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Event-triggered control for multi-agent systems with randomly occurring nonlinear dynamics and time-varying delay
Available online at www.sciencedirect.com
Journal of the Franklin Institute 351 (2014) 2582–2599 www.elsevier.com/locate/jfranklin
Event-triggered control for multi-agent systems with randomly occurring nonlinear dynamics and time-varying delay Hongjie Lia, Chen Minga, Shigen Shena, W.K. Wongb,n a
College of Mathematics and Information and Engineering, Jiaxing University, Zhejiang 314001, PR China b Institute of Textile and Clothing, The Hong Kong Polytechnic University, Hong Kong, PR China Received 29 June 2012; received in revised form 6 December 2013; accepted 2 January 2014 Available online 16 January 2014
Abstract The paper investigates the consensus problem for multi-agent systems with randomly occurring nonlinear dynamics and time-varying delay. A novel event-triggered scheme has been proposed, which can lead to a significant reduction in information communication in a network. By utilizing stochastic analysis and properties of the Kronecker product, consensus criteria are derived in the form of linear matrix inequalities, which can be readily solved using the standard numerical software. Finally, an illustrative example is used to show the effectiveness of the event-triggered scheme. & 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction The decentralized coordinated control of multi-agent systems has witnessed an increasing interest owing to its many applications in biology, computer science and control engineering [1–6]. A critical problem in distributed coordinated control of multiple agents is the consensus problem, which is the key to find control laws such that all agents can reach an agreement and has been extensively studied in the past few years with many profound results [4,7–12]. The decentralized consensus control for multi-agent is facilitated by recent technological advances in computing and communication resources. Future design may equip each agent with a small embedded micro-processor, which collects information from neighboring nodes and n
Corresponding author. E-mail address: [email protected] (W.K. Wong).
0016-0032/$32.00 & 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2014.01.004
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actuates the controller updates according to some rules. The control update scheduling can be done in a time-driven or an event-driven fashion. The first case involves the traditional approach to sampling at pre-specified time instances, usually separated by a specific period and all the sampled signals are transmitted through communication networks regardless of the state of the system to be controlled. Clearly, it may be conservative to guarantee the stability of a network in the worst cases, as stated in [13], the worst cases are seldom happen in practical systems. Therefore, this kind of sampled control method can lead to sending of many unnecessary signals to a network, which may cause a huge wastage of communication bandwidth. In real-world application, embedded processors are resources-limited, and thus it seems that an event-triggered approach is more favorable [13–20], where sampled signals whether or not to be instantly transmitted to the controller are determined by certain events that are triggered depending on some rules. The rules provide a useful way of determining when the sampled signals are sent out. In [13], the event-triggered H 1 control design is investigated for networked control systems with uncertainties and transmission delays, the novel event-triggered scheme can lead to a larger average release period. Peng et al. [18] proposes a discrete event-triggered communication scheme for a class of networked Takagi–Sugeno fuzzy systems. Compared with a time-triggered periodic communication scheme, fewer communication resources are utilized. Hu et al. [19] concerns the control design of event-triggered networked systems with both state and control input quantizations by taking the characteristics of an event-triggered mechanism into account, and a novel interval delay analysis technique is developed. In [20], the event-based H 1 filtering problem is studied for networked systems with communication delay. Recently, event-triggered control strategies are used to achieve the consensus for multi-agent systems [21–26], Dimarogonas et al. [21–23] investigates the consensus for a single integrator model-based event-triggered control and actuation updates depended on the ratio of a certain measurement error with respect to the norm of a function of the state. Liu et al. [26] studies average-consensus protocols for multi-agent systems, two average consensus protocols with different trigger functions are proposed. But in [21,22,26], the agent dynamics are assumed to be first-order, and the effect of transmission delay is not considered. As is well known, a wide class of practical systems is influenced by additive nonlinear disturbances that are caused by environmental circumstances. such nonlinear disturbances themselves may be subject to random abrupt changes, which may result from abrupt phenomena such as random failures and repairs of the components. In other words, the nonlinear disturbances may occur in a probabilistic way. The randomly occurring nonlinearities, also called stochastic nonlinearities, have recently received some interest in the literature [27,28]. To date, very little research effort has been done about consensus problem for multi-agent systems with randomly occurring nonlinear dynamics and time varying-delay. The purpose of our study is to fill the gap. Motivated by the above discussion, the paper studies the consensus problem for multi-agent systems with randomly occurring nonlinear dynamics and time-varying delay. To reduce communication load in a network, a novel event-triggered scheme has been proposed. Whether the sampled state signal of each agent should be transmitted to its neighbors is determined by the error between the current sampled state and the latest transmitted one. Based on the Lyapunov stability theory, consensus criteria can be obtained in the form of linear matrix inequalities. A numerical example is given to illustrate the effectiveness of the proposed method. The main contributions of this paper are highlighted as follows (1) An novel event-triggered scheme is proposed. Compared with time-triggered periodic communication scheme, the number of transmitted state signals through multi-agent networks
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is reduced since the proposed communication scheme only depends on the state at the sampled instant and the state error between the current sampled instant and the latest transmitted state. Therefore, communication resources embedded in agents can be saved and communication can be reduced. (2) Consensus criteria are obtained in the form of linear matrix inequalities which can be readily solved by using the LMI toolbox in Matlab. The solvability of derived conditions depends not only on trigger parameters and the probability of switching in nonlinear functions, but also the size of delay.
2. Preliminaries Let g ¼ fv; ɛ; Ag be a weighted diagraph of order N, with the set of nodes v ¼ fv1 ; v2 ; …; vN g, an edge set ɛDv v, and a weighted adjacency matrix A ¼ ðaij ÞNN with nonnegative elements. Directed edge eij in network g is denoted by the ordered pair of nodes ðvi ; vj Þ, which means that node vj can receive information from node vi. The elements of adjacency matrix A are defined as aij 40 if and only if there is a directed edge ðvj ; vi Þ in g, otherwise aij ¼ 0. We assume that aii ¼ 0 for all iA v. The neighbor set of node i is defined by N i ¼ fjA vjðvj ; vi ÞA ɛg, and the in-degree and out-degree of node i are defined as degin ðiÞ ¼
N
∑
j ¼ 1;j a i
aij ;
degout ðiÞ ¼
N
∑
j ¼ 1;j a i
ð1Þ
aji
A diagraph is balanced if degin ðiÞ ¼ degout ðiÞ for all iA v. The Laplacian matrix L ¼ ðlij ÞNN associated with adjacency matrix A is defined as lij ¼ aij lii ¼
ði; j ¼ 1; 2; …; N; ia jÞ
ð2Þ
N
∑
j ¼ 1;j a i
ð3Þ
aij
which ensures the diffusion property that ∑Nj¼ 1 lij ¼ 0. Consider the following multi-agent systems with stochastic nonlinearities: x_ i ðtÞ ¼ Axi ðtÞ þ βðtÞf ðxi ðtÞÞ þ ð1 βðtÞÞgðxi ðtÞÞ þ ui ðtÞ ði ¼ 1; 2; …; NÞ
ð4Þ
is a where xi ðtÞ ¼ ðxi1 ðtÞ; xi2 ðtÞ; …; xin ðtÞÞ A R is the position state of the ith agent. A A R constant matrix. f ðxi ðtÞÞ ¼ ðf 1 ðxi1 ðtÞÞ; f 2 ðxi2 ðtÞÞ; …; f n ðxin ðtÞÞÞT A Rn and gðxi ðtÞÞ ¼ ðg1 ðxi1 ðtÞÞ; g2 ðxi2 ðtÞÞ; …; gn ðxin ðtÞÞÞT A Rn are nonlinear vector-valued functions satisfying certain conditions to be given later. ui ðtÞ ¼ ðui1 ðtÞ; ui1 ðtÞ; …; uin ðtÞÞT A Rn is control input to be designed, and βðtÞ is a Bernoulli stochastic variable that describes the following random events of system (4): ( 1 if f ðxi ðtÞÞ occurs βðtÞ ¼ ð5Þ 0 if gðxi ðtÞÞ occurs T
n
nn
it follows that βðtÞ satisfies ProbfβðtÞ ¼ 1g ¼ EfβðtÞg ¼ β0
ð6Þ
ProbfβðtÞ ¼ 0g ¼ 1 EfβðtÞg ¼ 1 β0
ð7Þ
where constant β0 A ½0; 1 reflects the occurrence probability of nonlinear functions f ðÞ and gðÞ.
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Remark 1. Inspired by references [27,28], the random variable βðtÞ is used to model the probability distribution of nonlinear functions, to the best of the author0 s knowledge. It is the first attempt to consider the occurrence of different nonlinear functions in a probabilistic for multiagent systems. In this paper, control input ui(t) is sampled before entering networks based on the sampling technique and zero-order hold circuit. That is ui ðtÞ ¼ c ∑ aij ðxj ðt k hÞ xi ðt k hÞÞ j A Ni
t A ½t k h; t kþ1 hÞ
ð8Þ
where ui(t) is a discrete-time control signal, tk denotes the sampling instance of the ith agent and satisfies limk-1 t k ¼ 1, h is the sampling period. In practical systems, periodic sampling mechanisms often lead to sending of many unnecessary signals through networks, which in turn increases the load of network transmission and wastage of the network bandwidth. Therefore, it is significant to introduce a mechanism to determine which sampled signal should be sent out. As stated in [19], the event-triggered sampling scheme is effective because they can reduce traffic and power consumption. The sampled data xðt kþj hÞ is released by the event generators only when the current sampled value xðt kþj hÞ and the previously transmitted one xðt k hÞ satisfy the following judgment algorithm: ½xi ðt kþj hÞ xi ðt k hÞT W½xi ðt kþj hÞ xi ðt k hÞosxT ðt kþj hÞWxðt kþj hÞ
ð9Þ
where W AR is a positive matrix, s A ½0; 1Þ. The sampled state xðt kþj hÞ satisfying the inequality (9) will not be transmitted. Only the one that exceeds the threshold in Eq. (9) is sent to its local neighbors. When s ¼ 0, inequality (9) is not satisfied for almost all the sampled state xðt kþj hÞ, and the event-triggered scheme reduces to a periodic release scheme. nn
Remark 2. Different from the continuous event generator, the event generator with the algorithm (9) only supervises the difference between the states sampled in discrete instants having no interest in what happens in between updates. That is, according to the event-triggering condition in Eq. (9), it can be concluded that only parts of the sampled data will be sent to the remote agents. Remark 3. The set of the release instants, i.e., ft 0 ; t 1 ; t 2 ; …g, is a subset of f0; 1; 2; …g. The amount of ft 0 ; t 1 ; t 2 ; …g depends on not only the value of s, but also the variation of the system state. When s ¼ 0, ft 0 ; t 1 ; t 2 ; …g ¼ f0; 1; 2; …g , it reduces to the case with periodic release times. It is supposed that time-varying delay in network communication is τk A ½0; τ, and the controller ui(t) in Eq. (8) is rewritten as ui ðtÞ ¼ c ∑ aij ðxj ðt k hÞ xi ðt k hÞÞ; j A Ni
t A ½t k h þ τk ; t kþ1 h þ τkþ1 Þ
ð10Þ
For technical convenience, similar to [13,19], consider the following intervals: ½t k h þ τk ; t k h þ h þ τÞ;
½t k h þ dh þ τ; t k h þ dh þ h þ τÞ
ð11Þ
where d is a positive integer, and τ ¼ maxfτk g. Case (A): If t k h þ h þ τ4t kþ1 h þ τkþ1 , define function τðtÞ as follows: τðtÞ ¼ t t k h;
t A ½t k h þ τk ; t kþ1 h þ τkþ1 Þ
ð12Þ
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obviously, it can be obtained τk r τðtÞr ðt kþ1 t k Þh þ τkþ1 r h þ τ
ð13Þ
Case (B): If t k h þ h þ τot kþ1 h þ τkþ1 , there exists a positive integer l, such that t k h þ lh þ τot k h þ τkþ1 ot k h þ lh þ h þ τ
ð14Þ
It can be easily shown that ½t k h þ τk ; t kþ1 h þ τkþ1 Þ ¼ I 1 [ I 2 [ I 3
ð15Þ
where I 1 ¼ ½t k h þ τk ; t k h þ h þ τÞ l1
I 2 ¼ ⋃ fI d2 g d¼1
I d2 ¼ ½t k h þ dh þ τ; t k h þ dh þ h þ τÞ I 3 ¼ ½t k h þ lh þ τ; t kþ1 h þ τkþ1 Þ Define function τðtÞ as 8 > < t tk h τðtÞ ¼ t t k h dh > : t t h lh k
t A I1 t A I d2 t A I3
From Eq. (16), we have 8 > < 0r τk r τðtÞoh þ τ 0r τk r τ r τðtÞoh þ τ > : 0r τ r τ r τðtÞoh þ τ k
ðd ¼ 1; 2; …; l 1Þ
ð16Þ
t A I1 t A I d2
ðd ¼ 1; 2; …; l 1Þ
ð17Þ
t A I3
therefore, we have 0r τðtÞoτM , where τM ¼ h þ τ. For t A ½t k h þ τk ; t kþ1 h þ τkþ1 Þ, we define 8 t A I1 > <0 d ð18Þ eki ðtÞ ¼ xi ðt k h þ dhÞ xi ðt k hÞ t A I 2 ðd ¼ 1; 2; …; l 1Þ > : x ðt h þ lhÞ x ðt hÞ t A I i k
i k
3
For Case (A), we define eki ðtÞ ¼ 0. By combining the definitions of τðtÞ and ek(t), Eqs. (4) and (9) can be rewritten as x_ i ðtÞ ¼ Axi ðtÞ þ βðtÞf ðxi ðtÞÞ þ ð1 βðtÞÞgðxi ðtÞÞ c ∑ lij ðxj ðt τðtÞÞ xi ðt τðtÞÞÞ c ∑ lij ðekj ðtÞ eki ðtÞÞ
ð19Þ
eTki ðtÞWeki ðtÞr sxTi ðt τðtÞÞWxi ðt τðtÞÞ
ð20Þ
j A Ni
j A Ni
Remark 4. Inspired by [13,19], the main property of event- triggered condition is that it relies on intermittent supervision of sampled measurement rather than continuous supervision of the state0 s measurement.
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By utilizing Kronecker product of the matrices, Eqs. (19)–(20) can be written in a compact form as x_ ðtÞ ¼ ðI N AÞxðtÞ þ βðtÞFðxðtÞÞ þ ð1 βðtÞÞGðxðtÞÞ cðL I n Þxðt τðtÞÞ cðL I n Þek ðtÞ
ð21Þ
eTk ðtÞðI N WÞek ðtÞr sxT ðt τðtÞÞðI N WÞxðt τðtÞÞ
ð22Þ
where xðtÞ ¼ ½xT1 ðtÞ; xT2 ðtÞ; …; xTN ðtÞT A RNn FðxðtÞÞ ¼ ½f T ðx1 ðtÞÞ; f T ðx2 ðtÞÞ; …; f T ðxN ðtÞÞT A RNn GðxðtÞÞ ¼ ½gT ðx1 ðtÞÞ; gT ðx2 ðtÞÞ; …; gT ðxN ðtÞÞT A RNn ek ðtÞ ¼ ½eTk1 ðtÞ; eTk2 ðtÞ; …; eTkN ðtÞT A RNn Before giving the main results, some definitions and lemmas are given as follows. Definition 1 (Olfati-Saber [7]). The multi-agent systems (4) are said to achieve global consensus for any initial conditions, if the state of each agent i ði ¼ 1; 2; …; NÞ satisfies lim Jxi ðtÞ xj ðtÞ J ¼ 0
t-1
ði; j ¼ 1; 2; …; NÞ
Lemma 1 (Schur complement). The following linear matrix inequality (LMI) " # QðxÞ SðxÞ 40 ST ðxÞ RðxÞ
ð23Þ
ð24Þ
where QðxÞ ¼ QT ðxÞ; RðxÞ ¼ RT ðxÞ, is equivalent to either of the following conditions: ð1Þ
QðxÞ40;
RðxÞ ST ðxÞQ 1 ðxÞSðxÞ40
ð2Þ
RðxÞ40;
QðxÞ SðxÞR 1 ðxÞST ðxÞ40
Lemma 2. For matrices A; B; C; D with appropriate dimensions and a positive scalar α, the following properties can be proved: ðαAÞ B ¼ A ðαBÞ ðA þ BÞ C ¼ A C þ B C ðA BÞðC DÞ ¼ ðACÞ ðBDÞ ðA BÞT ¼ AT BT Lemma 3. Suppose τðtÞA ½τm ; τM , Qi ði ¼ 1; 2; 3Þ are some constant matrices with appropriate dimensions, then Q1 þ ðτM τðtÞÞQ2 þ ðτðtÞ τm ÞQ3 o0
ð25Þ
if and only if the following inequalities hold: Q1 þ ðτM τm ÞQ2 o0
ð26Þ
Q1 þ ðτM τm ÞQ3 o0
ð27Þ
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Proof. (I) If τðtÞ ¼ τm or τðtÞ ¼ τM , the conclusion is obvious. (II) If τm oτðtÞoτM (Sufficiency): define a function as f ðτðt ÞÞ ¼ Q1 þ ðτM τðt ÞÞQ2 þ ðτðt Þ τm ÞQ3 τM τðtÞ τðtÞ τm ½Q1 þ ðτM τm ÞQ2 þ ½Q þ ðτM τm ÞQ3 ¼ τM τm τM τm 1 from Eqs. (26) and (27), we have f ðτðtÞÞo0 It is equal to Eq. (25). (Necessity): Letting τðtÞ ¼ τm and τðtÞ ¼ τM respectively in Eq. (25), we can easily obtain Eqs. (26) and (27). □ Assumption 1 (Wang et al. [27,28]). For 8 u; v A Rn , the nonlinear functions f ðÞ and gðÞ satisfy the following sector-bounded condition: ½f ðuÞ f ðvÞ F 1 ðu vÞT ½f ðuÞ f ðvÞ F 2 ðu vÞ r 0
ð28Þ
½gðuÞ gðvÞ G1 ðu vÞT ½gðuÞ gðvÞ G2 ðu vÞ r 0
ð29Þ
where F 1 ; F 2 and G1 ; G2 are real constant matrices with F 2 F 1 Z 0 and G2 G1 Z 0. Remark 5. Note that the sector-like description of Assumption 1 is more general than the usual Lipschitz functions. It would be possible to reduce the conservatism of results caused by quantifying nonlinear functions via matrix inequality. 3. Main results In this section, we present the consensus criteria for multi-agent systems (4) with stochastic nonlinearities. The system (21) can be rewritten as x_ ðtÞ ¼ ðI N AÞxðtÞ þ β0 FðxðtÞÞ þ ð1 β0 ÞGðxðtÞÞ cðL I n Þxðt τðtÞÞ cðL I n Þek ðtÞ þ ðβðtÞ β0 ÞðFðxðtÞÞ GðxðtÞÞÞ
ð30Þ
Let ξðtÞ ¼ ½xT ðtÞ; xT ðt τðtÞÞ; xT ðt τM Þ; F T ðxðtÞÞ; GT ðxðtÞÞ; eTk ðtÞT A ¼ ½I N A; cðL I n Þ; 0; β0 I Nn ; ð1 β0 ÞI Nn ; cðL I n Þ B ¼ ½0; 0; 0; I Nn ; I Nn ; 0 then system (30) can be expressed as x_ ðtÞ ¼ AξðtÞ þ ðβðtÞ β0 ÞBξðtÞ
ð31Þ
Theorem 1. Suppose that Assumption 1 holds, the global consensus is achieved under the consensus protocol (10). For given some positive scalar β0, s, c and τM , there exist appropriate dimensional matrices P40, Q40, R40 and Si, Ti ði ¼ 1; 2; …; 6Þ, and two positive scalars α, β, such that the following linear matrix inequalities hold: " # τM ðI N SÞ Π þ Σ 11 þ Σ T11 o0 ð32Þ n τM ðI N RÞ
H. Li et al. / Journal of the Franklin Institute 351 (2014) 2582–2599
"
# τM ðI N T Þ o0 τM ðI N RÞ
Π þ Σ 11 þ Σ T11 n
where
2
6 6 6 6 6 6 6 Π¼6 6 6 6 6 6 6 4
2589
ð33Þ
Π 11
cðL PÞ
0
Π 14
Π 15
cðL PÞ
Π 17
n
sðI N WÞ
n
n
0 IN Q
0 0
0 0
0 0
Π 27 0
n
n
n
2αI Nn
0
0
Π 47
n
n
n
n
2βI Nn
n
n
n
n
n
0 IN W
Π 57 Π 67
n
n
n
n
n
n
Π 77
n
n
n
n
n
n
n
h Σ 11 ¼ I N S I N ðT SÞ I N T 0 h iT T T T T T T S ¼ S1 S2 S3 S4 S5 S6 0 0 h iT T T T T T T T ¼ T1 T2 T3 T4 T5 T6 0 0
0
3
7 7 7 7 7 7 Π 48 7 7 Π 58 7 7 7 0 7 7 0 7 5 Π 88 0 0
i 0 0
0
0
Π 11 ¼ I N PA þ I N AT P þ I N Q αI N ðF T1 F 2 þ F T2 F 1 Þ βI N ðGT1 G2 þ GT2 G1 Þ Π 14 ¼ β0 ðI N PÞ þ αI N ðF T1 þ F T2 Þ Π 15 ¼ ð1 β0 ÞðI N PÞ þ βI N ðGT1 þ GT2 Þ Π 17 ¼ τM ðI N AT RÞ Π 27 ¼ cτM ðLT RÞ Π 47 ¼ τM β0 ðI N RÞ Π 48 ¼ τM β0 ð1 β0 ÞðI N RÞ Π 57 ¼ τM ð1 β0 ÞðI N RÞ Π 58 ¼ τM β0 ð1 β0 ÞðI N RÞ Π 67 ¼ τM cðLT RÞ Π 77 ¼ τM ðI N RÞ Π 88 ¼ τM β0 ð1 β0 ÞðI N RÞ Proof. Consider the following Lyapunov function candidate: Vðt; xðtÞÞ ¼ V 1 ðt; xðtÞÞ þ V 2 ðt; xðtÞÞ þ V 3 ðt; xðtÞÞ where V 1 ðt; xðtÞÞ ¼ xT ðtÞðI N PÞxðtÞ Z t V 2 ðt; xðtÞÞ ¼ xT ðsÞðI N QÞxðsÞ ds Zt t τM Z t V 3 ðt; xðtÞÞ ¼ x_ T ðsÞðI N RÞ_x ðsÞ ds dθ t τM
θ
ð34Þ
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The infinitesimal operator L of Vðt; xðtÞÞ is defined as follows: LV ðt; xðt ÞÞ ¼ limþ Δ-0
EðVðt þ Δ; xðt þ ΔÞÞÞ Vðt; xðtÞÞ Δ
ð35Þ
By simple calculation from Eqs. (34) and (35), the following equalities can be obtained: LV 1 ðt; xðtÞÞ ¼ 2xT ðtÞðI N PÞAξðtÞ
ð36Þ
LV 2 ðt; xðtÞÞ ¼ xT ðtÞðI N QÞxðtÞ xT ðt τM ÞðI N QÞxðt τM Þ Z t T x_ T ðsÞðI N RÞ_x ðsÞ ds LV 3 ðt; xðtÞÞ ¼ τM Ef_x ðtÞðI N RÞ_x ðtÞg
ð37Þ
t τM
¼ τM ξT ðtÞAT ðI N RÞAξðtÞ þ τM β0 ð1 β0 ÞξT ðtÞBT ðI N RÞBξðtÞ Z t x_ T ðsÞðI N RÞ_x ðsÞ ds
ð38Þ
t τM
Employing the free matrix method [29,30], it is easily derived that Z t T 2ξ ðtÞðI N SÞ xðtÞ xðt τðtÞÞ x_ ðsÞ ds ¼ 0
ð39Þ
t τðtÞ
Z 2ξT ðtÞðI N TÞ xðt τðtÞÞ xðt τM Þ
t τðtÞ t τM
x_ ðsÞ ds ¼ 0
ð40Þ
where
h iT T T T T T T S ¼ S1 S2 S 3 S4 S5 S6 h iT T T T T T T T ¼ T1 T2 T3 T4 T5 T6
It follows that from Eqs. (39)–(40) Z t x_ ðsÞ ds 2ξT ðtÞðI N SÞ t τðtÞ
r τðtÞξ ðtÞðI N SÞðI N RÞ 1 ðI N SÞT ξðtÞ þ
Z
t
T
Z 2ξ ðtÞðI N TÞ T
t τðtÞ t τM
t τðtÞ
x_ T ðsÞðI N RÞ_x ðsÞ ds
ð41Þ
x_ ðsÞ ds
r ðτM τðtÞÞξT ðtÞðI N TÞðI N RÞ 1 ðI N TÞT ξðtÞ þ
Z
t τðtÞ t τM
x_ T ðsÞðI N RÞ_x ðsÞ ds ð42Þ
Considering Eqs. (38)–(42), we can easily obtain LV 3 ðt; xðtÞÞ r τM ξT ðtÞAT ðI N RÞAξðtÞ þ τM β0 ð1 β0 ÞξT ðtÞBT ðI N RÞBξðtÞ þ2ξT ðtÞðI N SÞ½xðtÞ xðt τðtÞÞ þ 2ξT ðtÞðI N TÞ½xðt τðtÞÞ xðt τM Þ þτðtÞξT ðtÞðI N SÞðI N RÞ 1 ðI N SÞT ξðtÞ
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2591
þðτM τðtÞÞξT ðtÞðI N TÞðI N RÞ 1 ðI N TÞT ξðtÞ
ð43Þ
For 8 α; β40, it can be derived that #" # " #T " xðtÞ xðtÞ IN F I N F~ Z0 α T FðxðtÞÞ FðxðtÞÞ I N F~ 2I Nn "
xðtÞ β GðxðtÞÞ
#T "
I N G~
IN G ~T IN G
2I Nn
#"
# xðtÞ Z0 GðxðtÞÞ
ð44Þ
ð45Þ
where F ¼ F T1 F 2 þ F T2 F 1 ; G
¼ GT1 G2
þ
GT2 G1 ;
F~ ¼ F T1 þ F T2 G~ ¼ GT þ GT 1
2
It follows that from Eqs. (36)–(45) LVðt; xðtÞÞ r ξT ðtÞ½Π þ Σ 11 þ Σ T11 ξðtÞ þ τðtÞξT ðtÞðI N SÞðI N RÞ 1 ðI N SÞT ξðtÞ þðτM τðtÞÞξT ðtÞðI N TÞðI N RÞ 1 ðI N TÞT ξðtÞ
ð46Þ
where Π and Σ11 are defined in Theorem 1. By the Schur complement in Eqs. (32)–(33), the following inequalities hold: Π þ Σ 11 þ Σ T11 þ τM ðI N TÞðI N RÞ 1 ðI N TÞT r 0
ð47Þ
Π þ Σ 11 þ Σ T11 þ τM ðI N SÞðI N RÞ 1 ðI N SÞT r 0
ð48Þ
By Lemma 3, it can be concluded that LVðt; xðtÞÞo0. That is, JξðtÞJ -0 as t-1. So the consensus in multi-agent systems (4) is achieved. □ Remark 6. The trigger parameters s, W and τM are involved in Eqs. (32)–(33). For given s, the corresponding trigger parameter W and the upper bound of τM can be obtained by using the LMI toolbox in Matlab. Note that τM ¼ h þ τ, selecting a sampling period hoτM , the upper bound of time delay τ can be obtained, if the effect of the transmission delay can be ignored, that is τ ¼ 0, the obtained τM the maximum sampling period. Let us now consider two special cases, and the corresponding results are still believed to be new. Case 1: We first specialize system (4) to case without stochastic switch in nonlinearities, that is βðtÞ ¼ 1, and system (4) is degenerated as follows: x_ i ðtÞ ¼ Axi ðtÞ þ f ðxi ðtÞÞ þ ui ðtÞ
ð49Þ
similar to Theorem 1, the consensus criteria can be derived for multi-agent systems (49) with nonlinear dynamics. Corollary 1. Suppose that Assumption 1 holds, a global consensus is achieved. For given some positive scalars s, c and τM , there exist appropriate dimensional matrices P40, Q40, R40 and Si,Ti ði ¼ 1; 2; …; 5Þ, and positive scalar α, such that the following linear matrix inequalities
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hold:
H. Li et al. / Journal of the Franklin Institute 351 (2014) 2582–2599
"
T Π~ þ Σ~ 11 þ Σ~ 11
~ τM ðI N SÞ
n
τM ðI N RÞ
# o0 ð50Þ
"
T Π~ þ Σ~ 11 þ Σ~ 11
τM ðI N T~ Þ
n
τM ðI N RÞ
# o0 ð51Þ
where
2
6 6 6 6 6 Π~ ¼ 6 6 6 6 4
Π~ 11
cðL PÞ
0
Π~ 14
cðL PÞ
n
sðI N WÞ
n
n
0 IN Q
0 0
0 0
n
n
n
2αI Nn
0
n
n
n
n
IN W
n
n
n
n
n
h ~ I N T~ Σ~ 11 ¼ I N S~ I N ðT~ SÞ h iT T T T T T S~ ¼ S1 S2 S3 S4 S5 0 h iT T T T T T T~ ¼ T 1 T 2 T 3 T 4 T 5 0
τM ðI N AT RÞ
3
7 cτM ðLT RÞ 7 7 7 0 7 7 τM ðI N RÞ 7 7 cτM ðLT RÞ 7 5 τM ðI N RÞ
i 0
0
0
Π~ 11 ¼ I N PA þ I N AT P þ I N Q αI N ðF T1 F 2 þ F T2 F 1 Þ Π~ 14 ¼ I N P þ αI N ðF T1 þ F T2 Þ Case 2: Let us assume that system (4) evolves with neither stochastic switching nor nonlinearities and system (4) reduces to x_ i ðtÞ ¼ Axi ðtÞ þ ui ðtÞ
ð52Þ
similar to Theorem 1, consensus criteria can be derived for multi-agent systems (52) with linear dynamics. Corollary 2. For given positive scalars s, c and τM , a global consensus is achieved. If there exist appropriate dimensional matrices P40, Q40, R40 and Si,Ti ði ¼ 1; 2; 3; 4Þ, such that the following linear matrix inequalities hold: " # T ^ Π^ þ Σ^ 11 þ Σ^ 11 τM ðI N SÞ o0 n τM ðI N RÞ ð53Þ "
T Π^ þ Σ^ 11 þ Σ^ 11
τM ðI N T^ Þ
n
τM ðI N RÞ
# o0
ð54Þ
H. Li et al. / Journal of the Franklin Institute 351 (2014) 2582–2599
where
2
Π^ 11
6 n 6 6 Π^ ¼ 6 6 n 6 4 n h
cðL PÞ
0
cðL PÞ
sðI N WÞ
0
0
n
IN Q
n
n
0 IN W
n
n
n
n
^ Σ^ 11 ¼ I N S^ I N ðT^ SÞ h iT T T T T S^ ¼ S1 S2 S3 S4 0 h iT T T T T T^ ¼ T 1 T 2 T 3 T 4 0
I N T^
i
τM ðI N AT RÞ
2593
3
cτM ðLT RÞ 7 7 7 7 0 7 7 T cτM ðL RÞ 5 τM ðI N RÞ
0 0
Π^ 11 ¼ I N PA þ I N AT P þ I N Q
4. Numerical results In this section, a numerical example is given to verify the effectiveness of the proposed control techniques for multi-agent systems to achieve a global consensus. Example 1. Consider multi-agent systems with a topology structure shown in Fig. 1, where the weights are indicated on the edges. Multi-agent systems consisting of seven agents are described as follows: x_ i ðtÞ ¼ Axi ðtÞ þ βðtÞf ðxi ðtÞÞ þ ð1 βðtÞÞgðxi ðtÞÞ þ ui ðtÞ ði ¼ 1; 2; …; 7Þ
ð55Þ
where 7
7
j¼1
j¼1
ui ðtÞ ¼ c ∑ lij xj ðt τðtÞÞ c ∑ lij ekj ðtÞ pffiffiffi pffiffiffiffiffi and A ¼ 1, f ðxi ðtÞÞ ¼ ð 5=5Þxi ðtÞ cos ðxi ðtÞÞ and gðxi ðtÞÞ ¼ ð 10=5Þxi ðtÞ sin ðxi ðtÞÞ. It is easy to verify that nonlinearpffiffifunctions fp ðÞffiffiffi and gðÞ satisfy ffi pffiffiffiffiffi Assumption pffiffiffiffiffi1, by some calculation, we can obtain F 1 ¼ 5=5, F 2 ¼ 5=5, G1 ¼ 10=5, G2 ¼ 10=5. Setting c ¼ 0.1, β0 ¼ 0:1, h¼ 0.01, s ¼ 0:1, we have τM ¼ 0:7136 by applying Theorem 1, that is, the maximum allowable delay τ ¼ τM h ¼ 0:7036. Suppose that τ ¼ 0, then τM ¼ h, it can be
Fig. 1. The topology structure of the agents with the weights on the connections.
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known that the maximum sampling period is 0.7136. More detailed calculation results for different values of s are given in Table 1. It can be seen form the calculation results in Table 1 that s becomes larger when τ becomes smaller. For given c¼ 0.1, β0 ¼ 0:1, h ¼ 0.01, s ¼ 0:1 and τ ¼ 0:1, using the algorithm (9), the simulation results, for t A ½0; 10, show that only (37) sampled signals need to be sent out, which takes 3.76% of sampled signals. Moreover, it can be computed that average release period h ¼ 0:2659, Table 2 shows the relationship trigger parameter s, trigger times, average release period h and the percentage of data transmissions among the agents, it can be seen that the larger s, the smaller trigger times, the larger average release period h, the smaller percentage of data transmission γ, all of which are reasonable results. When s ¼ 0, event-triggered reduces to timetriggered scheme, the curves of position states xi(t) ði ¼ 1; 2; …; 7Þ are depicted in Fig. 2, from which we can see that consensus is achieved. If setting s ¼ 0:1, the corresponding trigger parameter W ¼ 91.8608. The response of position states is depicted in Fig. 3. Figs. 2 and 3 show that the simulation results are almost the same, but the percentage of data transmission α under
Table 1 The maximum allowable delay for given c ¼0.1, β0 ¼ 0:1, h¼ 0.01, t¼ 10. s
0
0.1
0.2
0.3
0.4
0.5
τ
0.9706
0.7036
0.5650
0.4240
0.2697
0.1257
Table 2 The computation results for given c¼ 0.1, β0 ¼ 0:1, h¼0.01, τ ¼ 0:1, t¼ 10. s
0
0.1
0.2
0.3
0.4
0.5
Trigger times Trigger matrix W Average release period h Data transmission γ
1000 36.6741 0.01 100%
37 91.8608 0.2659 3.76%
29 2.2226 0.3362 2.97%
23 4.8967 0.4322 2.31%
22 1.5921 0.4350 2.30%
20 12.1210 0.4905 2.04%
Table 3 The maximum allowable delay for given c ¼0.1, β0 ¼ 0:8, h¼ 0.01, t¼ 10. s
0
0.1
0.2
0.3
0.4
0.5
τ
1.1513
0.9011
0.7710
0.6688
0.5682
0.4591
Table 4 The computation results for given c¼ 0.1, β0 ¼ 0:8, h¼0.01, τ ¼ 0:1, t¼ 10. s
0
0.1
0.2
0.3
0.4
0.5
Trigger times Trigger matrix W Average release period h Data transmission γ
1000 1.2372 0.01 100%
28 37.9507 0.3557 2.81%
22 0.6877 0.4495 2.22%
16 298.1335 0.5950 1.68%
15 37.4459 0.6207 1.61%
13 244.7115 0.7246 1.38%
H. Li et al. / Journal of the Franklin Institute 351 (2014) 2582–2599
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1 0.8 0.6
xi(t) (i=1,2,¡-,7)
0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0
2
4
6
8
10
t
Fig. 2. The position curves xi(t) ði ¼ 1; 2; …; 7Þ with switching probability β0 ¼ 0:1 and trigger parameter s ¼ 0 (timetriggered scheme).
1 0.8 0.6
xi(t) (i=1,2,¡-,7)
0.4 0.2 0 −0.2 −0.4 −0.6 −0.8
0
2
4
6
8
10
t
Fig. 3. The position curves xi(t) ði ¼ 1; 2; …; 7Þ with switching probability β0 ¼ 0:1 and trigger parameter s ¼ 0:1 (eventtriggered scheme).
the event-triggered scheme is much smaller than time-triggered scheme. To show the release instants and release intervals, which are illustrated in Fig. 4. From these results, it can be concluded that event-triggered scheme has advantages over time-triggered scheme in terms of improving the resource utilization.When β0 ¼ 0:8, we have similar results. From Tables 3 and 4 and Figs. 5–7, similar conclusions can be drawn.
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event−based release instants and release interval
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
2
4
6
8
10
t(s)
Fig. 4. Release instants and release interval by event-triggered scheme.
0.8
0.6
xi(t) (i=1,2,¡-,7)
0.4
0.2
0
−0.2
−0.4
−0.6
0
2
4
6
8
10
t
Fig. 5. The position curves xi(t) ði ¼ 1; 2; …; 7Þ with switching probability β0 ¼ 0:8 and trigger parameter s ¼ 0 (timetriggered scheme).
5. Conclusions This paper proposes a novel event-triggering scheme to achieve consensus for multi-agent systems with randomly occurring nonlinear dynamics and time-varying delay. Less communication resources are utilized while preserving the consensus performance in the proposed scheme. Due to it is practically impossible to apply control actions to all agents in large-scale multi-agent systems, future work will combine pinning control schemes and event-triggering scheme to study the consensus problem of multi-agent systems, a very challenging problem is how to choose a set of pinned agents.
H. Li et al. / Journal of the Franklin Institute 351 (2014) 2582–2599
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0.6 0.4
xi(t) (i=1,2,¡-,7)
0.2 0 −0.2 −0.4 −0.6 −0.8 −1
0
2
4
6
8
10
t
Fig. 6. The position curves xi(t) ði ¼ 1; 2; …; 7Þ with switching probability β0 ¼ 0:8 and trigger parameter s ¼ 0:1 (eventtriggered scheme).
event−based release instants and release interval
0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
0
2
4
6
8
10
t(s)
Fig. 7. Release instants and release interval by event-triggered scheme.
Acknowledgements The authors would like to express their sincere appreciation to the editor and reviewers for their helpful comments which help to improve quality of the paper. This work was jointly supported by the National Natural Science Foundation of China under Grant nos. 61272034 and 11301226, Zhejiang Provincial Natural Science Foundation of China under Grant nos. LY13F030012 and LQ13A010017.
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References [1] L. Consolini, F. Morbidi, D. Prattichizzo, M. Tosques, Leader–follower formation control of nonholonomic mobile robots with input constraints, Automatica 44 (2008) 1343–1349. [2] S. Loizou, K. Kyriakopoulos, Navigation of multiple kinematically constrained robots, IEEE Trans. Robot. 24 (2008) 221–231. [3] D. Dimarogonas, K. Kyriakopoulos, A connection between formation infeasibility and velocity alignment in kinematic multi-agent systems, Automatica 44 (2008) 2648–2654. [4] W. Ren, E. Atkins, Distributed multi-vehicle coordinated control via local information exchange, Int. J. Robust Nonlinear Control 17 (2007) 1002–1033. [5] W. Ren, R. Beard, E. Atkins, Information consensus in multivehicle cooperative control, IEEE Control Syst. 27 (2007) 71–82. [6] T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen, O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett. 75 (1995) 1226–1229. [7] R. Olfati-Saber, Flocking for multi-agent dynamic systems: algorithms and theory, IEEE Trans. Autom. Control 51 (2006) 401–420. [8] W. Ren, On consensus algorithms for double-integrator dynamics, IEEE Trans. Autom. Control 53 (2008) 1503–1509. [9] W. Yu, G. Chen, M. Cao, Some necessary and sufficient conditions for second-order consensus in multi-agent dynamical systems, Automatica 46 (2010) 1089–1095. [10] W. Yu, G. Chen, M. Cao, J. Kurths, Second-order consensus for multiagent systems with directed topologies and nonlinear dynamics, IEEE Trans. Syst. Man Cybern. Part B 40 (2010) 881–891. [11] Y. Sun, Average consensus in networks of dynamic agents with uncertain topologies and time-varying delays, J. Frankl. Inst. 349 (2012) 1061–1073. [12] P. Lin, K. Qin, H. Zhao, A new approach to average consensus problems with multiple time-delays and jointlyconnected topologies, J. Frankl. Inst. 349 (2012) 293–304. [13] D. Yue, E. Tian, Q. Han, A delay system method to design of event-triggered control of networked control systems, in: 50th IEEE Conference on Decision and Control and European Control Conference, 2011, pp. 16681673. [14] J. Sijs, M. Lazar, On event based state estimation, Hybrid Syst. Comput. Control 336 (2009) 881–891. [15] P. Tabuada, Event-triggered real-time scheduling of stabilizing control tasks, IEEE Trans. Autom. Control 52 (2007) 1680–1685. [16] X. Wang, M. Lemmon, Self-triggered feedback control systems with finite-gain l2 stability, IEEE Trans. Autom. Control 54 (2009) 452–467. [17] X. Wang, M. Lemmon, Self-triggering under state-independent disturbances, IEEE Trans. Autom. Control 55 (2010) 1494–1500. [18] C. Peng, Q. Han, D. Yue, A discrete event-triggered communication scheme for networked ts fuzzy systems, in: IECON 2011-37th Annual Conference on IEEE Industrial Electronics Society, 2011, pp. 2282–2287. [19] S. Hu, D. Yue, Event-triggered control design of linear networked systems with quantizations, ISA Trans. 51 (2012) 153–162. [20] S. Hu, D. Yue, Event-based H 1 filtering for networked system with communication delay, Signal Process. 92 (2012) 2029–2039. [21] D. Dimarogonas, K. Johansson, Event-triggered control for multi-agent systems, in: CDC/CCC 2009, 2009, pp. 7131–7136. [22] D. Dimarogonas, E. Frazzoli, Distributed event-triggered control strategies for multi-agent systems, in: Communication, Control, and Computing, Allerton 2009, 2009, pp. 906–910. [23] D. Dimarogonas, E. Frazzoli, K. Johansson, Distributed event-triggered control for multi-agent systems, IEEE Trans. Autom. Control 57 (2012) 1291–1297. [24] G.S. Seyboth, D.V. Dimarogonas, K.H. Johansson, Event-based broadcasting for multi-agent average consensus, Automatica 49 (2012) 245–252. [25] D. Xie, D. Yuan, J. Lu, Consensus control of second-order leader–follower multi-agent systems with event-triggered strategy, Trans. Inst. Meas. Control 35 (2012) 426–436. [26] Z. Li, Z. Chen, Event-triggered average-consensus for multi-agent systems, in: 2010 29th Chinese Control Conference, 2010, pp. 4506–4511. [27] Y. Wang, Z. Wang, J. Liang, Global synchronization for delayed complex networks with randomly occurring nonlinearities and multiple stochastic disturbances, J. Phys. A: Math. Theoret. 42 (2009) 135101-1–135101-19.
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Journal of the Franklin Institute 351 (2014) 2582–2599 www.elsevier.com/locate/jfranklin
Event-triggered control for multi-agent systems with randomly occurring nonlinear dynamics and time-varying delay Hongjie Lia, Chen Minga, Shigen Shena, W.K. Wongb,n a
College of Mathematics and Information and Engineering, Jiaxing University, Zhejiang 314001, PR China b Institute of Textile and Clothing, The Hong Kong Polytechnic University, Hong Kong, PR China Received 29 June 2012; received in revised form 6 December 2013; accepted 2 January 2014 Available online 16 January 2014
Abstract The paper investigates the consensus problem for multi-agent systems with randomly occurring nonlinear dynamics and time-varying delay. A novel event-triggered scheme has been proposed, which can lead to a significant reduction in information communication in a network. By utilizing stochastic analysis and properties of the Kronecker product, consensus criteria are derived in the form of linear matrix inequalities, which can be readily solved using the standard numerical software. Finally, an illustrative example is used to show the effectiveness of the event-triggered scheme. & 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction The decentralized coordinated control of multi-agent systems has witnessed an increasing interest owing to its many applications in biology, computer science and control engineering [1–6]. A critical problem in distributed coordinated control of multiple agents is the consensus problem, which is the key to find control laws such that all agents can reach an agreement and has been extensively studied in the past few years with many profound results [4,7–12]. The decentralized consensus control for multi-agent is facilitated by recent technological advances in computing and communication resources. Future design may equip each agent with a small embedded micro-processor, which collects information from neighboring nodes and n
Corresponding author. E-mail address: [email protected] (W.K. Wong).
0016-0032/$32.00 & 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2014.01.004
H. Li et al. / Journal of the Franklin Institute 351 (2014) 2582–2599
2583
actuates the controller updates according to some rules. The control update scheduling can be done in a time-driven or an event-driven fashion. The first case involves the traditional approach to sampling at pre-specified time instances, usually separated by a specific period and all the sampled signals are transmitted through communication networks regardless of the state of the system to be controlled. Clearly, it may be conservative to guarantee the stability of a network in the worst cases, as stated in [13], the worst cases are seldom happen in practical systems. Therefore, this kind of sampled control method can lead to sending of many unnecessary signals to a network, which may cause a huge wastage of communication bandwidth. In real-world application, embedded processors are resources-limited, and thus it seems that an event-triggered approach is more favorable [13–20], where sampled signals whether or not to be instantly transmitted to the controller are determined by certain events that are triggered depending on some rules. The rules provide a useful way of determining when the sampled signals are sent out. In [13], the event-triggered H 1 control design is investigated for networked control systems with uncertainties and transmission delays, the novel event-triggered scheme can lead to a larger average release period. Peng et al. [18] proposes a discrete event-triggered communication scheme for a class of networked Takagi–Sugeno fuzzy systems. Compared with a time-triggered periodic communication scheme, fewer communication resources are utilized. Hu et al. [19] concerns the control design of event-triggered networked systems with both state and control input quantizations by taking the characteristics of an event-triggered mechanism into account, and a novel interval delay analysis technique is developed. In [20], the event-based H 1 filtering problem is studied for networked systems with communication delay. Recently, event-triggered control strategies are used to achieve the consensus for multi-agent systems [21–26], Dimarogonas et al. [21–23] investigates the consensus for a single integrator model-based event-triggered control and actuation updates depended on the ratio of a certain measurement error with respect to the norm of a function of the state. Liu et al. [26] studies average-consensus protocols for multi-agent systems, two average consensus protocols with different trigger functions are proposed. But in [21,22,26], the agent dynamics are assumed to be first-order, and the effect of transmission delay is not considered. As is well known, a wide class of practical systems is influenced by additive nonlinear disturbances that are caused by environmental circumstances. such nonlinear disturbances themselves may be subject to random abrupt changes, which may result from abrupt phenomena such as random failures and repairs of the components. In other words, the nonlinear disturbances may occur in a probabilistic way. The randomly occurring nonlinearities, also called stochastic nonlinearities, have recently received some interest in the literature [27,28]. To date, very little research effort has been done about consensus problem for multi-agent systems with randomly occurring nonlinear dynamics and time varying-delay. The purpose of our study is to fill the gap. Motivated by the above discussion, the paper studies the consensus problem for multi-agent systems with randomly occurring nonlinear dynamics and time-varying delay. To reduce communication load in a network, a novel event-triggered scheme has been proposed. Whether the sampled state signal of each agent should be transmitted to its neighbors is determined by the error between the current sampled state and the latest transmitted one. Based on the Lyapunov stability theory, consensus criteria can be obtained in the form of linear matrix inequalities. A numerical example is given to illustrate the effectiveness of the proposed method. The main contributions of this paper are highlighted as follows (1) An novel event-triggered scheme is proposed. Compared with time-triggered periodic communication scheme, the number of transmitted state signals through multi-agent networks
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is reduced since the proposed communication scheme only depends on the state at the sampled instant and the state error between the current sampled instant and the latest transmitted state. Therefore, communication resources embedded in agents can be saved and communication can be reduced. (2) Consensus criteria are obtained in the form of linear matrix inequalities which can be readily solved by using the LMI toolbox in Matlab. The solvability of derived conditions depends not only on trigger parameters and the probability of switching in nonlinear functions, but also the size of delay.
2. Preliminaries Let g ¼ fv; ɛ; Ag be a weighted diagraph of order N, with the set of nodes v ¼ fv1 ; v2 ; …; vN g, an edge set ɛDv v, and a weighted adjacency matrix A ¼ ðaij ÞNN with nonnegative elements. Directed edge eij in network g is denoted by the ordered pair of nodes ðvi ; vj Þ, which means that node vj can receive information from node vi. The elements of adjacency matrix A are defined as aij 40 if and only if there is a directed edge ðvj ; vi Þ in g, otherwise aij ¼ 0. We assume that aii ¼ 0 for all iA v. The neighbor set of node i is defined by N i ¼ fjA vjðvj ; vi ÞA ɛg, and the in-degree and out-degree of node i are defined as degin ðiÞ ¼
N
∑
j ¼ 1;j a i
aij ;
degout ðiÞ ¼
N
∑
j ¼ 1;j a i
ð1Þ
aji
A diagraph is balanced if degin ðiÞ ¼ degout ðiÞ for all iA v. The Laplacian matrix L ¼ ðlij ÞNN associated with adjacency matrix A is defined as lij ¼ aij lii ¼
ði; j ¼ 1; 2; …; N; ia jÞ
ð2Þ
N
∑
j ¼ 1;j a i
ð3Þ
aij
which ensures the diffusion property that ∑Nj¼ 1 lij ¼ 0. Consider the following multi-agent systems with stochastic nonlinearities: x_ i ðtÞ ¼ Axi ðtÞ þ βðtÞf ðxi ðtÞÞ þ ð1 βðtÞÞgðxi ðtÞÞ þ ui ðtÞ ði ¼ 1; 2; …; NÞ
ð4Þ
is a where xi ðtÞ ¼ ðxi1 ðtÞ; xi2 ðtÞ; …; xin ðtÞÞ A R is the position state of the ith agent. A A R constant matrix. f ðxi ðtÞÞ ¼ ðf 1 ðxi1 ðtÞÞ; f 2 ðxi2 ðtÞÞ; …; f n ðxin ðtÞÞÞT A Rn and gðxi ðtÞÞ ¼ ðg1 ðxi1 ðtÞÞ; g2 ðxi2 ðtÞÞ; …; gn ðxin ðtÞÞÞT A Rn are nonlinear vector-valued functions satisfying certain conditions to be given later. ui ðtÞ ¼ ðui1 ðtÞ; ui1 ðtÞ; …; uin ðtÞÞT A Rn is control input to be designed, and βðtÞ is a Bernoulli stochastic variable that describes the following random events of system (4): ( 1 if f ðxi ðtÞÞ occurs βðtÞ ¼ ð5Þ 0 if gðxi ðtÞÞ occurs T
n
nn
it follows that βðtÞ satisfies ProbfβðtÞ ¼ 1g ¼ EfβðtÞg ¼ β0
ð6Þ
ProbfβðtÞ ¼ 0g ¼ 1 EfβðtÞg ¼ 1 β0
ð7Þ
where constant β0 A ½0; 1 reflects the occurrence probability of nonlinear functions f ðÞ and gðÞ.
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Remark 1. Inspired by references [27,28], the random variable βðtÞ is used to model the probability distribution of nonlinear functions, to the best of the author0 s knowledge. It is the first attempt to consider the occurrence of different nonlinear functions in a probabilistic for multiagent systems. In this paper, control input ui(t) is sampled before entering networks based on the sampling technique and zero-order hold circuit. That is ui ðtÞ ¼ c ∑ aij ðxj ðt k hÞ xi ðt k hÞÞ j A Ni
t A ½t k h; t kþ1 hÞ
ð8Þ
where ui(t) is a discrete-time control signal, tk denotes the sampling instance of the ith agent and satisfies limk-1 t k ¼ 1, h is the sampling period. In practical systems, periodic sampling mechanisms often lead to sending of many unnecessary signals through networks, which in turn increases the load of network transmission and wastage of the network bandwidth. Therefore, it is significant to introduce a mechanism to determine which sampled signal should be sent out. As stated in [19], the event-triggered sampling scheme is effective because they can reduce traffic and power consumption. The sampled data xðt kþj hÞ is released by the event generators only when the current sampled value xðt kþj hÞ and the previously transmitted one xðt k hÞ satisfy the following judgment algorithm: ½xi ðt kþj hÞ xi ðt k hÞT W½xi ðt kþj hÞ xi ðt k hÞosxT ðt kþj hÞWxðt kþj hÞ
ð9Þ
where W AR is a positive matrix, s A ½0; 1Þ. The sampled state xðt kþj hÞ satisfying the inequality (9) will not be transmitted. Only the one that exceeds the threshold in Eq. (9) is sent to its local neighbors. When s ¼ 0, inequality (9) is not satisfied for almost all the sampled state xðt kþj hÞ, and the event-triggered scheme reduces to a periodic release scheme. nn
Remark 2. Different from the continuous event generator, the event generator with the algorithm (9) only supervises the difference between the states sampled in discrete instants having no interest in what happens in between updates. That is, according to the event-triggering condition in Eq. (9), it can be concluded that only parts of the sampled data will be sent to the remote agents. Remark 3. The set of the release instants, i.e., ft 0 ; t 1 ; t 2 ; …g, is a subset of f0; 1; 2; …g. The amount of ft 0 ; t 1 ; t 2 ; …g depends on not only the value of s, but also the variation of the system state. When s ¼ 0, ft 0 ; t 1 ; t 2 ; …g ¼ f0; 1; 2; …g , it reduces to the case with periodic release times. It is supposed that time-varying delay in network communication is τk A ½0; τ, and the controller ui(t) in Eq. (8) is rewritten as ui ðtÞ ¼ c ∑ aij ðxj ðt k hÞ xi ðt k hÞÞ; j A Ni
t A ½t k h þ τk ; t kþ1 h þ τkþ1 Þ
ð10Þ
For technical convenience, similar to [13,19], consider the following intervals: ½t k h þ τk ; t k h þ h þ τÞ;
½t k h þ dh þ τ; t k h þ dh þ h þ τÞ
ð11Þ
where d is a positive integer, and τ ¼ maxfτk g. Case (A): If t k h þ h þ τ4t kþ1 h þ τkþ1 , define function τðtÞ as follows: τðtÞ ¼ t t k h;
t A ½t k h þ τk ; t kþ1 h þ τkþ1 Þ
ð12Þ
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obviously, it can be obtained τk r τðtÞr ðt kþ1 t k Þh þ τkþ1 r h þ τ
ð13Þ
Case (B): If t k h þ h þ τot kþ1 h þ τkþ1 , there exists a positive integer l, such that t k h þ lh þ τot k h þ τkþ1 ot k h þ lh þ h þ τ
ð14Þ
It can be easily shown that ½t k h þ τk ; t kþ1 h þ τkþ1 Þ ¼ I 1 [ I 2 [ I 3
ð15Þ
where I 1 ¼ ½t k h þ τk ; t k h þ h þ τÞ l1
I 2 ¼ ⋃ fI d2 g d¼1
I d2 ¼ ½t k h þ dh þ τ; t k h þ dh þ h þ τÞ I 3 ¼ ½t k h þ lh þ τ; t kþ1 h þ τkþ1 Þ Define function τðtÞ as 8 > < t tk h τðtÞ ¼ t t k h dh > : t t h lh k
t A I1 t A I d2 t A I3
From Eq. (16), we have 8 > < 0r τk r τðtÞoh þ τ 0r τk r τ r τðtÞoh þ τ > : 0r τ r τ r τðtÞoh þ τ k
ðd ¼ 1; 2; …; l 1Þ
ð16Þ
t A I1 t A I d2
ðd ¼ 1; 2; …; l 1Þ
ð17Þ
t A I3
therefore, we have 0r τðtÞoτM , where τM ¼ h þ τ. For t A ½t k h þ τk ; t kþ1 h þ τkþ1 Þ, we define 8 t A I1 > <0 d ð18Þ eki ðtÞ ¼ xi ðt k h þ dhÞ xi ðt k hÞ t A I 2 ðd ¼ 1; 2; …; l 1Þ > : x ðt h þ lhÞ x ðt hÞ t A I i k
i k
3
For Case (A), we define eki ðtÞ ¼ 0. By combining the definitions of τðtÞ and ek(t), Eqs. (4) and (9) can be rewritten as x_ i ðtÞ ¼ Axi ðtÞ þ βðtÞf ðxi ðtÞÞ þ ð1 βðtÞÞgðxi ðtÞÞ c ∑ lij ðxj ðt τðtÞÞ xi ðt τðtÞÞÞ c ∑ lij ðekj ðtÞ eki ðtÞÞ
ð19Þ
eTki ðtÞWeki ðtÞr sxTi ðt τðtÞÞWxi ðt τðtÞÞ
ð20Þ
j A Ni
j A Ni
Remark 4. Inspired by [13,19], the main property of event- triggered condition is that it relies on intermittent supervision of sampled measurement rather than continuous supervision of the state0 s measurement.
H. Li et al. / Journal of the Franklin Institute 351 (2014) 2582–2599
2587
By utilizing Kronecker product of the matrices, Eqs. (19)–(20) can be written in a compact form as x_ ðtÞ ¼ ðI N AÞxðtÞ þ βðtÞFðxðtÞÞ þ ð1 βðtÞÞGðxðtÞÞ cðL I n Þxðt τðtÞÞ cðL I n Þek ðtÞ
ð21Þ
eTk ðtÞðI N WÞek ðtÞr sxT ðt τðtÞÞðI N WÞxðt τðtÞÞ
ð22Þ
where xðtÞ ¼ ½xT1 ðtÞ; xT2 ðtÞ; …; xTN ðtÞT A RNn FðxðtÞÞ ¼ ½f T ðx1 ðtÞÞ; f T ðx2 ðtÞÞ; …; f T ðxN ðtÞÞT A RNn GðxðtÞÞ ¼ ½gT ðx1 ðtÞÞ; gT ðx2 ðtÞÞ; …; gT ðxN ðtÞÞT A RNn ek ðtÞ ¼ ½eTk1 ðtÞ; eTk2 ðtÞ; …; eTkN ðtÞT A RNn Before giving the main results, some definitions and lemmas are given as follows. Definition 1 (Olfati-Saber [7]). The multi-agent systems (4) are said to achieve global consensus for any initial conditions, if the state of each agent i ði ¼ 1; 2; …; NÞ satisfies lim Jxi ðtÞ xj ðtÞ J ¼ 0
t-1
ði; j ¼ 1; 2; …; NÞ
Lemma 1 (Schur complement). The following linear matrix inequality (LMI) " # QðxÞ SðxÞ 40 ST ðxÞ RðxÞ
ð23Þ
ð24Þ
where QðxÞ ¼ QT ðxÞ; RðxÞ ¼ RT ðxÞ, is equivalent to either of the following conditions: ð1Þ
QðxÞ40;
RðxÞ ST ðxÞQ 1 ðxÞSðxÞ40
ð2Þ
RðxÞ40;
QðxÞ SðxÞR 1 ðxÞST ðxÞ40
Lemma 2. For matrices A; B; C; D with appropriate dimensions and a positive scalar α, the following properties can be proved: ðαAÞ B ¼ A ðαBÞ ðA þ BÞ C ¼ A C þ B C ðA BÞðC DÞ ¼ ðACÞ ðBDÞ ðA BÞT ¼ AT BT Lemma 3. Suppose τðtÞA ½τm ; τM , Qi ði ¼ 1; 2; 3Þ are some constant matrices with appropriate dimensions, then Q1 þ ðτM τðtÞÞQ2 þ ðτðtÞ τm ÞQ3 o0
ð25Þ
if and only if the following inequalities hold: Q1 þ ðτM τm ÞQ2 o0
ð26Þ
Q1 þ ðτM τm ÞQ3 o0
ð27Þ
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Proof. (I) If τðtÞ ¼ τm or τðtÞ ¼ τM , the conclusion is obvious. (II) If τm oτðtÞoτM (Sufficiency): define a function as f ðτðt ÞÞ ¼ Q1 þ ðτM τðt ÞÞQ2 þ ðτðt Þ τm ÞQ3 τM τðtÞ τðtÞ τm ½Q1 þ ðτM τm ÞQ2 þ ½Q þ ðτM τm ÞQ3 ¼ τM τm τM τm 1 from Eqs. (26) and (27), we have f ðτðtÞÞo0 It is equal to Eq. (25). (Necessity): Letting τðtÞ ¼ τm and τðtÞ ¼ τM respectively in Eq. (25), we can easily obtain Eqs. (26) and (27). □ Assumption 1 (Wang et al. [27,28]). For 8 u; v A Rn , the nonlinear functions f ðÞ and gðÞ satisfy the following sector-bounded condition: ½f ðuÞ f ðvÞ F 1 ðu vÞT ½f ðuÞ f ðvÞ F 2 ðu vÞ r 0
ð28Þ
½gðuÞ gðvÞ G1 ðu vÞT ½gðuÞ gðvÞ G2 ðu vÞ r 0
ð29Þ
where F 1 ; F 2 and G1 ; G2 are real constant matrices with F 2 F 1 Z 0 and G2 G1 Z 0. Remark 5. Note that the sector-like description of Assumption 1 is more general than the usual Lipschitz functions. It would be possible to reduce the conservatism of results caused by quantifying nonlinear functions via matrix inequality. 3. Main results In this section, we present the consensus criteria for multi-agent systems (4) with stochastic nonlinearities. The system (21) can be rewritten as x_ ðtÞ ¼ ðI N AÞxðtÞ þ β0 FðxðtÞÞ þ ð1 β0 ÞGðxðtÞÞ cðL I n Þxðt τðtÞÞ cðL I n Þek ðtÞ þ ðβðtÞ β0 ÞðFðxðtÞÞ GðxðtÞÞÞ
ð30Þ
Let ξðtÞ ¼ ½xT ðtÞ; xT ðt τðtÞÞ; xT ðt τM Þ; F T ðxðtÞÞ; GT ðxðtÞÞ; eTk ðtÞT A ¼ ½I N A; cðL I n Þ; 0; β0 I Nn ; ð1 β0 ÞI Nn ; cðL I n Þ B ¼ ½0; 0; 0; I Nn ; I Nn ; 0 then system (30) can be expressed as x_ ðtÞ ¼ AξðtÞ þ ðβðtÞ β0 ÞBξðtÞ
ð31Þ
Theorem 1. Suppose that Assumption 1 holds, the global consensus is achieved under the consensus protocol (10). For given some positive scalar β0, s, c and τM , there exist appropriate dimensional matrices P40, Q40, R40 and Si, Ti ði ¼ 1; 2; …; 6Þ, and two positive scalars α, β, such that the following linear matrix inequalities hold: " # τM ðI N SÞ Π þ Σ 11 þ Σ T11 o0 ð32Þ n τM ðI N RÞ
H. Li et al. / Journal of the Franklin Institute 351 (2014) 2582–2599
"
# τM ðI N T Þ o0 τM ðI N RÞ
Π þ Σ 11 þ Σ T11 n
where
2
6 6 6 6 6 6 6 Π¼6 6 6 6 6 6 6 4
2589
ð33Þ
Π 11
cðL PÞ
0
Π 14
Π 15
cðL PÞ
Π 17
n
sðI N WÞ
n
n
0 IN Q
0 0
0 0
0 0
Π 27 0
n
n
n
2αI Nn
0
0
Π 47
n
n
n
n
2βI Nn
n
n
n
n
n
0 IN W
Π 57 Π 67
n
n
n
n
n
n
Π 77
n
n
n
n
n
n
n
h Σ 11 ¼ I N S I N ðT SÞ I N T 0 h iT T T T T T T S ¼ S1 S2 S3 S4 S5 S6 0 0 h iT T T T T T T T ¼ T1 T2 T3 T4 T5 T6 0 0
0
3
7 7 7 7 7 7 Π 48 7 7 Π 58 7 7 7 0 7 7 0 7 5 Π 88 0 0
i 0 0
0
0
Π 11 ¼ I N PA þ I N AT P þ I N Q αI N ðF T1 F 2 þ F T2 F 1 Þ βI N ðGT1 G2 þ GT2 G1 Þ Π 14 ¼ β0 ðI N PÞ þ αI N ðF T1 þ F T2 Þ Π 15 ¼ ð1 β0 ÞðI N PÞ þ βI N ðGT1 þ GT2 Þ Π 17 ¼ τM ðI N AT RÞ Π 27 ¼ cτM ðLT RÞ Π 47 ¼ τM β0 ðI N RÞ Π 48 ¼ τM β0 ð1 β0 ÞðI N RÞ Π 57 ¼ τM ð1 β0 ÞðI N RÞ Π 58 ¼ τM β0 ð1 β0 ÞðI N RÞ Π 67 ¼ τM cðLT RÞ Π 77 ¼ τM ðI N RÞ Π 88 ¼ τM β0 ð1 β0 ÞðI N RÞ Proof. Consider the following Lyapunov function candidate: Vðt; xðtÞÞ ¼ V 1 ðt; xðtÞÞ þ V 2 ðt; xðtÞÞ þ V 3 ðt; xðtÞÞ where V 1 ðt; xðtÞÞ ¼ xT ðtÞðI N PÞxðtÞ Z t V 2 ðt; xðtÞÞ ¼ xT ðsÞðI N QÞxðsÞ ds Zt t τM Z t V 3 ðt; xðtÞÞ ¼ x_ T ðsÞðI N RÞ_x ðsÞ ds dθ t τM
θ
ð34Þ
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The infinitesimal operator L of Vðt; xðtÞÞ is defined as follows: LV ðt; xðt ÞÞ ¼ limþ Δ-0
EðVðt þ Δ; xðt þ ΔÞÞÞ Vðt; xðtÞÞ Δ
ð35Þ
By simple calculation from Eqs. (34) and (35), the following equalities can be obtained: LV 1 ðt; xðtÞÞ ¼ 2xT ðtÞðI N PÞAξðtÞ
ð36Þ
LV 2 ðt; xðtÞÞ ¼ xT ðtÞðI N QÞxðtÞ xT ðt τM ÞðI N QÞxðt τM Þ Z t T x_ T ðsÞðI N RÞ_x ðsÞ ds LV 3 ðt; xðtÞÞ ¼ τM Ef_x ðtÞðI N RÞ_x ðtÞg
ð37Þ
t τM
¼ τM ξT ðtÞAT ðI N RÞAξðtÞ þ τM β0 ð1 β0 ÞξT ðtÞBT ðI N RÞBξðtÞ Z t x_ T ðsÞðI N RÞ_x ðsÞ ds
ð38Þ
t τM
Employing the free matrix method [29,30], it is easily derived that Z t T 2ξ ðtÞðI N SÞ xðtÞ xðt τðtÞÞ x_ ðsÞ ds ¼ 0
ð39Þ
t τðtÞ
Z 2ξT ðtÞðI N TÞ xðt τðtÞÞ xðt τM Þ
t τðtÞ t τM
x_ ðsÞ ds ¼ 0
ð40Þ
where
h iT T T T T T T S ¼ S1 S2 S 3 S4 S5 S6 h iT T T T T T T T ¼ T1 T2 T3 T4 T5 T6
It follows that from Eqs. (39)–(40) Z t x_ ðsÞ ds 2ξT ðtÞðI N SÞ t τðtÞ
r τðtÞξ ðtÞðI N SÞðI N RÞ 1 ðI N SÞT ξðtÞ þ
Z
t
T
Z 2ξ ðtÞðI N TÞ T
t τðtÞ t τM
t τðtÞ
x_ T ðsÞðI N RÞ_x ðsÞ ds
ð41Þ
x_ ðsÞ ds
r ðτM τðtÞÞξT ðtÞðI N TÞðI N RÞ 1 ðI N TÞT ξðtÞ þ
Z
t τðtÞ t τM
x_ T ðsÞðI N RÞ_x ðsÞ ds ð42Þ
Considering Eqs. (38)–(42), we can easily obtain LV 3 ðt; xðtÞÞ r τM ξT ðtÞAT ðI N RÞAξðtÞ þ τM β0 ð1 β0 ÞξT ðtÞBT ðI N RÞBξðtÞ þ2ξT ðtÞðI N SÞ½xðtÞ xðt τðtÞÞ þ 2ξT ðtÞðI N TÞ½xðt τðtÞÞ xðt τM Þ þτðtÞξT ðtÞðI N SÞðI N RÞ 1 ðI N SÞT ξðtÞ
H. Li et al. / Journal of the Franklin Institute 351 (2014) 2582–2599
2591
þðτM τðtÞÞξT ðtÞðI N TÞðI N RÞ 1 ðI N TÞT ξðtÞ
ð43Þ
For 8 α; β40, it can be derived that #" # " #T " xðtÞ xðtÞ IN F I N F~ Z0 α T FðxðtÞÞ FðxðtÞÞ I N F~ 2I Nn "
xðtÞ β GðxðtÞÞ
#T "
I N G~
IN G ~T IN G
2I Nn
#"
# xðtÞ Z0 GðxðtÞÞ
ð44Þ
ð45Þ
where F ¼ F T1 F 2 þ F T2 F 1 ; G
¼ GT1 G2
þ
GT2 G1 ;
F~ ¼ F T1 þ F T2 G~ ¼ GT þ GT 1
2
It follows that from Eqs. (36)–(45) LVðt; xðtÞÞ r ξT ðtÞ½Π þ Σ 11 þ Σ T11 ξðtÞ þ τðtÞξT ðtÞðI N SÞðI N RÞ 1 ðI N SÞT ξðtÞ þðτM τðtÞÞξT ðtÞðI N TÞðI N RÞ 1 ðI N TÞT ξðtÞ
ð46Þ
where Π and Σ11 are defined in Theorem 1. By the Schur complement in Eqs. (32)–(33), the following inequalities hold: Π þ Σ 11 þ Σ T11 þ τM ðI N TÞðI N RÞ 1 ðI N TÞT r 0
ð47Þ
Π þ Σ 11 þ Σ T11 þ τM ðI N SÞðI N RÞ 1 ðI N SÞT r 0
ð48Þ
By Lemma 3, it can be concluded that LVðt; xðtÞÞo0. That is, JξðtÞJ -0 as t-1. So the consensus in multi-agent systems (4) is achieved. □ Remark 6. The trigger parameters s, W and τM are involved in Eqs. (32)–(33). For given s, the corresponding trigger parameter W and the upper bound of τM can be obtained by using the LMI toolbox in Matlab. Note that τM ¼ h þ τ, selecting a sampling period hoτM , the upper bound of time delay τ can be obtained, if the effect of the transmission delay can be ignored, that is τ ¼ 0, the obtained τM the maximum sampling period. Let us now consider two special cases, and the corresponding results are still believed to be new. Case 1: We first specialize system (4) to case without stochastic switch in nonlinearities, that is βðtÞ ¼ 1, and system (4) is degenerated as follows: x_ i ðtÞ ¼ Axi ðtÞ þ f ðxi ðtÞÞ þ ui ðtÞ
ð49Þ
similar to Theorem 1, the consensus criteria can be derived for multi-agent systems (49) with nonlinear dynamics. Corollary 1. Suppose that Assumption 1 holds, a global consensus is achieved. For given some positive scalars s, c and τM , there exist appropriate dimensional matrices P40, Q40, R40 and Si,Ti ði ¼ 1; 2; …; 5Þ, and positive scalar α, such that the following linear matrix inequalities
2592
hold:
H. Li et al. / Journal of the Franklin Institute 351 (2014) 2582–2599
"
T Π~ þ Σ~ 11 þ Σ~ 11
~ τM ðI N SÞ
n
τM ðI N RÞ
# o0 ð50Þ
"
T Π~ þ Σ~ 11 þ Σ~ 11
τM ðI N T~ Þ
n
τM ðI N RÞ
# o0 ð51Þ
where
2
6 6 6 6 6 Π~ ¼ 6 6 6 6 4
Π~ 11
cðL PÞ
0
Π~ 14
cðL PÞ
n
sðI N WÞ
n
n
0 IN Q
0 0
0 0
n
n
n
2αI Nn
0
n
n
n
n
IN W
n
n
n
n
n
h ~ I N T~ Σ~ 11 ¼ I N S~ I N ðT~ SÞ h iT T T T T T S~ ¼ S1 S2 S3 S4 S5 0 h iT T T T T T T~ ¼ T 1 T 2 T 3 T 4 T 5 0
τM ðI N AT RÞ
3
7 cτM ðLT RÞ 7 7 7 0 7 7 τM ðI N RÞ 7 7 cτM ðLT RÞ 7 5 τM ðI N RÞ
i 0
0
0
Π~ 11 ¼ I N PA þ I N AT P þ I N Q αI N ðF T1 F 2 þ F T2 F 1 Þ Π~ 14 ¼ I N P þ αI N ðF T1 þ F T2 Þ Case 2: Let us assume that system (4) evolves with neither stochastic switching nor nonlinearities and system (4) reduces to x_ i ðtÞ ¼ Axi ðtÞ þ ui ðtÞ
ð52Þ
similar to Theorem 1, consensus criteria can be derived for multi-agent systems (52) with linear dynamics. Corollary 2. For given positive scalars s, c and τM , a global consensus is achieved. If there exist appropriate dimensional matrices P40, Q40, R40 and Si,Ti ði ¼ 1; 2; 3; 4Þ, such that the following linear matrix inequalities hold: " # T ^ Π^ þ Σ^ 11 þ Σ^ 11 τM ðI N SÞ o0 n τM ðI N RÞ ð53Þ "
T Π^ þ Σ^ 11 þ Σ^ 11
τM ðI N T^ Þ
n
τM ðI N RÞ
# o0
ð54Þ
H. Li et al. / Journal of the Franklin Institute 351 (2014) 2582–2599
where
2
Π^ 11
6 n 6 6 Π^ ¼ 6 6 n 6 4 n h
cðL PÞ
0
cðL PÞ
sðI N WÞ
0
0
n
IN Q
n
n
0 IN W
n
n
n
n
^ Σ^ 11 ¼ I N S^ I N ðT^ SÞ h iT T T T T S^ ¼ S1 S2 S3 S4 0 h iT T T T T T^ ¼ T 1 T 2 T 3 T 4 0
I N T^
i
τM ðI N AT RÞ
2593
3
cτM ðLT RÞ 7 7 7 7 0 7 7 T cτM ðL RÞ 5 τM ðI N RÞ
0 0
Π^ 11 ¼ I N PA þ I N AT P þ I N Q
4. Numerical results In this section, a numerical example is given to verify the effectiveness of the proposed control techniques for multi-agent systems to achieve a global consensus. Example 1. Consider multi-agent systems with a topology structure shown in Fig. 1, where the weights are indicated on the edges. Multi-agent systems consisting of seven agents are described as follows: x_ i ðtÞ ¼ Axi ðtÞ þ βðtÞf ðxi ðtÞÞ þ ð1 βðtÞÞgðxi ðtÞÞ þ ui ðtÞ ði ¼ 1; 2; …; 7Þ
ð55Þ
where 7
7
j¼1
j¼1
ui ðtÞ ¼ c ∑ lij xj ðt τðtÞÞ c ∑ lij ekj ðtÞ pffiffiffi pffiffiffiffiffi and A ¼ 1, f ðxi ðtÞÞ ¼ ð 5=5Þxi ðtÞ cos ðxi ðtÞÞ and gðxi ðtÞÞ ¼ ð 10=5Þxi ðtÞ sin ðxi ðtÞÞ. It is easy to verify that nonlinearpffiffifunctions fp ðÞffiffiffi and gðÞ satisfy ffi pffiffiffiffiffi Assumption pffiffiffiffiffi1, by some calculation, we can obtain F 1 ¼ 5=5, F 2 ¼ 5=5, G1 ¼ 10=5, G2 ¼ 10=5. Setting c ¼ 0.1, β0 ¼ 0:1, h¼ 0.01, s ¼ 0:1, we have τM ¼ 0:7136 by applying Theorem 1, that is, the maximum allowable delay τ ¼ τM h ¼ 0:7036. Suppose that τ ¼ 0, then τM ¼ h, it can be
Fig. 1. The topology structure of the agents with the weights on the connections.
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known that the maximum sampling period is 0.7136. More detailed calculation results for different values of s are given in Table 1. It can be seen form the calculation results in Table 1 that s becomes larger when τ becomes smaller. For given c¼ 0.1, β0 ¼ 0:1, h ¼ 0.01, s ¼ 0:1 and τ ¼ 0:1, using the algorithm (9), the simulation results, for t A ½0; 10, show that only (37) sampled signals need to be sent out, which takes 3.76% of sampled signals. Moreover, it can be computed that average release period h ¼ 0:2659, Table 2 shows the relationship trigger parameter s, trigger times, average release period h and the percentage of data transmissions among the agents, it can be seen that the larger s, the smaller trigger times, the larger average release period h, the smaller percentage of data transmission γ, all of which are reasonable results. When s ¼ 0, event-triggered reduces to timetriggered scheme, the curves of position states xi(t) ði ¼ 1; 2; …; 7Þ are depicted in Fig. 2, from which we can see that consensus is achieved. If setting s ¼ 0:1, the corresponding trigger parameter W ¼ 91.8608. The response of position states is depicted in Fig. 3. Figs. 2 and 3 show that the simulation results are almost the same, but the percentage of data transmission α under
Table 1 The maximum allowable delay for given c ¼0.1, β0 ¼ 0:1, h¼ 0.01, t¼ 10. s
0
0.1
0.2
0.3
0.4
0.5
τ
0.9706
0.7036
0.5650
0.4240
0.2697
0.1257
Table 2 The computation results for given c¼ 0.1, β0 ¼ 0:1, h¼0.01, τ ¼ 0:1, t¼ 10. s
0
0.1
0.2
0.3
0.4
0.5
Trigger times Trigger matrix W Average release period h Data transmission γ
1000 36.6741 0.01 100%
37 91.8608 0.2659 3.76%
29 2.2226 0.3362 2.97%
23 4.8967 0.4322 2.31%
22 1.5921 0.4350 2.30%
20 12.1210 0.4905 2.04%
Table 3 The maximum allowable delay for given c ¼0.1, β0 ¼ 0:8, h¼ 0.01, t¼ 10. s
0
0.1
0.2
0.3
0.4
0.5
τ
1.1513
0.9011
0.7710
0.6688
0.5682
0.4591
Table 4 The computation results for given c¼ 0.1, β0 ¼ 0:8, h¼0.01, τ ¼ 0:1, t¼ 10. s
0
0.1
0.2
0.3
0.4
0.5
Trigger times Trigger matrix W Average release period h Data transmission γ
1000 1.2372 0.01 100%
28 37.9507 0.3557 2.81%
22 0.6877 0.4495 2.22%
16 298.1335 0.5950 1.68%
15 37.4459 0.6207 1.61%
13 244.7115 0.7246 1.38%
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1 0.8 0.6
xi(t) (i=1,2,¡-,7)
0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0
2
4
6
8
10
t
Fig. 2. The position curves xi(t) ði ¼ 1; 2; …; 7Þ with switching probability β0 ¼ 0:1 and trigger parameter s ¼ 0 (timetriggered scheme).
1 0.8 0.6
xi(t) (i=1,2,¡-,7)
0.4 0.2 0 −0.2 −0.4 −0.6 −0.8
0
2
4
6
8
10
t
Fig. 3. The position curves xi(t) ði ¼ 1; 2; …; 7Þ with switching probability β0 ¼ 0:1 and trigger parameter s ¼ 0:1 (eventtriggered scheme).
the event-triggered scheme is much smaller than time-triggered scheme. To show the release instants and release intervals, which are illustrated in Fig. 4. From these results, it can be concluded that event-triggered scheme has advantages over time-triggered scheme in terms of improving the resource utilization.When β0 ¼ 0:8, we have similar results. From Tables 3 and 4 and Figs. 5–7, similar conclusions can be drawn.
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event−based release instants and release interval
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
2
4
6
8
10
t(s)
Fig. 4. Release instants and release interval by event-triggered scheme.
0.8
0.6
xi(t) (i=1,2,¡-,7)
0.4
0.2
0
−0.2
−0.4
−0.6
0
2
4
6
8
10
t
Fig. 5. The position curves xi(t) ði ¼ 1; 2; …; 7Þ with switching probability β0 ¼ 0:8 and trigger parameter s ¼ 0 (timetriggered scheme).
5. Conclusions This paper proposes a novel event-triggering scheme to achieve consensus for multi-agent systems with randomly occurring nonlinear dynamics and time-varying delay. Less communication resources are utilized while preserving the consensus performance in the proposed scheme. Due to it is practically impossible to apply control actions to all agents in large-scale multi-agent systems, future work will combine pinning control schemes and event-triggering scheme to study the consensus problem of multi-agent systems, a very challenging problem is how to choose a set of pinned agents.
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0.6 0.4
xi(t) (i=1,2,¡-,7)
0.2 0 −0.2 −0.4 −0.6 −0.8 −1
0
2
4
6
8
10
t
Fig. 6. The position curves xi(t) ði ¼ 1; 2; …; 7Þ with switching probability β0 ¼ 0:8 and trigger parameter s ¼ 0:1 (eventtriggered scheme).
event−based release instants and release interval
0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
0
2
4
6
8
10
t(s)
Fig. 7. Release instants and release interval by event-triggered scheme.
Acknowledgements The authors would like to express their sincere appreciation to the editor and reviewers for their helpful comments which help to improve quality of the paper. This work was jointly supported by the National Natural Science Foundation of China under Grant nos. 61272034 and 11301226, Zhejiang Provincial Natural Science Foundation of China under Grant nos. LY13F030012 and LQ13A010017.
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