this article as: J. Han, H. Zhang and X. Liang et al., Distributed impulsive control for heterogeneous multi-agent systems based on event-triggered scheme, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.01.055

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J. Han, H. Zhang and X. Liang et al. / Journal of the Franklin Institute xxx (xxxx) xxx 1 2 N tk+1 − tk1 = tk+1 − tk2 = · · · = tk+1 − tkN = k , k = 1, 2, . . . .

(32)

According to this case, we have the following conclusion. Theorem 3. Assume that Assumptions 1–2 and condition (32) hold, the heterogeneous multiagent systems (1)–(5) can achieve output consensus with compensator (8)–(9) and controller (10) if there exist matrices Ki ∈ Rni ×ni , Fi ∈ Rn0 ×n0 , symmetric positive definite matrix Q ∈ ∗ ∗ Rn ×n , and positive t2 > 0, t1,i > 0 for i = 1, 2, . . . , N such that

T (t2 )Q Q > 0, (33) Q(t2 ) Q

t1 =

N−1 

t1,i < t2 ,

(34)

i=1

where ∗ ∗ ∗ ∗ (t ) = Wˆ N eA t1,N−1 Wˆ N−1 eA t1,N−2 · · · Wˆ 2 eA t1,1 Wˆ 1 eA (t−t1 ) ,

Kˆi −F H i Wˆ i = , O Fˆi      T T T T Kˆi = [I1T , I2T , . . . , Ii−1 , K˜iT , Ii+1 , . . . , INT ]T , Fˆi = [I1T , I2T , . . . , Ii−1 , F˜iT , Ii+1 , . . . , INT ]T ,

T

T T  F H i = [O1T , O2T , . . . , Oi−1 , F H i , Oi+1 , . . . , ONT ],

Ii = rowi

n j −ni +1,

K˜i = rowi

n j −ni +1,

 F H i = rowi

n j −ni +1,

j=1

j=1

j=1

i j=1

i j=1

i j=1

n j (I ), n j (I



Ii = row(i−1)n0 +1,i·n0 (I ), Oi = rowi

j=1

n j −ni +1,

i j=1

n j (O),

+ K ), F˜i = row(i−1)n0 +1,i·n0 (I + F˜ H˜ ),

˜ ˜ ˜

n j (F H ).

H˜ , A˜ 0 , F˜ , K, A∗ have the same definitions given in Theorem 2, and k is designed as follows: k = min{T , inf {k > t2 | ε T (tk1 )T T (k )Q(k )ε(tk1 ) < δε T (tk1 )T Qε(tk1 )}, (35) where  = Wˆ N eA t1,N−1 Wˆ N−1 eA given in Theorem 1. ∗

∗

t1,N−2

· · · Wˆ 2 eA

∗

t1,1

, 0 < δ < 1 and T has the same meaning as

Proof. By similar analysis of Theorems 1 and 2, heterogeneous multi-agent systems (1)–(5) can achieve output consensus if the following system ε˙1i (t ) = Ai ε1i (t ), t = tki ,

(36) ⎡

ε1i (tki ) = (I + Ki )ε1i (tki− ) − i Fi ⎣



⎤ ai j (ε2i (tki− ) − ε2 j (tki− )) + bi ε2i (tki− )⎦,

(37)

j∈Ni

ε˙2i (t ) = A0 ε2i (t ), t = tki ,

(38)

Please cite this article as: J. Han, H. Zhang and X. Liang et al., Distributed impulsive control for heterogeneous multi-agent systems based on event-triggered scheme, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.01.055

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⎡ ε2i (tki ) = ε2i (tki− ) + Fi ⎣



11

⎤ ai j (ε2i (tki− ) − ε2 j (tki− )) + bi ε2i (tki− )⎦,

(39)

j∈Ni

such that limt→∞ ε1i (t ) + ε2i (t ) = 0. Since tki such that condition (34), we have tkN = 1 tk1 + t1 < tk1 + t2 ≤ tk+1 , which means that every agent undergoes once impulsive control 1 N 1 in t ∈ [tk , tk+1 ] and no impulsive control occurs in t ∈ (tkN , tk+1 ). Then we can get that 1      ˜ ˜ = Kˆ1 eA(k −t1 ) ε1 tkN − F H 1 eA0 (k −t1 ) ε2 tkN , ε1 tk+1 2  1  1  ˜ ˜ = Kˆ2 eAt1,1 ε1 tk+1 − F H 2 eA0 t1,1 ε2 tk+1 , ε1 tk+1 ··· N   N−1   N−1  ˜ ˜ ε1 tk+1 = KˆN eAt1,N−1 ε1 tk+1 − F H N eA0 t1,N−1 ε2 tk+1 , 1  ˜ 0 (tk −t1 )  N  A ε2 tk+1 = Fˆ1 e ε2 tk , 2    ˜ 1 , ε2 tk+1 = Fˆ2 eA0 t1,1 ε2 tk+1 ··· N    ˜ ε2 tk+1 = FˆN eA0 t1,N−1 ε2 t N−1 , k+1

which means that N    = (k )ε tkN . ε tk+1

(40)

It can be found out that ε(tkN ) = ε(tk1 ), so ε T (tk1 )T T (k )Q(k )ε(tk1 ) < T 1 δε (tk )T Qε(tk1 ) is equivalent to ε T (tkN )T (k )Q(k )ε(tkN ) < δε T (tkN )Qε(tkN ). Then, by utilizing similar analysis of Theorem 1, limk→∞ ε(tkN ) = 0 if k is designed by Eq. (35). Moreover, we can get that N  ∗ 1 . ε(tk+2 ) = Wˆ 1 eA (k+1 −t1 ) ε tk+1

(41)

Therefore, limk→∞ ε(tk1 ) = 0 if limk→∞ ε(tkN ) = 0. By similar analysis, we have limk→∞ ε(tki ) = 0 for i = 1, 2, . . . , N . Consider that     ∗ i ε(t ) = eA (t −tk ) ε tki , t ∈ tki , tki+1 , i = 1, 2, . . . , N − 1, (42) ε(t ) = eA

∗

(t −tkN )

    1 , ε tkN , t ∈ tkN , tk+1

(43)

we have limt→∞ ε(t ) = 0. Therefore, the proof is completed.  Remark 4. It can be found that the output consensus conditions of Theorems 2 and 3 are different even though t1,i → 0 is set. This is because the impulsive control changes agents’ states in an instant, which means that the variation of each agent’s state, before and after impulsive time instant, is quite large. Therefore, for each agent, the information obtained from one of the in-neighbor agents at its impulsive time instant is quite different whether this in-neighbor agent’s impulsive time instant is same with it or not, which leads to the different cases of central impulsive control and distributed impulsive control. Remark 5. Since impulsive control make the system’s state have a great change at the impulsive instant, the power needed for each agent is quite large at this instant. Moreover, Please cite this article as: J. Han, H. Zhang and X. Liang et al., Distributed impulsive control for heterogeneous multi-agent systems based on event-triggered scheme, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.01.055

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considering the case that the energy of all controllers is provided by only one machine, the power of it can hardly large enough to realize the impulsive control if all the agent systems occur impulsive control at the same time. Therefore, the distributed control scheme of this part cannot only make the information exchange occur at discrete time, but also make the impulsive time instants of all agents be different, which effectively reduces the minimum power making the whole multi-agent achieve consensus by using impulsive control scheme. Although Theorem 3 provides an approach to make the heterogeneous multi-agent systems (1)–(5) achieve consensus, it is hard to get the values of unknown parameters by using condition (33). In order to make the calculation more simple, we provide an extended approach of Theorem 3. Since  Assumption 1 holds, without loss of generality, we can realignment the agents such that j >i, j ∈Ni ai j > 0 for i = 1, 2, . . . , N − 1 and bN > 0. Then a novel dynamic compensator is designed as follows z˙i (t ) = A0 zi (t ), t = tki , ⎡

(44) ⎤



zi (tki ) = zi (tki− ) + Fi ⎣

ai j (zi (tki− ) − z j (tki− )) + bi (zi (tki− ) − x0 (tki− ))⎦.

(45)

j >i, j ∈Ni

According to this compensator, we have the following conclusion. Corollary 1. The heterogeneous multi-agent systems (1)–(5) can achieve output consensus with compensator (44)–(45), controller (10) and event-triggered condition (35) if there exist matrices Ki ∈ Rni ×ni , Fi ∈ Rn0 ×n0 , symmetric positive definite matrices Si ∈ Rni ×ni , Pi ∈ Rn0 ×n0 and positive t1,i , t2 , for i = 1, 2, . . . , N such that

[(I + Ki )eAi t2 ]T Si Si > 0, (46) Si (I + Ki )eAi t2 Si

Pi Pi i (t2 )

t1 =

N−1 

Ti (t2 )Pi Pi

> 0,

(47)

t1,i < t2 ,

(48)

j=1

where i (t ) = eAi

N−1 j=i−1

t1, j

N−1

(I + hii Fi )eAi (t−

j=i−1

t1, j )

, t1,0 = 0,

(49)

and hii represents ith diagonal element of H. Proof. By using the proof of Theorem 3, the heterogeneous multi-agent systems (1)–(5) can achieve output consensus if conditions (33) and (48) hold. Consider the compensator given in Eqs. (44)–(45), we have ⎛ ⎞ I O O Fˆi = ⎝O col(i−1)n0 +1,i·n0 (F˜i ) coli·n0 +1,Nn0 (F˜i )⎠. (50) O O I Please cite this article as: J. Han, H. Zhang and X. Liang et al., Distributed impulsive control for heterogeneous multi-agent systems based on event-triggered scheme, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.01.055

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13

Fig. 1. The topology structure of heterogeneous multi-agent systems.

By utilizing simple analysis, we can find out that (t) with condition (50) such that ⎛ ⎞ 11 (t ) 12 (t ) · · · 12N (t ) ⎜ O 22 (t ) · · · 22N (t ) ⎟ ⎜ ⎟ (t ) = ⎜ . (51) ⎟, .. .. .. ⎝ .. ⎠ . . . O

···

O

2N2N (t )

where ii (t ) = (I + Ki )e for i = 1, 2, . . . , N and ii (t ) = i−N (t ) for i = N + 1, N + 2, . . . , 2N . Therefore, ρ((t)) < 1 if and only if ρ((I + Ki )eAi t ) < 1 and ρ( i (t)) < 1 for i = 1, 2, . . . , N, which means that condition (33) is equivalent to conditions (46)–(47) if we use compensator (44)–(45). The proof is completed.  Ai t

4. Numerical example Consider the heterogeneous multi-agent systems (1)–(5) with one leader and four agents. The topology structure is shown in Fig 1 and the detail parameters are given as follows:





2 4 0 −1 2.5 2 , A1 = , A2 = , A0 = −2 0 −2 1 −2 −2 ⎛ ⎞

0 2 0 2 −2 , A4 = ⎝−2 0 1⎠, A3 = 4 −2 0 0 1           C0 = 1 1 , C1 = 1 0.5 , C2 = 1 2 , C3 = 0.4 0.2 , C4 = 1 1 2 , ⎛ ⎞





1 0 1 0.5 1 −1 1 2 , 2 = , 3 = , 4 = ⎝ 0 1 ⎠ , 1 = 0 1 0 1 3 1 0 0 T = 3. (52) (i) Think about the case that this system is controlled by dynamic compensator (6) and controller (7). By utilizing Theorem 1, choose





0 0 0 −0.5 −0.45 −0.5 , K2 = , K3 = , K1 = 0 −0.5 0 −0.45 0 −0.6 ⎛ ⎞ −0.4 0 0 0 ⎠, K4 = ⎝−0.35 −0.4 −0.4 −0.3 −0.5 Please cite this article as: J. Han, H. Zhang and X. Liang et al., Distributed impulsive control for heterogeneous multi-agent systems based on event-triggered scheme, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.01.055

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1.5

y (t)−y (t) 1

the output errors of all agents

1

0

y2(t)−y0(t) y3(t)−y0(t)

0.5

y4(t)−y0(t) 0 −0.5 −1 −1.5 −2 −2.5

0

2

4

6

8

10

time (s) Fig. 2. The output errors of all agents with dynamic compensator (6) and controller (7).







0 −5 −2 −4 −4.5 −2 −5.5 , F2 = , F3 = , F4 = F1 = 0 −5 2 −6 0 −3 2 t1 = t2 = t3 = t4 = 0.5, δ1 = 0.2, δ2 = 0.1, δ3 = 0.1, δ4 = 0.3.

0 , −4

According to Fig 2 and Fig 3, this system has achieved output consensus. (ii) Think about the case that this system is controlled by dynamic compensator (8)–(9) and controller (10). Choose K1 K4 F1 F4



0 −0.6 −0.55 , K2 = = 0 −0.6 0 ⎛ ⎞ −0.7 0 0 0 ⎠, = ⎝−0.35 −0.8 −0.4 −0.3 −0.9

−2 −0.5 −0.64 , F2 = = 0 −0.5 2

0 −0.75 . = 2 −0.9

0 −0.5 , K3 = −0.55 0

0 , −0.6

0 −0.65 , F3 = −0.7 0

−2 , −0.8

(ii-a) Consider the triggered time instants of all agents are the same. By utilizing Theorem 2, set t = 0.05 and δ = 0.7. It can be found out that this system has achieved output consensus according to Fig 4 and Fig 5. (ii-b) Consider the triggered time instants of all agents are different. By Fig 1, (9) and (45) are equivalent in this example. Set δ = 0.9, according to Theorem 3 and Corollary 1, Please cite this article as: J. Han, H. Zhang and X. Liang et al., Distributed impulsive control for heterogeneous multi-agent systems based on event-triggered scheme, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.01.055

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6

agent 1 agent 2 agent 3 agent 4

triggered time instant

5

4

3

2

1

0

0

2

4

6

8

10

time (s) Fig. 3. The triggered time instants of all agents with dynamic compensator (6) and controller (7).

6

y1(t)−y0(t) y2(t)−y0(t)

the output errors of all agents

4

y (t)−y (t) 3

0

y4(t)−y0(t)

2

0

−2

−4

−6

0

2

4

6

8

10

time (s) Fig. 4. The output errors of all agents with dynamic compensator (8)–(9) and controller (10).

Please cite this article as: J. Han, H. Zhang and X. Liang et al., Distributed impulsive control for heterogeneous multi-agent systems based on event-triggered scheme, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.01.055

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3 2.8

triggered time instant

triggered time instant

2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1

0

2

6

4

8

10

time (s) Fig. 5. The triggered time instant of system with dynamic compensator (8)–(9) and controller (10). Table 1 The cases of triggered frequency for Scheme b and condition (35) with t2 = 0.05, 0.04, 0.03, 0.02, respectively. The value of t2

Triggered times

Mean time interval

0.05 0.04 0.03 0.02

185 227 222 199

0.053 0.044 0.045 0.05

set t1,1 = t1,2 = t1,3 = 0.01 and t2 = 0.05. Through Figs. 6–8, this system has achieved output consensus. Moreover, a table listing triggered times and mean time interval with different values of t2 is given by Table 1. According to this table, we can find out that the triggered frequency is different with change of t2 but will not increase with the decrease of t2 , which means that t2 can affect the triggered frequency but the triggered times also be influenced by the real time information of system. It can be found out that, the triggered frequency of Scheme a is much lower than that of Scheme b. This is because of the difference of the triggered conditions. More specially, for Scheme a, impulsive control is only used for each agent, which means that triggered condition (13) only needs to achieve limt→∞ ε1i (t ) = 0, however, for Scheme b, impulsive control is used for both each agent and its dynamic compensator, which means that the triggered conditions (25) and (35) should make limt→∞ ε1i (t ) = 0 and limt→∞ ε2i (t ) = 0 both Please cite this article as: J. Han, H. Zhang and X. Liang et al., Distributed impulsive control for heterogeneous multi-agent systems based on event-triggered scheme, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.01.055

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4

y (t)−y (t) 1

3

2

the output errors of all agents

0

y (t)−y (t) 0

y3(t)−y0(t)

2

y (t)−y (t) 4

0

1

0

−1

−2

−3

0

2

4

6

8

10

time (s) Fig. 6. The output errors of all agents with dynamic compensator (8), (45) and controller (10).

6

agent 1 agent 2 agent 3 agent 4

triggered time instant

5

4

3

2

1

0

0

2

4

6

8

10

time (s) Fig. 7. The triggered time instant of all agents with dynamic compensator (8), (45) and controller (10).

Please cite this article as: J. Han, H. Zhang and X. Liang et al., Distributed impulsive control for heterogeneous multi-agent systems based on event-triggered scheme, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.01.055

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agent 1 agent 2 agent 3 agent 4

triggered time instant

5

4

3

2

1

0

0

0.2

0.4

0.6

0.8

1

time (s) Fig. 8. The triggered time instant of all agents with dynamic compensator (8), (45) and controller (10).

hold. Therefore, if Aj is not Huwritz for j = 0, 1, 2, . . . , N, conditions (25) and (35) may be much stricter than condition (13), which can lead to high triggered frequency. 5. Conclusion In this paper, the research on the consensus of heterogeneous multi-agent systems with event-triggered impulsive control has been given. By using some basic results of impulsive control and LMI theory, two types of controllers have been designed and corresponding distributed event-triggered schemes have been proposed, respectively. According to the conclusions of Theorems 1, 2 and the numerical example, we have proved that our control schemes can make heterogeneous multi-agent systems achieve output consensus and avoid the Zenobehavior. The event-triggered control for heterogeneous multi-agent systems with unknown disturbance and time-delay will be studied in our future works. Acknowledgment This work was supported by the National Natural Science Foundation of China (61433004, 61627809, 61621004). References [1] J. Toner, Y. Tu, Flocks, herds, and schools: a quantitative theory of flocking, Phys. Rev. E 58 (4) (1998) 4828–4858. [2] J.G. Bender, An overview of systems studies of automated highway systems, IEEE Trans. Veh. Technol. 40 (1) (1991) 82–99. Please cite this article as: J. Han, H. Zhang and X. Liang et al., Distributed impulsive control for heterogeneous multi-agent systems based on event-triggered scheme, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.01.055

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Please cite this article as: J. Han, H. Zhang and X. Liang et al., Distributed impulsive control for heterogeneous multi-agent systems based on event-triggered scheme, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.01.055